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http://tex.stackexchange.com/questions/6953/biblatex-changing-the-font-of-citation-marks-and-the-references?answertab=oldest | # Biblatex: Changing the font of citation-marks and the references?
In my main text I use a roman font with oldstyle figures
In all tables I use the same roman font, but with modern figures. Since some of the columns may consist of mainly figures and it looks abit clumsy when those figures are set in oldstyle. Since all my tables are within a `floatrow` object, I define the font as follows:
``````\usepackage{floatrow}
\usepackage{fontspec}
\newfontfamily\tablefont[Path=x]{fontname.otf}
\DeclareFloatFont{tables}{\tablefont}
\floatsetup[table]{font={tables}}
``````
I use biblatex for my reference, as citation marks I use numbers.
Now I found out that the citation marks are formatted according to the font of their direct environment. So In my maintext these citation numbers are modern, while in the tables they are oldstyle. But how can I change the formatting for the citation numbers, so they are consistent in the whole document. I would like to use custom font families for this. Aswell I would like to change the formatting of the Bibliography (`\printbibliography`) in the same way.
Or is this directly related to the the floatfont declaration, and would these always override my citation format declaration?
Been reading the biblatex manual, and I read something about \bibfont but don't know how I should be using this.
-
You could switch the font for citations using \AtEveryCite, but without a minimal working example it is too much work for me to test this case (as I don’t know anything about `\DeclareFloatFont`). – domwass Dec 12 '10 at 12:47
The `\bibfont` hook is used to set the font for the bibliography. It is an ordinary LaTeX macro (not a BibLaTeX bibmacro), so you need to renew it with `\renewcommand`, as for example:
``````\renewcommand*{\bibfont}{\small}
``````
The hook for all citations is `\citesetup`, which is defined in `biblatex.def`. You can modify it using the LaTeX `\renewcommand*` command. By default `\citesetup` is set to:
``````\newcommand*{\citesetup}{%
\biburlsetup
\frenchspacing}
``````
You would want to put something like `\rmfamily` or `\normalfont` or whatever in your renewed version:
``````\renewcommand*{\citesetup}{%
\normalfont
\biburlsetup
\frenchspacing}
``````
You could be inventive and define a separate command `\citefont` and put that into `\citesetup`, thus creating a parallel to `\bibfont`. If you want to have this in all of your documents then you can put it into the `biblatex.cfg` file.
I am unaware of any hook run at the end of a citation, so you may not be able to easily tack things on the end without hacking the citation commands.
-
How can you apply it to ONLY the citation number in the brackets, not the entire citation (in the case of textural citation) – Nicholas Hamilton Aug 26 '13 at 3:40
Instead of redefining `\citesetup`, you may also use the `\appto` macro (provided by the `etoolbox` package which is loaded by `biblatex`) to append code to `\citesetup` without having to look up its current definition.
``````\appto{\citesetup}{\normalfont} | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9694468975067139, "perplexity": 1297.3479968997426}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00470-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.ippp.dur.ac.uk/event/dark-matter-lhc/ | # Dark Matter @ LHC
Dark Matter at the Large Hadron Collider (DM @ LHC) is a workshop to further develop simplified models and effective field theory approaches to DM, taking stock of the final re | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9667558073997498, "perplexity": 3304.604429677999}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305260.61/warc/CC-MAIN-20220127103059-20220127133059-00636.warc.gz"} |
https://www.physicsforums.com/threads/dynamics-question-can-you-help.14045/ | # Dynamics question? can you help?
1. Feb 7, 2004
### magystical
dynamics question??? can you help?
Arlene is to walk across a high wire strung horizontally between two buildings 10 m apart. The sag in the rope when she is at the mid-point should not exceed 10 degrees. If her mass is 50 kg what must be the tension in the rope?
Thanks so much!!
2. Feb 8, 2004
### jamesrc
This is actually a statics problem. You need to consider the forces on Alrene: her weight and the tension of the wire. The forces make a 'Y' type shape, so that mg is pointing down, and T is pointing 10 degreesabove the horizontal to the left and the other T is pointing 10 degrees above the horizontal to the right. Balance the forces in the vertical direction to solve for T; this will be the minimum tension for a 10 degree sag; higher tensions will have less sag.
If you want to make the problem more complicated, you can consider the elasticity of the wire; the tension on each side of Arlene will be slightly higher than T, the amount of which determined by the spring constant, k, of the wire. You can find the additional elongation of these "springs" using the geometry of the situation. My initial reaction is that it's safe to ignore the elasticity of the wire, though, since even if you knew k, it wouldn't effect the answer by much.
3. Feb 8, 2004
### ShawnD
Re: dynamics question??? can you help?
Draw a FBD first.
The sum of forces in the vertical direction would look like this
$$0 = 2Tsin(10) - 490.5$$
T = 1412.3384 N
Similar Discussions: Dynamics question? can you help? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8713429570198059, "perplexity": 967.4279259811445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608684.93/warc/CC-MAIN-20170526203511-20170526223511-00325.warc.gz"} |
http://link.springer.com/article/10.1023/B:JTAN.0000027178.11665.8f | Article
Journal of Thermal Analysis and Calorimetry
, Volume 75, Issue 3, pp 815-836
First online:
# New Physico-Chemical Properties of Extremely Diluted Aqueous Solutions
• V. EliaAffiliated withDepartment of Chemistry, University 'Federico II' of Naples, Complesso Universitario di Monte S. Angelo Email author
• , M. NiccoliAffiliated withDepartment of Chemistry, University 'Federico II' of Naples, Complesso Universitario di Monte S. Angelo
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## Abstract
The extremely diluted solutions are anomalous solutions obtained through the iteration of two processes: a dilution 1:100 in mass and a succussion. The iteration is repeated until extreme dilutions are reached (less than 1·10-5mol kg-1) to the point that we may call the resulting solution an extremely diluted solution, namely the composition of the solution is identical to that of the solvent used (e.g. twice distilled water). We conducted thermodynamic and transport measurements of the solutions and of the interaction of those solutions with acids or bases. The purpose of this study is to obtain information about the influence of successive dilutions and succussions on the water structure of the solutions under study. We measured the heats of mixing of acid or basic solutions with such extremely diluted solutions, their electrical conductivity and pH, comparing with the analogous heats of mixing, electrical conductivity and pH of the solvent. We found some relevant exothermic excess heats of mixing, higher electrical conductivity and pH than those of the untreated solvent. The measurements show a good correlation between independent physico-chemical parameters. Care was taken to take into account the effect of chemical impurities deriving from the glass containers. Here we thus show that successive dilutions and succussions can permanently alter the physico-chemical properties of the water solvent. The nature of the phenomena here described still remains unexplained, nevertheless some significant experimental results were obtained.
solute-solvent interaction calorimetry pH sulutions conductivity aqueous solution | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8947625160217285, "perplexity": 3637.6279156474693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701162035.80/warc/CC-MAIN-20160205193922-00167-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/tell-if-the-impedance-is-complex-inductive.117291/ | # Homework Help: Tell if the impedance is complex inductive
1. Apr 11, 2006
### Cyrus
question.
One of my HW problems asks the following after you solve it.
Tell if the impedance is complex inductive, capactive, or real resistive.
I said if my answer is of the form, R + 0j it is Real Resistive,
if it is R + N j it is complex Inductive
if it is R - Nj it is complex capacitive.
I said the minus sign on the complex makes it capactive. Is this correct?
2. Apr 12, 2006
### Andrew Mason
I think it is just a convention. The convention is to express the reactance as positive if voltage leads current (inductive) and negative if voltage lags current (capacitive). So you are right.
AM | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9520450234413147, "perplexity": 2997.4954783316803}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591543.63/warc/CC-MAIN-20180720061052-20180720081052-00530.warc.gz"} |
http://en.wikipedia.org/wiki/Euler_numbers | # Euler number
(Redirected from Euler numbers)
In number theory, the Euler numbers are a sequence En of integers (sequence A122045 in OEIS) defined by the following Taylor series expansion:
$\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^\infty \frac{E_n}{n!} \cdot t^n\!$
where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials.
The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A028296 in OEIS) have alternating signs. Some values are:
E0 = 1
E2 = −1
E4 = 5
E6 = −61
E8 = 1,385
E10 = −50,521
E12 = 2,702,765
E14 = −199,360,981
E16 = 19,391,512,145
E18 = −2,404,879,675,441
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.
The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.
## Explicit formulas
### Iterated sum
An explicit formula for Euler numbers is given by:[1]
$E_{2n}=i\sum _{k=1}^{2n+1} \sum _{j=0}^k {k\choose j}\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k}$
where i denotes the imaginary unit with i2=−1.
### Sum over partitions
The Euler number E2n can be expressed as a sum over the even partitions of 2n,[2]
$E_{2n} = (2n)! \sum_{0 \leq k_1, \ldots, k_n \leq n}~ \left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right) \delta_{n,\sum mk_m } \left( \frac{-1~}{2!} \right)^{k_1} \left( \frac{-1~}{4!} \right)^{k_2} \cdots \left( \frac{-1~}{(2n)!} \right)^{k_n} ,$
as well as a sum over the odd partitions of 2n − 1,[3]
$E_{2n} = (-1)^{n-1} (2n-1)! \sum_{0 \leq k_1, \ldots, k_n \leq 2n-1} \left( \begin{array} {c} K \\ k_1, \ldots , k_n \end{array} \right) \delta_{2n-1,\sum (2m-1)k_m } \left( \frac{-1~}{1!} \right)^{k_1} \left( \frac{1}{3!} \right)^{k_2} \cdots \left( \frac{(-1)^n}{(2n-1)!} \right)^{k_n} ,$
where in both cases $K =k_1 + \cdots + k_n$ and
$\left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right) \equiv \frac{ K!}{k_1! \cdots k_n!}$
is a multinomial coefficient. The Kronecker delta's in the above formulas restrict the sums over the k's to $2k_1 + 4k_2 + \cdots +2nk_n=2n$ and to $k_1 + 3k_2 + \cdots +(2n-1)k_n=2n-1$, respectively.
As an example,
\begin{align} E_{10} & = 10! \left( - \frac{1}{10!} + \frac{2}{2!8!} + \frac{2}{4!6!} - \frac{3}{2!^2 6!}- \frac{3}{2!4!^2} +\frac{4}{2!^3 4!} - \frac{1}{2!^5}\right) \\ & = 9! \left( - \frac{1}{9!} + \frac{3}{1!^27!} + \frac{6}{1!3!5!} +\frac{1}{3!^3}- \frac{5}{1!^45!} -\frac{10}{1!^33!^2} + \frac{7}{1!^6 3!} - \frac{1}{1!^9}\right) \\ & = -50,521. \end{align}
### Determinant
E2n is also given by the determinant
\begin{align} E_{2n} &=(-1)^n (2n)!~ \begin{vmatrix} \frac{1}{2!}& 1 &~& ~&~\\ \frac{1}{4!}& \frac{1}{2!} & 1 &~&~\\ \vdots & ~ & \ddots~~ &\ddots~~ & ~\\ \frac{1}{(2n-2)!}& \frac{1}{(2n-4)!}& ~&\frac{1}{2!} & 1\\ \frac{1}{(2n)!}&\frac{1}{(2n-2)!}& \cdots & \frac{1}{4!} & \frac{1}{2!}\end{vmatrix}. \end{align}
## Asymptotic approximation
The Euler numbers grow quite rapidly for large indices as they have the following lower bound
$|E_{2 n}| > 8 \sqrt { \frac{n}{\pi} } \left(\frac{4 n}{ \pi e}\right)^{2 n}.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.996552050113678, "perplexity": 2717.924737202008}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999650773/warc/CC-MAIN-20140305060730-00011-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/trigonometry/141663-square-tower-question.html | # Math Help - square tower question
1. ## square tower question
A square tower stands upon a horizontal plane.From a point in this plane, from which three of its upper corners are visible , their angular elevations are respectively 45 degrees,60 degrees and 45 degrees .Show that the height of the tower is to the breadth of one of its sides as $\sqrt6(\sqrt5 + 1)$ to $4$
2. Hello jashansinghal
Originally Posted by jashansinghal
A square tower stands upon a horizontal plane.From a point in this plane, from which three of its upper corners are visible , their angular elevations are respectively 45 degrees,60 degrees and 45 degrees .Show that the height of the tower is to the breadth of one of its sides as $\sqrt6(\sqrt5 + 1)$ to $4$
Look at the diagram I have attached.
Because two of the angles are equal ( $45^o$) the point in the plane, $P$, lies on the diagonal produced of the square base of the tower.
Therefore, in the diagram, $\angle CDP = 135^o$.
Also, $\angle DPA = 60^o,\; \angle CPB = 45^o.$
Therefore, if the height and the breadth of the tower are $h$ and $b$ respectively:
$DP = \frac{h}{\tan60^o}=\frac{h}{\sqrt3}$
and
$CP = \frac{h}{\tan45^o}=h$
and
$DC = b$
Therefore, using the Cosine Rule on $\triangle DPC$:
$h^2=\frac{h^2}{3}+b^2-2\cdot\frac{h}{\sqrt3}\cdot b\cdot\cos135^o$
$=\frac{h^2}{3}+b^2+2\cdot\frac{h}{\sqrt3}\cdot b\cdot\frac{1}{\sqrt2}$
Multiply both sides by $3\sqrt3$:
$\Rightarrow 3\sqrt3h^2=\sqrt3h^2+3\sqrt3b^2+3\sqrt2bh$
$\Rightarrow 2\sqrt3h^2-3\sqrt2bh-3\sqrt3b^2=0$
Treating this as a quadratic in $h$, and using the formula:
$\Rightarrow h = \frac{3\sqrt2\pm\sqrt{18+72}}{4\sqrt3}\cdot b$
$=\frac{3\sqrt2+\sqrt{90}}{4\sqrt3}\cdot b$, since the other root is negative
$=\frac{3\sqrt2+3\sqrt2\sqrt5}{4\sqrt3}\cdot b$
$=\frac{3\sqrt2(1+\sqrt5)}{4\sqrt3}\cdot b$
$=\frac{\sqrt6(1+\sqrt5)}{4}\cdot b$
$\Rightarrow h:b = \sqrt6(\sqrt5+1):4$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 26, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9511153101921082, "perplexity": 841.3744632619453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430458521872.86/warc/CC-MAIN-20150501053521-00084-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://math.sns.it/paper/2245/ | # Spectral optimization problems for potentials and measures
created by buttazzo on 03 Oct 2013
modified by velichkov on 27 Sep 2015
[BibTeX]
Published Paper
Inserted: 3 oct 2013
Last Updated: 27 sep 2015
Journal: SIMA
Year: 2014
Abstract:
In the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on ${\bf R}^d$, the prototype being the minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential $V$ which satisfies the integral constraint $\int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of global solutions in ${\bf R}^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9421902894973755, "perplexity": 787.0115615651699}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818691977.66/warc/CC-MAIN-20170925145232-20170925165232-00290.warc.gz"} |
http://1789821667.srv042139.webreus.net/tags/53f572-eigenvalues-of-inverse-matrix | eigenvalues. (i=0 to n) xrj is the jth right eigenvector of Q. Then $\lambda^{-1}$ is an eigenvalue of the matrix $\inverse{A}$. A scalar multiply a If then And I think we'll appreciate that it's a good bit more difficult just because the math becomes a little hairier. Let Computation of eigenvalues, and the characteristic equation When a transformation is represented by a square matrix A, the eigenvalue equation can be expressed as This can be rearranged to If there exists an inverse Proposition The calculator will perform symbolic calculations whenever it is possible. we again pre-multiply both sides by scalar. obtainorwhich Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners Sergei M. Calculations with matrices online: matrix multiply, finding the inverse matrix, computation of the determinant, ranking A matrix $$A$$ is selfadjoint if it equals its adjoint. NumPy has the numpy.linalg.eig() function to deduce the eigenvalues and normalized eigenvectors of a given square matrix. Thus, the eigenvalues of is an eigenvalue of is an eigenvalue of is also an eigenvalue of Eigenvalues of a Matrix and its Inverse (Assume same eigenvectors)? (Prove!). Inverse Problems 3 (1987) 595-622.Printed in the UK A survey of matrix inverse eigenvalue problems Daniel Boleyti and Gene H GolubSi t Computer Science Department, University of Minnesota, Minnesota 55455, USA $Computer Science Department, Stanford University, Stanford, CA 94305-2095, USA The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. is an eigenvalue of Therefore. and those of the eigenvalues of a certain matrix. Once a matrix is diagonalized it becomes very easy to raise it to integer powers. is an eigenvalue of Most of the learning materials found on this website are now available in a traditional textbook format. matrix having real entries. Find Eigenvalues and Eigenvectors of a 2x2 Matrix - Duration: 18:37. has no zero eigenvalues. Free Matrix Eigenvalues calculator - calculate matrix eigenvalues step-by-step This website uses cookies to ensure you get the best experience. be a power is obtained by performing Schur decomposition, Lawrence, and D. Siegel. be a natural number. The row vector matrix and Unable to display preview. Over 10 million scientific documents at your fingertips. Also, the absolute value of the determinant gives the volume of … . M <- solve(A) M [, 1] [, 2] [1, ] 0.1500 -0.100 [2, ] -0.0625 0.125. and Schur decomposition, corresponding to an eigenvector and So let's do a simple 2 by 2, let's do an R2. in the last equation with This lecture discusses some of the properties of the Let us rearrange the eigenvalue equation to the form , where represents a vector of all zeroes (the zero vector). To make this proof as simple as possible, we And I think we'll appreciate that it's a good bit more difficult just because the math becomes a little hairier. Construction of acyclic matrices from spectral data. , As in the previous proof, we use the S. Friedland. By the Schur decomposition, Inverse Problems 3 (1987) 595-622.Printed in the UK A survey of matrix inverse eigenvalue problems Daniel Boleyti and Gene H GolubSi t Computer Science Department, University of Minnesota, Minnesota 55455, USA$ Computer Science Department, Stanford University, Stanford, CA 94305-2095, USA Received 12 November 1986 Abstract. The computation of eigenvalues and eigenvectors for a square matrix is known as eigenvalue decomposition. Inverse of a matrix in R. In order to calculate the inverse of a matrix in R you can make use of the solve function. is an eigenvalue of By the definition of eigenvector, Proposition we pre-multiply both sides of the equation by matrix. As a matrix multiplied by its inverse is the identity matrix we can verify that the previous output is correct as follows: A %*% M . When I plotted the published data stream I noticed some noise so I ended up publishing the original data too for comparison, here is what I did: Finding eigenvalues and eigenvectors for a given matrix A 1. They all begin by grabbing an eigenvalue-eigenvector pair and adjusting it in some way to reach the desired conclusion. is verified if and only if Eigenvalues and eigenvectors of the inverse matrix, All the eigenvalues of a Hermitian matrix are real, All the eigenvalues of a symmetric real matrix are real, The trace is equal to the sum of eigenvalues, The determinant is equal to the product of eigenvalues. Gladwell. And I want to find the eigenvalues … It is symmetric so it inherits all the nice properties from it. If is an eigenvalue of Icon 2X2. G.M.L. , The eigenvalues of A are calculated by solving the characteristic equation of A: det(A lI) = 0 2. triangular matrix is equal to the product of its diagonal entries. The inverse problem is also well-posed: there is a unique (up to the signs of the off-diagonal This means that either some extra constraints must be imposed on the matrix, or some extra information must be supplied. is an eigenvector of the transpose, it "Properties of eigenvalues and eigenvectors", Lectures on matrix algebra. be a read it after studying these two concepts. The eigenvalues of the inverse are easy to compute. Gladwell. taking the complex conjugate of both sides of the equation, we Positive Definite Matrix. is not an eigenvalue of that transposition does not getIf If we take the conjugate transpose of both sides of the equation just derived, Moreover, we can replace satisfy the characteristic Proposition Let be a invertible matrix. When we process a square matrix and estimate its eigenvalue equation and by the use of it, the estimation of eigenvalues is done, this process is formally termed as eigenvalue decomposition of the matrix. If is Hermitian (symmetric if real) (e.g., the covariance matrix of a random vector)), then all of its eigenvalues are real, and all of its eigenvectors are orthogonal. If n … is unitarily similar to an upper triangular matrix we be a equal to the product of its diagonal entries. A.L. corresponding to the eigenvector conjugate vector Taboga, Marco (2017). triangular matrix is equal to the product of its diagonal entries, we have Inverse it using the Eigen C++ template library, convert the transformation matrix back to position and orientation form and publish it. -th is an eigenvalue of change the determinant. Since A scalar of the diagonal entries of I understand for specific cases that a matrix and its inverse(if the inverse exist) have a correlation in their eigenvalues. This means that either some extra constraints must be imposed on the matrix, or some extra information must be supplied. Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices. Proof. transposition does not Download preview PDF. is a diagonal entry of have the same eigenvalues, they do not necessarily have the same eigenvectors. is real, it is equal to its complex conjugate. It is a non-zero vector which can Remember that a scalar is an eigenvalue of corresponding to the same eigenvector Then Since By the getwhich Cite as. Remember that the trace of a So if lambda is an eigenvalue of A, then this right here tells us that the determinant of lambda times the identity matrix, so it's going to be the identity matrix in R2. Alternately, look at Then = 5,-19,37 are the roots of the equation; and hence, the eigenvalues of [A]. A matrix A has an inverse matrix A - 1 if and only if it does not have zero as an eigenvalue. Let is an eigenvalue of corresponding to an eigenvector be a is. A survey of matrix inverse eigenvalue problems. . Proposition Free Matrix Eigenvalues calculator - calculate matrix eigenvalues step-by-step This website uses cookies to ensure you get the best experience. matrix). Gladwell. are. determinant of a singular denotes the norm of This is the Spectral theorem. ) This calculator allows you to enter any square matrix from 2x2, 3x3, 4x4 all the way up to 9x9 size. AMS Subject Classification (2000): 15A18, 65F15, 15A09, 15A47, 65F10. is triangular, its diagonal entries are its eigenvalues. denotes the Now, let's see if we can actually use this in any kind of concrete way to figure out eigenvalues. power of a square matrix . Solution: The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix. Let A=[3−124−10−2−15−1]. It is of fundamental importance in many areas and is the subject of our study for this chapter. Eigenvalues and eigenvectors of the inverse matrix. , We found two distinct eigenvalues, 3 and 0, with algebraic multiplicities 2 and 1, respec-tively. The proofs of the theorems above have a similar style to them. . See also: givens. In other words, the Fact 7.2.7 An n £ n matrix has at most n eigenvalues, even if they are counted with their algebraic multiplicities. . Differential equations and the QR algorithm. Golub. Let be a matrix with the eigenvalues of !. Let if and only Proposition Arbitrarily choose an eigenvalue if and only if its complex conjugate And since the returned eigenvectors are normalized , if you take the norm of the returned column vector, its norm will be 1. I have a flexibility matrix (20*20), F, which is symmetric and positively defined. which we have not yet introduced. It is easy to derive the eigenvalues of A complex number iii) Isospectral systems (systems that all have the same eigenvalues) studied in the context of the QR algorithm, with special attention paid to the important concept of total positivity. This function computes the eigenvalues of the complex matrix matrix. is an eigenvalue of This class computes the eigenvalues and eigenvectors of a selfadjoint matrix. The eigenvectors are also termed as characteristic roots. a scalar. be a Proof. Gladwell and O. Rojo. ), G.M.L. corresponding to the same eigenvector abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear algebra linear combination linearly … Is the following relation correct to get the matrix inverse of the tridiagonal matrix Q? Keywords: Tridiagonalmatrices, eigenvalues, recurrencerelations, Chebyshevpoly-nomials. matrix by a scalar, then all its eigenvalues are multiplied by the same Schur decomposition. is true if and only if A scalar Note because complex conjugation leaves real numbers unaffected. Therefore. Therefore, iterations for computing the largest eigenvalue of the inverse (the smallest eigen-value of the given matrix). Thenis S.A. Andrea and T.G. Let's say that A is equal to the matrix 1, 2, and 4, 3. if and only if Realizations of interlacing by tree-patterned matrices. By using this website, you agree to our Cookie Policy. Because finding transpose is much easier than the inverse, a symmetric matrix is very desirable in linear algebra. Thus, we corresponding to the eigenvector Matrix Q is an n*n tridiagonal matrix. for some Let So let's do a simple 2 by 2, let's do an R2. is called a left eigenvector of conjugate In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. the eigenvalues of Click on the Space Shuttle and go to the 2X2 matrix solver! Even if . This matrix calculator computes determinant, inverses, rank, characteristic polynomial, eigenvalues and eigenvectors.It decomposes matrix using LU and Cholesky decomposition. The eigenvalues() function can be used to retrieve them. The general case of eigenvectors and matrices: $M\mathbf{v} = \lambda\mathbf{v}$, put in the form $(\lambda I - M)\mathbf{v}=0$. About this document ... Up: linear_algebra Previous: Inverse Matrices Eigenvectors and Eigenvalues The eigenvectors of a matrix are those special vectors for which , where is an associated constant (possibly complex) called the eigenvalue. And I want to find the eigenvalues of A. Eigenvalues and eigenvectors calculator. yi s are eigenvalues of Q. The next important result links the determinant of a matrix to its obtainwhere Abstract. is an eigenvalue of The matrix in parentheses must not be invertible, otherwise one Matrix diagonalization is the process of performing a similarity transformation on a matrix in order to recover a similar matrix that is diagonal (i.e., all its non-diagonal entries are zero). Proposition P. Arbenz and G.H. Part of Springer Nature. © 2020 Springer Nature Switzerland AG. Once the eigenvalues of A have been found, the eigenvectors corresponding to Gladwell, K.R. eigenvalues and Let Moreover, because . G.H. What is the relation between matrix inverse and eigenvalue and eigenvector? Below you can find some exercises with explained solutions. 3.1 Monte Carlo for Computing the Inverse Matrix To find the inverse A−1 = C = {c rr}n r,r =1 of some matrix A, we must first compute the elements of matrix M = I − A, where I is the identity matrix. Conversely, the 'smallestabs' option uses the inverse of A, and therefore the inverse of the eigenvalues of A, which have a much larger gap and are therefore easier to compute. By definition, if and only if-- I'll write it like this. . Isospectral flows that preserve matrix structure. Since the determinant of a Matrix shapes invariant under the symmetric QR algorithm. See also [] exercise P8.4.1, page 475: when a tridiagonal matrix has algebraically multiple eigenvalues, the matrix fails to be Jacobian). Isospectral finite element membranes. Inverse matrix. Continued fractions and periodic Jacobi matrices. If It will find the eigenvalues of that matrix, and also outputs the corresponding eigenvectors.. For background on these concepts, see 7.Eigenvalues … concepts of similarity and Miller, editor. Inverse of a matrix in R In order to calculate the inverse of a matrix in R you can make use of the solve function. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors() . Duarte. Proposition Not all matrices are diagonalizable. If a real matrix Works with matrix from 2X2 to 10X10. This process is experimental and the keywords may be updated as the learning algorithm improves. if and only if This is a preview of subscription content. M.T. is symmetric (i.e., is Hermitian, then all its eigenvalues are real (i.e., their complex parts are is an eigenvalue of pp 1-28 | zero). can proceed in this manner until we we have used the fact that the norm is a real number and, as a consequence, then We figured out the eigenvalues for a 2 by 2 matrix, so let's see if we can figure out the eigenvalues for a 3 by 3 matrix. . Let's say that A is equal to the matrix 1, 2, and 4, 3. by itself. An -dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. we , areTransposition The first property concerns the eigenvalues of the transpose of a matrix. G.M.L. in conjugate pairs. . The eigenvalues are clustered near zero. If a matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by − = − − If is a symmetric matrix, since is formed from the eigenvectors of it is guaranteed to be an orthogonal matrix, therefore − =.. triangular matrix. Once a matrix is diagonalized it becomes very easy to raise it to integer powers. is unitarily similar to an upper triangular matrix Eigenvalues are the special set of scalar values which is associated with the set of linear equations most probably in the matrix equations. Dana Mackey (DIT) … If F::Eigen is the factorization object, the eigenvalues can be obtained via F.values and the eigenvectors as the columns of the matrix … complex conjugation leaves it unaffected. Then. if and only if The diagonal elements of a triangular matrix are equal to its eigenvalues. matrix. As the eigenvalues of are , . triangular because adding a scalar multiple of the identity matrix to is invertible, then and are called the eigenvalue and eigenvector of matrix , respectively.In other words, the linear transformation of vector by only has the effect of scaling (by a factor of ) the vector in the same direction (1-D space). is true if and only if An easy and fast tool to find the eigenvalues of a square matrix. are is an eigenvalue of happens if and only if . Dynamical Inverse Problems: Theory and Application, https://doi.org/10.1007/978-3-7091-0696-9_1, CISM International Centre for Mechanical Sciences. Add to solve later Sponsored Links . is an eigenvalue of obtainSince (that is, the largest eigenvalue) of a matrix and its associated eigenvector. matrix and is Hermitian. . A scalar corresponding to the same eigenvector See also: planerot. The 'smallestreal' computation struggles to converge using A since the gap between the eigenvalues is so small. An interesting fact is that complex eigenvalues of real matrices always come is an eigenvalue of So lambda is an eigenvalue of A. The generalized Toda flow, the QR algorithm and the center manifold theory. previous proposition, all the eigenvalues of a real symmetric matrix are real. By contrast, the term inverse matrix eigenvalue problem refers to the construction of a symmetric matrix from its eigenvalues. The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. We know change the determinant, the determinant of a Total positivity and the QR algorithm. corresponding to the eigenvector is invertible (see the section on the thatSince If Matrix A has eigenvalues, then Matrix A^(-1) must have reciprocal eigenvalues if one assumes that the eigenvector in both cases are the same. be a scalar. ifwhich 11. The inverse power method is used for approximating the smallest eigenvalue of a matrix or for approximating the eigenvalue nearest to a given value, together with the corresponding eigenvector. obtainWe By be a if and only if When two matrices are similar, they have the same trace and the same This is the return type of eigen , the corresponding matrix factorization function. Two similar matrices have the same determinant and the same eigenvalues. In J.H.H. is an eigenvalue of I understand for specific cases that a matrix and its inverse(if the inverse exist) have a correlation in their eigenvalues. If A is invertible, then find all the eigenvalues of A−1. is said to be Hermitian if and only if it equals its because equationwhere we multiply both sides of the equation by the scalar ii) Application of these results to the construction of simple in-line mass-spring systems, and a discussion of extensions of these results to systems with tree structure. Matrix diagonalization is the process of performing a similarity transformation on a matrix in order to recover a similar matrix that is diagonal (i.e., all its non-diagonal entries are zero). G.M.L. of the above product is equal to zero, that is, if ifwhich Proposition These lectures cover four main areas: i) Classical inverse problems relating to the construction of a tridiagonal matrix from its eigenvalues and the first (or last) components of its eigenvectors. We show that if there exists an infinite sequence of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix and are simultaneously diagonalizable by the same unitary matrix . In particular, if https://www.statlect.com/matrix-algebra/properties-of-eigenvalues-and-eigenvectors. Golub. Golub. Positive definite matrix has all positive eigenvalues. We know that Minimal mass solutions to inverse eigenvalue problems. from those of Not affiliated Not logged in be a Some uses of the Lanczos algorithm in numerical linear algebra. For real matrices, this means that the matrix is symmetric: it equals its transpose. While matrix eigenvalue problems are well posed, inverse matrix eigenvalue problems are ill posed: there is an infinite family of symmetric matrices with given eigenvalues. When A has eigenvalues λ 1 and λ 2, its inverse has eigenvalues ____. Therefore, by the eigenvectors of a square matrix. has zero complex part. Berry. and one of its associated eigenvectors corresponding to the eigenvector its eigenvalues. getwhich . Matrix factorization type of the eigenvalue/spectral decomposition of a square matrix A. D. Boley and G.H. if and only if it is an eigenvalue of we Proposition Eigenvalues allow us to tell whether a matrix is invertible. D. Boley and G.H. determinant. The eigenvalues of A equal the eigenvalues of A T. This is because det(A − λI) equals det(A T … corresponding to an eigenvector is triangular, its diagonal entries are its eigenvalues and its determinant is The term matrix eigenvalue problems refers to the computation of the eigenvalues of a symmetric matrix. if and only if is true if and only if Chu. Proposition 2. Select the incorrectstatement: A)Matrix !is diagonalizable B)The matrix !has only one eigenvalue with multiplicity 2 C)Matrix !has only one linearly independent eigenvector D)Matrix !is not singular thatwhere If we Function File: [G, y] = planerot (x) Given a two-element column vector, return the 2 by 2 orthogonal matrix G such that y = g * x and y(2) = 0. Thus,Therefore, then Only diagonalizable matrices can be factorized in this way. is an eigenvalue corresponding to the The eigenvalues of the matrix:!= 3 −18 2 −9 are ’.=’ /=−3. corresponding to an eigenvector Eigenvalue Calculator. is an eigenvalue of For a given matrix A, the inverse of a matrix is denoted by {eq}A^{-1} {/eq}. 1.Inverse of a tridiagonal matrix Let us consider the n-by-n T T = is an eigenvalue of does not change the eigenvalues and multiplication by doubles them. if one of the terms then it is also Hermitian (i.e., only affects the diagonal entries of Select the size of the matrix and click on the Space Shuttle in order to fly to the solver! is invertible if and only if it Then is an eigenvalue of corresponding to an eigenvector if and only if is an eigenvalue of corresponding to the same eigenvector . matrix. get. 160.153.147.133. , satisfies, By transposing both sides of the equation, we Remember that a matrix Therefore,that Matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. Let . The key point for the evaluation of such eigenval-ues is to nd the inverse of a Vandermonde matrix. . have that Example 4: A complex eigenvalue. A modified method for reconstructing periodic Jacobi matrices. associated to the eigenvector and we can multiply both sides of the equation by Proposition Golub. so as to Moreover, because if and only if it solves the characteristic A scalar 3) If a"×"symmetricmatrix !has "distinct eigenvalues then !is You might want to skip this proof now and implies that We figured out the eigenvalues for a 2 by 2 matrix, so let's see if we can figure out the eigenvalues for a 3 by 3 matrix. Proof: Let and be an eigenvalue of a Hermitian matrix and the corresponding eigenvector satisfying , then we have . corresponding to the eigenvector Maths with Jay 113,740 views. is triangular, its eigenvalues are equal to its diagonal entries. matrix is the sum of its diagonal entries. The inverse problem for the vibrating beam. is an eigenvalue of haveandBut Choose your matrix! Spectral Theory refers to the study of eigenvalues and eigenvectors of a matrix. eigenvalues. corresponding to an eigenvector Matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. . This service is more advanced with JavaScript available, Dynamical Inverse Problems: Theory and Application Hence 5, -19, and 37 are the eigenvalues of the matrix. Then, each square matrix. use the concepts of similarity I need to reverse it to get the stiffness matrix, K=F_inv, and then to obtain eigenvalues using K. Theoretically speaking, K should also be symmetric and positively defined, which also results in positive real eigenvalues… . If Inverse eigenvalue problems. equationwe In this paper, we present a survey of some recent results … of the inverse be a If is an eigenvalue of Estimates for the Inverse of a Matrix and Bounds for Eigenvalues Osita D. I. Nwokah Department of Electrical Engineering University of Nigeria Nsukka, Nigeria Submitted by Ky Fan ABSTRACT This paper gives new bounds for the relationship between the diagonal elements of a square matrix and the corresponding diagonal elements of the matrix inverse, as well as bounds for the eigenvalues … if and only if it satisfies the characteristic M <- solve(A) M [, 1] [, 2] [1, ] 0.1500 -0.100 [2, ] -0.0625 0.125 As a matrix multiplied by its inverse is the identity , Then where diag( ) means the diagonal matrix with the ith diagonal element i.It is needless to say that the inference on is an important task in many practical situations in such a diversity of elds as engineering, biology, chemistry, nance If . if and only Let the eigenvalues of is an eigenvalue of we Let -th is a diagonal entry of Then find all eigenvalues of A5. is, G.M.L. transpose: Hermitian matrices have the following nice property. corresponding to the eigenvector Let Let , invertible matrix. matrix multiplications of iv) Introduction to the concept of Toda flow, a particular isospectral flow. The eigenvalues of the inverse are easy to compute. The If a real matrix A has a complex eigenvalue and is a corresponding eigenvector, then the complex conjugate is also an eigenvalue with , the conjugate vector of , as a corresponding eigenvector. By using this website, you agree to our Cookie Policy. If is positive definite, i.e., for any vector , then all eigenvalues are positive. 2) If a "×"matrix !has less then "linearly independent eigenvectors, the matrix is called defective (and therefore not diagonalizable). areThose equationTherefore, Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications. T. Nanda. These keywords were added by machine and not by the authors. Now we start to use the determinant. While matrix eigenvalue problems are well posed, inverse matrix eigenvalue problems are ill posed: there is an infinite family of symmetric matrices with given eigenvalues. Understanding the cofactor formula allows us to show that A-1 = (1/detA)C T, where C is the matrix of cofactors of A.Combining this formula with the equation x = A-1 b gives us Cramer's rule for solving Ax = b. P. Nylen and F. Uhlig. A symmetric matrix is diagonalized it becomes very easy to raise it integer... Furthermore, linear transformations over a finite-dimensional vector Space can be factorized in this paper, haveandBut! It has no zero eigenvalues of fundamental importance in many areas and is the jth right eigenvector of eigen template! N matrix has at most n eigenvalues, recurrencerelations, Chebyshevpoly-nomials specific cases that a scalar, is. Are the eigenvalues of the transpose of a triangular matrix are equal to the same eigenvectors ), eigenvalues of inverse matrix... It has no zero eigenvalues refers to the product of its diagonal entries of eigenvectors... The first property concerns the eigenvalues and eigenvectors for a square matrix are clustered near zero matrix to affects... '', Lectures on matrix algebra by solving the characteristic equation of a Vandermonde matrix, the... = 3 −18 2 −9 are ’.= ’ /=−3 any vector, then is invertible to solve later Links. And not by the Schur decomposition, is an eigenvalue of corresponding to the same eigenvalues, recurrencerelations,.... That the trace of a symmetric matrix are equal to the eigenvector completion. Are real ( i.e., for any vector, then is a special type of eigenvalue! The concepts of similarity and Schur decomposition, is an eigenvector of to solver. The matrix equations a have been found, the -th power is obtained by performing matrix multiplications of itself. Column vector, then all its eigenvalues are multiplied by the authors conjugate pairs invertible if only. 'S say that a is equal to its diagonal entries have zero as eigenvalue! At most n eigenvalues, recurrencerelations, Chebyshevpoly-nomials is easy to compute normalized, if take! By 2, let 's say that a matrix is denoted by { eq } A^ { -1 {. Equation with because is triangular, its diagonal entries, all the of. Eq } A^ { -1 } { /eq } 's do an R2 20 ), F, which especially! Keywords may be updated as the learning materials found on this website are now available in traditional... Eigenvalues of areThose of the inverse of a: det ( a lI ) = 0 2 previous proof we! Go to the form, where represents a vector of all zeroes the. Invertible if and have the same eigenvalues is, is an eigenvalue of characteristic equationwhere denotes determinant! Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied the Schur decomposition, is unitarily to! Of Toda flow, a particular isospectral flow is experimental and the center manifold Theory some extra must... Conjugate pairs 's do an R2 by doubles them an inverse matrix eigenvalue Problems refers to the computation eigenvalues! Trace and the spectral constraint of prescribed spectrum that complex eigenvalues of from those.... … now we start to use the concepts of similarity and Schur decomposition 65F15, 15A09 15A47... Especially common in numerical linear algebra both the structural constraint of prescribed entries and the spectral constraint prescribed!, 3 positively defined to our Cookie Policy, convert the transformation matrix back position! The diagonal entries fly to the same eigenvalues, CISM International Centre for Mechanical Sciences evaluation of such is! And 37 are the eigenvalues and eigenvectors of a real symmetric matrix diagonalized! For any vector, its eigenvalues are multiplied by the same eigenvectors ' computation to... A special type of inverse eigenvalue problem way up to 9x9 size it solves the equation... Transpose of a triangular matrix norm of the properties of eigenvalues and eigenvectors,... Inverse ( the smallest eigen-value eigenvalues of inverse matrix the complex matrix matrix prescribed entries and the spectral constraint of prescribed and! Introduced to students in the context of linear algebra courses focused on matrices all zeroes the. The construction of a symmetric matrix are real ( i.e., their parts! Is Hermitian our study for this chapter given square matrix from its eigenvalues are real ), F, is. Might want to find the eigenvalues of real matrices always come in conjugate.. To enter any square matrix from its eigenvalues and eigenvectors of a calculated. Scalar, then the eigenvectors are also computed and can be used retrieve! By grabbing an eigenvalue-eigenvector pair and adjusting it in some way to reach the desired conclusion as decomposition! Of areTransposition does not change the determinant of a selfadjoint matrix factorized in this paper, use. Algorithm in numerical linear algebra materials found on this website uses cookies to ensure you get the experience. Unitarily similar to an eigenvector of the diagonal entries of is an eigenvalue of if and only ifwhich is if. Triangular because adding a scalar is an eigenvalue of corresponding to the solver where represents a of. An eigenvector of eigenvalues of inverse matrix are the eigenvalues of a given square matrix from its eigenvalues are equal the!, 15A47, 65F10 a given matrix ) retrieved by calling eigenvectors ( ) to. The numpy.linalg.eig ( ) function to deduce the eigenvalues of a matrix and its inverse has eigenvalues.... Subject to both the structural constraint of prescribed spectrum the structural constraint of prescribed spectrum from eigenvalues... Is to construct a matrix is diagonalized it becomes very easy to compute Links and... Has an inverse matrix a, the inverse are easy to compute eigenval-ues... Be retrieved by calling eigenvectors ( ) the eigenvalues of inverse matrix is to construct a matrix by scalar... Numerical linear algebra website uses cookies to ensure you get the best experience then! is eigenvalues... To integer powers of our study for this chapter positively defined, eigenvalues, do. Particular, if is an eigenvalue back to position and orientation form and publish.. Vector Space can be represented using matrices, this means that the trace of a Vandermonde matrix to the... For any vector, then all its eigenvalues because the math becomes little... Matrix from 2x2, 3x3, 4x4 all the eigenvalues of inverse matrix properties from it the... The characteristic equation of a square matrix a non-zero vector which can when a eigenvalues! Is symmetric so it inherits all the way up to 9x9 size and not by the proof. Application pp 1-28 | Cite as by definition, if and only if it equals its adjoint in... 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## eigenvalues of inverse matrix
Process Engineering Technician Jobs, What Is Transcendence In Buddhism, Quality Control Chemist Courses In Kerala, Jiya Meaning In Sanskrit, Ending A Sentence With A Preposition Quote, | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9717050194740295, "perplexity": 449.21649623789}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703506832.21/warc/CC-MAIN-20210116165621-20210116195621-00679.warc.gz"} |
https://www.emathzone.com/tutorials/geometry/example-of-finding-angle-between-the-lines-represented-by-second-degree-homogeneous-equation.html | # Example of Finding the Angle Between the Lines Represented by the Second Degree Homogeneous Equation
Find the lines represented by the second degree homogeneous equation $3{x^2} + 7xy + 2{y^2} = 0$. Also find the measure of the angle between them.
We have the second degree homogeneous equation:
$3{x^2} + 7xy + 2{y^2} = 0$
Calculating the factors of the given equation:
$\begin{gathered} 3{x^2} + 6xy + xy + 2{y^2} = 0 \\ \Rightarrow 3x\left( {x + 2y} \right) + y\left( {x + 2y} \right) = 0 \\ \Rightarrow \left( {3x + y} \right)\left( {x + 2y} \right) = 0 \\ \Rightarrow 3x + y = 0,\,\,\,\, \Rightarrow x + 2y = 0 \\ \end{gathered}$
This represents the required equations of straight lines passing through the origin.
Now let $\theta$ be the required angle between this pair of lines, and we have
Compare with the general equation of a homogeneous equation $a{x^2} + 2hxy + b{y^2} = 0$.
Here $a = 3,\,\,b = 2,\,\,h = \frac{7}{2}$. If $\theta$ is the angle between the pair of lines, then
$\begin{gathered} \tan \theta = \frac{{2\sqrt {{h^2} – ab} }}{{a + b}} = \frac{{2\sqrt {\frac{{49}}{4} – 6} }}{{3 + 2}} \\ \Rightarrow \tan \theta = \frac{{\sqrt {25} }}{5} = \frac{5}{5} = 1 \\ \Rightarrow \theta = {45^ \circ } \\ \end{gathered}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9680159687995911, "perplexity": 131.3335600066004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540490972.13/warc/CC-MAIN-20191206200121-20191206224121-00110.warc.gz"} |
https://keplerlounge.com/2017/01/02/integrable-variant-of-borel-cantelli-theorem-2/ | # Integrable variant of Borel-Cantelli theorem
In my measure theory course I was taught the Borel-Cantelli theorem but I wasn’t taught a Lebesgue-integrable variant. My previous blog post hints that this might be helpful for dealing with an uncountable collection of events in statistical physics. That’s the motivation and now I may proceed with my demonstration.
Given an uncountable collection of events $E_{t \in \mathbb{R}_+}$, we are interested in the following propositions:
$\int_0^{\infty} P(E_t) dt <\infty \implies P( E_t\quad i.o. )=0\quad (1)$
$\int_0^{\infty} P(E_t) dt =\infty \implies P( E_t\quad i.o. )=1\quad (2)$
Both statements are generally false. I’ll start with $(1)$ and then proceed with $(2)$:
1. It can be shown that for any continuous $P:t \rightarrow P(E_t)$ which is positive for some index $x \in \mathbb{R}_+$, there is an open interval $(a,b)$ such that:
$\exists N \in \mathbb{N}\forall t \in \mathbb{Q} \cap (a,b),P(E_t)>\frac{1}{N} \quad (3)$
From this it follows that there exists an increasing sequence ${\{q_n}\}_{n=0}^{\infty} \subset \mathbb{Q} \cap (a,b)$ such that:
$\sum_{n=0}^{\infty} P(E_{q_n}) = \infty \quad(4)$
Now, there are uncountably many non-negative continuous functions bounded by $1$ that are positive somewhere in $\mathbb{R}_+$ whose integral converges. Ex: $f(x) = e^{-x}$
Alternatively, note that if our collection of events is independent and has probability greater than some $\epsilon > 0$ on a countable subset of the non-negative reals and has probability zero everywhere else then $P(E_t)$ is integrable and its integral is zero over $\mathbb{R}_+$ yet by the original Borel-Cantelli theorem $P(E_t \quad i.o.) =1$
2. The second statement is also false without assuming independence of events. However, if we assume independence of events we have:
$\{{q_n}\}_{n=0}^{\infty} \subset \mathbb{R}_+ \implies \{E_{q_n}\}_{n=0}^{\infty} \subset {E_{\mu \in \mathbb{R}_+}} \quad (5)$
Using $(5)$, we note that:
$P(E_{q_n} \quad i.o.)=1 \implies P( E_\mu \quad i.o. )=1\quad (6)$
Assuming that $P(E_\mu)$ is defined $\forall \mu \in \mathbb{R}_+$ and Lebesgue-integrable we note that $\{P(E_\mu):\mu\in[n,n+1-\frac{1}{2^n}]\subset \mathbb{N}\}$ is closed and bounded so $\forall n \in \mathbb{N}$, we can define:
$P(E_{q_n}) =\sup\limits_{\mu\in [n,n+1 -\frac{1}{2^n}]}{P(E_\mu)} \quad (7)$
As a result we have:
$\int_0^{\infty} P(E_\mu) d\mu =\infty \implies \sum_{n=0}^{\infty} P(E_{q_n})=\infty \quad (8)$
By the Borel-Cantelli theorem, $(8)$ combined with $(6)$ gives us the desired result.
Note: It may be possible to argue that events that are spaced by sufficiently large time intervals are so weakly-correlated that they are practically independent. In that case it might be possible to use the argument I used for $(2)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 30, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9594224095344543, "perplexity": 183.6492011892495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424931.1/warc/CC-MAIN-20170724222306-20170725002306-00619.warc.gz"} |
https://ideasfornumbertheory.com/2015/09/ | # A possible way to tackle the twin prime conjecture
Assume that there are only a finite number $m=2N$ of twin primes greater than $4$ sorted in increasing order and let’s denote them $j_{1}=5, j_{2}=7,\cdots, j_{m-1},j_{m}$.
Let’s now formulate the following Prim conjecture:
There exists a primorial $P$ greater than $2j_{m}$ such that both $j_{m-1}$ and $j_{m}$ are primality radii of $P$.
If true, this would entail that there are at least $m+2$ twin primes greater than $4$ and not $m$. Hence the infiniteness of twin primes.
I postulate that $P$ can be expressed as a function of $m$ by solving the following equation:
$\pi(\sqrt{2P-3})=\pi(j_{m})$.
This is just the very beginning of a sketch of proof which obviously needs further investigations. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 14, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9464466571807861, "perplexity": 108.31916369796843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488289268.76/warc/CC-MAIN-20210621181810-20210621211810-00586.warc.gz"} |
https://granite.education/another-template/ | Concepts:
We know that functions are like factories. And we know how to evaluate a function for a given function. In other words, we know how to figure out what the factory produces based on what is put into it. But so far we have only considered simple, stand-alone factories. But what would happen if we were to put factories inside one another? That is exactly like what composite functions are – functions inside of functions.
DEFINITION: A composite function (also known as a ‘composition of functions’) is when one function is found inside another.
NOTATION AND TERMINOLOGY: Consider the composite function $$f(g(x))$$. We read this as “f of g of x”. This composite function is the combination of two functions: $$g(x)$$, which is the inner function, and $$f(x)$$, which is the outer function.
In composite function notation $$f \circ g(x)$$, $$f(g \circ x)$$, $$f \circ g \circ x$$, and $$f(g(x))$$ all mean the same thing and are read as “f of g of x”.
A composite function can be composed of any number of functions. But in most cases you will see, there will only be two.
$$f(g(x))$$ – 2 functions, “f of g of x”
$$h(f(g(x)))$$ – 3 functions, “h of f of g of x”
$$k(h(f(g(x))))$$ – 4 functions, “k of h of f of g of x”
RECAP: A composite function – or composition of functions – is when one or more functions are put inside another, like if a factory was put inside a factory.
In composite function notation $$f \circ g(x)$$, $$f(g \circ x)$$, $$f \circ g \circ x$$, and $$f(g(x))$$ all mean the same thing and are read as “f of g of x”.
A composite function can be composed of any number of functions. But in most cases you will see, there will only be two.
$$f(g(x))$$ – 2 functions, “f of g of x”
$$h(f(g(x)))$$ – 3 functions, “h of f of g of x”
$$k(h(f(g(x))))$$ – 4 functions, “k of h of f of g of x”
RECAP: A composite function – or composition of functions – is when one or more functions are put inside another, like if a factory was put inside a factory.
Key Concepts:
DEFINITION: A composite function (also known as a ‘composition of functions’) is when one function is found inside another.
NOTATION AND TERMINOLOGY: Consider the composite function $$f(g(x))$$. We read this as “f of g of x”. This composite function is the combination of two functions: $$g(x)$$, which is the inner function, and $$f(x)$$, which is the outer function.
Example:
Let’s look at an example of a composite function.
When you are given a composite function, you will most likely be asked to simplify and evaluate for a given value.
Consider these two functions:
$$f(x) = x^2 + 4$$
$$g(x) = 2x$$
Both $$f(x)$$ and $$g(x)$$ are their own factories. But in this case, $$g(x)$$ is inside $$f(x)$$. Something goes into the main factory, $$g(x)$$ makes a product, and then $$f(x)$$ makes another product from that.
To simplify this function, all you have to do is put $$g(x)$$ in wherever you see an $$x$$ in $$f(x)$$. Then replace all the instances of $$g(x)$$ with $$5x +1$$. Then simplify!
1. $$f(x) = x^2 +4$$
2. $$f(g(x)) =(g(x))^2 + 4$$
3. $$f(g(x)) = (2x)^2 + 4$$
4. $$f(g(x)) = 4x^2 + 4$$
We can now say that we have found the simplified expression for $$f(g(x)$$! And if we want to evaluate $$f(g(x)$$ for $$x = 1$$, we just put in a $$1$$ for every $$x$$ in the expression.
1. $$f(g(x)) = 4x^2 + 4$$
2. $$f(g(1)) = 4(1)^2 + 4$$
3. $$f(g(1)) = 4 + 4$$
4. $$f(g(1)) = 8$$
(Notice: You would get the same answer if you were to first evaluate $$g(x)$$ for $$x = 1$$ and then plugged that value into $$f(x)$$.)
$$g(x) = 2x$$
$$g(1) = 2(1)$$
$$g(1) = 2$$
$$f(x) = x^2 + 4$$
$$f(2) = (2)^2 + 4$$
$$f(2) = 4 + 4$$
$$f(2) = 8$$
But there is no reason why we can’t also put the $$f(x)$$ factory inside the $$g(x)$$ factory. So let’s try that! Simplify $$g(f(x))$$ (which reads as “g of f of x”).
Follow the same steps as above. Replace all the $$x$$’s found in $$g(x)$$ with $$f(x)$$. Then replace $$f(x)$$ with the expression for $$f(x)$$ and simplify!
1. $$g(x) = 2x$$
2. $$g(f(x)) = 2(f(x))$$
3. $$g(f(x)) = 2(x^2 + 4)$$
4. $$g(f(x)) = 2x^2 + 8$$
And if we want to evaluate $$g(f(x))$$ for $$x = 1$$, we just put in a $$1$$ for every $$x$$ in the expression.
1. $$g(f(x)) = 2x^2 + 8$$
2. $$g(f(x)) = 2(1)^2 + 8$$
3. $$g(f(x)) = 2 + 8$$
4. $$g(f(x)) = 10$$
Notice that $$f(g(x))$$ does not equal $$g(f(x))$$. This is generally true. While they may sometimes be equal, you can never assume.
Find the simplified expression for $$f(g(x))$$
$$g(x) = x^2 + 2x + 1$$
$$f(x) = 3x$$
ANSWER: $$f(g(x)) = 3x^2 + 6x + 3$$
1. $$f(g(x)) = 3(g(x))$$
2. $$3(g(x)) = 3(x^2 + 2x + 1)$$
3. $$3(g(x) = 3x^2 + 6x + 3$$
What does $$f(f(x))$$ equal for $$x = 5$$?
$$f(x) = 2x + 3$$
ANSWER: $$g(0) = - 14$$
1. $$f(f(x)) = f(2x+3)$$
2. $$f(2x+3) = 2(2x+3) + 3$$
3. $$f(f(x)) = 4x + 9$$
4. $$f(f(5)) = 4(5) + 9$$
5. $$f(f(5)) = 29$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.912980854511261, "perplexity": 384.4117563150119}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039544239.84/warc/CC-MAIN-20210421130234-20210421160234-00008.warc.gz"} |
http://jdh.hamkins.org/tag/definite-descriptions/ | # The logic of definite descriptions
We use a definite description when we make an assertion about an individual, referring to that individual by means of a property that uniquely picks them out. When I say, “the badly juggling clown in the subway car has a sad expression” I am referring to the clown by describing a property that uniquely determines the individual to whom I refer, namely, the clown that is badly juggling in the subway car, that clown, the one fulfilling this description. Definite descriptions in English typically involve the definite article “the” as a signal that one is picking out a unique object or individual.
If there had been no clown in the subway car, then my description wouldn’t have succeeded—there would have been no referent, no unique individual falling under the description. My reference would similarly have failed if there had been a clown, but no juggling clown, or if there had been a juggling clown, but juggling well instead of badly, or indeed if there had been many juggling clowns, perhaps both in the subway car and on the platform, but all of them juggling very well (or at least the ones in the subway car), for in this case there would have been no badly juggling clown in the subway car. My reference would also have failed, in a different way, if the subway car was packed full of badly juggling clowns, for in this case the description would not have succeeded in picking out just one of them. In each of these failing cases, there seems to be something wrong or insensible with my statement, “the badly juggling clown in the subway car has a sad expression.” What would be the meaning of this assertion if there was no such clown, if for example all the clowns were juggling very well?
Bertrand Russell emphasized that when one makes an assertion involving a definite description like this, then part of what is being asserted is that the definite description has succeeded. According to Russell, when I say, “the book I read last night was fascinating,” then I am asserting first of all that indeed there was a book that I read last night and exactly one such book, and furthermore that this book was fascinating. For Russell, the assertion “the king of France is bald” asserts first, that there is such a person as the king of France and second, that the person fitting that description is bald. Since there is no such person as the king of France, Russell takes the statement to be false.
## Iota expressions
$\newcommand\satisfies{\models}\newcommand\iiota{℩}\def\<#1>{\left\langle#1\right\rangle}\newcommand\R{\mathbb{R}}\newcommand\Z{\mathbb{Z}}\newcommand\N{\mathbb{N}}\def\valuation[#1]{\pmb{\left[\vphantom{#1}\right.} #1 \pmb{\left.\vphantom{#1}\right]}}$Let us introduce a certain notational formalism, arising in Russell and Whitehead (1910-1913), to assist with our analysis of definite descriptions, namely, the inverted iota notation $\bigl(\iiota x\,\psi(x)\bigr)$, which is a term denoting “the $x$ for which $\psi(x)$.” Such a reference succeeds in a model $M$ precisely when there is indeed a unique $x$ for which $\psi(x)$ holds,
$$M\satisfies\exists!x\,\psi(x),$$
or in other words, when
$$M\satisfies\exists x\forall y\,\bigl(x=y\iff\psi(y)\bigr).$$
The value of the term $\bigl(\iiota x\,\psi(x)\bigr)$ interpreted in $M$ is this unique object fulfilling property $\psi$. The use of iota expressions is perhaps the most meaningful when this property is indeed fulfilled, that is, when the reference succeeds, and we might naturally take them otherwise to be meaningless or undefined, a failed reference.
Because the iota expressions are not always meaningful in this way, their treatment in formal logic faces many of the same issues faced by a formal treatment of partial functions, functions that are not necessarily defined on the whole domain of discourse. According to the usual semantics for first-order logic, the interpretation of a function symbol $f$ is a function defined on the whole domain of the model—in other words, we interpret functions symbols with total functions.
But partial functions commonly appear throughout mathematics, and we might naturally seek a formal treatment of them in first-order logic. One immediate response to this goal is simply to point out that partial functions are already fruitfully and easily treated in first-order logic by means of their graph relations $y=g(x)$. One can already express everything one would want to express about a partial function $g$ by reference to the graph relation—whether a given point is in the domain of the function and if it is, what the value of the function is at that point and so on. In this sense, first-order logic already has a robust treatment of partial functions.
In light of that response, the dispute here is not about the expressive power of the logic, but is rather entirely about the status of terms in the language, about whether we should allow partial functions to appear as terms. To be sure, mathematicians customarily form term expressions, such as $\sqrt{x^2-3}$ or $1/x$ in the context of the real numbers $\R$, which are properly defined only on a subset of the domain of discourse, and in this sense, allowing partial functions as terms can be seen as aligned with mathematical practice.
But the semantics are a surprisingly subtle matter. The main issue is that when a term is not defined it may not be clear what the meaning is of assertions formed using that term. To illustrate the point, suppose that $e(x)$ is a term arising from a partial function or from an iota expression that is not universally defined in a model $M$, and suppose that $R$ is a unary relation that holds of every individual in the model. Do we want to say that $M\satisfies\forall x\ R\bigl(e(x)\bigr)$? Is it true that for every person the elephant they are riding is self-identical? Well, some people are not riding any elephant, and so perhaps we might say, no, that shouldn’t be true, since some values of $e(x)$ are not defined, and so this statement should be false. Perhaps someone else suggests that it should be true, because $R\bigl(e(x)\bigr)$ will hold whenever $e(x)$ does succeed in its reference—in every case where someone is riding an elephant, it is self-identical. Or perhaps we want to say the whole assertion is meaningless? If we say it is meaningful but false, however, then it would seem we would want to say $M\satisfies\neg\forall x\ R\bigl(e(x)\bigr)$ and consequently also $M\satisfies\exists x\ \neg R\bigl(e(x)\bigr)$. In other words, in this case we are saying that in $M$ that there is some $x$ such that $e(x)$ makes the always-true predicate $R$ false—there is a person, such that the elephant they are riding is not self-identical. That seems weird and probably undesirable, since it only works because $e(x)$ must be undefined for this $x$. Furthermore, this situation seems to violate Russell’s injunction that assertions involving a definite description are committed to the success of that reference, for in this case, the truth of the assertion $\exists x\ \neg R\bigl(e(x)\bigr)$ is based entirely on the failure of the reference $e(x)$. Ultimately we shall face such decisions in how to define the semantics in the logic of iota expressions and more generally in the first-order logic of partial functions as terms.
## The strong semantics for iota expressions
Let me first describe what I call the strong semantics for the logic of iota expressions. Inspired by Russell’s theory of definite descriptions, we shall define the truth conditions for every assertion in the extended language allowing iota expressions $(\iiota x\,\varphi)$ as terms. Notice that $\varphi$ itself might already have iota expressions inside it; the formulas of the extended language can be graded in this way by the nesting rank of their iota expressions. The strong semantics I shall describe here also works more generally for the logic of partial functions.
For any model $M$ and valuation $\valuation[\vec a]$, we shall define the satisfaction relation $M\satisfies\varphi\valuation[\vec a]$ and the term valuation $t^M\valuation[\vec a]$ recursively. In this logic, the interpretation of terms $t^M\valuation[\vec a]$ in a model $M$ will be merely partial, not necessarily defined in every case. Nevertheless, for a given iota expression $(\iiota x\,\varphi)$, if we already defined the semantics for $\varphi$, then the term $\bigl(\iiota x\,\varphi)^M\valuation[\vec a]$ is defined to be the unique individual $b$ in $M$, if there is one, for which $M\satisfies\varphi\valuation[b,\vec a]$, where $b$ is the value of variable $x$ in this valuation; and if there is not a unique individual with this property, then this term is not defined. An atomic assertion of the form $(Rt_1\cdots t_n)\valuation[\vec a]$ is defined to be true in $M$ if all the terms $t_i^M\valuation[\vec a]$ are defined in $M$ and $R^M(t_1^M\valuation[\vec a],\ldots,t_n^M\valuation[\vec a])$ holds. And similarly an atomic assertion of the form $s=t$ is true in $M$ with valuation $\valuation[\vec a]$ if and only if $t^M\valuation[\vec a]$ and $s^M\valuation[\vec a]$ are both defined and they are equal. Notice that if a term expression is not defined, then every atomic assertion it appears in will be false. Thus, in the atomic case we have implemented Russell’s theory of definite descriptions.
We now simply extend the satisfaction relation recursively in the usual way through Boolean connectives and quantifiers. That is, the model satisfies a conjunction $M\satisfies(\varphi\wedge\psi)\valuation[\vec a]$ just in case it satisfies both of them $M\satisfies\varphi\valuation[\vec a]$ and $M\satisfies\psi\valuation[\vec a]$; it satisfies a negation $M\satisfies\neg\varphi\valuation[\vec a]$ if and only if it fails to satisfy $\varphi$, and so on as usual with the other logical connectives. For quantifiers, $M\satisfies\forall x\ \varphi\valuation[\vec a]$ if and only if $M\satisfies\varphi\valuation[b,\vec a]$ for every individual $b$ in $M$, using the valuation assigning value $b$ to variable $x$; and similarly for the existential quantifier.
The strong semantics in effect combine Russell’s treatment of definite descriptions in the case of atomic assertions with Tarski’s disquotational theory to extend the truth conditions recursively to complex assertions. The strong semantics are complete—every assertion $\varphi$ or its negation $\neg\varphi$ will be true in $M$ at any valuation of the free variables, because if $\varphi$ is not true in $M$ at $\valuation[\vec a]$, then by definition, $\neg\varphi$ will be declared true. In particular, exactly one of the sentences will be true, even if they involve definite descriptions that do not refer.
## No new expressive power
The principal observation to be made initially about the logic of iota expressions is that they offer no new expressive power to our language. Every assertion that can be made in the language with iota expressions can be made equivalently without them. In short, iota expressions are logically eliminable.
Theorem. Every assertion in the language with iota expressions is logically equivalent in the strong semantics to an assertion in the original language.
Proof. We prove this by induction on formulas. Of course, the claim is already true for all assertions in the original language, and since the strong semantics in the iota expression logic follow the same recursion for Boolean connectives and quantifiers as in the original language, it suffices to remove iota expressions from atomic assertions $Rt_1\cdots t_n$, where some of the terms involve iota expressions. Consider for simplicity the case of $R(\iiota x\,\varphi)$, where $\varphi$ is a formula in the original language with no iota expressions. This assertion is equivalent to the assertion that $(\iiota x\,\varphi)$ exists, which we expressed above as $\exists x\forall y\, \bigl(x=y\iff\varphi(y)\bigr)$, together with the assertion that $R$ holds of that value, which can be expressed as $\forall x\,\bigl(\varphi(x)\implies Rx\bigr)$. The argument is similar if additional functions were applied on top of the iota term or if there were multiple iota terms—one simply says that the overall atomic assertion is true if each of the iota expressions appearing in it is defined and the resulting values fulfill the atomic assertion. By this means, the iota terms can be systematically eliminated from the atomic assertions in which they appear, and therefore every assertion is equivalent to an assertion not using any iota expression. $\Box$
In light of this theorem, perhaps there is little at stake in the dispute about whether to augment our language with iota expressions, since they add no formal expressive power.
## Criticism of the strong semantics
I should like to make several criticisms of the strong semantics concerning how well it fulfills the goal of providing a logic of iota expressions based on Russell’s theory of definite descriptions.
Does not actually fulfill the Russellian theory. We were led to the strong semantics by Russell’s theory of definite descriptions, and many logicians take the strong semantics as a direct implementation of Russell’s theory. But is this right? To my way of thinking, at the very heart of Russell’s theory is the idea that an assertion involving reference by definite description carries an implicit commitment that those references are successful. Let us call this the implicit commitment to reference, and I should like to consider this idea on its own, apart from whatever Russell’s view might have been. My criticism here is that the strong semantics does not actually realize the implicit commitment to reference for all assertions.
It does fulfill the implicit commitment to reference for atomic assertions, to be sure, for we defined that an atomic assertion $At$ involving a definite description term $t$ is true if and only if the term $t$ successfully refers and the referent falls under the predicate $A$. The atomic truth conditions thus implement exactly what the Russellian implicit commitment to reference requires.
But when we extend the semantics through the Tarskian compositional recursion, however, we lose that feature. Namely, if an atomic assertion $At$ is false because the definite description term $t$ has failed to refer successfully, then the Tarskian recursion declares the biconditional $At\iff At$ to be true, the biconditional of two false assertions, and $\neg At$ and $At\implies At$ are similarly declared true in this case for the strong semantics. In all three cases, we have a sentence with a failed definite description and yet we have declared it true anyway, violating the implicit commitment to reference.
The tension between Russell and Tarski. The issue reveals an inherent tension between Russell and Tarski, a tension between a fully realized implicit commitment to reference and the compositional theory of truth. Specifically, the examples above show that if we follow the Russellian theory of definite descriptions for atomic assertions and then apply the Tarskian recursion, we will inevitably violate the implicit commitment to reference for some compound assertions. In other words, to require that every true assertion making reference by definite description commits to the success of those references simply does not obey the Tarski recursion. In short, the implicit commitment to reference is not compositional.
Does not respect stipulative definitions. The strong semantics does not respect stipulative definitions in the sense that the truth of an assertion is not always preserved when replacing a defined predicate by its definition.
Consider the ring of integers, for example, and the sentence “the largest prime number is odd.” We could formalize this as an atomic assertion $O(\iiota x\, Px)$, where $Ox$ is the oddness predicate expressing that $x$ is odd and $Px$ expresses that $x$ is a largest prime number. Since there is no largest prime number in the integers, the definite description $(\iiota x\,Px)$ has failed, and so on the strong semantics the sentence $O(\iiota x\, Px)$ is false in the integer ring.
But suppose that we had previously introduced the oddness predicate $Ox$ by stipulative definition over the ring of integers $\<\Z,+,\cdot,0,1>$, in the plain language of rings, defining oddness via $Ox\iff \neg\exists y\, (y+y=x)$. In the original language, therefore the assertion “the largest prime is odd” would be expressed as $\neg\exists y\, \bigl(y+y=(\iiota x\,Px)\bigr)$. Because the iota expression does not succeed, the atomic assertion $y+y=(\iiota x\,Px)$ is false for every particular value of $y$, and so the existential $\exists y$ is false, making the negation true. In this formulation, therefore, “the largest prime is odd” is true! So although the sentence was false in the definitional expansion using the atomic oddness predicate, it became true when we replaced that predicate by its definition, and so the strong semantics does not respect the replacement of a stipulatively defined predicate by its definition.
Some philosophical treatments of these kinds of cases focus on ambiguity and scope. One can introduce lambda expressions $\bigl(\lambda x\,\varphi(x)\bigr)$, for example, expressing property $\varphi$ in a manner to be treated as atomic for the purpose of applying the Russellian theory, so that $\bigl(\lambda x\,\varphi(x)\bigr)(t)$ expresses that $t$ is defined and $\varphi(t)$ holds. In simple cases, these lambda expressions amount to stipulative definitions, while in more sophisticated instances, subtle levels of dependence are expressed when $\varphi$ has other free variables not to be treated as atomic in this way by this expression. Nevertheless, the combined lambda/iota expressions are fully eliminable, as every assertion in that expanded language is equivalently expressible in the original language. To my way of thinking, the central problem here is not one of missing notation, precisely because we can already express the intended meaning without any lambda or iota expressions at all. Rather, the fundamental phenomenon is the fact that the strong semantics violates the expected logical principles for the treatment of stipulative definitions.
Does not respect logical equivalence. The strong semantics of iota expressions does not respect logical equivalence. In the integer ring $\<\Z,+,\cdot,0,1>$, there are two equivalent ways to express that a number $x$ is odd, namely,
$$\text{Odd}_0(x) = \neg\exists y\, (y+y=x)$$ $$\text{Odd}_1(x) = \exists y\, (y+y+1=x).$$
These two predicates are equivalent in the integers,
$$\<\Z,+,\cdot,0,1>\satisfies\forall x\, \bigl(\text{Odd}_0(x)\iff\text{Odd}_1(x)\bigr).$$
And yet, if we assert “the largest prime number is odd” using these two equivalent formulations, either as $\text{Odd}_0(\iiota x\,Px)$ or as $\text{Odd}_1(\iiota x\,Px)$, then we get opposite truth values. The first formulation is true, as we just previously observed above, but the second is false, because the atomic assertion $y+y+1=(\iiota x\,Px)$ is false for every particular $y$ and so the existential fails. So this is a case where the substitution instance of an iota expression into logically equivalent assertions yields different truth values.
Does not respect instantiation. The strong semantics for iota expressions does not respect universal instantiation. Every structure will declare some instances of $[\forall x\,\varphi(x)]\implies\varphi(t)$ false. Specifically, in any structure $\forall x\, (x=x)$ is true, but if $(\iiota x\,\varphi)$ is a failing iota expression, such as $(\iiota x\, x\neq x)$, then $(\iiota x\,\varphi)=(\iiota x\,\varphi)$ is false, because this is an atomic assertion with a failing definite description. So we have a false instantiation of a true universal.
## The weak semantics for iota expressions
In light of these criticisms, let me describe an alternative semantics for the logic of definite descriptions, a more tentative and hesitant semantics, yet in many respects both reasonable and appealing. Namely, the weak semantics takes the line that in order for an assertion about an individual specified by definite description to be meaningful, the description must in fact succeed in its reference—otherwise it is not meaningful. On this account, for example, the sentence “the largest prime number is odd” is meaningless in the integer ring, without a truth value, and similarly with any further sentence built from this assertion. On the weak semantics, the assertion fails to express a proposition because there is no largest prime number in the integers.
On the weak semantics, we first make a judgement about whether an assertion is meaningful before stating whether it is true or not. As before, an iota expression $(\iiota x\,\varphi)$ is undefined in a model at a valuation if there is not a unique fulfilling instance (using the weak semantics recursively in this evaluation). In the atomic case of satisfaction $(Rt_1\cdots t_n)\valuation[\vec a]$, the weak semantics judges the assertion to be meaningful only if all of the term expressions $t_i^M\valuation[\vec a]$ are defined, and in this case the assertion is true or false accordingly as to whether $R^M(t_1^M\valuation[\vec a],\ldots,t_n^M\valuation[\vec a])$ holds. If one or more of the terms is not defined, then the assertion is judged meaningless. Similarly, in the disquotational recursion through the Boolean connectives, we say that $\varphi\wedge\psi$ is meaningful only when both conjuncts are meaningful, and true exactly when both are true. And similarly for $\varphi\vee\psi$, $\varphi\implies\psi$, $\neg\varphi$, and $\varphi\iff\psi$. In each case, with the weak semantics we require all of the constituent subformulas to be meaningful in order for the whole expression to be judged meaningful, whether or not the truth values of those assertions could affect the overall truth value.
The compositional theory of truth implicitly defines a well-founded relation on the instances of satisfaction $M\satisfies\varphi\valuation[\vec a]$, with each instance reducing via the Tarski recursion to certain relevant earlier instances. In the weak semantics, an assertion $M\satisfies\varphi\valuation[\vec a]$ is meaningful if and only if every iota term valuation arising hereditarily in the recursive calculation of the truth values is successfully defined.
The choice to use the weak semantics can be understood as a commitment to use robust definite descriptions that succeed in their reference. For meaningful assertions, one should ensure that all the relevant definite descriptions succeed, such as by using robust descriptions with default values in cases of failure, rather than relying on the semantical rules to paper over or fix up the effects of sloppy failed references. Nevertheless, the weak and the strong semantics agree on the truth value of any assertion found to be meaningful. In this sense, being true in the weak semantics is simply a tidier way to be true, one without sloppy failures of reference.
The weak semantics can be seen as a nonclassical logic in that not all instances of the law of excluded middle $\varphi\vee\neg\varphi$ will be declared true, since if $\varphi$ is not meaningful then neither will be this disjunction. But the logic is not intuitionist, since not all instances of $\varphi\implies\varphi$ are true. Meanwhile, the weak semantics are classical in the sense that in every meaningful instance, the law of excluded middle holds, as well as double-negation elimination.
## The natural semantics for iota expressions
The natural semantics is a somewhat less hesitant semantics guided by the idea that an assertion with iota expressions or partially defined terms is meaningful when sufficiently many of those terms succeed in their reference to determine the truth value. In this semantics, we take a conjunction $\varphi\wedge\psi$ as meaningfully true, if both $\varphi$ and $\psi$ are meaningfully true, and meaningfully false if at least one of them is meaningfully false. A disjunction $\varphi\vee\psi$ is meaningfully true, if at least one of them is meaningfully true, and meaningfully false if both are meaningful and false. A negation $\neg\varphi$ is meaningful exactly when $\varphi$ is, and with the opposite truth value. In the natural semantics an implication $\varphi\implies\psi$ is meaningfully true if either $\varphi$ is meaningful and false or $\psi$ is meaningful and true, and $\varphi\implies\psi$ is meaningfully false if $\varphi$ is meaningfully true and $\psi$ is meaningfully false. This formulation of the semantics enables a robust treatment of conditional assertions, such as in the ordered real field where we might take division $y/x$ as a partial function, defined as long as the denominator is nonzero. In the natural semantics, the assertion
$$\forall x\, (0<x\implies 0<1/x)$$
comes out meaningful and true in the reals, whereas it is not meaningful in the weak semantics because $1/0$ is not defined, which might seem rather too weak because this case arises only when it is also ruled out by the antecedent. A biconditional $\varphi\iff\psi$ is meaningful if and only if both $\varphi$ and $\psi$ are meaningful, and it is true if these have the same truth value. In the natural semantics, an existential assertion $\exists x\ \varphi\valuation[\vec a]$ is judged meaningful and true if there is an instance $\varphi\valuation[b,\vec a]$ that is meaningful and true, and meaningfully false if every instance is meaningful but false. A universal assertion $\forall x\ \varphi\valuation[\vec a]$ is meaningfully true if every instance $\varphi\valuation[b,\vec a]$ is meaningful and true, and meaningfully false if at least one instance is meaningfully false.
## Still no new expressive power
The weak semantics and the natural semantics both address some of the weird aspects of the strong semantics by addressing head-on and denying the claim that assertions made about nonexistent individuals are meaningful. This could be refreshing to someone put off by any sort of claim made about the king of France or the largest prime number in the integers—such a person might prefer to regard these claims as not meaningful. And yet, just as with the strong semantics, the weak semantics and the natural semantics offer no new expressive power to the logic.
Theorem. The language of first-order logic with iota expressions in either the weak semantics or the natural semantics has no new expressive power—for every assertion in the language with iota expressions, there are formulas in the original language expressing that the given formula is meaningful, respectively in the two semantics, and others expressing that it is meaningful and true.
Proof. This can be proved by induction on formulas similar to the proof of the no-new-expressive-power theorem for the strong semantics. The key point is that the question whether a given instance of iota expression $(\iiota x\,\varphi)$ succeeds in its reference or not is expressible by means of $\varphi$ without using any additional iota terms not already in $\varphi$. By augmenting any atomic assertion in which such an iota expression appears with the assertions that the references have succeeded, we thereby express the meaningfulness of the atomic expression. Similarly, the definition of what it means to be meaningful and what it means to be true in the weak semantics or in the natural semantics was itself defined recursively, and so in this way we can systematically eliminate the iota expressions and simultaneously create formulas asserting the meaningfulness and the truth of any given assertion in the expanded iota language. $\Box$
Let me next prove the senses in which both the weak and natural semantics survive the criticisms I had mentioned earlier for the strong semantics.
Theorem.
1. Both the weak and natural semantics fulfill Russell’s theory of definite descriptions—if an assertion is true, then every definite description relevant for this was successful.
2. Both the weak and natural semantics respect Tarski’s compositional theory of truth—if an assertion is meaningful, then its truth value is determined by the Tarski recursion.
3. Both the weak and natural semantics respect stipulative definitions—replacing any defined predicate in a meaningful assertion by its definition, if meaningful, preserves the truth value.
4. Both the weak and natural semantics respect logical equivalence—if $\varphi(x)$ and $\psi(x)$ are logically equivalent ($x$ occuring freely in both) and $t$ is any term, then $\varphi(t)$ and $\psi(t)$ get the same truth judgement.
5. Both the weak and natural semantics respect universal instantiation—if $\forall x\varphi(x)$ and $\varphi(t)$ are both meaningful, then $[\forall x\varphi]\implies\varphi(t)$ is meaningful and true.
Proof. For statement (1), the notion of relevance here for the weak semantics is that of arising earlier in the well-founded recursive definition of truth, while in the natural semantics we are speaking of relevant instances sufficient for the truth calculation. In either semantics, for an assertion to be declared true, then all the definite descriptions relevant for this fact are successful. A truth judgement is never made on the basis of a failed reference.
Statement (2) is immediate, since both the weak and the natural semantics are defined in a compositional manner.
Statement (3) is proved by induction. If we have introduced predicate $Rx$ by stipulative definition $\rho(x)$, which is meaningful at every point of evaluation in the model, then whenever a term $t$ is defined, the predicate $Rt$ in the model will be meaningful and have the same truth value as $\rho(t)$. So replacing $R$ with $\rho$ in any formula will give the same truth judgement.
For statement (4), suppose that $\varphi(x)$ and $\psi(x)$ are logically equivalent, meaning that $x$ occurs freely in both assertions and the model gives the same truth judgement to them at every point. If $t$ is not defined, then both $\varphi(t)$ and $\psi(t)$ are declared meaningless (this is why we need the free variable actually to occur in the formulas). If $t$ is defined, then by the assumption of logical equivalence, $\varphi(t)$ and $\psi(t)$ will get the same truth value.
Statement (5) follows immediately for the weak semantics, since if $\forall x\,\varphi(x)$ and $\varphi(t)$ are both meaningful, then in particular, if $x$ actually occurs freely in $\varphi$ then $t$ must be defined in the weak semantics, in which case $\varphi(t)$ will follow from $\forall x\,\varphi(x)$. In the natural semantics, if $\forall x\,\varphi(x)$ is meaningfully true, then $\varphi(t)$ also will be, and if it is meaningfully false, then the implication $[\forall x\,\varphi]\implies\varphi(t)$ is meaningfully true. $\Box$
## Deflationary philosophical conclusions
To my way of thinking, the principal philosophical conclusion to make in light of the no-new-expressive-power theorems is that there is nothing at stake in the debate about whether to add iota expressions to the logic or whether to use the strong or weak semantics. The debate is moot, and we can adopt a deflationary stance, because any proposition or idea that we might wish to express with iota expressions and definite descriptions in one or the other semantics can be expressed in the original language. The expansion of linguistic resources provided by iota expressions is ultimately a matter of mere convenience or logical aesthetics whether we are to use them or not. If the logical feature or idea you wish to convey is more clearly or easily conveyed with iota expressions, then go to town! But in principle, they are eliminable.
Similarly, there is little at stake in the dispute between the weak and the strong semantics. In fact they agree on the truth judgements of all meaningful assertions. In light of this, there seems little reason not to proceed with the strong semantics, since it provides coherent truth values to the assertions lacking a truth value in the weak semantics. The criticisms I had mentioned may be outweighed simply by having a complete assignment of truth values. The question of whether assertions made about failed definite descriptions are actually meaningful can be answered by the reply: they are meaningful, because the strong semantics provide the meaning. But again, because every idea that can be expressed in these semantics can also be expressed without it, there is nothing at stake in this decision.
This blog post is adapted from my book-in-progress, Topics in Logic, an introduction to a wide selection of topics in logic for philosophers, mathematicians, and computer scientists. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8910034894943237, "perplexity": 426.63265145596813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710918.58/warc/CC-MAIN-20221203011523-20221203041523-00243.warc.gz"} |
https://thehousecarpenter.wordpress.com/2014/09/10/the-axioms-of-set-theory/ | # The axioms of set theory, part 1
An axiom is traditionally defined as a self-evident truth. However, a more accurate modern description is that an axiom is a statement that is accepted as true, without proof, within some logical system, and used as a starting point to allow other statements to be proved within the system. The most famous examples of axioms are those of Euclid, but the more relevant ones to modern mathematician are the Zermelo-Fraenkel (ZF) axioms of set theory. The ZF axioms are powerful enough to prove the vast majority of theorems that mathematicians are interested in.
This precise concept of a set is captured by the first ZF axiom, which is called the axiom of extensionality. It says that for all intents and purposes, two sets are exactly the same object if and only if each of them contains every object in the other one. In formal logical language, given two sets $A$ and $B$,
$A = B \Leftrightarrow \forall x \, (x \in A \Leftrightarrow x \in B).$
The ancient Greeks would probably have been satisfied with this axiom; it’s self-evident in that it’s essentially a statement of what the word “set” means.
Early set theorists completed their set theory by adding an axiom schema, called the axiom schema of comprehension. Before I explain what an axiom schema is, I’ll tell you what this axiom says: it says that for every conceivable logical formula, there is a set whose members are precisely the objects that satisfy the formula. Basically, for every collection of objects you might want to work with, there’s a set containing just those objects. The reason why this isn’t a true axiom is that an axiom is supposed to be a logical formula itself, and you can’t talk about logical formulae within logical formulae! Well, actually, there are systems of logic where you can do this, but it gets really complicated and it would certainly be better if we didn’t have to resort to such advanced systems of logic. Thankfully, there’s an easy way to get around the problem. We just say that for every logical formula, there’s an individual axiom which says that there’s a set whose members are the objects that satisfy the formula. This means our theory has an infinite number of axioms, which Euclid probably wouldn’t be too happy with. This doesn’t faze modern mathematicians too much though; I guess they leave it up to the philosophers of mathematics to argue over whether infinity is real.
[There is, actually, a way to finitely axiomatise NBG set theory, which is essentially the same as ZF but slightly stronger. Look it up if you’re interested.]
Note that we haven’t actually including any other axioms stating that objects exist. If we only have these two axioms, the only objects in our logical universe are, in fact, sets created by the axiom schema of comprehension. You might wonder, then, how there can be any objects at all, since sets have to contain objects themselves, don’t they? The axiom schema of comprehension doesn’t prove the existence of any sets which contain themselves, because there isn’t a logical formula that talks about the set if it doesn’t actually exist. The possibility that a set containing itself could exist isn’t ruled out, we just can’t prove that they exist. However, the axiom schema of comprehension does prove the existence of a set containing no members, which is called the empty set and denoted $\emptyset$. To see this, just let the logical formula be any one which involves a contradiction and is therefore false applied to every object. Once the empty set’s existence is proved, we can prove the existence of an infinite number of more complex sets, such as the set which contains only the empty set, the set which contains both of the afore-mentioned sets, and so on.
It would be very nice if all set theory required was these two axioms. Unfortunately, Bertrand Russell came along in 1901 and ruined everyone’s fun by showing that the axiom of comprehension proves a contradiction. More on that tomorrow. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 4, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8802058696746826, "perplexity": 202.5770192818675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608870.18/warc/CC-MAIN-20170527055922-20170527075922-00403.warc.gz"} |
https://www.smashingmagazine.com/2022/08/fluid-sizing-multiple-media-queries/?ref=sidebar | # Fluid Sizing Instead Of Multiple Media Queries?
Ruslan Yevych is a senior researcher in the Research Institute of Physics and Chemistry of Solid States of Uzhhorod National University (Uzhhorod, Ukraine) and … More about Ruslan ↬
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Quick summary ↬ If you like math and CSS, you’ll love this one. We don’t need to use media queries to change the values of some CSS properties — font-size, padding, margin etc. — depending on the viewport width, with the CSS clamp function. But: we still need to use media queries for changing colors, borders, shadows and other CSS styles. This article is an enhanced release of the tutorial published earlier.
Media queries are a great concept. Building a complex structure in an HTML document and adapting it for various devices is not often possible without them (for the moment at least). Here we will talk about the fact that one of the drawbacks of fluid typography, namely the appearance of a large number of media queries, can be avoided. Or, at least, the number of records in @media rule can be reduced.
The advent of vw and vh relative units, the calc function, and later the min, max, clamp functions in CSS gave us a lot of power. An exhaustive review of Modern Fluid Typography Using CSS Clamp has been recently published by Adrian Bece. I advise everyone to get acquainted with it.
The advantages of the clamp function are obvious. We can define a change, for example, of a font size in a certain range of viewport (screen size) and, at the same time, limit it by maximum and minimum values. In such a simple case, we automatically (thanks to clamp) do not need to use media queries for changing sizes on breakpoints.
So, the following block in the CSS:
.block{
font-size: 2rem;
}
@media (max-width: 1200px) {
.block{
font-size: calc(-1rem + 4vw);
}
}
@media (max-width: 800px) {
.block{
font-size: 1rem;
}
}
…can be easily replaced using clamp with a single line:
.block{
font-size: clamp(1rem, -1rem + 4vw, 2rem);
}
But what if we need to set a more complex behavior which is determined by variety of unique behavior on different ranges of the screen width? Look at the following modified code:
.block{
font-size: calc(-4rem + 8vw);
}
@media (max-width: 1200px) {
.block{
font-size: calc(-1rem + 4vw);
}
}
@media (max-width: 800px) {
.block{
font-size: calc(0.5rem + 0.8vw);
}
}
Can the clamp help us again?
Let’s take a closer look: from simple to complex.
## Brief Digression Into Mathematics
As we all know, there is only one way to draw a straight line through two points. There are several ways to write equations to define the straight line. Below, it will be convenient for us to write the equation in the form:
$$(1)\;\;\; y=y_0 + k*x$$
where y0 is a point of intersection of the line with the Y-axis (fig.1), k parameter defines the slope of the straight line to the X-axis and represents the growth/fall rate. Using the following canonical representation of the equation of a straight line:
$$\frac{y - y_1}{y_1 - y_2}=\frac{x - x_1}{x_1 - x_2}$$
It is easy to connect the y0 and k parameters with the coordinates of the two points that belong to this line:
$$(1a)\;\;\; k=\frac{y_2 - y_1}{x_2 - x_1} ,\;\;\; y_0=y_1 - k*x_1$$
It should be noted that in such problems it is convenient to define the input parameters (point coordinates) in pixels. But at the output the relative unit rem is preferable. We should also remember the fact that viewport width is equal to 100vw (this follows from the definition of vw unit). So, in equation (1) we must replace x variable by just 100vw. Hence, we’ll have:
$$(1b)\;\;\; y=y_0 + k*100vw$$
Now it becomes clear the origin of expressions like 1rem + 2.5vw as one of the arguments of clamp or calc function. The first term (1rem) is the y0 parameter expressed in relative units (rem), and the second one (2.5vw) is the parameter k multiplied by 100vw since x=100vw. The choice of such units — relative unit (rem) for values of output variable and viewport unit (vw) for screen size — has been made due to accessibility and responsiveness, respectively.
So, now we know how to determine the parameters in the equation of the form (1) or (1b) of a straight line drawn through two points. This will be used in the next section.
More after jump! Continue reading below ↓
## Main Formula Derivation For General Case
Now we’ll try to obtain an equation for F(x) function that will follow the behavior of property in general case (fig.2). Some information below may be omitted for readers who do not like pure math. You can look at the final equation (3) and the simple example for explanation how to use it.
Let’s look at fig. 2a (below). As you can see from the figure, the behavior of the property is determined (tabulated) by N points with coordinates (xi, yi). Let fi(x) be a function that defines the straight line drawn through (xi, yi) and (xi+1, yi+1) points. We have (N-1) such functions (fig2b). So, how to get a general F(x) function that will be exactly equal to fi(x) function on the corresponding [xi, xi+1] range, notably completely repeat the behavior of property from fig.2?
Let’s define the collection of functions gi(x) as:
${{g}}_{{i}}{\left(}{x}{\right)}{=}{c}{l}{a}{m}{p}\left({{y}}_{{i}}{,}{{f}}_{{i}}{\left(}{x}{\right)}{,}{{y}}_{{i}{+}{1}}\right){.}$
According to clamp function definition:
Let’s sum the gi(x) functions, and denote the result as G(x) function,
Now we can calculate the values of this function for different ranges of x variable:
or
${G}{\left(}{x}{\right)}{=}{{f}}_{{i}}{\left(}{x}{\right)}{+}{C}{o}{n}{s}{t}{,}$
for
${{x}}_{{i}}{\le }{x}{<}{{x}}_{{i}{+}{1}}{,}$${i}{=}{1,}{\dots }{,}\left({N}{-}{1}\right){,}$${C}{o}{n}{s}{t}{=}{\sum }_{{j}{=}{2}}^{{N}{-}{1}}{{y}}_{{j}}$
It follows that G(x) function is equal to corresponding fi(x) function for each [xi, xi+1] ranges, after deduction of Const constant term, or…
${F}\left({x}\right){=}{G}\left({x}\right){-}{C}{o}{n}{s}{t}{=}{\sum }_{{i}{=}{1}}^{{N}{-}{1}}{c}{l}{a}{m}{p}\left({{y}}_{{i}}{,}{{f}}_{{i}}{\left(}{x}{\right)}{,}{{y}}_{{i}{+}{1}}\right){-}{\sum }_{{i}{=}{2}}^{{N}{-}{1}}{{y}}_{{i}}$
Equation (3) represents the final result and is the solution to the problem.
Several remarks should be made here:
• If we have the range with yi= yi+1 then fi(x)= yi, and clamp(yi, yi, yi) = yi properly. This fact will give us some simplification in the final expression for F(x) function.
• If we have the range where the yi>yi+1 inequality is satisfied, then we should write the corresponding gi(x) function in the following form:
… because of clamp function definition.
• Let’s consider the ranges where х<х1 and х>хN. From equation (3) it follows that the values of property inside these ranges will be constant and equal to y1 and yN, correspondingly. We can do nothing and leave it like that. But in these intervals, I prefer that the values continue to change according to the same laws as in the [x1, x2] and [xN-1, xN] ranges. Therefore, g1(x) and gN-1(x) functions should be written not by clamp function but through min or max functions depending on the increasing or decreasing of the values in the given ranges. If we take the behavior of the property from fig.2 as an example, then we will have the following two redefinitions:
• It is possible to set an abrupt change in the property. In this case, the maximum difference between хi and хi+1 is set to 1px (fig.3) or, which is even more convenient in practice, even less than 1px.
• Obviously, the more complex (more ranges) the behavior of the property is, the longer the resulting function will be and vice versa.
• Because of the possible complex structure of the function by equation (3), it must be used as an argument of the CSS calc function in the CSS stylesheet.
### Simple Example
Let’s simulate the following simple case. We have a header with a logo and menu items on some page. For mobile devices we need to create a hamburger menu. In this case, the font size of menu items, for example, is equal to 18px at 1920px of screen size and decreases to 12px at the viewport width of 768px. In the range from 320px to 767.98px of viewport width, the font-size is fixed at 20px (fig 4a.). This behavior of font size can be described by equation (3). Let’s start.
1. We need to calculate the parameters for f1, f2 and f3 lines according to equation (1a) for its representation in the (1b) form. For f1 function we have (using the coordination of 1 and 2 points):
So, using equation (1b):
The same procedure for another two lines gives us:
and
and
2. Now we can construct the gi functions using equation (2):
3. The last term (constant) in equation (3) can be determined:
4. Desired form of equation (3) is…
or, in the final form:
5. Final record in CSS file will be (by remark 6):
.block{
font-size: calc(clamp(0.75rem, 19200.75rem – 40000vw, 1.25rem) + max(0.75rem, 0.5rem + 0.5208333vw) – 0.75rem);
}
… and is fully equivalent to:
.block{
font-size: calc(0.5rem + 0.5208333333vw);
}
@media (max-width: 767px) {
.block {
font-size: 1.25rem;
}
}
Evaluations for modeling right margin behavior (see fig. 4b) give the following equation (everyone can verify this):
You can check this example here:
### Another Example
The following dependence of padding:
.block {
}
@media (max-width: 1199px) {
.block {
}
}
@media (max-width: 991px) {
.block {
}
}
@media (max-width: 767px) {
.block {
}
}
@media (max-width: 575px) {
.block {
}
}
… can be also easily replaced by:
.block {
padding: calc(clamp(1rem, -14398.5rem + 40000vw, 1.5rem) + clamp(1.5rem, -19198rem + 40000vw, 2rem) + clamp(2rem, -24797.5rem + 40000vw, 2.5rem) + clamp(2.5rem, -29997rem + 40000vw, 3rem) - 6rem);
}
So, equation (3) gives us the possibility to describe any arbitrary behavior of the property depending on the viewport width. And it should be pointed out, we achieved this by completely eliminating the use of media queries.
Obviously, formula (3) can be used to describe not only the font size, but also every other property whose dimension is a unit of length. This limitation is due to a limitation in CSS itself, namely the lack of support for mathematical operations on values that have dimensions. For example, we cannot calculate something like (2px*6px)/4px in CSS at the date. I hope that such restrictions will disappear in future, and the equation (3) will be applied to properties specified as a percentage or dimensionless.
Also, it should be noted that it makes sense to apply the equation (3) only when the dependence is more complex than in primitive cases shown below:
1. Continuous unlimited change, the behavior is determined by two points only, k parameter of f function can be positive, negative or even zero (see fig.5a, k>0, for example). Instead of the equation(3) we use a function of the form calc(f(x)). A font size of some text that monotonically changes with viewport width can be as an example.
2. Behavior is determined by three points, for k parameters of f1 and f2 functions the inequality k1>k2 is satisfied (see fig.5b, k1>0 and k2<0, for example). In such case, min(f1,f2) function should be used instead of equation (3). A font size of some text that monotonically changes with viewport width to a certain value and then keeps a constant value can be used as an example.
3. The same as the second case, but the inequality k1<k2 is satisfied (see fig.5c, k1<0 and k2>0, for example). In such case, max(f1,f2) function should be used instead of equation (3).
Everyone can write their own mixin or function based on the equation (3) or use my implementation as CSS functions that take into account the above remarks and primitive cases.
You can also find a simple example with the flex structure here. In this example, the width of flex elements is calculated using the equation (3). This is an attempt to avoid media queries with such a commonly used transformation — initially the width of each of the four flex items is set to 25% (convert to px units, of course), and then, as the viewport width decreases, it changes to 50%, and then to 100%. It even seems possible to take into account the value of gap, and make it also responsive.
But the example will break as soon as there is a vertical scroll. To avoid example breaking and to compensate for scrollbar, we can replace vw units by % in expressions for width of flex elements. But while CSS restrictions do not allow us to use the equation (3) directly for values given in percentages, it is better to abandon such idea.
## Summary
I’d like to mention that the idea was just to proving the possibility of such an approach, and the suitability of its application is your own choice. Of the obvious disadvantage of this approach, I have to highlight that the resulting code becomes less readable. The advantage is that media queries remain solely for describing the logic of changes while adaptive, structural changes and are not clogged with technical lines of changing font sizes, paddings, margins etc.
Editor’s note: It would be fantastic to build up a little tool that would allow developers to define changes and construct these complex queries in no time. Still, the maintainance would become quite a hassle — unless we use a Sass-mixing for it, maybe? We kindly thank Ruslan for taking the time to work on this, and oh my, what a journey it was!
### Acknowledgements
Thanks to Evgeniy Andrikanich and his YouTube channel “FreelancerStyle” for creation of free high-quality educational content (HTML, CSS, JS).
(vf, yk, il) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.810738205909729, "perplexity": 1339.728869022068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573533.87/warc/CC-MAIN-20220818215509-20220819005509-00007.warc.gz"} |
https://mrchasemath.com/2010/10/12/powerful-problem-hint/?shared=email&msg=fail | # Powerful Problem (hint)
A few weeks ago I posted this “powerful” problem:
Solve $\left(x^2-5x+5\right)^{\left(x^2-9x+20\right)}=1$
Now, allow me to give you a major hint. Consider the simpler equation
$a^b=1$
What are the possible values of $a$ and $b$? Here are the possible combinations:
• $a=1$ and $b$ is anything
• $b=0$ and $a$ is any nonzero number
And here’s the tricky one that most people forget:
• $a=-1$ and $b$ is even
You now have enough information to solve the original equation. I think you’ll be delighted with the solution! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 10, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9270566701889038, "perplexity": 460.6811680247124}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571993.68/warc/CC-MAIN-20220814022847-20220814052847-00055.warc.gz"} |
https://jayryablon.wordpress.com/category/five-dimensions/ | # Lab Notes for a Scientific Revolution (Physics)
## April 22, 2012
### Back to Blogging, Uploaded a paper I wrote in 1986 about Preonic Grand Unification
It has been almost 3 years since my last Blog post. Much of my time has been diverted into a condo hotel project in Longboat Key Florida, and the focus I need to do good physics has been impossible to come by. Then, the other day, Ken Tucker, a frequent participant at sci.physics.foundations, emailed me about some new research showing that electrons have constituent substructure. That brought me back immediately to the half a year I spent back in 1986 developing a 200-page paper about a preonic substructure for quarks and leptons, which culminated six years of study from 1980 to 1986. I finished that paper in August 1986, and then took an 18 year hiatus from physics, resuming again in late-2004.
Ken’s email motivated me to dig out this 1986 paper which I manually typed out on an old-fashioned typewriter, scan it into electronic form, and post it here. Links to the various sections of this paper are below. This is the first time I have ever posted this.
Keep in mind that I wrote this in 1986. I tend to study best by writing while I study, and in this case, what I wrote below was my “study document” for Halzen and Martin’s book “Quarks and Leptons” which had just come out in 1984 and was the first book to pull together what we now think of as modern particle physics and the (then, still fairly new) electroweak unification of Weinberg-Salam.
What is in this paper that I still to this day believe is fundamentally important, and has not been given the attention it warrants, is the isospin redundancy between (left-chiral) quarks and leptons. This to me is an absolute indication that these particles have a substructure, so that a neutrino and an up quark both have contain the same “isospin up” preon, and an electron and a down quark both contain the same “isospin down” preon. Section 2.11 below is the key section, if you want to cut to the chase with what I was studying some 26 years ago. I did post about this in February 2008 at https://jayryablon.wordpress.com/2008/02/02/lab-note-4-an-interesting-left-chiral-muliplet-perhaps-indicative-of-preonic-structure-for-fermions/, though that post merely showed a 1988 summary I had assembled based on my work in 1986, at the behest of the late Nimay Mukhopadhyay, who at the time was teaching at RPI and had become a good friend and one of my early sources of encouragement. This is the first time I am posting all of that early up-to-1986 work, in complete detail.
Lest you think me crazy, note that seventeen years later, G. Volovik, in his 2003 book “The Universe in a Helium Droplet,” took a very similar tack, see Figure 12.2 in this excerpt: Volovik Excerpt on Quark and Lepton Preonic Structure.
The other aspect of this 1986 paper that I still feel very strongly about, is taking the Dirac gamma-5 as a fifth-dimension indicator. I know I have been critiqued by technical arguments as to why this should not be taken as a sign of a fifth dimension, but this fits seamlessly with Kaluza Klein which geometrizes the entirely of Maxwell’s theory and is still the best formal unification of classical electromagnetism and gravitation ever developed. For those who maintain skepticism of Kaluza-Klein and ask “show me the fifth dimension,” just look to chirality which is well-established experimentally. Why do we have to assume that this fifth dimension will directly manifest in the same way as space and time, if its effects are definitively observable in the chiral structure of fermions? Beyond this, I remain a very strong proponent of the 5-D Space-Time-Matter Consortium, see http://astro.uwaterloo.ca/~wesson/, which regards matter itself as the most direct manifestation of a fifth physical dimension. Right now, most folks think about 4-D spacetime plus matter. These folks correctly think about 5-D space-time-matter, no separation. And Kaluza-Klein, which historically predated Dirac’s gamma-5, is the underpinning of this.
After my hiatus of the past couple of years, I am going to try in the coming months to write some big-picture materials about physics, which will pull together all I have studied so far in my life. I am thinking of doing a “Physics Time Capsule for 2100” which will try to explore in broad strokes, how I believe physics will be understood at the end of this century, about 88 years from now.
Anyway, here is my entire 1986 paper:
Preonic Grand Unification and Quantum Gravitation: Capsule Outline and Summary
Abstract and Contents
Section 1.1: Introduction
Section 1.2: Outline and Summary
Section 2.1: A Classical Spacetime Introduction to the Dirac Equation, and the Structure of Five-Dimensional Spacetime with a Chiral Dimension
Section 2.2: Particle/Antiparticle and Spin-Up/Spin-Down Degrees of Quantum Mechanical Freedom in Spacetime and Chirality, Gauge Invariance and the Dirac Wavefunction
Section 2.3: Determination and Labeling of the Spinor Eigensolutions to the Five-Dimensional Dirac Equation, and the High and Low Energy Approximations
Section 2.4: The Fifth-Dimensional Origin of Left and Right Handed Chiral Projections and the Continuity equation in Five Dimensions: Hermitian Conjugacy, Adjoint Spinors, and the Finite Operators Parity (P) and Axiality (A)
Section 2.5: Conjugate and Transposition Symmetries of the Dirac Equation in Five Dimensions, the Finite Operators for Conjugation (C) and Time Reversal (T), and Abelian Relationships Among C, P, T and A
Section 2.6: Charge Conjugation, and the Definitions and Feynman Diagrams for “Electron” and “Positron” Spinors
Section 2.7: Simple Unpolarized s,t,u Scattering Channels with a Covariant Propagator, and the Covariant (Real and Virtual) Polarization States of Massive and Massless Vector Bosons
Section 2.8: Prelude to Preons: The Spinor Decomposition of Four Real Spacetime Dimensions ct,x,y,z into Two Complex Spinor Dimensions Using the Covariant Polarization States of Vector Bosons
Section 2.9: Introduction to Isospin Preons in Electroweak Theory: The Preonic Decomposition of Four Real Electroweak Bosons A, W+, W-, Z into Two Complex Preons Denoting “Isospin Up” and “Isospin Down”
Section 2.10: Summarization of Prior Discussion, and on the Fundamental Importance of Preons in Particle Physics
Section 2.11: The Four-Preon Flavor SU(4) Unification of the Electromagnetic, Weak and Colorless Strong Interactions Excluding Quantum Gravitation; and the Colorless Flavor Classification of Left Handed Real Fermion and Boson Chiral Projections, for a Single Fermion Generation
Section 2.12: The Four-Preon Flavor SU(4)xU(1) Unification of Electromagnetic, Weak, Colorless Strong and Quantum Gravitational Interactions; and the Colorless Flavor Classification of Left and Right Handed Real Fermion and Boson Chiral Projections, for a Single Fermion Generation
Section 2.13: The Six-Preon Unification of Flavor SU(4)xU(1) with High Energy Color SU(4)xU(1) and Two Overlapping Degrees of Freedom; the Flavor and Color Classification of Real Fermions and Vector Bosons for a Single Generation; and the Derivation of Electroweak and Strong/Hyperweak Massless and Massive Neutral Current Vector Bosons
Section 2.14: On the Replication of Fermion Generations: Four Generational Grand Unification with Eighteen Preons and Nine Independent Flavor/Color/Generation Degrees of Freedom, and a Preonic Discussion of Mesons and Meson Decay
References and Bibliography
## April 13, 2008
### Thesis Defense of the Kaluza-Klein, Intrinsic Spin Hypothesis — EARLY DRAFT
I have been engaged in a number of Usenet and private discussions about the paper Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0 which I posted here on this blog on March 30.
A number of critiques have been raised, which you can see if you check out the recent Usenet threads I started related to intrinsic spin under the heading “Query about intrinsic verus [sic] orbital angular momentum,” over at sci.physics.foundations, sci.physics.relativity and sci.physics.research. These are among the “links of interest” provided in the right-hand pane of this weblog.
I believe that these critiques can be overcome, and that this hypothesis relating to Kaluza-Klein and intrinsic spin and the spatial isotropy of the square of the spin will survive and be demonstrated, ultimately, to be in accord with the physical reality of nature.
I have begun a new paper which is linked at: Thesis Defense of the Kaluza-Klein, Intrinsic Spin Hypothesis, Rev 1.0, which will respond thoroughly and systematically to the various critiques. What is here so far is the introductory groundwork. But, I would appreciate continued feedback as this development continues.
Note that the links within the PDF file unfortunately do not work, so to get the intrinsic spin paper, you need to go to Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0. Also, to get Wheeler’s paper which is referenced, go to Wheeler Geometrodynamics.
## March 30, 2008
### Revised Paper on Kaluza-Klein and Intrinsic Spin, Based on Spatial Isotropy
I have now prepared an updated revision of a paper demonstrating how the compact fifth dimension of Kaluza-Klein is responsible for the observed intrinsic spin of the charged leptons and their antiparticles. The more global, underlying view, is that all intrinsic spins originate from motion through the curled up, compact $x^5$ dimension. This latest draft is linked at:
Intrinsic Spin and the Kaluza-Klein Fifth Dimension, Rev 3.0
Thanks to some very helpful critique from Daryl M. on a thread at sci.physics.relativity, particularly post #2, I have entirely revamped section 5, which is the heart of the paper wherein we establish the existence of intrinsic spin in $x^5$, on the basis of “fitting” oscillations around a $4\pi$ loop of the compact fifth dimension (this is to maintain not only orientation but entanglement version). From this approach, quantization of angular momentum in $x^5$ naturally emerges, it also emerges that the intrinsic $x^5$ angular momentum in the ground state is given by $(1/2) \hbar$.
In contrast to my earlier papers where I conjectured that the intrinsic spin in $x^5$ projects out into the three ordinary space dimensions by virtue of its orthogonal orientation relative to the $x^5$ plane, which was critiqued in several Usenet posts, I now have a much more direct explanation of how the intrinsic spin projects out of $x^5$ to where we observe it.
In particular, we recognize that one of the objections sometimes voiced with regards to a compactified fifth dimension is the question: how does one “bias” the vacuum toward one of four space dimensions, over the other three, by making that dimension compact? This was at the root of some Usenet objections also raised earlier by DRL.
In this draft — and I think this will overcome many issues — we require that at least as regards intrinsic angular momentum, the square of the $J^5 = (1/2) \hbar$ obtained for the intrinsic angular momentum in $x^5$, must be isotropically shared by all four space dimensions. That is, we require that there is to be no “bias” toward any of the four space dimensions insofar as squared intrinsic spin is concerned. Because $J^5 = (1/2) \hbar$ emanates naturally from the five dimensional geometry, we know immediately that $\left(J^{5} \right)^{2} ={\textstyle\frac{1}{4}} \hbar ^{2}$, and then, by the isotropic requirement, that $\left(J^{1} \right)^{2} =\left(J^{2} \right)^{2} =\left(J^{3} \right)^{2} ={\textstyle\frac{1}{4}} \hbar ^{2}$ as well. We then arrive directly at the Casimir operator $J^{2} =\left(J^{1} \right)^{2} +\left(J^{2} \right)^{2} +\left(J^{3} \right)^{2} ={\textstyle\frac{3}{4}} \hbar ^{2}$ in the usual three space dimensions, and from there, continue forward deductively.
For those who have followed this development right along, this means in the simplest terms, that rather than use “orthogonality” to get the intrinsic spin out of $x^5$ and into ordinary space, I am instead using “isotropy.”
There is also a new section 8 on positrons and Dirac’s equations which has not been posted before, and I have made other editorial changes throughout the rest of the paper.
## March 29, 2008
### Stepping Back from Kaluza-Klein: Planned Revisions
Those who have followed my Weblog are aware that I have been putting in a lot of work on Kaluza-Klein theory. This post is to step back from the canvas, lay out the overall picture of what I am pursuing, and summarize what I plan at present to change or correct in the coming days and weeks. This is in keeping with the concept of this Weblog as “Lab Notes,” or as a public “scientific diary.”
There are really two main aspects to this Kaluza-Klein work:
First, generally, I have found that 5-D Kaluza-Klein theory is most simply approached by starting with (classical) Lorentz force motion, and requiring the Lorentz force motion to be along the geodesics of the five dimensional geometry. I am far from the first person who recognizes that the Lorentz force can be represented as geodesic motion in a 5-D model. But I have found, by starting with the Lorentz force, and by requiring the 5-D electromagnetic field strength tensor to be fully antisymmetric, that all of the many “special assumptions” which are often employed in Kaluza-Klein theory energy very naturally on a completely deductive basis, with no further assumptions required. I also believe that this approach leads to what are perhaps some new results, especially insofar as the Maxwell tensor is concerned, and insofar as QED may be considered in a non-linear context. The latest draft of this global work on Kaluza-Klein may be seen at Kaluza-Klein Theory and Lorentz Force Geodesics.
Second, specifically, within this broader context, is the hypothesis that the fifth-dimensional “curled” motion is the direct mainspring of intrinsic spin. More than anything else, the resistance by many physicists to Kaluza-Klein and higher-dimensional theories, rests on the simple fact that this fifth dimension — and any other higher dimensions — are thought to not be directly observable. In simplest form, “too small” is the usual reason given for this. Thus, if it should become possible to sustain the hypothesis that intrinsic spin is a directly-observable and universally-pervasive outgrowth of the fifth dimension, this would revitalize Kaluza-Klein as a legitimate and not accidental union of gravitation and electrodynamics, and at the same time lend credence to the higher-dimensional efforts also being undertaken by many researchers. The latest draft paper developing with this specific line of inquiry is at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.
Now, the general paper at Kaluza-Klein Theory and Lorentz Force Geodesics is very much a work in progress and there are things in this that I know need to be fixed or changed. If you should review this, please keep in mind the following caveats:
First, sections 1-4 are superseded by the work at Intrinsic Spin and the Kaluza-Klein Fifth Dimension and have not been updated recently.
Second, sections 5-7 are still largely OK, with some minor changes envisioned. Especially, I intend to derive the “restriction” $\Gamma^u_{55}=0$ from $F^{{\rm M} {\rm N} } =-F^{{\rm N} {\rm M} }$ rather than impose it as an ad hoc condition.
Third, sections 8-11 needs some reworking, and specifically: a) I want to start with an integration over the five-dimensional volume with a gravitational constant $G_{(5)}$ suited thereto, and relate this to the four dimensional integrals that are there at present; and b) I have serious misgivings about using a non-symmetric (torsion) energy tensor and am inclined to redevelop this to impose symmetry on the energy tensor — or at least to explore torsion versus no torsion in a way that might lead to an experimental test. If we impose symmetry on the energy tensor, then the Maxwell tensor will be the $J^{\mu } =0$ special case of a broader tensor which includes a $J^\mu A^\nu + J^\nu A^\mu$ term and which applies, e.g., to energy flux densities (Poynting components) $T^{0k}$, k=1,2,3 for “waves” of large numbers of electrons.
Fourth, I am content with section 12, and expect it will survive the next draft largely intact. Especially important is the covariant derivatives of the electrodynamic potentials being related to the ordinary derivatives of the gravitational potentials, which means that the way in which people often relate electrodynamic potentials to gravitational potentials in Kaluza-Klein theory is valid only in the linear approximation. Importantly, this gives us a lever in the opposite direction, into non-linear electrodynamics.
Fifth, I expect the development of non-linear QED in section 13 to survive the next draft, but for the fact that the R=0 starting point will be removed as a consequence of my enforcing a symmetric energy tensor in sections 8-11. Just take out all the “R=0” terms and leave the rest of the equation alone, and everything else is more or less intact.
Finally, the experiment in Section 15, if it stays, would be an experiment to test a symmetric, torsionless energy tensor against a non-symmetric energy tensor with torsion. (Basically, metric theory versus Cartan theory.) This is more of a “back of the envelope” section at present, but I do want to pursue specifying an experiment that will test the possible energy tensors which are available from variational principles via this Kaluza-Klein theory.
The paper at Intrinsic Spin and the Kaluza-Klein Fifth Dimension dealing specifically with the intrinsic spin hypothesis is also a work in progress, and at this time, I envision the following:
First, I will in a forthcoming draft explore positrons as well as electrons. In compactified Kaluza-Klein, these exhibit opposite motions through $x^5$, and by developing the positron further, we can move from the Pauli spin matrices toward the Dirac $\gamma^\mu$ and Dirac’s equation.
Second, I have been engaged in some good discussion with my friend Daryl M. on a thread at sci.physics.relativity. Though he believes I am “barking up the wrong tree,” he has provided a number of helpful comments, and especially at the bottom of post #2 where he discusses quantization in the fifth dimension using a wavelength $n \lambda = 2 \pi R$. (I actually think that for fermions, one has to consider orientation / entanglement issues, and so to secure the correct “version,” one should use $n \lambda = 4 \pi R$ which introduces a factor of 2 which then can be turned into a half-integer spin.) I am presently playing with some calculations based on this approach, which you will recognize as a throwback to the old Bohr models of the atom.
Third, this work of course uses $x^5 = R\phi$ to define the compact fifth dimension. However, in obtaining $dx^5$, I have taken $R$ to be a fixed, constant radius. In light of considering a wavelength $n \lambda = 4 \pi R$ per above, I believe it important to consider variations in $R$ rather than fixed $R$, and so, to employ $dx^5 = Rd\phi + \phi dR$.
There will likely be other changes along the way, but these are the ones which are most apparent to me at present. I hope this gives you some perspective on where this “work in flux” is at, and where it may be headed.
Thanks for tuning in!
Jay.
## March 28, 2008
### Further Considerations on the Energy Tensor: Metric versus Torsion
After reviewing some very helpful discussion on sci.physics.foundations and sci.physics.relativity in the threads I initiated regarding non-symmetric energy tensors and a suggested Kaluza-Klein experiment, and going back to Misner, Thorne and Wheeler’s “Gravitation,” I am starting to shift my viewpoint to be in opposition to the idea of using a non-symmetric (Cartan / Torsion) energy tensor because of the adverse impact this has on formulating a metric theory of gravitation.
There is a non-symmetric energy tensor in equations (15.11) to (15.13) of:
Kaluza-Klein Theory and Lorentz Force Geodesics Rev. 6.0
which is based upon the *non-symmetric* energy tensor of trace matter derived in (11.6). What I have been turning over, is whether I ought to be comfortable with this result, and my sense runs against it.
However, at the point of original derivation in sections 8-11, there is actually a choice: one can construct the variation of the Lagrangian density of matter with respect to $g_{\mu \nu}$ such that a symmetric tensor will result, or one can choose not to, by creating a symmetric term or not. This is actually a form of gravitational “symmetry breaking” that occurs in the process of taking the variation of the matter Lagrangian density with respect to the metric tensor. I think both paths need to be developed, because they lead to on the one hand to a symmetric energy tensor, and on the other to a non-symmetric energy tensor. In either case, the key term distinguishing this energy tensor from the Maxwell tensor is $J^\mu A^\nu$.
If we impose symmetry on the energy tensor, then the Maxwell tensor will be the $J^u=0$ special case of a broader tensor which includes a $J^\mu A^\nu + J^\nu A^\mu$ term and which applies, e.g., to energy flux densities (Poynting components) $T^{0k}$, $k=1,2,3$ for “waves” of large numbers of electrons.
Then, the experiment becomes — not a test of the torsion tensor — but a test as between an energy tensor with and without torsion. That is, the experiment as reformulated, becomes a test of metric-style versus Cartan-style theories of gravitation.
Dealing with the currents $J^\mu$ is clear. Regarding how to deal with the potential $A^\mu$ in doing the experiment, think about a beam of electrons. They of course will all repel, so the beam will emerge conically from the electron gun if nothing is done to force them onto a parallel path. Now, take a circular cross section of electrons from the beam striking an energy flux detector. One can think of the cross-sectional surface where the electron stream meets the detector as a “disk,” not unlike a charged, flat, frisbee, which is also productively thought about as a dielectric. I would submit that one can assign a “zero” potential to the center of the cross section, and a varying non-zero potential to the periphery. That is, if one were to take a circular dielectric disk and fill it with electric charge, then float some positive charge nearby, the positive charge — I believe — would be attracted toward and seat itself at the center of the disk, and so that would be a natural place to define the zero of potential.
This would also mean that regional detections of flux toward the fringes of the detector will be different than toward the center, assuming uniformity of charge distribution, because the energy created by the potentials among the electrons are different in different regions. So, there is a way to assign potentials even without applying an external voltage, though someone conducting this experiment may want to also apply an external voltage simply to vary the range of experiments.
Now, to the main point: one should do the experiment with random, unpolarized electrons, and then again with spins aligned with and against the direction of propagation, merely to test the symmetric versus non-symmetric energy tensors one to the other. One will win, the other not. Metric versus torsion.
I am planning a restructuring of the paper at the above link. In the near future, I will outline the proposed restruturing — what I would plan to keep and what I plan to change.
Jay.
## March 22, 2008
### A Possible Kaluza-Klein Experiment
It has been suggested — appropriately so — that I consider whether there might be one or more experiments which can be designed to validate or falsify some of the Klauza-Klein results which I have been posting of late. I believe that one possible experiment resides in the non-symmetric energy tensor of trace matter derived in (11.6) of my latest posted paper. Thus, I have added a new section 15 to this paper, and reposted the entire paper, with this new section 15, at Kaluza-Klein Theory and Lorentz Force Geodesics Rev. 6.0 Because this is of particular interest as it may open some new experimental windows, I have posted section 15 below as well. Please note: the specific discussion of the connection between the compactified fifth dimension, and intrinsic spin, is not updated in this paper, and the latest discusssion I have written up on this topic, is at Intrinsic Spin and the Kaluza-Klein Fifth Dimension.
Section 15: At this juncture, we have enough information to propose an experiment to validate or falsify some of the results derived thus far. We turn for this purpose to the stress energy tensor of matter (11.6), which we raise into contravariant notation as follows:
$\kappa T^{\nu \mu } =-\kappa \left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)+{\textstyle\frac{\sqrt{2} }{2}} \overline{\kappa }g^{5\mu } J^{\nu } =\kappa T^{\mu \nu } _{Maxwell} +{\textstyle\frac{\sqrt{2} }{2}} \overline{\kappa }g^{5\mu } J^{\nu }$. (15.1)
The Maxwell tensor $T^{\mu \nu } _{Maxwell} =T^{\nu \mu } _{Maxwell}$ is, of course, a symmetric tensor. But the added trace matter term $g^{5\mu } J^{\nu }$ is not necessarily symmetric, that is, there is no a priori reason why $g^{5\mu } J^{\nu }$ must be equal to $g^{5\nu } J^{\mu }$. The origin of this non-symmetry was discussed earlier in Section 9.
With an eye toward conducting an experiment, let us now consider (15.1) in the linear approximation of (13.6) where ${\rm L}_{QCD} \approx -A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau }$. In the linear approximation, as used to reach (13.3), (12.11) reduces to $g^{5\mu } \approx \overline{\kappa }\left(\frac{\phi ^{5\mu } -{\textstyle\frac{1}{2}} bA^{\mu } }{1+{\textstyle\frac{1}{2}} \overline{\kappa }\phi } \right)\approx -{\textstyle\frac{1}{2}} \overline{\kappa }bA^{\mu }$, and (15.1) becomes:
$T^{\nu \mu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-2J^{\nu } A^{\mu } =T^{\mu \nu } _{Maxwell} -2J^{\nu } A^{\mu }$, (15.2)
where we have also used $b^{2} =8$ and $2\kappa =\overline{\kappa }^{2}$, and divided out $\kappa$. The transpose of this non-symmetric energy tensor is:
$T^{\mu \nu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-2J^{\mu } A^{\nu } =T^{\mu \nu } _{Maxwell} -2J^{\mu } A^{\nu }$, (15.3)
Now, it is known that a non-symmetric energy tensor, physically, is indicative of a non-zero spin density. In particular, using (15.2) and (15.3), the non-symmetry of the energy tensor is related to a non-zero spin density tensor $S^{\mu \nu \alpha }$ according to: [A good, basic discussion of the spin tensor is at http://en.wikipedia.org/wiki/Spin_tensor.]
$S^{\mu \nu \alpha } _{;\alpha } =T^{\mu \nu } -T^{\nu \mu } =-2J^{\mu } A^{\nu } +2J^{\nu } A^{\mu }$. (15.4)
For such a non-symmetric tensor, the “energy flux” is not identical to the “momentum density, as these differ by (15.4), for $\mu =0$, $\nu =k=1,2,3$ and vice versa. If the spin density $S^{\mu \nu \alpha } =0$, then in this special case, (15.4) yields:
$J^{\mu } A^{\nu } =J^{\nu } A^{\mu }$. (15.5)
So, for $S^{\mu \nu \alpha } =0$, (15.3) may be written using (15.5) as the explicitly-symmetric tensor:
$T^{\mu \nu } \approx -\left(F^{\mu \tau } F^{\nu } _{\tau } -{\textstyle\frac{1}{4}} g^{\mu \nu } F^{\sigma \tau } F_{\sigma \tau } \right)-J^{\mu } A^{\nu } -J^{\nu } A^{\mu } =T^{\mu \nu } _{Maxwell} -J^{\mu } A^{\nu } -J^{\nu } A^{\mu }$. (15.6)
Now, let’s consider a experiment which is entirely classical. The $T^{0k}$ “Poynting” components of (15.4), (15.6) represent the energy flux across a two-dimensional area, for a flux of matter which we will take to be a stream of electrons, while the $T^{k0}$ components represent the momentum density. The proposed experiment, then, will be to fire a stream of a very large number of electrons thereby constituting an electron “wave,” and to detect the aggregate flux of energy across a two-dimensional surface under various spin preparations, in precisely the same manner that one might test the flow of luminous energy across a surface when using light waves rather than electron waves. Specifically, we propose in test I to fire electrons without doing anything to orient their spins, so that, statistically, the number of electrons flowing through the flux surface with positive helicity is equal to the number with negative helicity and so the spin density is zero, and (15.6) applies. In test II, we fire electrons, but apply a magnetic field before detecting the flux, to ensure that all of the electrons are aligned to positive helicity. In this event, the spin density, by design, is non-zero, and one of (15.2) or (15.3) will apply. In test III, we do the same, but now apply the magnetic field to ensure that all of the electrons have negative helicity, before detecting the flux.
## March 20, 2008
### Derivation of Heisenberg Uncertainty from Kaluza Klein Geometry
For those who have followed my Kaluza-Klein (KK) work, I believe that it is now possible to derive not only intrinsic spin, but Heisenberg uncertainty directly from a fifth, compactified dimension in Kaluza Klein. This would put canonical quantum mechanics on a strictly Riemannian geometric foundation which — as a side benefit — unites gravitation and electromagnetism.
I need to consolidate over the next few days and will of course make a more expanded post when I am ready, but here is the basic outline. First, take a look at:
Intrinsic Spin and the Kaluza-Klein Fifth Dimension
where I show how intrinsic spin is a consequence of the compactified fifth dimension. This paper, at present, goes so far as to show how the Pauli spin matrices emerge from KK.
Next, go to the two page file:
Spin to Uncertainty
This shows how one can pop Heisenberg out of the spin matrices.
Finally, go to the latest draft paper on KK generally, at:
Kaluza-Klein Theory and Lorentz Force Geodesics with Non-linear QED
This lays out the full context in which I am developing this work. Please note that the discussion on intrinsic spin in the third link is superseded by the discussion thereof in the first link.
More to follow . . .
Jay.
## March 6, 2008
### Electrodynamic Potentials and Non-Linear QED in Kaluza-Klein
I have now added new sections 12, 13 and 14 to the Kaluza-Klein paper earlier posted. These sections examine the relationship between the electrodynamic potentials and the gravitational potentials, and the connection to QED. You may view this all at:
Electrodynamic Potentials and Non-Linear QED
Most significantly, these three new sections not only connect to the QED Lagrangian, but, they show how the familiar QED Lagrangian density
${\rm L}_{QCD} =-A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau }$
emerges in the linear approximation of 5-dimensional Kaluza-Klein gravitational theory.
Then, we go in the opposite direction, to show the QED Lagrangian density / action for non-linear theory, based on the full-blown apparatus of gravitational theory.
Expressed in terms of the electrodynamic field strength $F^{\sigma \tau }$ and currents $J_{\beta }$, this non-linear result is:
${\rm L}_{QCD} =0={\textstyle\frac{1}{8\kappa }} b\overline{\kappa }g^{5\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau } \approx -A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau }$, (13.6)
where the approximation $\approx$ shows the connection to the linear approximation. Re-expressed solely in terms of the fifth-dimensional gravitational metric tensor components $g_{5\sigma }$ and energy tensor source components $T_{\beta 5}$, this result is:
$\kappa {\rm L}_{QCD} =0={\textstyle\frac{1}{2}} g^{5\beta } \kappa T_{\beta 5} +{\textstyle\frac{1}{8}} g^{\sigma \alpha } \partial ^{\beta } g_{5\alpha } \left[\partial _{\sigma } g_{5\beta } -\partial _{\beta } g_{5\sigma } \right]$. (14.4)
You may also enjoy the derivations in section 12 which decompose the contravariant metric tensor into gravitons, photons, and the scalar trace of the graviton.
Again, if you have looked at earlier drafts, please focus on the new sections 12, 13 and 14. Looking for constructive feedback, as always.
## March 3, 2008
### Intrinsic Spin and the Kaluza-Klein Fifth Dimension: Journal Submission
I mentioned several days ago that I had submitted a Kaluza Klein paper to one of the leading journals. That lengthy paper was not accepted, and you can read the referee report and some of my comments here at sci.physics.foundations or here, with some other folks’ comments, at sci.physics.relativity. The report actually was not too bad, concluding that “the author must have worked a considerable amount to learn quite a few thing in gravitation theory, and a number of the equations are correctly written and they do make sense, however those eqs. do not contain anything original.” I would much rather hear this sort of objection, than be told — as I have been in the past — that I don’t know anything about the subject I am writing about.
In fact, there is one finding in the above-linked paper which, as I thought about it more and more, is quite original, yet I believe it was lost in the mass of this larger paper. And, frankly, it took me a few days to catch on to the full import of this finding, and so I downplayed it in the earlier paper. Namely: that the compactified fifth dimension of Kaluza-Klein theories is the mainspring of the intrinsic spins which permeate particle physics.
I have now written and submitted for publication, a new paper which only includes that Kaluza-Klein material which is necessary to fully support this particular original finding. You may read the submitted paper at Intrinsic Spin and the Kaluza-Klein Fifth Dimension. I will, of course, let you know what comes from the review of this paper.
Jay.
## February 29, 2008
### Lab Note 2: Why the Compactified, Cylindrical Fifth Dimension in Kaluza Klein Theory may be the “Intrinsic Spin” Dimension
FOR SOME REASON, THESE EQUATIONS APPEAR CORRUPTED. I AM CHECKING WITH WORDPRESS TECHNICAL SUPPORT AND HOPE TO HAVE THIS FIXED IN THE NEAR FUTURE — JAY.
I am posting here a further excerpt from my paper at Kaluza-Klein Theory, Lorentz Force Geodesics and the Maxwell Tensor with QED. Notwithstanding some good discussion at sci.physics.relativity, I am coming to believe that the intrinsic spin interpreation of the compacified, hypercylindrical fifth dimension presented in section 4 of this paper may be compelling. The math isn’t too hard, and you can follow it below: The starting point for discussion equation is (3.2) below,
$frac{dx^{5} }{dtau } equiv -frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =-frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m}$. (3.2)
which is used to connect the $q/m$ ratio from the Lorentz law to geodesic motion in five dimensions, and $b$ is a numeric constant of proportionality. Section 4 below picks up from this.
Excerpt from Section 4:
Transforming into an “at rest” frame, $dx^{1} =dx^{2} =dx^{3} =0$, the spacetime metric equation $d/tau ^{2} =g_{/mu /nu } dx^{/mu } dx^{/nu }$ reduces to $dtau =pm sqrt{g_{00} } dx^{0}$, and (3.2) becomes:
$frac{dx^{5} }{dx^{0} } =pm frac{1}{b} sqrt{frac{g_{00} }{4pi G} } frac{q}{m}$. (4.1)
For a timelike fifth dimension, $x^{5}$ may be drawn as a second axis orthogonal to $x^{0}$, and the physics ratio $q/m$ (which, by the way, results in the $q/m$ material body in an electromagnetic field actually “feeling” a Newtonian force in the sense of $F=ma$ due to the inequivalence of electrical and inertial mass) measures the “angle” at which the material body moves through the $x^{5} ,x^{0}$ “time plane.”
For a spacelikefifth dimension, where one may wish to employ a compactified, hyper-cylindrical $x^{5} equiv Rphi$ (see [Sundrum, R., TASI 2004 Lectures: To the Fifth Dimension and Back, http://arxiv.org/abs/hep-th/0508134 (2005).], Figure 1) and $R$ is a constant radius (distinguish from the Ricci scalar by context), $dx^{5} equiv Rdphi$. Substituting this into (3.2), leaving in the $pm$ ratio obtained in (4.1), and inserting $c$ into the first term to maintain a dimensionless equation, then yields:
$frac{Rdphi }{cdtau } =pm frac{1}{b} frac{sqrt{hbar calpha } }{sqrt{G} m} =pm frac{1}{b} frac{1}{sqrt{4pi G} } frac{q}{m}$. (4.2)
We see that here, the physics ratio $q/m$ measures an “angular frequency” of fifth-dimensional rotation. Interestingly, this frequency runs inversely to the mass, and by classical principles, this means that the angular momentum with fixed radius is independent of the mass, i.e., constant. If one doubles the mass, one halves the tangential velocity, and if the radius stays constant, then so too does the angular momentum. Together with the $pm$ factor, one might suspect that this constant angular momentum is, by virtue of its constancy independently of mass, related to intrinsic spin. In fact, following this line of thought, one can arrive at an exact expression for the compactification radius $R$, in the following manner:
Assume that $x^{5}$ is spacelike, casting one’s lot with the preponderance of those who study Kaluza-Klein theory. In (4.2), move the $c$ away from the first term and move the $m$ over to the first term. Then, multiply all terms by another $R$. Everything is now dimensioned as an angular momentum $mcdot vcdot R$, which we have just ascertained is constant irrespective of mass. So, set this all to $pm {textstylefrac{1}{2}} nhbar$, which for $n=1$, represents intrinsic spin. The result is as follows:
$mfrac{Rdphi }{dtau } R=pm frac{1}{b} frac{sqrt{hbar c^{3} alpha } }{sqrt{G} } R=pm frac{1}{b} frac{c}{sqrt{4pi G} } qR=pm frac{1}{2} nhbar$. (4.3)
Now, take the second and fourth terms, and solve for $R$ with $n=1$, to yield:
$R=frac{b}{2sqrt{alpha } } sqrt{frac{Ghbar }{c^{3} } } =frac{b}{2sqrt{alpha } } L_{P}$, (4.4)
where $L_{P} =sqrt{Ghbar /c^{3} }$ is the Planck length. This gives a definitive size for the compactification radius, and it is very close to the Planck length. (more…)
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http://mathoverflow.net/questions/123482/is-there-a-constructive-proof-of-cantor-bernstein-schroeder-theorem | # Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ?
An important feature of the Cantor-Schroeder-Bernstein theorem is that it does not rely on the axiom of choice. However, its various proofs are non-constructive, as they depend on the law of excluded middle.
There are several well-known proof strategies. For example, there is a simple proof which uses Tarski's fixed point theorem.
My question pertains to the well known proof attributed to Julius Konig, and given here: Proof by Konig. What I have not been able to grasp is, how this is not a constructive proof? Given functions $f$ and $g$, isn't it possible to actually arrive at the bijection by forming biinfinite sequences as given in the proof?
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There is no constructive proof.
To show this, let $A = \mathbb{N}$ and let $B = \{ n \in \mathbb{N}\; |\; \varphi_n(0) \uparrow \}$, where $\varphi_n(0) \uparrow$ means that the $n$th Turing machine is undefined on input 0. There are computable injections $f:A \rightarrow B$ and $g:B \rightarrow A$ (the identity), but no computable surjection from $A$ to $B$ because $B$ is not a c.e. set.
To see why Konig's proof is nonconstructive, let's say that we want to know what the image of $n \in A$ is under the bijection. In order to do this we need to know whether we apply $f$ or $g^{-1}$. But to know this we need to decide whether $n$ is an $A$-stopper, a $B$-stopper, or neither. However we can't do this constructively because it would require checking the entire possibly infinite sequence to see if it ever stops. In fact, in the example above, being able to decide whether or not $g^{-1}$ is defined is the same as deciding the halting problem, so at odd stages we don't even know whether or not the sequence extends any further back at all.
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Meanwhile, in the realm of the purely computable, the Cantor-Schröder-Bernstein theorem holds again, in the form of the Myhill isomorphism theorem en.wikipedia.org/wiki/Myhill_isomorphism_theorem, which asserts that if $A$ and $B$ are subsets of $\mathbb{N}$ and there are computable injective functions $f,g:\mathbb{N}\to\mathbb{N}$ such that $f$ carries $A$ to $B$ and the complement to the complement, and the same for $g$ from $B$ to $A$, then there is a computable bijection of $\mathbb{N}$ taking $A$ exactly to $B$. – Joel David Hamkins Mar 4 '13 at 0:49
The inclusion $g$ seems the more obvious to me. For $f$, I imagine you would start with a Turing machine which loops on input 0, and then for each $n$ attach a subroutine which effectively lets it sit idle that many steps before looping? – Adam Epstein Mar 4 '13 at 22:20
@Joel: very cool, thanks for pointing out the Myhill isomorphism theorem. I shall constructivise it, i.e., it should be just the CBS theorem with extra assumptions, valid in the effective topos. – Andrej Bauer Mar 5 '13 at 11:11
@Adam: yes, I think that way would work. Other ways would be to set $\varphi_{f(n)}(1) = n$, or to add a block of "dead code" that never gets executed, eg loop forever and then return $n$. – aws Mar 5 '13 at 12:44
The Cantor-Bernstein-Schröder theorem (did I read somewhere that Dedekind was the first to prove it?) is not valid constructively. Here are a couple of reasons why.
### There are models in which the theorem fails
Here is a pretty one. In the realizability topos over the infinite-time Turing machines there is an injection $\mathbb{R} \to \mathbb{N}$. Because the topos satisfies countable choice there is no surjection $\mathbb{N} \to \mathbb{R}$ (hence no bijection), but there clearly is an injection $\mathbb{N} \to \mathbb{R}$.
Todd mentioned a presheaf model in his answer.
### The usual proofs are non-constructive
You asked what is non-constructive about König's proof. It uses excluded middle several times:
1. "the sequence can be extended on the left, depending on whether $a$ is in the image of $g$ or not",
2. "the sequence constructed in the proofs either terminates or not",
3. "the sequences so constructed partition the disjoint union of the two sets".
For a proof which uses excluded middle in a more controled way, see the one using Tarski's fixed point theorem, as outlined here. Excluded middle is used "only" in the last step, when the bijection $i$ is defined, depending on whether an element is in $C$ or not.
### The theorem implies strange things, constructively speaking
Here is a strange consequence of the theorem, I will try to improve on it but do not have the time right now. Consider the Cantor space $C = 2^\mathbb{N}$ and the subspace of those sequences which contain a 1, $D = \lbrace \alpha \in C \mid \exists n . \alpha n = 1\rbrace$. As there are injections $C \to D$ and $D \to C$ there is a bijection $h : C \to D$. That is very strange, for it is consistent to assume that all maps on $C$ are uniformly continuous, but such a bijection cannot be uniformly continuous. I will try to improve on this.
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I am intrigued as to what countable choice has to do with no surjection $\Bbb{N\to R}$. – Asaf Karagila Mar 3 '13 at 23:20
@Andrej Yes. It is an interesting history: The first proof of the result dates back to 1887, and is due to Dedekind, who never published it. Jean Cavaillès found the manuscript, and included it in Dedekind’s Collected Works, published in 1932. Dedekind also discussed the proof in a letter to Cantor, dated August 29, 1899. Schröder announced a proof in 1896, details appearing in 1898, but his argument was flawed. Korselt pointed this out, and gave a correct proof, his paper was submitted in 1902, but did not appear until 1911. Burali-Forti found in 1896 a proof for countable sets. – Andrés Caicedo Mar 4 '13 at 0:37
Bernstein found a (correct) proof in 1897, while a student in Cantor’s seminar. This argument was communicated by Cantor to Borel, who published it in a complex analysis book in 1898. Bernstein proof used the countable axiom of choice. Jourdain, in 1907, saw how to make do without it. – Andrés Caicedo Mar 4 '13 at 0:40
Andrej, I really like your infinite-time Turing machine realizability topos! – Joel David Hamkins Mar 4 '13 at 1:23
@Andres, what is the precise statement of the result? I understand answers given so far in the following way: "if by a (model of) (constructive) set theory we mean an elementary topos, then there are models of (constructive) set theory where CSB does not hold". However, the theory of elementary toposes is quite weak, so it is perhaps not the sharpest possible interpretation of the question. Of course Dedekind in 1887 could not interpret the question in this way. – Michal R. Przybylek Mar 4 '13 at 15:10
If you accept that toposes are models of constructive set theory, then another way to answer the question is to give a (non-Boolean) topos where the CBS theorem fails; that would show that this theorem can't possibly have a constructive proof.
A simple example of such a topos is the arrow category $Set^\to$, whose objects are functions $X_0 \to X_1$ between sets and whose morphisms are commutative squares. Let $X$ be the object $f: \mathbb{N} \to \mathbb{N}$ that takes $n \in \mathbb{N}$ to $\mathrm{int}(n/2)$, where $\mathrm{int}(x)$ is the greatest integer less than or equal to $x$; let $Y$ be the object $g: \mathbb{N} \to \mathbb{N}$ that takes $n$ to $\mathrm{Int}((n+1)/2)$, where $\mathrm{Int}(x)$ is the least integer greater than or equal to $x$. It is pretty clear that $X$ and $Y$ are non-isomorphic, because $g^{-1}(0)$ has cardinality $1$ where all fibers of $f$ have cardinality $2$. But, just by drawing pictures of these objects, it is easy to construct monomophisms $i: X \to Y$ and $j: Y \to X$ (e.g., define $i_0(n) = n+1$ and $i_1(n) = n+1$ for all $n$, and define $j_0(n) = n+1$ for $n \gt 0$, $j_0(0) = 0$, and $j_1(n) = n$ for all $n$).
For people who are not used to thinking of topos theory as "constructive set theory", there is another way of considering the example above in terms of "$H$-valued sets", where $H$ is the Heyting algebra with three elements, $H = \{0 < 1/2 < 1\}$. The law of the excluded middle does not hold; one can easily calculate $\neg \neg (1/2) = 1$.
An $H$-valued set is a set $X$ together with a function $e_X: X \times X \to H$ which measures the extent to which elements of $X$ are considered "equal", subject to transitivity and symmetry axioms:
$$e_X(x, y) \wedge e_X(y, z) \leq e_X(x, z), \qquad e_X(x, y) = e_X(y, x).$$
We can think of $e_X(x, x)$ as measuring the extent to which $x$ "exists". An $H$-valued relation is a function $r: X \times Y \to H$ such that
$$e_X(x', x) \wedge r(x, y) \leq r(x', y), \qquad r(x, y) \wedge e_Y(y, y') \leq r(x, y'), \qquad r(x, y) \leq e_X(x, x) \wedge e_Y(y, y).$$
An $H$-valued function $f: X \to Y$ is an $H$-valued relation such that
$$f(x, y) \wedge f(x, y') \leq e_Y(y, y'), \qquad e_X(x, x) \leq \bigvee_{y \in Y} f(x, y)$$
where the first condition is an analogue of well-definedness and the second roughly says that $f(x)$ is defined to the extent $x$ exists. It turns out that the category of $H$-valued sets is equivalent to the topos $Set^\to$.
Under this equivalence, $X$ in the example above is identified with the pair $(\mathbb{N}, e_X: \mathbb{N} \times \mathbb{N} \to H)$ where $e_X(n, n) = 1$, where $e_X(m, n) = 1/2$ if $m \neq n$ but $f(m) = f(n)$, and otherwise $e_X(m, n) = 0$. There is a similar description of $Y$ in terms of $H$-valued sets. For such $H$-valued sets where $e_X(x, x') = 1$ for all precisely when $x = x'$, functional $H$-relations between them can be described as actual functions $f: X \to Y$ subject to the condition $e_X(x, x') \leq e_Y(f(x), f(x'))$ for all $x, x'$ in $X$.
The monomorphic functional relations $i: (\mathbb{N}, e_X) \to (\mathbb{N}, e_Y)$ and $j: (\mathbb{N}, e_Y) \to (\mathbb{N}, e_X)$ turn out to be given by $i(n) = n+1$ and $j(0) = 0$, $j(n) = n+1$ for $n > 0$ (i.e., the functions $i_0$ and $j_0$ in the example above). The monomorphicity amounts to the condition that $e_Y(i(x), i(x')) = e_X(x, x')$ for all $x, x'$ in $X$ (and similarly for $j$). This is easily checked.
Now one can try to run through the König proof to see what goes kaflooey. Following the Wikipedia description, every element of the disjoint union $X \sqcup Y$ unambiguously belongs to an "$X$-stopper" or to a "$Y$-stopper":
$$1_X \stackrel{i}{\mapsto} 2_Y \stackrel{j}{\mapsto} 3_X \stackrel{i}{\mapsto} \ldots$$
$$0_Y \stackrel{j}{\mapsto} 0_X \stackrel{i}{\mapsto} 1_Y \stackrel{j}{\mapsto} \ldots$$
But: when one attempts to define an isomorphism $\phi: X \to Y$ out of this by following the prescription, it immediately goes wrong. Look at what this putative function $\phi$ does to the "half-equal" elements $0_X$ and $1_X$. It sends them respectively to the not-at-all-equal elements $0_Y$ and $2_Y$. Thus it fails to respect
$$e_X(0_X, 1_X) \leq e_Y(\phi(0_X), \phi(1_X))$$
as required by law.
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And internal existence of a bijection implies the external one? – Andrej Bauer Mar 3 '13 at 21:20
It's true that I elided over the internal/external distinction in this answer. However, if you're asking whether $Iso(X, Y) \to 1$ epic implies there exists an isomorphism $X \to Y$, that's of course true for this topos since the terminal object $1$ is projective there. – Todd Trimble Mar 3 '13 at 23:15
It is well-known that Grothendieck toposes and realizability toposes (for different reasons) are models of Intuitionistic Zermelo–Fraenkel set theory. Therefore both Andrej and Todd showed in their answers (in essentially different way) that Cantor–Bernstein–Schroeder cannot be proved in IZF.
Of course, this does not mean that Cantor–Bernstein–Schroeder property is incompatible with constructive mathematics --- it just shows that IZF is too weak to prove CBS. Therefore, a complementary question could be: what are the implications of IZF+CBS; does IZF+CBS make the logic collapse to the boolean case?
If I am not mistaken, the answer is no, and the counterexample is constructed below. However, I shall start with a negative observation.
Let $\Omega$ be a Heyting algebra with countable unions. An element $v \in \Omega$ is complementable if there exists an element $w \in \Omega$ such that $v \vee w = 1$ and $v \wedge w = 0$.
We say that $\Omega$ is a boolean algebra if its every element is a finite union of complementable elements (equivalently, if every element is complementable).
We say that $\Omega$ is a pro-boolean algebra if its every element is a countable union of complementable elements.
The claim is that every elementary topos with countable colimits satisfying Cantor–Bernstein–Schroeder propery is pro-boolean (i.e. its subobject classifier is a pro-boolean algebra).
Let $v \colon V \rightarrow 1$ be a "truth" value (a monomorphism into terminal object). We shall construct two objects $X = \coprod_{\mathbb{N}} 1$ and $Y = V \sqcup X$. There are obvious monomorphisms ${\iota_X}\colon{X}\rightarrow{Y}$ given by the coproduct injection, and ${v \sqcup \mathit{id}}\colon{Y}\rightarrow{X}$. Therefore, by Cantor–Bernstein–Schroeder there is an isomorphism ${b}\colon{Y}\rightarrow{X}$. Since coproducts in a topos are extensive, we may divide each component ${\iota_0}\colon V \rightarrow{Y}$, ${\iota_k}\colon 1 \rightarrow{Y}$ of the coproduct $Y$ along $b$ through coproduct's injections ${\iota_l} \colon 1 \rightarrow{X}$ obtaining elements $\alpha_{k,l}$ such that $Y \approx \coprod_k \coprod_l \alpha_{k,l} \approx \coprod_l \coprod_k \alpha_{k,l}$ and $b = \coprod_l b_l$ with each ${b_l}\colon{\coprod_l \alpha_{l,k}}\rightarrow{1}$ being an isomorphism (using again extensivity of coproducts and the fact that pullback of an iso is iso). Because unions in a topos are effective (and coproducts are disjoint) $1 \approx \coprod_l \alpha_{l,k} = \bigcup_l \alpha_{l,k}$ and so each $\alpha_{l,k}$ is complemented by $\bigcup_{x \neq l} \alpha_{x,k}$. Since every subterminal value can sit in exactly one way in the terminal object, $V = \bigcup_l \alpha_{l,0}$ is a countable disjoint union of complementable elements.
(We cannot get more from this construction; for example in the category of sheaves over rational numbers with the usual topology, $\coprod_{\mathbb{N}} 1 \approx V \sqcup \coprod_{\mathbb{N}} 1$ for (non-complementable) truth value $V$ corresponding to the open ball $(-1, 1)$; unfortunately there are other objects in this category that can serve as counterexamples for CBS. Let me also point that the standard procedure of constructing an isomorphism from two monomorphisms would not work in this case. However, by the above argument it is clear that such an isomorphism may be constructed for any pro-boolean topos. The solution is to not shift uniformly the whole $V$ (or its pseudocomplement), but to move each of (countably many) complementable parts of $V$ separately.)
[I am terribly sorry, now I see that there is an error in the following argument; I will try to fix it (on condition it is possible --- I am not sure now). I should have checked all relevant detail before posted this as an answer.]
For a positive result, consider set: $$\mathcal{D} = \lbrace 0, 1, \frac12, \frac13, \frac14, \dotsc\rbrace$$ with topology inherited from $\mathbb{R}$ and construct the category $\mathit{Sh}(\mathcal{D})$ of sheaves over $\mathcal{D}$. Every open set in $\mathcal{D}$ can be build from singletons $\lbrace\frac1n\rbrace$ and a set of the form $[0, \frac1k]$.
Let $F, G \colon \mathcal{D}^{op} \rightarrow \mathbf{Set}$ be any sheaves and assume there are monomorphisms $m \colon F \rightarrow G$ and $n \colon G \rightarrow F$. A monomorphism between sheaves is an injection on its each component, therefore by CBS theorem for sets $F(U) \approx G(U)$ for every open set $U$. Let $\phi_U \colon F(U) \approx G(U)$ be a collection of such isomorphisms. We shall construct an isomorphism $\alpha \colon F \rightarrow G$ between sheaves inductively:
• $\lambda_{[0, 1]} = \phi_{[0, 1]}$
• for every nonempty $F(\lbrace\frac1k\rbrace)$ choose an element $1_{\frac1k} \in F(\lbrace\frac1k\rbrace)$; if $F(\lbrace\frac1k\rbrace)$ is empty then $\lambda_{[0, \frac1{k+1}]} = \phi_{[0, \frac1{k+1}]}$, otherwise $\lambda_{[0, \frac1{k+1}]} = G([0, \frac1{k+1}] \subset [0, \frac1{k}]) \circ h_{\frac1k}$, where $h_{\frac1k} \colon F([0, \frac1{k+1}]) \rightarrow F([0, \frac1{k}])$ is the unique morphism to the product $F([0, \frac1{k}]) = F([0, \frac1{k+1}]) \times F(\lbrace\frac1k\rbrace)$ induced by $F([0, \frac1{k+1}]) \overset{!}\rightarrow 1 \overset{1_{\frac1k}}\rightarrow F(\lbrace\frac1k\rbrace)$ and the identity on $F([0, \frac1{k+1}])$
• similarly, for every nonempty $F([0, \frac1{k+1}])$ choose an element $1_{[0, \frac1{k+1}]} \in F([0, \frac1{k+1}])$; if $F([0, \frac1{k+1}])$ is empty then $\lambda_{\lbrace\frac1{k}\rbrace} = \phi_{\lbrace\frac1{k}\rbrace}$, otherwise $\lambda_{\lbrace\frac1{k}\rbrace} = G(\lbrace\frac1{k}\rbrace \subset [0, \frac1{k}]) \circ h_{{[0, \frac1{k+1}]}}$
• if $U$ is a disjoint union of the form $[0, \frac1k] \sqcup \bigcup_i\lbrace\frac{1}{n_i}\rbrace$, where $[0, \frac1k]$ is the largest interval contained in $U$, then $\lambda_{U} = \lambda_{[0, \frac1k]} \times \prod_i \lambda_{\lbrace\frac{1}{n_i}\rbrace}$, where the products are determined by structures of the sheaves.
In the second and third step we have chosen the components of $\lambda$ to be upward compatible, and in the fourth step the naturality condition follows from the universal property of products. Thus $F \approx G$.
[EDIT: Let me argue that $\phi_U$ may be chosen in such a way that each $\lambda_U$ is really an isomorphism. Assume, that all $F(\lbrace \frac1k\rbrace)$ are nonempty. Define $\mathit{colim}F([0, \frac1k])$ to be the colimit of the diagram: $$F([0, 1]) \rightarrow F([0, \frac12]) \rightarrow \cdots \rightarrow F([0, \frac1k]) \rightarrow \cdots$$ We have: $$(\mathit{colim}_kF([0, \frac1k])) \times (\prod_i F(\lbrace\frac1i\rbrace)) \approx F([0, 1]) \times \mathit{colim}\_k \prod\_{i > k} F(\lbrace\frac1i\rbrace) \approx F([0,1])$$ and similarly for $G$. Since in a locally presentable category monomorphisms are stable under directed colimits, both: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(m_{[0, \frac1k]}\right)}\rightarrow \mathit{colim}G([0, \frac1k])$$ and: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(n_{[0, \frac1k]}\right)}\leftarrow \mathit{colim}G([0, \frac1k])$$ are monomorphisms, thus by CBS for sets $\mathit{colim}F([0, \frac1k]) \overset{\phi_0}\approx \mathit{colim}G([0, \frac1k])$. Therefore, $\phi_{[0, 1]}$ may be assumed to be of the form $\phi_0 \times \prod \phi_{\lbrace\frac1k\rbrace}$. Likewise every $\phi_{[0, \frac1k]}$. ]
(BTW, I think we are not really that far from the inverse of the above theorem, but that is for another story...)
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Failure of the law of the excluded middle is hard for me to imagine. But here is a try...
That König proof has this:
For any particular $a$, this sequence may terminate to the left or not
But perhaps we also have to consider the case of both: it terminates to the left and also does not terminate to the left... ???
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This answer seems orthogonal to the original question. The OP states that the proof requires the law of excluded middle. – John Pardon Mar 3 '13 at 18:48
@unknown: I think the question is precisely "where does Koenig's proof fail to be constructive" and Gerald is flagging precisely where it fails. – user30035 Mar 3 '13 at 20:34
Gerald is certainly not instilling a lot of confidence in his answer. – Andrej Bauer Mar 3 '13 at 20:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9638230800628662, "perplexity": 245.3698115009984}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049279525.46/warc/CC-MAIN-20160524002119-00173-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://lucene.472066.n3.nabble.com/Probability-from-log-likelihood-in-LDA-output-td2028173.html | Probability from log likelihood in LDA output
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Probability from log likelihood in LDA output
Hello, As I understand it, the output for LDA is a log likelihood value for each word/topic pair, which is a function of log(P(w|t)). Is it possible to invert that function to obtain P(w|t)? I have a feeling it is not, since it looks like the final value is obtained as a sum of log probabilities, but I just wanted to check, since an output as a probability is more readable than the likelihood value given. Thanks, Andres
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Re: Probability from log likelihood in LDA output
Yes. I should be possible to use exp to get the actual probability. The fact that it is a sum of log probabilities just means that the probability is a product of probabilities. It is possible that the probabilities are not normalized, but that would be a bit surprising for this kind of algorithm. On Mon, Dec 6, 2010 at 8:02 AM, Quiroz Hernandez, Andres < [hidden email]> wrote: > Hello, > > As I understand it, the output for LDA is a log likelihood value for > each word/topic pair, which is a function of log(P(w|t)). Is it possible > to invert that function to obtain P(w|t)? I have a feeling it is not, > since it looks like the final value is obtained as a sum of log > probabilities, but I just wanted to check, since an output as a > probability is more readable than the likelihood value given. > > Thanks, > > Andres > | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9006781578063965, "perplexity": 623.3526036621205}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669183.52/warc/CC-MAIN-20191117142350-20191117170350-00241.warc.gz"} |
https://math.stackexchange.com/questions/2164407/on-the-theorem-of-the-cube-and-the-semicontinuity-theorem-intuition-for-proof-o | # On the theorem of the cube and the semicontinuity theorem: intuition for proof of proposition?
Consider the following result from p. 89 of Mumford's Abelian Varieties.
Proposition. Let $$X$$ be a complete variety, $$Y$$ any scheme and $$L$$ a line bundle on $$X \times Y$$. Then there exists a unique closed subscheme $$Y_1 \hookrightarrow Y$$ having the following properties:
(a) if $$L_1$$ is the restriction of $$L$$ to $$X \times Y_1$$, there is a line bundle $$M_1$$ on $$Y$$ and an isomorphism $$p_2^* M_1 \cong L_1$$ on $$X \times Y_1$$;
(b) if $$f: Z \to Y$$ is any morphism such that there exists a line bundle $$K$$ on $$Z$$ and an isomorphism $$p_2^*(K) \cong (1_X \times f)^*(L)$$ on $$X \times Z$$, $$f$$ can be factored as $$Z \to Y_1 \hookrightarrow Y$$.
Is it possible somebody could give me their intuitions behind Mumford's proof of this result? The proof is as follows. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9830428957939148, "perplexity": 57.92914922665848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703513144.48/warc/CC-MAIN-20210117174558-20210117204558-00261.warc.gz"} |
https://www.physicsforums.com/threads/black-body-radiation-entropy-question.422069/ | # Homework Help: Black body radiation entropy question
1. Aug 13, 2010
1. The problem statement, all variables and given/known data
An evacuated container with volume V and at a temperature T contains black body radiation with an energy density equal to 4$$\sigma$$T4/c
I Determine the heat capacity at constant volume of the radiation
II Hence show that the entropy of the radiation is given by
S(T,V)=16$$\sigma$$VT3/3c (can't seem to get the sigma to display correctly here)
III The container is placed in thermal contact with a heat bath at temperature Tr. If the heat capacity of the cavity material itself is negligible, show that the overall change in entropy of the universe after the system and heat bath have reached thermal equilibrium is
$$\Delta$$Stot=(4$$\sigma$$VTr3/3c)(1-t3(4-3t))
t= T/Tr
IV comment on the sign of $$\Delta$$Stot as a function of t
2. Relevant equations
3. The attempt at a solution
Have done I and II correctly. IV looks fine.
Just need a little nudge as to how I should start III, I just can't think of the appropriate formulae at the moment. So any links to references etc are greatly appreciated.
2. Aug 13, 2010
### diazona
Well, think about it this way: the total change in entropy of the universe comes from only two systems, the radiation and the heat bath. So you can calculate ΔStot as
$$\Delta S_{\text{tot}} = \Delta S_{\text{rad}} + \Delta S_{\text{bath}}$$
To calculate the entropy change of the heat bath, recall the (well, a) definition of entropy:
$$\Delta S = \int_{T_i}^{T_f} \frac{\mathrm{d}Q}{T}$$
Remember that the heat bath is at a constant temperature - how does that fact allow you to simplify the equation?
Calculating the entropy change of the radiation can be done in the patently obvious manner Recall the definition of the Δ symbol. Other than that, you already have the only formula you need.
3. Aug 13, 2010
### Andrew Mason
the latex "tex" tokens are intended be used on a separate line. For inline latex use "itex".
Keep in mind that entropy change is defined as the integral $\int dQ/T$ over the reversible path between two states. The reversible path would be one in which the cavity is cooled by means of a carnot heat engine operating between the cavity and the heat bath. In that case, the carnot engine absorbs heat at the cavity temperature (which decreases with heat flow) and the heat bath absorbs heat from the carnot engine at constant temperature.
AM
4. Aug 14, 2010 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9393284916877747, "perplexity": 446.8091624075426}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376826856.91/warc/CC-MAIN-20181215131038-20181215153038-00541.warc.gz"} |
https://dlmf.nist.gov/search/search?q=connection%20formula | # connection formula
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## 1—10 of 60 matching pages
##### 1: 31.18 Methods of Computation
Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of $z$; see Laĭ (1994) and Lay et al. (1998). …
##### 2: 25.13 Periodic Zeta Function
25.13.2 $F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$ $0, $\Re s>1$,
25.13.3 $\zeta\left(1-s,x\right)=\frac{\Gamma\left(s\right)}{(2\pi)^{s}}\left(e^{-\pi is% /2}F\left(x,s\right)+e^{\pi is/2}F\left(-x,s\right)\right),$ $0, $\Re s>0$.
##### 3: 14.21 Definitions and Basic Properties
###### §14.21(iii) Properties
This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
##### 6: 19.21 Connection Formulas
###### §19.21 ConnectionFormulas
The complete cases of $R_{F}$ and $R_{G}$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … Connection formulas for $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ are given in Carlson (1977b, pp. 99, 101, and 123–124). …
##### 8: Bibliography O
• F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
• F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
• F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 39, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9532444477081299, "perplexity": 4162.662231814247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657129517.82/warc/CC-MAIN-20200712015556-20200712045556-00435.warc.gz"} |
https://zombiedoc.com/1993-multiple-choice-screening-test-30.html | # 1993 MULTIPLE CHOICE SCREENING TEST 30 …
1993 MULTIPLE CHOICE SCREENING TEST 30 QUESTIONS—40 MINUTES DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN ... spring with force constant k a dista...
PHYSICS OLYMPIAD (ΠΗΨΣΙΧΣ ΟΛΨΜΠΙΑ∆) 1993 MULTIPLE CHOICE SCREENING TEST 30 QUESTIONS—40 MINUTES
me
= 9.1 x 10-31 kg
mass of proton
mp
= 1.7 x 10-27 kg
electronic charge
e
= 1.6 x 10-19 C
speed of light
c
= 3.0 x 108 m/s
Coulomb's constant
k
= 9.0 x 109 Nm2/C2
permittivity constant
ε0
= 8.9 x 10-12 C2/Nm2
permeability constant
µ0
=
gravitational constant
G
= 6.7 x 10-11 Nm2/kg2
acceleration due to gravity
g
= 10 m/s2
speed of sound (20 oC)
vs
= 340 m/s
4π x 10-7 Tm/A
DO NOT OPEN THIS TEST UNTIL YOU ARE INSTRUCTED TO BEGIN Copyright © 1993, AAPT
1. A ball of mass m is thrown vertically upward. Air resistance is not negligible. Assume the force of air resistance has magnitude proportional to the velocity, and direction opposite to the velocity's. At the highest point, the ball's acceleration is: (A) 0
(B) less than g
(C) g
(D) greater than g
(E) upward
2. A train is moving forward at a velocity of 2.0 m/s. At the instant the train begins to accelerate at 0.80 m/s2, a passenger drops a quarter which takes 0.50 s to fall to the floor. Relative to a spot on the floor directly under the quarter at release, it lands: (A) 1.1 m toward the rear of the train. (B) 1.0 m toward the rear of the train. (C) 0.10 m toward the rear of the train. (D) directly on the spot. (E) 0.90 m toward the front of the train. 3. The dropped quarter in the preceding question (#2) is viewed by an observer standing next to the tracks. Relative to this observer, the quarter moves before landing. (A) forward 1.1 m (B) forward 1.0 m (C) forward 0.10 m (D) straight down (E) backward 0.10 m x
4. The accompanying graph of position x versus time t represents the motion of a particle. If b and c are both positive constants, which of the following expressions best describes the acceleration a of the particle? (A) a = 0
t
(B) a = +b (C) a = -c (D) a = b + ct (E) a = b - ct 5. In the system shown to the right, a force F pushes on block A, giving the system an acceleration a. The coefficient of static friction between the blocks is µ. The correct equation for block B not to slip is: (A) a > µg
(B) a < µg
(C) a > g
(D) a > g/µ Page 1 of 7
F
(E) a < g/µ
A
B
6. A block of mass m starts at rest at height h on a frictionless inclined plane. The block slides down the plane, travels a total distance d across a rough surface h with coefficient of kinetic friction µ, and compresses a spring with force constant k a distance x before momentarily coming to rest. Then the spring extends and the block travels back across the rough surface, sliding up the plane. The correct expression for the maximum height h' that the block reaches on its return is: (A) mgh' = mgh - 2µmgd (B) mgh' = mgh + 2µmgd (C) mgh' = mgh + 2µmgd + kx2 (D) mgh' = mgh - 2µmgd- kx2 (E) mgh' = mgh - 2µmgd- kx2 - 1/2 mv2 7. Air track car A has mass m and velocity v. Air track car B has mass 2m and velocity 3v. The same constant force F is applied to each car until it stops. Car A is brought to rest in time t. The time required to stop car B is: (A) 2t
(B) 3t
(C) 6t
(D) 9t
(E) 18t
8. In the preceding question (#7), car A travels a distance d before coming to rest. The distance traveled by car B before coming to rest is: (A) 2d
(B) 3d
(C) 6d
(D) 9d
(E) 18d
9. Three air track cars are initially placed as shown in the A accompanying figure. Car A has mass m and initial velocity v to the right. Car B with mass m and car C with mass 4m are both m initially at rest. Car A collides elastically with car B, which in turn collides elastically with car C. After the collision, car C has a velocity of 0.4v to the right. The final velocities of cars A and B are: (A) Car A: 0.6v to the left
Car B: at rest
(B) Car A: 1.4v to the left
Car B: at rest
(C) Car A: v to the left
Car B: 0.6v to the left
(D) Car A: 0.4v to the left
Car B: v to the left
(E) Car A: 1.6v to the left
Car B: v to the right
Page 2 of 7
v
B
C
m
4m
10. Three cylinders, all of mass M, roll without slipping down an inclined plane of height H. The cylinders are described as follows: I. hollow of radius R II. solid of radius R/2 III. solid of radius R If all cylinders are released simultaneously from the same height, the cylinder (or cylinders) reaching the bottom first is (are): (A) I
(B) II
(C) III
(D) I & II
(E) II & III
11. The system shown to the right is free to rotate about a frictionless axis through its center and perpendicular to the page. All three forces are exerted tangent to their respective rims. The magnitude of the net torque acting on the system is:
37
R/2
F R
(A) 1.5 FR 2F
(B) 1.9 FR (C) 2.3 FR (D) 2.5 FR (E) 3.5 FR
12. Two identical disks are positioned on a vertical axis. The bottom disk is rotating at angular velocity ω0 and has rotational kinetic energy K0. The top disk is initially at rest. It is allowed to fall, and sticks to the bottom disk. What is the rotational kinetic energy of the system after the collision? (A) 1/4 K0
F
O
(B) 1/2 K0
(C) K0
(D) 2 K0
(E) 4 K0
13. If the sun were suddenly replaced by a black hole of one solar mass, what would happen to the earth's orbit immediately after the replacement? (A) The earth would spiral into the black hole. (B) The earth would spiral out away from the black hole. (C) The radius of the earth's orbit would be unchanged, but the period of the earth's motion would increase. (D) Neither the radius of the orbit nor the period would change. (E) The radius of the earth's orbit would be unchanged, but the period of the earth's motion would decrease.
Page 3 of 7
14. A hypothetical planet has density ρ, radius R, and surface gravitational acceleration g. If the radius of the planet were doubled, but the planetary density stayed the same, the acceleration due to gravity at the planet's surface would be: (A) 4g
(B) 2g
(C) g
(D) g/2
(E) g/4
15. An ideal organ pipe resonates at frequencies of 50 Hz, 150 Hz, 250 Hz,.... The pipe is: (A) open at both ends and of length 1.7 m. (B) open at both ends and of length 3.4 m. (C) open at both ends and of length 6.8 m. (D) closed at one end, open at the other, and of length 1.7 m. (E) closed at one end, open at the other, and of length 3.4 m. 16. A porpoise, whistle-clicking at a frequency f0, swims toward an underwater vertical cliff at a velocity that is 1.0% of the velocity of sound in sea water. The reflected frequency experienced by the swimming porpoise is: (A) 0.98 f0
(B) 0.99 f0
(C) f0
(D) 1.01 f0
17. Three processes compose a thermodynamic cycle shown in the accompanying pV diagram. Process 1→2 takes place at constant temperature. Process 2→3 takes place at constant volume, and process 3→1 is adiabatic. During the complete cycle, the total amount of work done is 10 J. During process 2→3, the internal energy decreases by 20 J; and during process 3→1, 20 J of work is done on the system. How much heat is added to the system during process 1→2?
(E) 1.02 f0
1
p
2 3 V
(A) 0 Q
(B) 10 J
1➝2 2➝3 3➝1 cycle
(C) 20 J (D) 30 J (E) 40 J
W
-20J -20J +10J
18. The root mean square velocity of oxygen gas is v at room temperature. What is the root mean square velocity of hydrogen gas at the same temperature? (A) 16 v
(B) 4 v
(C) v
(D) v/4
Page 4 of 7
(E) v/16
∆U
19. Monochromatic light of wavelength λ is shone on a grating consisting of six equally spaced slits. The first order interference maximum occurs at an angle of 0.00100 radians. If the outer two slits are covered, the first order maximum will occur at ______ radians. (A) 0.00025
(B) 0.00067
(C) 0.00100
(D) 0.00150
(E) 0.00400
20. You are given two lenses, a converging lens with focal length +10 cm and a diverging lens with focal length -20 cm. Which of the following would produce a virtual image that is larger than the object? (A) Placing the object 5 cm from the converging lens. (B) Placing the object 15 cm from the converging lens. (C) Placing the object 25 cm from the converging lens. (D) Placing the object 15 cm from the diverging lens. (E) Placing the object 25 cm from the diverging lens. 21. Two small identical conducting spheres are separated by a distance much larger than their diameter. They are initially given charges of -2.00 x 10-6 C and +4.00 x 10-6 C, and found to exert a force on each other of magnitude 1.000 N. Without changing their position, they are connected by a conducting wire. When the wire is removed, what is the magnitude of the force between them? (A) 0
(B) 0.125 N
(C) 0.250 N
(D) 1.000 N
(E) 1.125 N
22. Charges of ±Q, with Q = 2.0 x 10-7 C, are placed at three corners of a square whose sides are 0.10 m. The magnitude of the total field at the center of the square is:
+Q
(B) 2.5 x 104 V/m (C) 7.5 x
104
V/m
(D) 3.6 x
105
V/m
X
0.10m
(A) 5.1 x 103 V/m
-Q
+Q 0.10m
(E) 1.08 x 106 V/m 23. Three 60 W light bulbs are mistakenly wired in series and connected to a 120 V power supply. Assume the light bulbs are rated for single connection to 120 V. With the mistaken connection, the power dissipated by each bulb is: (A) 6.7 W
(B) 13.3 W
(C) 20 W
(D) 40 W
Page 5 of 7
(E) 60 W
24. Two thin spherical conducting shells are centered on the same point. The inner sphere has radius b and total charge q. The outer sphere has radius B and total charge Q. At a point a distance R from the common center, where b< R
80 Ω 0.1 A
25 Ω V
20 Ω
(A) 0.10 A (B) 0.20 A
20 Ω
(C) 0.30 A
60 Ω
(D) 0.40 A (E) 0.60 A 27. The circuit shown in the accompanying figure has been connected for a very long time. The voltage across the capacitor is: (A) 1.2 V (B) 2.0 V (C) 2.4 V (D) 4.0 V (E) 6.0 V
Page 6 of 7
100 Ω 6V 200 Ω
100 Ω 4 µF
28. The accompanying sketch represents a bubble chamber photograph. field is directed out of the plane of the sketch. All particles have the same velocity initially from left to . . . right. The one that is most likely an electron is: .A . . . . . (A) A . . . (B) B . . . . . . (C) C E . . . (D) D
The magnetic . . . . . . .
. .B . . . .D .
. . . .C . . .
(E) E 29. What is the correct expression for the magnetic field at point X in the diagram to the right? HINT: The magnitude of the magnetic field at the center of a circular current loop of radius R is µ 0i/(2R). (A) (µ 0i/4)(1/a + 1/b)
into the page
(B) (µ 0i/4)(1/a - 1/b)
out of the page
(C) (µ 0i/4)(1/a - 1/b) - µ 0i/(2πa)
out of the page
(D) (µ 0i/2)(1/a - 1/b)
out of the page
(E) (µ 0i/2)(1/a + 1/b) + µ 0i/(2πa)
into the page
b i
a X
30. You are given a bar magnet and a looped coil of wire. Which of the following would induce an emf in the coil? I. Moving the magnet away from the coil. II. Moving the coil toward the magnet. III. Turning the coil about a vertical axis. (A) I only
(B) II only
(C) I & II
(D) I & III
Page 7 of 7
(E) I, II, III
. . . . . . .
. . . . . . .
1993 MULTIPLE CHOICE SCREENING TEST ANSWER KEY 1. C
16. E
2. C
17. D
3. B
18. B
4. E
19. C
5. D
20. A
6. A
21. B
7. C
22. D
8. E
23. A
9. A
24. D
10. E
25. C
11. A
26. C
12. B
27. D
13. D
28. A
14. B
29. B
15. D
30. E
me = 9.1 x 10-31 kg
mass of proton
mp = 1.7 x 10-27 kg
electronic charge
e
= 1.6 x 10-19 C
speed of light
c
= 3.0 x 108 m/s
Coulomb's constant
k
= 9.0 x 109 N·m2/C2
permittivity constant
ε0
= 8.9 x 10-12 C2/N·m2
permeability constant
µ0
=
gravitational constant
G
= 6.7 x 10-11 N·m2/kg2
acceleration due to gravity
g
= 10 m/s2
speed of sound (20 oC)
vs
= 340 m/s
4π x 10-7 T·m/A
DO NOT OPEN THIS TEST UNTIL YOU ARE INSTRUCTED TO BEGIN Copyright © 1996, AAPT
1. An object is projected straight upward from ground level with a velocity of 50 m/s. Ignoring air resistance, it will return to ground level in approximately: A. 2.5 s
B. 5.0 s
C. 7.5 s
D. 10 s
E. 15 s
2. A jogger runs with constant speed v through a forest of pine trees. A pine cone starts to fall from a height h when the jogger is directly below it. How far behind the jogger will the pine cone land? 2hv2 g
A.
B.
hv2 2g
gh 2 C. 2v2
v2 E. 2g
2gh2 D. v2
3. A ball is thrown downward with speed 15 m/s from the roof of a 30 m building. At the same instant a ball is thrown upward with speed 15 m/s from ground level. Relative to ground level, the two balls pass each other at a height of: A. 0
B. 5.0 m
C. 10 m
D. 15 m
E. 20 m
4. A swimmer can swim with a velocity of 1.0 m/s in still water. The swimmer wishes to swim directly across a river with a current of 0.50 m/s directed from upstream to downstream. To end up directly across the river the swimmer must head at an angle of: A. B. C. D. E.
tan-1(1/2) upstream. sin-1(1/2) upstream. directly across the river. sin-1(1/2) downstream tan-1(1/2) downstream
5. What is the tension T in the rope if the 10-N weight is moving upward with constant velocity? A. 3.5 N B. 5.0 N C. 7.1 N D. 10 N E. 14 N
T
45o
10 N
page 1
6. As shown to the right, two Case I Case II blocks with masses m and M F (M > m) are pushed by a force F F M M m m in both Case I and Case II. The surface is horizontal and frictionless. Let RI be the force that m exerts on M in case I and RII be the force that m exerts on M in case II. Which of the following statements is true? A. B. C. D. E.
RI = RII = 0 RI = RII and is not equal to zero or F. RI = RII = F RI < RII RI > RII
7. Two blocks, with masses 17 kg and 15 kg, are connected by a light string that passes over a frictionless pulley of negligible mass as shown to the right. The surfaces of the planes are frictionless. The blocks are released from rest. T1 and T2 are the tensions in the strings. Which of the following statements is correct? A. B. C. D. E.
T2
T1
17 kg 15 kg 45o
60o
The 15-kg block accelerates down the plane. The 17-kg block accelerates down the plane. Both blocks remain at rest. T 1 > T2 T 1 < T2
8. A small block of mass m starts from rest at the top of a globe of radius R. At what angle θ does it slide off the surface of the globe? Assume the system is frictionless. A. θ = 0o B. θ = cos-1(1/3) C. θ = cos-1(2/3) D. θ = 60o E. θ = 90o
R θ
9. An object with mass m and initial velocity v is brought to rest by a constant force F acting for a time t and through a distance d. Possible expressions for the magnitude of the force F are: I. (mv2)/(2d) II. (2md)/t2 III. (mv)/t Which of these give(s) the correct expression for the magnitude of the force F? A. II only
B. III only
C. I and II only
D. II and III only
E. I, II, and III page 2
10. A small sphere is moving at a constant speed in a vertical circle. Below is a list of quantities that could be used to describe some aspect of the motion of the sphere. I – kinetic energy II – potential energy III – momentum Which of these quantities will change as this sphere moves around the circle? A. I and II only
B. I and III only
C. II only
D. III only
E. II and III only
11. A roller coaster travels with speed vA at point A. Point B is a height H above point A. Assuming no frictional losses and no work done by a motor, what is the speed at point B? vA 2 − 2gH
A.
B. v A − 2gH
C. v A − 2gH
D. v A + 2gH
E.
vA 2 + 2gH
12. Three air track cars are initially placed as shown in the A B C accompanying figure. Car A has mass m and initial velocity v to v the right. Car B with mass m and car C with mass 4m are both m m 4m initially at rest. Car A collides elastically with car B, which in turn collides elastically with car C. After the collision, car B is at rest. The final velocities of cars A and C are: A. B. C. D. E.
Car A: 0.6v to the left Car A: 2.6v to the left Car A: at rest Car A: at rest Car A: at rest
Car C: 0.4v to the right Car C: 0.4v to the right Car C: 0.5v to the right Car C: 0.25v to the right Car C: v to the right
13. A child with mass m is standing at the edge of a playground merry-goround with moment of inertia I, radius R, and initial angular velocity ω . See figure to the right. The child jumps off the edge of the merry-go-round with tangential velocity v with respect to the ground. The new angular velocity of the merry-go-round is:
v ω
A. ω Iω 2 − mv 2 I
B. C.
(I + mR 2 )ω 2 − mv2
I Iω − mvR D. I I + mR 2 ω − mvR E. I
(
)
page 3
14. As shown in the figure to the right, a spool has outer radius R and axle radius r. A string is wrapped around the axle of the spool and can be pulled in any of the directions labeled by I, II, or III. The spool will slide to the right without rolling on the horizontal surface if it is pulled in direction(s): A. I only B. II only C. III only D. I and II only E. II and III only
I II III
r R
15. A uniform flag pole of length L and mass M is pivoted on the ground with a frictionless hinge. The flag pole makes an angle θ with the horizontal. The moment of inertia of the flag pole about one end is (1/3)ML2. If it starts falling from the position shown in the accompanying figure, the linear acceleration of the free end of the flag pole – labeled P – would be: A. Β. C. D. E.
P L
θ
(2/3) g cos θ (2/3) g g (3/2) g cos θ (3/2) g
16. The root mean square velocity of oxygen gas (atomic mass 16) is v at room temperature. What is the root mean square velocity of helium (atomic mass 4) at the same temperature? A. 4 v
B. 2 v
C. v
D. v/2
E. v/4
17. Which of the accompanying PV diagrams best represents an adiabatic process (process where no heat enters or leaves the system)? A. P
C. P
B. P
V
V
E. P
D. P
V
V
V
18. String A and String B have the same mass and length. String A is under tension T and string B is under tension 2T. The speed of waves in B is _____ times the speed of waves in A. A. 0.50 B. 0.71 C. 1.00 D. 1.4 E. 2.0 1996 Physics Olympiad
page 4
19. On a day when the velocity of sound in air is v, a whistle moves with velocity u toward a stationary wall. The whistle emits sound with frequency f. What frequency of reflected sound will be heard by an observer traveling along with the whistle? A. f vv+-uu B. f v v+ u C. f D. f v-vu
u
E. f vv+-uu
20. You are given two lenses, a converging lens with focal length + 10 cm and a diverging lens with focal length - 20 cm. Which of the following would produce a real image that is larger than the object? A. B. C. D. E.
Placing the object 5 cm from the converging lens. Placing the object 15 cm from the converging lens. Placing the object 25 cm from the converging lens. Placing the object 15 cm from the diverging lens. Placing the object 25 cm from the diverging lens.
21. Two mirrors, labeled LM for left mirror and RM for right mirror in the accompanying figure, are parallel to each other and 3.0 m apart. A person standing 1.0 m from the right mirror (RM) looks into this mirror and sees a series of images. How far from the person is the second closest image seen in the right mirror (RM)? A. B. C. D. E.
2.0 m 4.0 m 6.0 m 8.0 m 10.0 m
LM
RM
2.0 m
1.0 m
22. A thin film of thickness t and index of refraction 1.33 coats a glass with index of refraction 1.50. What is the least thickness t that will strongly reflect light with wavelength 600 nm? Hint: Light undergoes a 180o phase shift when it is reflected off a material with a higher index of refraction.
t
A. 225 nm
E. 600 nm
B. 300 nm
C. 400 nm
D. 450 nm
n=1.00 n=1.33 n=1.50
page 5
23. The accompanying figure shows two concentric spherical shells isolated from each other. The smaller shell has radius b and net charge +Q. The larger shell has radius 2b and an equal net charge +Q. If R is the distance from the common center, the highest electric field magnitude E occurs: A. B. C. D. E.
+Q b +Q
only at R = 0, where E is infinite. anywhere R < b, where E is constant immediately outside the smaller (R = b) shell. immediately outside the larger (R = 2b) shell. far away from the shells; E increases with distance.
24. An infinite conducting plate of thickness 0.0200 m is surrounded by a uniform field E = 400 V/m directed left to right. See the figure to the right. Let the potential Vo = 0 a distance 0.0200 m to the right of the plate. What is V3, the potential 0.0300 m to the left of the plate? A. - 28 V B. - 20 V C. + 12 V D. + 20 V E. + 28 V
2b
E
E
V
V
3
1
0.020m
0.030m
25. A sphere of radius a has uniform charge density ρ. A spherical cavity of radius c is formed in the sphere. The cavity is centered a distance b (b>c) from the center of the sphere. What is the magnitude of the electric field at the center of the cavity?
B.
ρa 3
3ε b 2 o
ρ a3 − c3 C.
3ε o b2
c3 ρ b 2b2 D. 3ε o
26. As shown in the diagram to the right, two fixed charges, q1 = +1.00 µC and q2 = - 4.00 µC, are 0.200 m apart. Where is the total field zero? A. 0.40 m to the right of q1 B. 0.13 m to the right of q1 C. 0.1m to the right of q1 D. 0.067 m to the left of q1 E. 0.20 m to the left of q1
q
a c
c3 ρ b b2 E. 3ε o
0.200 m 1
0
0.020m
b
ρb A. 3ε o
V
V
2
q 2
page 6
27. The switch is closed in the circuit shown to the right. What is the charge on the capacitor when it is fully charged? A. B. C. D. E.
5.0 µC 10 µC 20 µC 40 µC 60 µC
100Ω 200Ω 6.00V 10.0µF
28. Two identical resistors with resistance R are connected in the two circuits drawn below. The battery in both circuits is a 12 volt battery. Which statement is correct?
I
II 12 V R
12 V
R
R
R
A. B. C. D. E.
More current will flow through each R in circuit I than in circuit II. More total power will be delivered by the battery in circuit II than in circuit I. The potential drop across each R will be greater in circuit I than in circuit II. The equivalent resistance will be greater in circuit II than in circuit I. The power dissipated in each R will be greater in circuit I than in circuit II.
29. The resistors – R1, R2, and R3 – have been adjusted so that the current in the ammeter (labeled A in the accompanying circuit diagram) is zero. What is R? A. R2 B. R3 C. R1R2/R3 D. R1R3/R2 E. R2R3/R1
30. A long cylindrical conducting wire – shown in cross section to the right – carries a conventional current out of the page. The wire has uniform current density J and radius a. What is the magnetic field inside the wire, a distance R (R
R3
R I=0 A
R1
R2
V
a R
page 7
1996 MULTIPLE CHOICE SCREENING TEST ANSWER KEY
1.
D
11. A
21. C
2.
A
12. A
22. A
3.
C
13. E
23. C
4.
B
14. B
24. D
5.
B
15. D
25. A
6.
D
16. B
26. E
7.
A
17. C
27. D
8.
C
18. D
28. B
9.
E
19. E
29. E
10. E
20. B
30. C
me = 9.1 x 10-31 kg
mass of proton
mp = 1.7 x 10-27 kg
electronic charge
e
= 1.6 x 10-19 C
speed of light
c
= 3.0 x 108 m/s
Coulomb's constant
k
= 9.0 x 109 N·m2/C2
permittivity constant
ε0
= 8.9 x 10-12 C2/N·m2
permeability constant
µ0
=
gravitational constant
G
= 6.7 x 10-11 N·m2/kg2
acceleration due to gravity
g
= 10 m/s2
speed of sound (20 oC)
vs
= 340 m/s
4¹ x 10-7 T·m/A
DO NOT OPEN THIS TEST UNTIL YOU ARE INSTRUCTED TO BEGIN Copyright © 1995, AAPT
1. The accompanying figure is a graph of an object’s position as a function of time. Which lettered segment corresponds to a time when the object has a negative acceleration? A. B. C. D. E.
+x
C B D
A B C D E
A t
E
2. A car is moving at a constant velocity to the right along a straight level highway. Just as the car passes a cliff, a rock falls straight down in the cliff’s reference system. Which of the accompanying curves best depicts the path the rock takes in the car’s reference system? A.
B.
D.
C.
E.
1 2 3 3. Three blocks – 1, 2, and 3 – rest on a horizontal F frictionless surface, as shown in the accompanying figure. m m m Each block has a mass m, and the blocks are connected by massless strings. Block 3 is pulled to the right by a force F. The resultant force on block 2 is: A. Zero
B. (1/3) F
C. (1/2) F
D. (2/3) F
E. F
4. Which solid vector in the accompanying figures best represents the acceleration of a pendulum mass at an intermediate point in its swing? A.
B.
C.
D.
E.
16 cm
5. A mass attached to a horizontal massless spring is displaced 16 cm to the right of its equilibrium position (see accompanying figure) and released from rest. At its 16 cm extension, the spring-mass system has 1.28 joules of potential energy. Upon release, it slides across a rough surface and comes momentarily to rest 8 cm to the left of its equilibrium position. How much mechanical energy was dissipated by friction? A. 0.16 J
B. 0.32 J
C. 0.64 J
D. 0.96 J
8 cm
E. 1.12 J
page 1
6. Two identical masses m are connected to a massless string which is hung over two frictionless pulleys as shown on the right. If everything is at rest, what is the tension in the cord? A. B. C. D. E.
Less than mg. Exactly mg. More than mg but less than 2mg. Exactly 2mg. More than 2mg.
m
m
7. Car A and car B are both traveling down a straight highway at 25 m/s (about 56 mph). Car A is only 6.0 m behind car B. The driver of car B brakes, slowing down with a constant acceleration of 2.0 m/s2. After a time 1.2 s, the driver of car A begins to brake, also at 2.0 m/s2. What is the relative velocity of the two cars when they collide? Hint: Both cars are still moving forward when they collide. A. 2.4 m/s
B. 5.0 m/s
C. 9.5 m/s
D. 21 m/s
E. 24 m/s
8. During the collision between car A and car B described in Problem # 7 above, which car experiences the greater change in momentum? A. B. C. D. E.
The more massive car. The less massive car. Car A because its velocity at the start of the collision is greater. Car B because its velocity at the start of the collision is less. Both cars experience the same magnitude of momentum change.
9. A 70 kg hunter ropes a 350 kg polar bear. Both are initially at rest, 30 m apart on a frictionless and level ice surface. When the hunter pulls the polar bear to him, the polar bear will move: A. 5 m
B. 6 m
C. 15 m
D. 24 m
E. 25 m
10. A bullet of mass m is fired at a block of mass M initially at rest. The bullet, moving at an initial speed v, embeds itself in the block. The speed of the block after the collision is: A.
Mv M +m
B.
(M + m )v m
M + m m v C. v D. M + m m
E.
mv M +m
11. The driver of a 1000 kg car tries to turn through a circle of radius 100 m on an unbanked curve at a speed of 10 m/s. The maximum frictional force between the tires and the slippery road is 900 N. The car will: A. B. C. D. E.
Slide into the inside of the curve. Make the turn. Slow down due to the centrifugal force. Make the turn only if it speeds up. Slide off to the outside of the curve.
page 2
12. A solid cylinder weighing 200 N has a fixed axis and a string wrapped around it. The string is pulled with a force equal to the weight of the cylinder. The acceleration of the string is approximately: (A) 10 m/s2
(B) 20 m/s2
(C) 30 m/s2
(D) 40 m/s2
(E) 50 m/s2
13. A spinning ice skater has an initial kinetic energy (1/2)Iω2. She pulls in her outstretched arms, decreasing her moment of inertia to (1/4)I. Her new angular speed is: A. ω/4
B. ω/2
C. ω
D. 2ω
E. 4ω 1
2 L/4
14. A heavy rod of length L and weight W is suspended horizontally by two vertical ropes as shown on the right. The first rope is attached to the left end of the rod while the second rope is attached a distance L/4 from the right end. The tension in the second rope is: A. (1/2)W
B. (1/4)W
C. (1/3)W
L
D. (2/3)W
E. W
15. A block of ice with mass m falls into a lake. After impact, a mass of ice m/5 melts. Both the block of ice and the lake have a temperature of 0oC. If L represents the heat of fusion, the minimum distance the ice fell is: L 5L gL mL 5gL A. B. C. D. E. 5g g 5m 5g m
16. Which of the accompanying PV diagrams best represents an isothermal (constant temperature) process? A. P
B. P
V
C. P
V
E. P
D. P
V
V
V
17. If the heat is added at constant volume, 6300 joules of heat are required to raise the temperature of an ideal gas by 150 K. If instead, the heat is added at constant pressure, 8800 joules are needed for the same temperature change. When the temperature of the gas changes by 150 K, the internal energy of the gas changes by: A. 2500 J
B. 6300 J
C. 8800 J
D. 11,300 J
E. 15,100 J
page 3
18. A firecracker exploding on the surface of a lake is heard as two sounds – a time interval t apart – by the paddler of a nearby canoe. Sound travels with a speed u in water and a speed v in air. The distance from the exploding firecracker to the canoe is: A.
uvt u +v
B.
t ( u + v) uv
C.
t ( u − v) uv
D. vt
E.
uvt u −v
19. Two interfering waves have the same wavelength, frequency, and amplitude. They are traveling in the same direction but are 90o out of phase. Compared to the individual waves, the resultant wave will have the same: A. B. C. D. E.
amplitude and velocity but different wavelength. amplitude and wavelength but different velocity. wavelength and velocity but different amplitude. amplitude and frequency but different velocity. frequency and velocity but different wavelength.
20. An organ pipe which is open at both ends resonates with fundamental frequency 300 Hz. If one end of the pipe is closed, it will resonate with a fundamental frequency of: A. 75 Hz
B. 150 Hz
C. 300 Hz
D. 600 Hz
E. 1200 Hz
21. You are given two lenses, a converging lens with focal length + 10 cm and a diverging lens with focal length - 20 cm. Which of the following would produce a virtual image that is larger than the object? A. B. C. D. E.
Placing the object 5 cm from the converging lens. Placing the object 15 cm from the converging lens. Placing the object 25 cm from the converging lens. Placing the object 15 cm from the diverging lens. Placing the object 25 cm from the diverging lens.
22. Light shining through two very narrow slits produces an interference maximum at point P when the entire apparatus is in air (see accompanying figure). For the interference maximum represented, light through the bottom slit travels one wavelength further than light through the top slit before reaching point P. If the entire apparatus is immersed in water, the angle θ to the interference maximum: A. B. C. D. E.
P x d
θ
is unchanged decreases because the frequency decreases. decreases because the wavelength decreases. increases because the frequency increases. increases because the wavelength increases.
page 4
23. The magnitude of the field due to an infinite plate of charge is σ/(2εo), where σ is the charge per unit area and εo is the vacuum + permittivity. The figure to the right depicts three infinite plates of charge perpendicular to the plane of the page with charge per unit area + + σ1, + 2σ1, and - σ1. The total field at the point labeled X is: + A. 4σ1/(2εo) to the left. + B. σ1/(2εo) to the left. C. 0 +σ 1 D. σ1/(2εo) to the right. E. 4σ1/(2εo) to the right.
+ + + + + + + +
X
-
- σ1
+ 2σ1
24. There are three conducting spheres of 2 1 identical diameter, shown to the right. Spheres 1 and 2 are separated by a distance 3 that is large compared to their diameter. They have equal charges and repel each other with an electrostatic force F. Sphere 3 is initially uncharged and has an insulated handle. It is touched first to sphere 1, then to sphere 2, and then removed. If the distance between spheres 1 and 2 has not changed, the force between these two spheres is: A. 0
B. (1/16)F
C. (1/4)F
D. (3/8)F
25. Two conducting spheres, A of radius a and B of radius b, are shown in the accompanying figure. Both are positively charged and isolated from their surroundings. They had been connected by a conducting wire but the wire has been removed. Assuming the potential an infinite distance away is zero, which A of the following statements about the spheres are true: I. Sphere A is at the higher potential. II. Sphere B is at the higher potential. III. Both spheres have the same potential. A. I and IV
B. I and VI
C. II and VI
D. III and IV
only at R = 0, where V is infinite. anywhere R ≤ b, where V is constant in the region b < R < B. immediately outside the larger shell; V is zero everywhere within it. far away from the shells; V increases with distance.
a
b B
IV. Sphere A has the larger charge. V. Sphere B has the larger charge. VI. Both spheres have the same charge.
26. The accompanying figure shows two concentric spherical shells isolated from each other. The smaller shell has radius b and net charge + q. The larger shell has radius B and net charge + Q. Assume that the potential is zero at an infinite distance from the shells. If R is the distance from the common center, the highest electric potential V occurs: A. B. C. D. E.
E. (1/2)F
E. III and V
+Q b
+q B
page 5
27. In the circuit represented to the right, the current I equals: A. B. C. D. E.
R
V/5R V/4R 2V/5R 2V/4R 2V/R
V
R
R
I
R
V
Use the circuit below to answer questions 28 and 29. L1, L2, L3, and L4 are identical light bulbs. There are six voltmeters connected to the circuit as shown. Assume that the voltmeters do not effect the circuit. L1 V0
V1
V2
L2
L3
V3
L4
V4
V5
28. Which of the following combinations are equal to V0? A. V1 + V3
B. V2 + V3 + V4 C. V1 + V5
D. V3 + V4
E. V1 + V4
29. If L3 were to burn out, opening the circuit, which voltmeter(s) would read zero volts? A. B. C. D. E.
None would read zero. only V3 only V4 only V3, V4, and V5 they would all read zero
30. Identical currents flow in two perpendicular wires, as shown in the accompanying figure. The wires are very close but do not touch. The magnetic field can be zero:
2
A. B. C. D. E.
3
at a point in region 1 only at a point in region 2 only at points in both regions 1 and 2 at points in both regions 1 and 4 at points in both regions 2 and 4
1 I
I 4
page 6
1995 MULTIPLE CHOICE SCREENING TEST ANSWER KEY
1.
C
11. E
21. A
2.
D
12. B
22. C
3.
B
13. E
23. C
4.
D
14. D
24. D
5.
D
15. A
25. D
6.
B
16. B
26. B
7.
A
17. B
27. A
8.
E
18. E
28. C
9.
A
19. C
29. C
10. E
20. B
30. E
me = 9.1 x 10-31 kg
mass of proton
mp = 1.7 x 10-27 kg
electronic charge
e
= 1.6 x 10-19 C
speed of light
c
= 3.0 x 108 m/s
Coulomb's constant
k
= 9.0 x 109 N·m2/C2
permittivity constant
ε0
= 8.9 x 10-12 C2/N·m2
permeability constant
µ0
=
gravitational constant
G
= 6.7 x 10-11 N·m2/kg2
acceleration due to gravity
g
= 10 m/s2
speed of sound (20 oC)
vs
= 340 m/s
4π x 10-7 T·m/A
DO NOT OPEN THIS TEST UNTIL YOU ARE INSTRUCTED TO BEGIN Copyright © 1994, AAPT
1. A motorist travels 320 km at 80 km/h and then 320 km at 100 km/h. What is the average speed of the motorist for the entire trip? A. 84 km/h
B. 89 km/h
C. 90 km/h
D. 91 km/h
E. 95 km/h
2. A sports car is stopped at a light. At t = 0 the light changes and the sports car accelerates at a constant 2.0 m/s2. At t = (10/3) s a station wagon traveling at a constant 15 m/s in the same direction passes the stop light. When does the station wagon catch up to the sports car? A. never B. t = 5.0 s C. t = 7.5 s D. t = 12 s E. t = 21 s 3. A ball is thrown from ground level with an initial velocity of vo in the upward direction. It reaches a maximum height y and returns to ground level at time t seconds after it was thrown. If its initial velocity is doubled, it will be in the air _____ seconds and reach a maximum height _____. A. t, 4y B. 2t, y C. 2t, 2y D. 2t, 4y E. 4t, 2y 4. The ball in the preceding question (#3) is taken to Mars where the acceleration due to gravity is approximately 4 m/s2 (0.4 ge). It is thrown from ground level with the same initial velocity as it originally had on Earth, vo in the upward direction. On Mars, it will be in the air _____ seconds and reach a maximum height _____. A. 0.4 t, 2.5 y B. 0.4 t, 6.25 y C. 2.5 t, 2.5y D. 6.25 t, 2.5 y E. 6.25 t, 6.25 y 5. A top is spinning in the direction shown in the accompanying figure. Its axis of rotation makes an angle of 15o with the vertical. Assume friction can be neglected. The magnitude of the top’s angular momentum will _____ while its direction will _____. A. increase, precess in the counterclockwise direction when seen from above. B. increase, not precess C. remain the same, not precess D. remain the same, precess in the clockwise direction when seen from above. E. remain the same, precess in the counterclockwise direction when seen from above. 6. Three air track cars, shown in the accompanying v 1 figure, all have the same mass m. Cars 2 and 3 are initially at rest. Car 1 is moving to the right with speed v. m Car 1 collides with car 2 and sticks to it. The 1-2 combination collides elastically with car 3. Which of the following is most nearly the final speed of the 1-2 combination? A. 0.17 v B. 0.50 v C. 0.67 v D. 0.80 v 7. A cube with mass M starts at rest at point 1 at a height 4R, where R is the radius of the circular part of the track. The cube slides down the frictionless track and around the loop. The force that the track exerts on the cube at point 2 is most nearly _____ times the cube’s weight Mg. A. 1 B. 2 C. 3 D. 4 E. 5
2
3
m
m
E. 1.0 v 1 4R
2 R
Page 1 of 6
8. An astronaut with weight W on Earth lands on a planet with mass 0.1 times the mass of Earth and radius 0.5 times the radius of Earth. The astronaut’s weight is _____ on the planet. A. 0.02 W B. 0.04 W C. 0.2 W D. 0.4 W E. W 9. A horizontal force F, represented by the arrow in the figure to the right, is used to push a block of weight mg up an inclined plane making an angle of θ with the horizontal. The coefficient of friction between the plane and the block is µ. The magnitude of the frictional force acting on the block is: A. µ mg cosθ B. µ mg/( cosθ ) C. µ ( mg cosθ + F sinθ ) D. µ ( F cosθ – mg sinθ ) E. µ F cosθ
θ
10. A child of mass M stands on the edge of a merry-go-round of radius R and moment of inertia I. Both the merry-go-round and child are initially at rest. The child walks around the circumference with speed v with respect to the ground. What is the magnitude of the angular velocity of the merry-go-round with respect to the ground? v MRv MRv MRv C. ω = D. ω = E. ω = A. 0 B. ω = 2 R I − MR I + MR 2 I 11. A rigid rod of mass M and length L has moment of 1 ML2 about its center of mass. A sphere of mass m inertia 12 2 2 5 mR
axis M
and radius R has moment of inertia about its center of mass. A combined system is formed by centering the sphere L at one end of the rod and placing an axis at the other (see accompanying figure). What is the moment of inertia of the combined system about the axis shown?
m
A. I = 121 ML2 + 25 mR2 B. I = 121 ML2 + 25 mR2 + mL2 C. I = 13 ML2 + 25 mR 2 + mL2 D. I = 121 ML2 + mL2 2 2 E. I = 13 ML + mL
12. A rocket is launched from the surface of a planet with mass M and radius R. What is the minimum velocity the rocket must be given to completely escape from the planet’s gravitational field? 2GM 2GM GM GM A. v = B. v = C. v = D. v = E. v = GM 2 2 R R R R
Page 2 of 6
13. A uniform rod of length L and weight WR is suspended as shown in the accompanying figure. A weight W is added to the end of the rod. The support wire is at an angle θ to the rod. What is the tension T in the wire? W A. T = sin θ B. T = W + W R
T θ W
C. T = W + 12 W R D. T =
W + 12 W R
sin θ W + WR E. T = sin θ
14. A hollow vertical cylinder of radius R is rotated with angular velocity ω about an axis through its center. What is the minimum coefficient of static friction necessary to keep the mass M suspended on the inside of the cylinder as it rotates?
M top view side view
A. µ =
gR ω2
B. µ =
ω 2g R
C. µ =
ω 2R g
D. µ =
ω2 gR
15. A block of mass M is attached to a relaxed spring with force constant k, placed on a frictionless inclined plane as shown in the accompanying figure, and released. What is the maximum extension of the spring? 2Mgsin θ Mgsin θ 2Mg Mg A. x = B. x = C. x = D. x = k k k k 16. A simple pendulum of length L and mass m is attached to a moving support. In order for the pendulum string to make a constant angle θ with the vertical, the support must be moving to the: A. right with constant acceleration a = g tanθ. B. left with constant acceleration a = g tanθ. C. right with constant acceleration a = g sinθ. D. right with constant velocity v = Lg tanθ. E. left with constant velocity v = Lg tanθ.
E. µ =
g ω 2R
θ E. x = 2gM
θ
L
17. A Carnot cycle takes in 1000 J of heat at a high temperature of 400 K. How much heat is expelled at the cooler temperature of 300 K? A. 0 B. 250 J C. 500 J D. 750 J E. 1000 J
Page 3 of 6
18. An ideal gas is expanded at constant pressure from initial volume Vi and temperature Ti to final volume Vf and temperature Tf . The gas has molar heat capacity CP at constant pressure. The amount of work done by n moles of the gas during the process can be expressed: A. 0 B. nRTi ln(Vf/Vi) C. CPn(Τf – Ti) D. nk(Vf – Vi) E. nR(Τf – Ti) 19. The average translational kinetic energy of any ideal gas depends only on: A. the absolute (Kelvin) temperature. B. the mass of the gas. C. the pressure of the gas. D. the amount of the gas. E. whether the gas is monatomic or diatomic. 20. A soap film of thickness t is surrounded by air (see accompanying figure). It is illuminated at near normal incidence by monochromatic light which has wavelength λ in the film. A film thickness _____ will produce maximum brightness of the reflected light. A. 1/4 λ B. 1/2 λ C. 1 λ D. 2 λ E. 4 λ
air t
air
21. You are given two lenses, a converging lens with focal length +10 cm and a diverging lens with focal length –20 cm. Which of the following would produce a real image that is smaller than the object? A. Placing the object 5 cm from the converging lens. B. Placing the object 15 cm from the converging lens. C. Placing the object 25 cm from the converging lens. D. Placing the object 15 cm from the diverging lens. E. Placing the object 25 cm from the diverging lens. 22. A source at rest emits waves with wavelength λ in a medium with velocity v. If the source moves to the right with velocity vs(see accompanying figure), the distance between adjacent crests λ’ directly behind the source is:
A.
λv v + vs
B.
λv v − vs
v C. λ 1 + vs
23. Two sources, in phase and a distance d apart, each emit a wave of wavelength λ. See accompanying figure. Which of the choices for the path difference ∆L = L1 – L2 will always produce constructive interference at point P? A. d sinθ B. x/L1 C. (x/L2)d D. λ/2 E. 2 λ Page 4 of 6
v D. λ 1 + s v
λ'
vs v E. λ 1 − s v P
L2
x d
θ L1
1
soap 2
Questions 24 and 25 are based on the following description: Two point charges of +4.00 µC and –9.00 µC are placed 1.00 m apart, as shown in the accompanying figure. Assume the potential goes to zero as R goes to infinity. +4.0 µC
24. The total electric field due to the two charges is zero at a point: A. 3.00 m to the right of the –9.00 µC charge. B. 0.40 m to the right of the +4.00 µC charge. C. 0.31 m to the right of the +4.00 µC charge. D. 0.80 m to the left of the +4.00 µC charge. E. 2.00 m to the left of the +4.00 µC charge.
–9.0 µC
1.00 m
25. How much work is done moving the –9.00 µC charge from its original position to a new position 2.00 m from the +4.00 µC charge? A. – 0.324 J B. –0.081 J C. +0.162 J D. +0.243 J E. +0.486 J 26. A point charge Q is placed at the center of a spherical conducting shell, the shaded part of the accompanying figure. A total charge of –q is placed on the shell. The magnitude of the electric field at point P1 a distance R1 from the center is _____. The magnitude of the electric field at point P2 a distance R2 from the center is _____. A. 0, 0 Q B. k 2 , 0 R1 Q− q C. k ,0 R12 Q− q D. 0, k R 22 Q Q− q E. k 2 , k R1 R 22
P
2
R2 Q
P1 R 1
–q
R1
27. R4, shown in the figure to the right, is a variable resistor. In order for there to be no current through the ammeter, R4 must be equal to: A. R2 B. R3 C. R1R2/R3 D. R1R3/R2 E. R2R3/R1
R2
R3
A R4
V 28. A resistor R dissipates power P when connected directly to a voltage source V, as shown in the accompanying figures. What resistance R′ must be connected in series with R to decrease the power dissipated in R to P/2? R R A. B. C. R D. R ( 2 − 1) E. R 2 2 2
Page 5 of 6
P V
P/2 V
R
R R'
29. The infinitely long straight wire carries a conventional current I as shown in the accompanying figure. The rectangular loop carries a conventional current I′ in the counterclockwise direction. The net force on the rectangular loop is: µ II' c 1 1 A. o − to the right 2π a b µ II' c 1 1 + to the left B. o 2π a b
I'
I
b
a
µ o II' c c 2(b − a ) + + to the left c 2π a b µ II' 2(b − a ) D. o to the right 2π c E. 0
c
C.
30. A spatially uniform magnetic field of 0.080 T is directed into the plane of the page and perpendicular to it, as shown in the accompanying figure. A wire loop in the plane of the page has constant area 0.010 m2. The magnitude of the magnetic field decreases at a constant rate of 3.0 x 10–4 T/s. What is the magnitude and direction of the induced emf? A. 3.0 x 10–6 V clockwise B. 3.0 x 10–6 V counterclockwise C. 2.4 x 10–5 V counterclockwise D. 8.0 x 10–4 V counterclockwise E. 8.0 x 10–4 V clockwise
Page 6 of 6
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
1994 MULTIPLE CHOICE SCREENING TEST ANSWER KEY
1.
B
11. C
21. C
2.
B
12. B
22. D
3.
D
13. D
23. E
4.
C
14. E
24. E
5.
E
15. A
25. C
6.
A
16. B
26. B
7.
C
17. D
27. E
8.
D
18. E
28. D
9.
C
19. A
29. A
10. B
20. A
30. A
me = 9.1 x 10-31 kg
mass of proton
mp = 1.7 x 10-27 kg
electronic charge
e
= 1.6 x 10-19 C
speed of light
c
= 3.0 x 108 m/s
Coulomb's constant
k
= 9.0 x 109 N·m2/C2
permittivity constant
ε0
= 8.9 x 10
µ
=
π
-7
C/ T·m/A
-11
G
2
N·m /kg
24 E = 6.0 x 10 kg
mass of Earth radius of Earth
RE = 6.4 x 106 m
gravitational field at Earth’s surface speed of sound (20 oC)
g = 9.8 m/s2 vs = 340 m/s
DO NOT OPEN THIS TEST UNTIL YOU ARE INSTRUCTED TO BEGIN Copyright © 1997, AAPT
1. Starting from rest at time t = 0, a car moves in a straight a in line with an acceleration given by the accompanying graph. m/s2 6 What is the speed of the car at t = 3 s? 5 A. 1.0 m/s 4 B. 2.0 m/s 3 C. 6.0 m/s 2 D. 10.5 m/s E. 12.5 m/s 1 0 0
1
2 3
4
5 6 t in s
2. A flare is dropped from a plane flying over level ground at a velocity of 70 m/s in the horizontal direction. At the instant the flare is released, the plane begins to accelerate horizontally at 0.75 m/s2. The flare takes 4.0 s to reach the ground. Assume air resistance is negligible. Relative to a spot directly under the flare at release, the flare lands A. B. C. D. E.
directly on the spot. 6.0 m in front of the spot. 274 m in front of the spot. 280 m in front of the spot. 286 m in front of the spot.
3. As seen by the pilot of the plane (in question #2) and measured relative to a spot directly under the plane when the flare lands, the flare lands A. B. C. D. E.
286 m behind the plane. 6.0 m behind the plane. directly under the plane. 12 m in front of the plane. 274 m in front of the plane
4. A force F is used to hold a block of mass m on an incline as shown in the diagram. The plane makes an angle of θ with the horizontal and F is perpendicular to the plane. The coefficient of friction between the plane and the block is µ. What is the minimum force, F, necessary to keep the block at rest? A. B. C. D. E.
µ mg mg cosθ mg sinθ (mg/µ) sinθ (mg/µ) ( sinθ - µ cosθ )
F
θ
5. You hold a rubber ball in your hand. The Newton’s third law companion force to the force of gravity on the ball is the force exerted by the A. ball on the Earth. B. ball on the hand. C. hand on the ball. D. Earth on the ball. E. Earth on your hand. 1997 Physics Olympiad
page 1
6. A ball of mass m is fastened to a string. The ball swings in a vertical circle of radius R with the other end of the string held fixed. Neglecting air resistance, the difference between the string’s tension at the bottom of the circle and at the top of the circle is A. mg
B. 2 mg
C. 4 mg
D. 6 mg
7. Three air track cars, shown in the accompanying figure, all have the same mass m. Cars 2 and 3 are initially at rest. Car 1 is moving to the right with speed v. Car 1 collides with car 2 and sticks to it. The 1-2 combination collides elastically with car 3. Which of the following is most nearly the final speed of car 3? A. 0.17 v
B. 0.50 v
C. 0.67 v
v
1 m
D. v E. 2v
2
3
m
m
D. 0.80 v
8. A point object of mass 2m is attached to one end of a rigid rod of negligible mass and length L. The rod is initially at rest but free to rotate about a fixed axis perpendicular to the rod and passing through its other end (see diagram to the right). A second point object with mass m and initial speed v collides and sticks to the 2m object. What is the tangential speed vt of the object immediately after the collision? A. v 3 B. v 2 C. v
E. 8 mg
E. 1.0 v
axis L
L vt
m
v
2m
3 2 3
9. Two artificial satellites I and II have circular orbits of radii R and 2R, respectively, about the same planet. The orbital velocity of satellite I is v. What is the orbital velocity of satellite II? v v A. B. C. v D. v 2 E. 2v 2 2 10. The gravitational acceleration on the surface of the moon is 1.6 m/s2. The radius of the moon is 1.7 x 106 m. The period of a satellite placed in a low circular orbit about the moon is most nearly A. 1.0 x 103 s
B. 6.5 x 103 s
C. 1.1 x 106 s
D. 5.0 x 106 s
E. 7.1 x 1012 s
page 2
11. A uniform ladder of length L rests against a smooth frictionless wall. The floor is rough and the coefficient of static friction between the floor and ladder is µ. When the ladder is positioned at angle θ, as shown in the accompanying diagram, it is just about to slip. What is θ? A. θ =
µ L
B. tan θ = 2µ
C. tan θ =
1 2µ
D. sin θ =
1 µ
L
E. cos θ = µ θ
12. Three objects, all of mass M, are released simultaneously from the top of an inclined plane of height H. The objects are described as follows I. a cube of side R. II. a solid cylinder of radius R III. a hollow cylinder of radius R Assume the cylinders roll down the plane without slipping and the cube slides down the plane without friction. Which object(s) reach(es) the bottom of the plane first? A. I
B. II
C. III
D. I & II
13. A massless rod of length 2R can rotate about a vertical axis through its center as shown in the diagram. The system rotates at an angular velocity ω when the two masses m are a distance R from the axis. The masses are simultaneously pulled to a distance of R/2 from the axis by a force directed along the rod. What is the new angular velocity of the system? A. ω/4
B. ω/2
C. ω
E. II & III R
m
D. 2ω
m
m
m R/2 E. 4ω
14. A meter stick moves with a velocity of 0.60 c relative to an observer. The observer measures the length of the meter stick to be L. Which of the following statements is always true? A. L = 0.60 m
B. L = 0.80 m
C. 0.80 m ² L ² 1.00 m
D. L = 1.00 m
E. L ³ 1.00m
15. A glowing ember (hot piece of charcoal) radiates power P in watts at an absolute temperature T. When the temperature of the ember has decreased to T/2, the power it radiates is most nearly A. P
B. P/2
C. P/4
D. P/8
E. P/16
page 3
16. Three processes compose a thermodynamic cycle shown in the accompanying pV diagram of an ideal gas. Process 1→2 takes place at constant temperature (300 K). During this process 60 J of heat enters the system. Process 2→3 takes place at constant volume. During this process 40 J of heat leaves the system. Process 3→1 is adiabatic. T3 is 275 K. What is the change in internal energy of the system during process 3→1? A. -40 J
B. -20 J
C. 0
p
1 2 3 V
D. +20 J
E. +40 J
17. What is the change in entropy of the system described in Question # 16 during the process 3→1? A. +5.0 K/J
B. +0.20 J/K
C.
0
D. -1.6 J/K
E. -6.9 K/J
18. A wave is described by the equation: y(x,t) = 0.030 ⋅ sin(5πx + 4πt) where x and y are in meters and t is in seconds. The +x direction is to the right. What is the velocity of the wave? A. B. C. D. E.
0.80 m/s to the left 1.25 m/s to the left 0.12 π m/s to the right 0.80 m/s to the right 1.25 m/s to the right
19. Two sources, in phase and a distance d apart, each emit a wave of wavelength λ. See accompanying figure. Which of the choices for the path difference ∆L = L1 – L2 will always produce destructive interference at point P? A. B. C. D. E.
d sinθ x/L1 (x/L2)d λ/2 2λ
P
L2
x d
θ L1
20. You are given two lenses, a converging lens with focal length + 10 cm and a diverging lens with focal length - 20 cm. Which of the following would produce a virtual image that is larger than the object? A. B. C. D. E.
Placing the object 5 cm from the converging lens. Placing the object 15 cm from the converging lens. Placing the object 25 cm from the converging lens. Placing the object 15 cm from the diverging lens. Placing the object 25 cm from the diverging lens.
page 4
21. You are given two identical plano-convex lenses, one of which is shown to the right. When you place an object 20 cm to the left of a single plano-convex lens, the image appears 40 cm to the right of the lens. You then arrange the two plano-convex lenses back to back to form a double convex lens. If the object is 20 cm to the left of this new lens, what is the approximate location of the image? A. B. C. D. E.
6.7 cm to the right of the lens. 10 cm to the right of the lens. 20 cm to the right of the lens. 80 cm to the right of the lens. 80 cm to the left of the lens.
Planoconvex
Double Convex
y
Q 22. Positive charge Q is uniformly distributed over a ring of radius a that lies in the y-z plane as shown in the diagram. The ring is centered at the origin. Which of the following graphs best represents the value of the electric field E as a function of x, the distance along the positive x axis? A
B E
C E
x
E
x
kq/a2 at an angle 45o above the +x axis. kq/a2 at an angle 45o below the -x axis. 3 kq/a2 at an angle 45o above the -x axis. 3 kq/a2 at an angle 45o below the +x axis. 9 kq/a2 at an angle 45o above the +x axis.
z x
D
E E
x
23. Four point charges are placed at the corners of a square with diagonal 2a as shown in the diagram. What is the total electric field at the center of the square? A. B. C. D. E.
a
E
x
x
y +3q
-q
2a x -2q
+3q
page 5
Both questions 24 and 25 refer to the system shown in the diagram. A spherical shell with an inner surface of radius a and an outer surface of radius b is made of conducting material. A point charge +Q is placed at the center of the spherical shell and a total charge -q is placed on the shell.
a -q
Q
b
24. How is the charge -q distributed after it has reached equilibrium? A. B. C. D. E.
Zero charge on the inner surface, -q on the outer surface. -Q on the inner surface, -q on the outer surface. -Q on the inner surface, -q+Q on the outer surface. +Q on the inner surface, -q-Q on the outer surface. The charge -q is spread uniformly between the inner and outer surface.
25. Assume that the electrostatic potential is zero at an infinite distance from the spherical shell. What is the electrostatic potential at a distance R from the center of the shell, where b ³ R ³ a? A. 0
B. k
Q a
C. k
Q R
D. k
Q-q R
E. k
Q-q b
Use the circuit below to answer questions 26 and 27. B1, B2, B3, and B4 are identical light bulbs. There are six voltmeters connected to the circuit as shown. All voltmeters are connected so that they display positive voltages. Assume that the voltmeters do not effect the circuit. B1 V0
V1
V2
B2
B3
V3
B4
V4
V5
26. If B2 were to burn out, opening the circuit, which voltmeter(s) would read zero volts? A. B. C. D. E.
none would read zero. only V2 only V3 and V4 only V3, V4, and V5 they would all read zero
27. If B2 were to burn out, opening the circuit, what would happen to the reading of V1? Let V be its original reading when all bulbs are functioning and let V’ be its reading when B2 is burnt out. A. V’ ³ 2V
B. 2V > V’ > V
C. V’ = V
D. V > V’ > V/2
E. V/2 ³ V’
page 6
28. A particle with positive charge q and mass m travels along a path perpendicular to a magnetic field. The particle moves in a circle of radius R with frequency f. What is the magnitude of the magnetic field? A.
mf q
B.
2πfm q
C.
m 2πfq
D.
mc qR
29. Two wires, each carrying a current i, are shown in the diagram to the right. Both wires extend in a straight line for a very long distance on both the right and the left. One wire contains a semi-circular loop of radius a centered on point X. What is the correct expression for the magnetic field at point X? HINT: The magnitude of the magnetic field at the center of a circular current loop of radius R is µ 0i/(2R). A. B. C. D. E.
µ 0i/(4a) + µ 0i/(2πa) µ 0i/(2a) - µ 0i/(2πa) + µ 0i/(2πa) µ 0i/(4a) + µ 0i/(2πa) µ 0i/(4a) + µ 0i/(2πa) + µ 0i/(2πa) µ 0i/(2a) - µ 0i/(2πa)
E.
µqf 2πR
i a
X a
i
out of the page out of the page into the page into the page into the page
30. You are given a bar magnet and a looped coil of wire. Which of the following would induce an emf in the coil? I. Moving the magnet toward the coil. II. Moving the coil away from the magnet. III. Turning the coil about a vertical axis. A. I only
B. II only
C. I & II
D. I & III
E. I, II, III
page 7
1997 MULTIPLE CHOICE SCREENING TEST ANSWER KEY
1.
D
11. C
21. B
2.
D
12. A
22. D
3.
B
13. E
23. B
4.
E
14. C
24. C
5.
A
15. E
25. E
6.
D
16. E
26. A
7.
C
17. C
27. D
8.
A
18. A
28. B
9.
B
19. D
29. C
10. B
20. A
30. E
me mp e c k e0 µ0 G ME RE g vs
= = = = = = = = = = = =
9.1 1.7 1.6 3.0 9.0 8.9 4π 6.7 6.0 6.4 9.8 340
x 10-31 x 10-27 x 10-19 x 108 x 109 x 10-12 x 10-7 x 10-11 x 1024 x 106 N/kg m/s
kg kg C m/s N.m2/C2 C2/N.m2 T.m/A N.m2/kg2 kg m
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN Copyright 1999, American Association of Physics Teachers. 1999 Physics Olympiad
1
1. A truck driver travels three-fourths the distance of his run at one velocity (v) and then completes his run at one half his original velocity (½v). What was the trucker’s average speed for the trip? [A] 0.85v
[B] 0.80v
[C] 0.75v
[D] 0.70v
[E] 0.65v
2. Hercules and Ajax horizontally push in the same direction on a 1200 kg crate. Hercules pushes with a force of 500 newtons and Ajax pushes with a force of 300 newtons. If a frictional force provides 200 newtons of resistance, what is the acceleration of the crate? [A] 1.3 m/s2
[B] 1.0 m/s2
[C] 0.87 m/s2
[D] 0.75 m/s2
[E] 0.5 m/s2
3. A uniform 2 kg cylinder rests on a laboratory cart as shown. The coefficient of static friction between the cylinder and the cart is 0.5. If the cylinder is 4 cm in diameter and 10 cm in height, which of the following is closest to the minimum acceleration of the cart needed to cause the cylinder to tip over? [A] 2 m/s2
[B] 4 m/s2
[C] 5 m/s2
[D] 6 m/s2
[E] The cylinder would slide at all of these accelerations.
4. At right is a graph of the distance vs. time for car moving along a road. According the graph, at which of the following times would the automobile have been accelerating positively? [A] [B] [C] [D] [E]
0, 20, 38, & 60 min. 5, 12, 29, & 35 min. 5, 29, & 57 min. 12, 35, & 41 min. at all times from 0 to 60 min.
5. A ball which is thrown upward near the surface of the earth with a velocity of 50 m/s will come to rest about 5 seconds later. If the ball were thrown up with the same velocity on Planet X, after 5 seconds it would still be moving upwards at nearly 31 m/s. The magnitude of the gravitational field near the surface of Planet X is what fraction of the gravitational field near the surface of the earth? [A] 0.16
[B] 0.39
[C] 0.53
[D] 0.63
[E] 1.59
2
6. Two light plastic shopping bags of negligible mass are placed 2 meters apart. Each bag contains 15 oranges. If 10 oranges were moved from one bag to the other, the gravitational force between the two bags would [A] [B] [C] [D] [E]
increase to 3/2 the original value. decrease to 2/5 the original value. increase to 5/3 the original value. decrease to 5/9 the original value. not change.
7. If a net force F applied to an object of mass m will produce an acceleration of a, what is the mass of a second object which accelerates at 5a when acted upon by a net force of 2F? [A] (2/5)m
[B] 2m
[C] (5/2)m
[D] 5m
[E] 10m
8. A large beach ball is dropped from the ceiling of a school gymnasium to the floor about 10 meters below. Which of the following graphs would best represent its velocity as a function of time?
[A]
[B]
[C]
[D]
[E]
9. A driver in a 1500 kg sports car wishes to pass a slow moving truck on a two lane road. What is the average power in watts required to accelerate the sports car from 20 m/s to 40 m/s in 3 seconds? [A] 10,000
[B] 20,000
[C] 100,000
[D] 300,000
[E] 400,000
10. A 40 kg mass is attached to a horizontal spring with a constant of 500 N/m. If the mass rests on a frictionlless horizontal surface, what is the total energy of this system when set into simple harmonic motion by an original displacement of 0.2 meters? [A] 10 J
[B] 20 J
[C] 50 J
[D] 4000 J
[E] 100,000 J
3
11. Two skaters on a frictionless pond push apart from one another. One skater has a mass M much greater than the mass m of the second skater. After some time the two skaters are a distance d apart. How far has the lighter skater moved from her original position? [A] d
[B] d
FG M IJ H mK
[C] d
FG m IJ H MK
[D] d
FG m IJ H M + mK
[E] d
FG M IJ H m+ MK
12. Given the three masses as shown in the diagram, if the coefficient of kinetic friction between the large mass (m2) and the table is µ, what would be the upward acceleration of the small mass (m3)? The mass and friction of the cords and pulleys are small enough to produce a negligible effect on the system. [A]
b
m1 g m1 + m2 + m3
g [D]
[B]
b
gµ m1 − m2 − m3 m1 + m2 + m3
b
g
g
b
b
g
g m1 + m2 µ m1 + m2 + m3
g
[C] [E]
b b
g m1 − m2 µ − m3 m1 + m2 + m3
g
g
b
gµ m1 + m2 + m3 m1 − m2 − m3
b
g
g
13. An object with a mass of 2 kilograms is accelerated from rest. The graph at right shows the magnitude of the net force in newtons as a function of time in seconds. At t = 4 seconds the object’s velocity would have been closest to which of the following: [A] [B] [C] [D] [E]
2.2 m/s 3.5 m/s 5.8 m/s 7.0 m/s 11.5 m/s
14. A thin ring of mass m and radius r rolls across the floor with a velocity v. Which of the following would be the best estimate of the ring’s total kinetic energy as it rolls across the floor? 1 1 1 mv 2 1 r2 [B] mv 2 [C] mv 2 [D] mv 2 + [E] mv 2 + m 2 [A] mv 2 2 4 2 r 2 t
4
15. A 4.0 kg mass is attached to one end of a rope 2 m long. If the mass is swung in a vertical circle from the free end of the rope, what is the tension in the rope when the mass is at its highest point if it is moving with a speed of 5 m/s? [A] 5.4 N
[B] 10.8 N
[C] 21.6 N
[D] 50 N
[E] 65.4 N
16. A mole of ideal gas at STP is heated in an insulated constant volume container until the average velocity of its molecules doubled. Its pressure would therefore increase by what factor? [A] 0.5
[B] 1
[C] 2
[D] 4
[E] 8
17. An ideal heat engine takes in heat energy at a high temperature and exhausts energy at a lower temperature. If the amount of energy exhausted at the low temperature is 3 times the amount of work done by the heat engine, what is its efficiency? [A] 0.25
[B] 0.33
[C] 0.67
[D] 0.9
[E] 1.33
18 . If 100 g of ice at 0o C is mixed with 100 g of boiling water at 100o C, which of the following graphs would best represent the temperature vs time of the two components of the mixture?
[A]
[B]
[C]
[D]
[E]
19. Two wave sources, G and H, produce waves of different wavelengths as shown in the diagram. Lines shown in the diagram represent wave crests. Which of the following would be true of the resulting interference pattern? [A] The perpendicular bisector of GH would be an antinodal line. [B] The interference pattern will be a stable pattern but not symetrical left to right. [C] The pattern of nodal and antinodal lines will sweep from left to right. [D] If G and H were moved farther apart, the fewer the number of nodal and antinodal lines. [E] There will not be an organized interference pattern because the sources are not producing the same frequency.
5
20. Given a wave produced by a simple harmonic oscillator whose displacement in meters is given by the equation: y = .3 sin (3πx + 24πt), what is the frequency of the wave in hertz? [A] 3 Hz
[B] 7.2 Hz
[C] 8 Hz
[D] 12 Hz
[E] 24 Hz
21. Which of the following wave properties cannot be demonstrated by all kinds of waves? [A] Polarization
[B] Diffraction
[C] Superposition [D] Refraction
[E] Frequency
22. A source when at rest in a medium produces waves with a velocity v and a wavelength of λ. If the source is set in motion to the left with a velocity vs, what would be the length of the wavelengths produced directly in front of the source?
FG H
[A] λ 1−
vs v
IJ K
FG H
[B] λ 1+
vs v
IJ K
FG H
[C] λ 1+
v vs
IJ K
[D]
λ −v v − vs
[E]
λv v + vs
23. A beam of light strikes one mirror of a pair of right angle mirrors at an angle of incidence of 45o as shown in the diagram. If the right angle mirror assembly is rotated such that the angle of incidence is now 60o, what will happen to the angle of the beam that emerges from the right angle mirror assembly? [A] It will move through an angle of 15o with respect to the original emerging beam. [B] It will move through an angle of 30o with respect to the original emerging beam. [C] It will move through an angle of 45o with respect to the original emerging beam. [D] It will move through an angle of 60o with respect to the original emerging beam. [E] It will emerge parallel to the original emerging beam.
24. What would be the total current being supplied by the battery in the circuit shown? [A] [B] [C] [D] [E]
3.0 amperes 2.25 amperes 2.0 amperes 1.5 amperes 1.0 amperes
6
25. Consider a simple circuit containing a battery and three light bulbs. Bulb A is wired in parallel with bulb B and this combination is wired in series with bulb C. What would happen to the brightness of the other two bulbs if bulb A were to burn out? [A] [B] [C] [D] [E]
only bulb B would get brighter. both would get brighter. bulb B would get brighter and bulb C would get dimmer. bulb B would get dimmer and bulb C would get brighter. There would be no change in the brightness of either bulb B or bulb C.
26. A coil of wire is moved vertically at a constant velocity through a horizontal magnetic field. Assume the plane of the coil is perpendicular to the magnetic field. Which of the following graphs would best represent the electric current induced in the coil if it started somewhat above the magnetic field and ended equally as far below the magnetic field ?
[A]
[B]
[C]
[D]
[E]
27. Three metal spheres A, B, and C are mounted on insulating stands. The spheres are touching one another, as shown in the diagram. A strong positively charged object is brought near sphere A and a strong negative charge is brought near sphere C. While the charged objects remain near spheres A and C, sphere B is removed by means of its insulating stand. After the charged objects are removed, sphere B is first touched to sphere A and then to sphere C. The resulting charge on B would be: [A] [B] [C] [D] [E]
the same sign but ½ the magnitude as originally on sphere A. the opposite sign but ½ the magnitude as originally on sphere A. the opposite sign but ¼ the magnitude as originally on sphere A. the same sign but ½ the magnitude as originally on sphere C. neutrally charged.
7
28. Which of the following graphs would best represent the electric field of a hollow Van de Graff sphere as a function of distance from its center when it is charged to a potential of 400,000 volts?
[A]
[B]
[C]
[D]
[E]
29. A mass spectrograph separates ions by weight using simple concepts from physics. Charged ions are given a specific kinetic energy by accelerating them through a potential difference. The ions then move through a perpendicular magnetic field where they are deflected into circular paths with differing radii. How would the radius of a singularly ionized common helium atom 2+ 4 1+ compare to the radius of a doubly ionized common oxygen atom 16 if they were 8 O 2 He
c
h
c
h
accelerated through the same potential difference and were deflected by the same magnetic field? [A] [B] [C] [D] [E]
The radius of the He ion path is 4 times the radius of the O ion path. The radius of the O ion path is 2 times the radius of the He ion path. The radius of the O ion path is 4 times the radius of the He ion path. The radius of the O ion path is 8 times the radius of the O ion path. The radius of the He ion path is equal to the radius of the O ion path.
30. A bar magnet is thrust into a coil of wire as indicated in the diagram. Which of the following statements concerning this experiment is correct? [A] The polarity of the coil at H is North, at G is South, and the induced current is indicated by arrow X. [B] The polarity of the coil at H is South, at G is North, and the induced current is indicated by arrow Y. [C] The polarity of the coil at H is North, at G is South, and the induced current is indicated by arrow Y. [D] The polarity of the coil at H is South, at G is North, and the induced current is indicated by arrow X. [E] There is no polarity to the coil and no current in the coil.
8
1. B
11. E
21. A
2. E
12. E
22. A
3. B
13. B
23. E
4. C
14. A
24. D
5. B
15. B
25. C
6. D
16. D
26. E
7. A
17. A
27. C
8. D
18. C
28. C
9. D
19. C
29. E
10. A
20. D
30. B
9
me = 9.1 x 10-31 kg
mass of proton
mp = 1.7 x 10-27 kg
electronic charge
e
= 1.6 x 10-19 C
speed of light
c
= 3.0 x 108 m/s
Coulomb's constant
k
= 9.0 x 109 N·m2/C2
permittivity constant
ε0
= 8.9 x 10-12 C2/N·m2
permeability constant
µ0
=
gravitational constant
G
= 6.7 x 10-11 N·m2/kg2
mass of Earth
ME = 6.0 x 1024 kg
RE = 6.4 x 106 m
gravitational field at Earth’s surface speed of sound (20oC)
g = 9.8 N/kg vs = 340 m/s
4¹ x 10-7 T·m/A
DO NOT OPEN THIS TEST UNTIL YOU ARE INSTRUCTED TO BEGIN Copyright © 1998, AAPT 1998 Physics Olympiad
page 1
1. The graph to the right is a plot of position versus time. For which labeled region is the velocity positive and the acceleration negative? A. A
B. B
C. C
D. D
x A B
E
C
E. E
D
t 2. A child left her home and started walking at a constant velocity. After a time she stopped for a while and then continued on with a velocity greater than she originally had. All of a sudden she turned around and walked very quickly back home. Which of the following graphs best represents the distance versus time graph for her walk? A.
B.
C.
D.
E.
s
s
s
s
s
t
t
t
t
t
3. In a rescue attempt, a hovering helicopter drops a life preserver to a swimmer being swept downstream by a river current of constant velocity v. The helicopter is at a height of 9.8 m. The swimmer is 6.0 m upstream from a point directly under the helicopter when the life preserver is released. It lands 2.0 m in front of the swimmer. How fast is the current flowing? Neglect air resistance. A. 13.7 m/s
B. 9.8 m/s
C. 6.3 m/s
D. 2.8 m/s
E. 2.4 m/s
4. A child tosses a ball directly upward. Its total time in the air is T. Its maximum height is H. What is its height after it has been in the air a time T/4? Neglect air resistance. A. (1/4) H
B. (1/3) H
C. (1/2) H
D. (2/3) H
E. (3/4) H
5. A whiffle ball is tossed straight up, reaches a highest point, and falls back down. Air resistance is not negligible. Which of the following statements are true? I. II. III.
A. I only
B. II only
The ball’s speed is zero at the highest point. The ball’s acceleration is zero at the highest point. The ball takes a longer time to travel up to the highest point than to fall back down. C. I & II only
D. I & III only
E. I, II, & III
page 2
6. A pendulum is attached to the ceiling of an elevator car. When the car is parked, the pendulum exhibits a period of 1.00 s. The car now begins to travel upward with an upward acceleration of 2.3 m/s2. During this part of the motion, what will be the approximate period of the pendulum? A. 0.80 s
B. 0.90 s
C. 1.00 s
D. 1.10 s
E. 1.20 s
7. Two identical blocks of weight W are placed one on top of the other as shown to the right. The upper block is tied to the wall. The lower block is pulled to the right with a force F. The coefficient of static friction between all surfaces in contact is µ. What is the largest force F that can be exerted before the lower block starts to slip? A. µW
B. (3/2) µW
C. 2 µW
W
F
W
D. (5/2) µW
E. 3 µW
8. An object placed on an equal arm balance requires 12 kg to balance it. When placed on a spring scale, the scale reads 120 N. Everything (balance, scale, set of masses, and the object) is now transported to the moon where the gravitational force is one-sixth that on Earth. The new readings of the balance and the spring scale (respectively) are: A. 12 kg, 20 N
B. 12 kg, 120 N
C. 12kg, 720 N
D. 2 kg, 20 N
E. 2 kg, 120 N
X
9. A large spool of rope lies on the ground. See the diagram to the right. The end, labeled X, is pulled a distance S in the horizontal direction. The spool rolls without slipping. The distance the spool’s center of mass moves is A. 2S
B. S
C. S/2
D. S/3
10. Air track car Z of mass 1.5 kg approaches and collides with air track car R of mass 2.0 kg. See the accompanying diagram. Car R has a spring attached to it and is initially at rest. When the separation between the cars has reached a minimum, then: A. B. C. D. E.
E. S/4
Z
R
car R is still at rest. car Z has come to rest. both cars have the same kinetic energy. both cars have the same momentum. the kinetic energy of the system has reached a minimum.
page 3
11. Two ice skaters, a 200 lb man and a 120 lb woman, are initially hugging on a frictionless level ice surface. Ten seconds after they push off from each other, they are 8.0 m apart. How far has the woman moved in that time? A. 8.0 m
B. 6.5 m
C. 5.0 m
D. 4.0 m
12. The diagram to the right shows the velocity-time graph for two masses R and S that collided elastically. Which of the following statements is true? I. II. III.
1.2 0.8
B. II only
C. I & II only
v(m/s) R
S
0.4
R and S moved in the same direction after the collision. The velocities of R and S were equal at the mid-time of the collision. The mass of S was greater than the mass of R.
A. I only
E. 3.0 m
1
D. II & III only
2
3
t(s)
4
E. I, II, & III
Questions 13 and 14 refer to the motion of two blocks along a frictionless level track. Block #1 (mass m1) is initially moving with speed vo. It collides with and sticks to an initially stationary block (#2) of mass m2 = 9 m1. 13. What is the speed of the two blocks after the collision? A. vo
B. (9/10) vo
C. (8/9) vo
D. (1/9) vo
E. (1/10) vo
14. What fraction of the initial kinetic energy of the system is converted to other forms (heat, sound, ...) as a result of the collision? A. 1 %
B. 10 %
C. 50 %
D. 90 %
15. Three identical objects of mass M are fastened to a massless rod of length L as shown. The array rotates about the center of the rod. Its rotational inertia is A. (1/2) ML2
B. ML2
C. (5/4) ML2
D. (3/2) ML2
16 A length of rope is wrapped around a spool of weight W with inner radius r and outer radius R as shown in the accompanying diagram. The rope is pulled with a tension T at an angle θ. Which of the following conditions must be satisfied for the spool to slide uniformly without rolling? A. cos θ = r/R
B. sin θ = r/R
E. 99 % M
M L/2
L/2
E. 3 ML2
T r
M
θ
R r
R
C. T = W
D. T = W sin θ
E. T = W cos θ
page 4
17. Two identical bricks of length L are piled one on top of the other on a table. See the diagram to the right. What is the maximum distance S the top brick can overlap the table with the system still balanced? A. (1/2) L
B. (2/3) L
C. (3/4) L
D. (7/8) L
L
E. L S
18. A gas contains a mixture of He4 and Ne20 atoms. If the average speed of the He4 atoms is vo, what is the average speed of the Ne20 atoms? B. (1/ 5 ) vo
A. (1/5) vo
C. vo
D.
5 vo
E. 5 vo
19. One end of a metal rod of length L and cross-sectional area A is held at a constant temperature T1. The other end is held at a constant T2. Which of the following statements about the amount of heat transferred through the rod per unit time are true? I. II. III. A. II only
The rate of heat transfer is proportional to 1/(T1 - T2). The rate of heat transfer is proportional to A. The rate of heat transfer is proportional to L. B. III only
C. I and II only
D. I and III only
E. II and III only
20. On all the pV diagrams shown below the lighter curve represents an isothermal process, a process for which the temperature remains constant. Which dark curve best represents an adiabatic process, a process for which no heat enters or leaves the system? A. p
B. p
C. p
V
D. p
V
E. p
V
V
V
21. The longest wavelength photon in the visible Balmer series for the hydrogen atom is A. 0.66 nm
B. 6.56 nm
C. 65.6 nm
D. 656 nm
E. 6560 nm X
22. Waves are produced by two point sources S and S’ vibrating in phase. See the accompanying diagram. X is a point on the second nodal line. The path difference SX – S’X is 4.5 cm. The wavelength of the waves is approximately A. 1.5 cm
B. 1.8 cm
C. 2.3 cm
D. 3.0 cm
E. 4.5 cm
S
S’
page 5
23. A thin film of thickness t and index of refraction 1.33 coats a glass with index of refraction 1.50 as shown to the right. Which of the following thicknesses t will not reflect light with wavelength 640 nm in air? Hint: Light undergoes a 180o phase shift when it is reflected off a material with a higher index of refraction. A. 160 nm
B. 240 nm
C. 360 nm
D. 480 nm
n=1.00 n=1.33
t
n=1.50
E. 640 nm
24. An object is placed 12 cm in front of a spherical mirror. The image is right side up and is two times bigger than the object. The image is: A. 6 cm in front of the mirror and real. B. 6 cm behind the mirror and virtual. C. 12 cm in front of the mirror and virtual. D. 24 cm in front of the mirror and virtual. E. 24 cm behind the mirror and virtual.
25. A charge is uniformly distributed through a volume of radius a. Which of the graphs below best represents the magnitude of the electric field as a function of distance from the center of the sphere? A.
B. E
C. E
a
r
D. E
a
r
E. E
a
r
E
a
26. A free electron and a free proton are placed between two oppositely charged parallel plates. Both are closer to the positive plate than the negative plate. See diagram to the right. Which of the following statements is true? I. II. III. A. I only
r
a
r
++ ++++ +++ + – – – – – – – –– –
The force on the proton is greater than the force on the electron. The potential energy of the proton is greater than that of the electron. The potential energy of the proton and the electron is the same. B. II only
C. III only
D. I & II only
E. I & III only
page 6
6.0Ω
27. In the electrical circuit shown to the right, the current through the 2.0 Ω resistor is 3.0 A. The emf of the battery is about
6.0Ω
A. 51 V
2.0Ω
B. 42 V
C. 36 V
D. 24 V
E. 21 V
3.0Ω
28. An ion with a charge q, mass m, and speed v enters a magnetic field B and is deflected into a path with a radius of curvature R. If an ion with charge q, mass 2m, and speed 2v enters the same magnetic field, it will be deflected into a path with a radius of curvature A. 4 R
B. 2 R
C. R
D. (1/2) R
E. (1/4) R
29. Which of the following wiring diagrams could be used to experimentally determine R using Ohm’s Law? Assume an ideal voltmeter and an ideal ammeter. A. R V
B.
C. A
R A
V
D.
R
R V A
E. A
V
R V
A
30. A vertical wire carries a current upward through a magnetic field directed to the north. The magnetic force on the wire points A. south
B. north
C. east
D. west
E. downward
page 7
1998 MULTIPLE CHOICE SCREENING TEST ANSWER KEY
1.
E
11. C
21. D
2.
B
12. C
22. D
3.
D
13. E
23. C
4.
E
14. D
24. E
5.
A
15. A
25. D
6.
B
16. A
26. B
7.
E
17. C
27. B
8.
A
18. B
28. A
9.
C
19. A
29. B
10. E
20. C
30. D
page 8
1994-1995 Physics Olympiad Preparation Program { University of Toronto { Problem Set 1: General Physics Due October 24, 1994
X
Y
Figure 1: Rectangular wire mesh of in nite extent. 4. Your Uncle Sal from Italy was in North America for the World Cup. He was terribly disappointed with their loss, and is in a blue funk (the blue that is found on the Brazilian ag, as a matter of fact). You want to make him a pesto pizza to cheer him up. While pouring the olive oil, you accidentally spill some into a glass of water, and it spreads out on the surface. Sal, who is watching, is a physics teacher in Italy and he is reminded of an experiment done by Lord Rayleigh. He tells you that he would be happy if only you would solve the following. Rayleigh1 found that 0.81 mg of olive oil on a water surface produced a mono-molecular layer 84 cm in diameter. What value of Avogadro's Number results? Note: The approximate composition of olive oil is H(CH2)18COOH, in a linear chain with one end (which?) hydrophilic and the other hydrophobic. Its density is 0:8 g=cm3 5. Ever the consummate student, you are at Woodbine doing some ||| betting studying. Of course, you brought your \$500 camera with a telephoto lens. Watching the thoroughbred racers Eric's Idle and Paul's Bunyon thunder down the home straight, you decide to take a picture. You are looking head on through the telephoto lens and you notice that the horses seem strangely foreshortened (not as long from front to back) as they gallop towards the camera. Explain this observation. 6. Your little brother has been playing with the wire screens and batteries again. Will he ever learn? He asks you the following. (a) A rectangular wire mesh of in nite extent in a plane has 1 A of current fed into it at point X, as in the diagram (Figure 1), and 1 A of current taken from it at point Y. Find the current in the wire XY. (b) Suppose the wire mesh was made of equilateral triangles joined in the obvious way. What is the current where X and Y are separated by one wire? 1 Rayleigh, Proc. Roy. Soc., 47, 364 (1890); Holy smokes, that's 104 years ago! And you're only learning about it now? Bonus points to anyone who can get me a photocopy of the article.
... Figure 2: In nite resistor ladder. (c) The screen has been cut up and now resembles the circuit in the diagram. What is the resistance of the circuit shown, an in nite ladder.
1994-1995 Physics Olympiad Preparation Program { University of Toronto {
Solution Set 1: General Physics
1. Dimensional analysis can be used to check to see if an expression is dimensionally correct or to get the form of an expression if we don't know it. We are going to do the latter. (a) Hmm... What could the period of a simple pendulum possibly depend on? There's the mass of the (ideal point) bob (or is that Bob, founder of the Church of the Sub-Genius?), m, the length of the (essentially massless ideal) string, l, the angle of the swing, , and the acceleration of gravity, g. We might think to include things like frictional forces, but they're small compared to gravity and besides, that would be too complicated. Let's assume the period T is a function of these four variables, each raised to some power: T = Cmw lx y gz : C is a dimensionless constant, and w, x, y, and z are exponents for which we wish to solve (no dangling prepositions here). The dimensional equation for this relationship is [T] = [M]w[L]x [L=T 2]z ; the angle (measured in radians) has no dimensions. Simplifying, [T ] = [M]w [L]x+z [T],2z : To have dimensional consistency, we must have 0 = w 0 = x+z 1 = ,2z which gives us w = 0, z = ,1=2, and x = 1=2. Thus the desired equation is p
T = C l=gf() It turns out that C = 2 and f() ' 1 for small angles, but we can't get that from this analysis. (b) The general equation for this problem is ac = Cmx ry vz , which gives us the dimensional equation [L=T 2] = [M]x[L]y [L=T ]z . Simplifying, [L][T],2 = [M]x[L]y+z [T ],z , which gives us the three equations 0 = x, 1 = y + z, and ,2 = ,z, which are solved by x = 0, z = 2, and y = ,1. Thus ac = Cv2 =r. It turns out that C = 1. (c) The general equation for this problem is T = Cpx y ez , which gives us the dimensional equation [T] = [M=LT 2 ]x[M=L3 ]y [ML2=T 2 ]z . Simplifying, [T] = [M]x+y+z [L],x,3y+2z [T ],2x,2z, which gives us the three equations 0 = x + y + z, 0 = ,x, 3y + 2z, and 1 = ,2x , 2z, which are solved 3 2 1=6 by x = ,5=6, y = 1=2, and z = 1=3. Thus T = C p5e . 2. For these ones, we quote numerical values of things from memory or make good guesses, multiply numbers in our head, rounding them o , and in general not worry about being too precise. It may also help to do the problem algebraically rst, then put in the numbers, since this will allow us to plug in more precise numbers if we wish. 1
2
clock. Thirty days later we check and the watch is ahead one minute. We have gained 1 minute over a period of 30 days. Thus the watch is about one minute per month fast. Strictly speaking the units can be canceled to get a unitless number, but it is more meaningful to people to use the above form. Also note that I have seen people with degrees in physics make the mistake of saying fast when they mean ahead! (c) The normal high and normal low are, in more precise language, the average high and low. That is, they are the averages of the daily highs and lows for a particular day of the year, averaged over the past 100 years or so for which there are recorded temperatures. What is left out is the variance of the highs and lows or rather the distribution of the temperatures. It is possible that the average temperature on August 28 is 25 C but there may be just as many August 28's which were 23, 24, 26, and 27. Does it then make sense to use the word normal to mean average? In such a case is 25 normal? Probably not. It is really a case of experts not wanting to use precise language because it may end up confusing people, or so they think. 4. We know that N=M = NA =MA where N is the number of molecules in the sample and M is the mass of the sample and the subscript refers to a mole of the substance. Thus Avogadro's Number is NA = NMA =M. To get MA , we take the formula and the molecular masses and nd the total; (38 1 + 19 12 + 2 16) = 298 g. To get N, we take the volume of the sample, V , and divide it by the volume of a single molecule, v. Getting v is the tricky part. The COOH end of the molecule is hydrophilic while the rest of the molecule is hydrophobic. Thus the molecules form a layer standing side-by-side vertically on the surface of the water. The diagram shows approximately what the molecule looks like if we assume that the carbons are at the centres of tetrahedrons with the neighbouring atoms on the corners. There is an end-on view and a side view. The distance between two atoms joined to a common carbon is x|this is the length of an edge of the tetrahedron. It turns out that the end-on area is nearly square|the zig-zag in the carbon chain compensates for the di erence between the edge length in one direction and the distance from edgecentre to opposite edge-centre in the other direction. In all three directions a distance of x is added to account for the fact that neighbouring molecules will not be butted up against one-another. The x puts the hydrogen atoms on one molecule a distance x from those on the neighbouring molecules. Thus the volume of each molecule is about v ' 4t3 =121 where t is the layer thickness. There is a measure of uncertainty in this last equation since we don't know how much space there is between molecules. The total volume is V = At, thus 1=v ' 121A3=4V 3 and N ' 121A3=4V 2 . Now A = (d=2)2 and V = M=; with a little algebra we get 32 d6 MA NA ' 121256M 3 : Putting in the numbers, 3 g=cm3)2 (84 cm)6 (298 g) NA ' 121 (0:8 256(0:81 10,3 g)3 := 2 1024: So Avogadro's number is about 1024 molecules per mole. We are o by a factor of 3 from the accepted value, which is ok considering how many uncertain numbers there were. 5. A telephoto lens, like a telescope, makes far objects appear near by giving rise to an angular magni cation. Let us approximate a horse (call him Boxy) by a rectangular body with a vertical leg at each corner. Suppose the separation of the forelegs and of the back legs is 0.5 m, and the distance between the front and back legs is 2.0 m. If we observe the horse head-on when it is 10 m away, the gap between 3
its forelegs will subtend an angle of 1/20 rad at the eye, while the gap between the hind legs subtends 1/24 rad. This di erence gives most of the impression of the length of the horse. If the same horse is 100 m away the same two angles become 1/200 rad and 1/204 rad, which are not very di erent. However, if we use a telescope with an angular magni cation of 10, the angles become 1/20 rad and 1/20.4 rad. The apparent size of the front is the same as that of a horse standing 10 m away, but now its hind legs appear to be only 0.2 m further back. A very short horse! 6. (a) Break the problem into two parts. Consider the mesh with 1 A going in at X and 1 A coming out at in nity. By symmetry, the wire XY must carry 1=4 A of current. Alternately, consider the mesh with 1 A going in at in nity and 1 A coming out at Y. Again, by symmetry, 1=4 A runs through the wire XY. Add the two cases together and you get the stated problem, with the solution that 1=2 A ows along wire XY. (b) Similarly, the two cases each have 1=6 A owing along XY, giving a solution of 1=3 A. (c) Referring to the diagram, you can see that because the circuit is in nite, it can be written as a circuit with itself being one of the components. If R0 is the resistance of the entire circuit, then, using the rules for combining resistors in series and parallel, we have 1 1 1 R0 = R + 2R + R0 : Multiplying by all three denominators and rearranging we end up with a quadratic equation for R0 , R0 2 + 2RR0 , 2R2 = p 0. We use the quadratic formula to solve, with the negative root thrown out. The result is R0 = ( 3 , 1)R.
4
1994-1995 Physics Olympiad Preparation Program { University of Toronto { Problem Set 2: Mechanics Due November 28, 1994
1. A crude estimate of the depth of a well can be obtained by dropping a stone down it and listening for the splash, then using s = gt2 =2. You are asked to make rst-order estimates of the corrections due to (a) the nite speed of sound (vs = 340 m=s), and (b) air drag. Which of these is the bigger correction for s = 30 m and a stone of radius r = 2 cm and density 3 103 kg=m3? Would your conclusion be signi cantly di erent if a table tennis ball were dropped in the same way (r = 2 cm, m = 1:7 g)? You may assume that the splash would still be audible! Note: the drag for a sphere of radius a depends on the dimensionless Reynolds number R = va=, where is the uid density and is its viscosity. For R < 10, Stokes' law F = 6rv is valid, but for R > 100 the drag force is given by F = CD r2 v2 ; the variation of CD with log10 R is given in the diagram. For air at 20 C, = 1:3 kg=m3 and = 1:8 10,5 kg=ms. 2. Your phys-ed teacher is a student of the physics of human motion, and asks you the following questions: (a) When told that the world record for the pole vault was about 5.5 metres, the fast rising athlete Rod Fibreglass told the press, \Give me a pole long enough, and I will raise the record to 9 metres." Could he manage it? How high might he raise it if he tried hard? (b) I obtained the following data one Sunday: i. There are 238 steps from the basement to the 10th oor of our building. ii. After measuring four steps, I believe the step height is 18:2 0:3 cm. iii. I jogged up the steps in 1 minute and 40 seconds, give or take 2 seconds. Well, actually started o jogging and ended up walking up with a not-so-steady pace. iv. My mass is 82:07 0:2 kg. The uncertainty is due to the fact that I drank some water, ate a banana, sweated and urinated in the time between doing the climb and measuring my mass. v. The value of g measured by some geophysicists at the base of the north stairwell is 9:804253 m=s2. They don't give an uncertainty estimate. Note that they say that g varies with height as g=g = ,2R=R, where1 R = 6371 km. How much (useful) work did I do? What was the average power output? Calculate the uncertainties in these values as well. (c) It is proposed to make a human-powered helicopter with a rotor 10 m in diameter. Assuming that the rotor blows a cylindrical column of air uniformly downwards, the cylinder diameter being the same as the rotor diameter, and that the mass of the pilot plus machine is 200 kg, calculate the minimum mechanical power (in Watts) that is necessary for the pilot to generate to remain airborne. Is the system practicable? Compare with my power output above. The density of air is 1:23 kg=m3. 3. NASA has just contracted you to make the following calculations, you lucky dog. As with many NASA contracts, you are being paid a handsome sum with bonuses for correctness. No \$10,000 toilet seats, please. 1 I don't know why this number, posted in our undergraduate labs, disagrees with others quoted here (6 38 103 km), except to say that the Earth isn't a sphere. :
1994-1995 Physics Olympiad Preparation Program { University of Toronto {
Solution Set 2: Mechanics
1. (a) If the real depth is s and the velocity of sound is vs, then the measured time would be tm = (2s=g)1=2 + s=vs and the measured depth would be sm = gt2m =2. Neglecting squares of small terms, we obtain s r sm , s ' 2gs = 2(9:8)(30) =: 7:2%: s vs2 3402 That is, the depth calculated without taking into account the nite speed of sound would be about 7 percent too large. (b) An approximate value for the nal velocity is1 (2gs)1=2 ' 24 m=s, showing that the Reynolds number is of the order of 104 most of the time; the drag force is therefore CD r2 v2 , with CD ' 0:4. The equation of motion is thus ma = mg , v2 , with = CD r2 . For the zeroth approximation, we neglect the last term so that a = g and thus v = gt. For the rst order approximation, we insert this value of v into the (small) last term to give ma = mg , g2 t2 , or a = g , g2 t2=m. Now, if you knew calculus, you would immediately integrate this to obtain s = gt2 =2 , g2 t4=12m. You might also just know that whenever you have a term Atn in the acceleration, it gives you an At(n+1)=(n + 1) term in the velocity and an At(n+2) =(n + 1)(n + 2) term in the displacement. You should really learn calculus. Anyway, neglecting the air drag thus introduces an error equal to the last term, g2 t4 =12m. The relative error is thus gt2 =6m ' s=3m. From the data given, we nd that this is about 6%, thus the air drag and the nite speed of sound give approximately equal errors. Both errors make the calculated depth too large. (c) For the table tennis ball, the drag force is much more important relative to the weight. The velocities are smaller, but still large enough to make R > 103, i.e. C:D is still about 0:4. In this case, the terminal velocity is given approximately by2 (mg= )1=2 = 5:3 m=s. The ball will reach this speed after a few meters, so that the zeroth approximation, sm = gt2 =2, is a gross overestimate of the depth|by a factor of about 2.6 for s = 30 m. A better rst approximation is to assume that it travels at its terminal velocity for the whole distance. In this approximation, the error due to the nite speed of sound is clearly given by the ratio of the terminal velocity to the speed of sound. The error due to the initial period of acceleration cannot be evaluated without using the exact solution of the equation of motion. This is too complicated to be evaluated in the present context. 2. (a) The dominant consideration is the conversion of the kinetic energy of the running man to gravitational potential energy with something less than 100% eciency. If, in his approach, he attains a speed of 10 m=s, the corresponding rise is 5 m. Smaller terms arise from i. the fact that his centre of mass is already 1 m above the ground when he starts, ii. the work done by his legs on take-o and by his arms in climbing up the pole (? = 0:5 m; consider the \vertical" of a standing person), iii. the fact that his centre of mass actually passes below the bar (? = 10 cm). Adding these terms together gives approximately 6.6 m. The di erence between this and the observed 5.5 m is due to an eciency of less than 100%|or to errors in the estimated quantities. In any case, there is clearly no hope of Mr. Fibreglass making good on his boast of 9 m since 10 m/s is an excellent sprint speed and the other terms can not be improved by much. Equate initial potential energy with nal kinetic energy and solve for speed. Write force equation for terminal body, including gravity and drag force but neglecting the small buoyant force, set acceleration to zero, then solve for speed. 1 2
1
(b) Work done is the change in potential energy of my body, W = mgh. Uncertainty is p
W=W = (m=m)2 + (g=g)2 + (h=h)2: The (g=g)2 is negligible. The result is 34:85 0:4 kJ. Note that the uncertainty in the stair height dominates. A measurement of the actual distance would improve things immensely. The average power output is P = W=t, while the uncertainty is p
P=P = (W=W)2 + (t=t)2 p = (m=m)2 + (g=g)2 + (h=h)2 + (t=t)2: The result is 348:5 8 W. This is nearly half a horsepower. Don't be impressed. I only did this for a couple of minutes. A horsepower is the average power a horse can put out over a work day. I've been told that for longer times (15 minutes to 2 or 3 hours) for people of average mass (65 to 75 kg), useful power output is roughly 275-300 W. By that criteria, given how winded I was, I am out of shape! (c) If the rotor, of radius R, gives a downward velocity v to a column of air initially at rest, the momentum transferred per second will be v where , the mass of the air accelerated per unit time, is R2v. If the helicopter is to be airborne, this must equal Mg, where M is the mass of the helicopter. Inserting the values given, we obtain v =: 4:5 m=s. Neglecting energy losses, the energy transferred per second to the bulk motion of the air is v2 =2 which equals 4.5 kW in this case. For the system to be practical, the pilot must thus generate about 5 kW continuously. From the answer to the previous question, it appears that the maximum rate of energy production by a human is of the order of 1=3 kW. Thus we are a far cry from being able to run a helicopter by human power. 3. (a) For the moon in an orbit of radius r, we have3 GMm=r2 = mr!2 , assuming M m, where G is the universal gravitational constant and ! = 2=T is the angular frequency with T being the period. For the rock of mass , in orbit of radius r , a, Newton's Law gives us GM Gm 2 (r , a)2 , a2 + F = (r , a)! where F is the force of contact between the rock and the moon. If the rock is to be lifted o , F = 0. Eliminating ! between the two equations, we obtain M = r3 (r , a)2 ' 1 r 3 m a3 3r2 , 3ra + a2 3 a as required. (b) For any spacecraft in a: circular orbit, GMm=r2 = mr!2 . For a body on the surface of the Earth, GME =rE2 = g = 10 m=s2. With the numbers given, we nd ! = 10:4 rad=s, i.e. the period is approximately =5 seconds|somewhat uncomfortable for the intrepid astronaut! We can also deduce that the linear velocity in orbit is approximately 107 m=s ' 0:03c|high, but not relativistic. i. For an astronaut of mass ma at the mass centre of the craft, GMma =r2 = ma r!2 is also true, i.e. she will, indeed, oat. ii. If r changes to r , x, the gravitational force becomes GMma =(r , x)2. If sideways motion across the vehicle is prevented, and if the vehicle is large compared with the mass of the person, then ! will remain constant, that is the ship will force the astronaut to orbit with 3
The gravitational force provides the centripetal force which is set equal to the mass times centripetal acceleration.
2
the same period. The force needed to maintain the astronaut in the new, smaller orbit will be ma (r , x)!2. This is no longer equal to the gravitational force. The di erence, approximately 3ma !2 x, will paccelerate the person towards the centre at a rate 3!2x. This will give rise to a velocity4 3!x. Starting from the end of the tunnel (x = 50 m), this gives rise to a maximum speed of 900 m=s (3240 km=h in more homely units). On the way to this unfortunate p conclusion there will be a Coriolis acceleration sideways across the tunnel equal to 2 3!2 x. This will reach the value g before the victim has moved more than a few centimetres. iii. The orbit as described is indeed stable. The di erential radial force calculated above|the so-called tide-raising force|will provide a restoring couple if the attitude is disturbed, as can be seen by drawing a diagram. Also, one can appeal to tides themselves; they are at a maximum when the graviational source is overhead. (c) Initially Newton's Law gives mv2 =r = Gm2 =4r2: (1) The nal system is most likely to break up if the explosion increases the velocity of one of the one star in the direction in which it is moving at the time of the explosion (consider the diagram where before and after denote the situation an instant before and after the explosion). Call the increased velocity vf . The centre of mass now has a velocity (v , vf )=2 and the velocity of either star with respect to the centre of mass is, say, (v + vf )=2 = v : (2) The system will break up if the new kinetic energy with respect to the centre of mass2 coordinates is greater than the work needed to separate the components to in nity, i.e. if mv > GM 2=2r. Using relations (1) and (2), the required condition becomes p p vf = (2 2 , 1)v = (2 2 , 1)(Gm=4r)1=2: The above argument assumes that the mass lost as a result of the explosion is so small that it has no e ect on the result. If this quantity is denoted by m, then an analysis along exactly the same lines gives the condition vf =v = 2(2 , m=m)1=2 , 1, thus showing that the assumption was valid. 4. We choose the positive sense of ! so that the velocity of the point of contact of the ball with the table, vc , is given by u + a!. The frictional force F = mg is assumed constant. The motion of the centre is thus given by u = u0 , gt and the rotation about the centre by ! = !0 , amgt=I. Since I = 2ma2 =5 for a sphere, the latter expression becomes ! = !0 , 5gt=2a. These results combine to give vc = u0 + a!0 , 7gt=2. This is equal to zero (i.e. slipping stops and rolling begins) when t = 2(u0 + a!0 )=7g = , say. The value of u at t = is (5u0 , 2a!0)=7; this gives the speed of rolling after slipping has stopped. If u0=a!0 < 2=5, the value is negative and the ball will roll backwards towards the cue. This corresponds to the imposition of \backspin" as speci ed. If u0 =a!0 > 2=5, then the ball will roll forwards. To enquire whether both motions are possible, we consider an impulse Q given to the ball at a point a , x above the table in the direction making an angle with the horizontal (see the diagram). Then mu0 = Q cos and I!0 = Qp, with p = a sin( + ) and sin = x=a. If we use the relation 5u0 = 2a!0 to determine the boundary condition between the two modes of behaviour, then, after a little algebra, we get cos = sin( + ), i.e. 1=2 1 , x=a tan = 1 + x=a : Real values of can be found for any x=a|except that x ! a would be a little dicult, in practice! 0
0
4 Think of simple harmonic motion with a spring constant over mass ! 2 = k=ma = 3! 2 ; the maximum speed, obtained while 0 passing through the centre, will be x!0 .
3
5. (a) The annihilation energy of a particle of rest mass m1 will make available an energy E = m1 c2. If this energy is given to another particle of rest mass m2 , which thereby acquires momentum p, we have the relation5 (E + m2 c2)2 = p2 c2 + (m2 c2 )2 : In the particular case envisaged in thepproblem, m1 = m2 = m is the mass of a proton or antiproton. This leads to the result p = 3mc. If there are initially N protons and N antiprotons, then this process can be repeated N times6. If all N ejected particles can be persuaded to leave p the spaceship in the same direction, the e ect will be to give a total momentum 3Nmc to the ship. (b) If we consider the mutual annihilation of two particles, each of rest mass m, a similar argument shows that the (scalar) total of the momentum thereby produced is 2mc. There will, in fact, be two photons proceeding in opposite directions (since the vector total must be zero). With the aid of a deep paraboloidal mirror made of a material that can re ect high-energy gamma rays (!) all such photons can be collimated into a beam carrying a momentum 2Nmc. The spaceship will have an equal momentum in the opposite direction. This is larger than the other result. The above arguments are strictly valid only if the increment in velocity of the spaceship is much less than c. If this is not the case, the total number N can be conceptually subdivided into small groups for which it is true, and the total e ect of the momentum increments between successive inertial frames can then be calculated as an integral (sum). The advantage of (5b) over (5a) applies at every stage and therefore also overall. 6. (a) As a rst guess, one thing we can do is to get a maximum on the period. We don't feel the e ect of the centripetal acceleration of our orbiting the sun very easily on Earth, so it must be less than, say, a percent of g. Since the acceleration goes as the inverse square of the period, the period must be less than 10 percent the Earth's year, that is less than a month. It is dicult to say anything more since we don't have an indication of how restrictive our maximum is. More quantitatively, the centripetal acceleration of a particle moving in a circle of radius r at a constant speed v is 2 2 1 ac = vr = 2r T r p where T is the period. Thus T = 2 r=ac . Using ac = g we get T =: 7:77 105 s =: 9:0 days. That seems like an awfully short time to go around the sun! Note that we've ignored the gravitation attraction of the star, which is small compared to g. (b) Consider a narrow section of the ring as shown in the diagram. The net force on the section is 2FT sin() towards the centre. This is the centripetal force and must equal mac = mg where m is the mass of the section. This is equal to 2rM=2r where r is the radius of the ring and M is the total mass. The total mass is the volume times the density, M = 2rA, where A is the cross-sectional area. Letting sin() ! for small angles, we nd that the tension is given by FT = rAg. This is just the \weight" of the ringworld on Earth (if we could weigh it) divided by 2! To get the tensile strength we just divide by the cross-sectional area; S = rg. Notice how we don't need to know A. If we take the density of aluminum as a possible density for the material, then the required tensile strength is 4 1015 N=m2. This is seven orders of magnitude greater than most metals. The left-hand side is the square of the total energy before, while the right-hand side is the square of the total energy after, using the relationship between energy and relativistic momentum. 6 Actually, the more complicated process of annihilating a proton-antiproton pair giving the energy to another proton and antiproton pair which doesn't annihilate but rather is ejected in the same direction is performed N=2 times, but the e ect is the same. 5
4
(c) The rotation of the ring will give it orientational stability, but not positional. The positional stability is depends only on the net gravitational attraction of the star. If the ring is pushed so that it moves perpendicularly to its plane, then one can see a net gravitational force attracting the ring back to equilibrium. If the ring is pushed in its plane, the situation is not as easy to see. Consider the force due to four pieces as shown in the diagram. The size of the force goes as 1=r2 which depends on the position of the piece on the circle. The \vertical" component (see gure) also depends on the position but in a opposing way. The result is that one can't nd out whether there is a net gravitational force in the plane by simple arguments. One must use math and it turns out there is a net restoring force. Remarkably enough, there are cases where one can have an instability. A Dyson Sphere (a spherical shell of similar dimensions surrounding a star) would have no net force on it due to a star inside, thus it would be unstable. The argument for this is mathematical as well. I got the idea for the ringworld question from a book by Larry Niven called, curiously enough, Ringworld. It is followed by Ringworld Engineers, and may be found in the science ction section of libraries and bookstores. The book describes the material as \Very dense, with a tensile strength on the order of the force that holds nuclei together." Anyone care to check on whether that would be enough?
5
1994-1995 Physics Olympiad Preparation Program { University of Toronto { Problem Set 3: Thermodynamics Due January
9th , 1995
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1994-1995 Physics Olympiad Preparation Program { University of Toronto {
Solution Set 3: Thermodynamics 1. First, calculate the heat released in freezing 2 kg of water at 0 C, Qc : 3 :0 kJ=mol) = 667 kJ Qc = (2 10 (18g) g =(6mol) The maximum coecient of performance, max , is given by: max WQc = T T,c T ; min h c where Wmin is the minimum work needed to e ect the cooling, and Tc and Th are the temperatures of the cold (inside fridge) and hot (outer air) reservoirs, respectively. Thus, Wmin = Th T, Tc Qc : c
The minimum time, min , is given by (P is the power, and note that absolute temperatures must be used): K , 273 K) : 3 3 min = WPmin = (300 (273 K)(50 W) (667 10 J) = 1:3 10 s: Assumptions can include: (a) The refrigerator, as suggested by the model name, is taken to be a Carnot heat engine, where Qc =Qh = Tc =Th . (b) The ice and water are in equilibrium; therefore, the liquid is at 0C. (c) Only the water is being cooled (i.e. no container material or air). (d) If the cold reservoir (interior of the fridge) were completely lled with ice water, then it would need to be expandable in some way, since water expands as it freezes. 2. (a) After a long enough time, and if there were no predator, the bacteria would amount to a huge number, N 10; 000. Since the predator eats either colour at random to keep the population at 10; 000, then it's the same thing as selecting n = 10; 000 surviving bacteria out of N total bacteria. Since N n, in every selection the probabilities of surviving beige or fuchsia bacteria are the same, namely 12 . In other words, there are 2n ways of selecting the n survivors. There are Cbn combinations of b beige bacteria to survive out of the n total surviving. The probability distribution of the number of beige E. coli is therefore: Cbn = 1 n! ; b = 0; 1; . . . ; n: 2n 2n b!(n , b)! (b) To answer this, you basically need to determine how long it takes for N n. Let us require N=n 102. Since N = 2t n, where t is the number of reproductive cycles, t = 6 or 7 hours would be sucient.
(c) Let p be the degree of preference of eating fuchsia The , bacteria over that of eating beige bacteria. , probability of eating beige bacteria is then 12 , p , and that of eating fuchsia bacteria is 12 + p . The result in (a) then becomes:
1 , p b 1 + p n,b C n = 1 , p b 1 + p n,b n! b 2 2 2 2 b!(n , b)!
The result in (b) is una ected by the predator preference p. (d) Sewage. 3. First of all, we assume that the room temperature falls between that of the milk and the black co ee. There are 2 principal competing e ects: (a) The addition of the milk cools the co ee directly by a simple law of mixtures. (b) \Newton's Law of Cooling", which basically says that the rate of heat loss is greater when the temperature di erence between the two heat reservoirs (i.e. the co ee or milk/co ee mixture and the surrounding air) is greater. For the simple case where the milk is colder than room temperature, it is in fact better to add the milk at the beginning of the 10 minutes in order that the milk/co ee mixture be as hot as possible at the end of the 10 minutes. By cooling with milk the cup's contents right at the beginning, the temperature di erence between the two reservoirs is lessened, and hence the rate of cooling slows. The \blackness" of co ee without milk also enables the co ee to be a better radiator of heat, but this e ect is rather smaller than those above. (Note: For a good article on this subject, see Rees, W.G., and C. Viney, Am. J. Phys. 56 (5) 434, May 1988). 4. We are dealing with the chemical equation: H2 + I2 * ) 2HI: P
We can write this in the form i iAi = 0, where Ai are the reacting substances and reaction products, and i are the coecients of the chemical equation. Hence, 2(HI ) , 1(H2) , 1(I2 ) = 0: Reaction equilibria can be determined by the Law of Mass Action, Y
i
Ci = Kc ; i
where Ci are the equilibrium concentrations of the various constituents of the reaction. Let x be the number of g-mol of H2 (or I2 ) used to produce HI . Then: :1 , x C(H2 ) = 811 :04 94 , x C(I2) = 2:11 :04 2 C(HI ) = 11:x04 (11.04 is the total number of g-mol present).
Now, using the Law of Mass Action gives:
2:94 , x ,1 8:10 , x ,1 2x 2 = 0:02: 11:04 11:04 11:04
Solving the quadratic equation, and choosing the root that doesn't violate the conservation laws, we get x = 0:319. Therefore, there will be 0.638 g-mol of HI . 5. Let the long axis of the dust grain be parallel with the z -axis. From the shape of the grain, we know that the principal moments of inertia satisfy:
Iz < Ix ; Iy : At thermal equilibrium: 1 I !2 = 1 I !2 = 1 I !2 ; 2z z 2x x 2y y where w the average angular velocity. This is due to the energy equipartition theorem of thermodynamics, which states that in thermal equilibrium each degree of freedom contributes an equal amount of kinetic energy to the total energy of a molecule. (Here the dust grain is being treated like a very big molecule.) Therefore, r
r
j!z j = IIx j!x j = IIy j!y j ; z z and Similarly,
r
jIz !z j = IIz jIx !x j < jIx !x j : x jIz !z j < jIy !y j :
So, the angular momentum vector is nearly perpendicular to the long axis of the dust grain. 6. Assume the initial length is lo . After heating, the average lengths of the 2 metallic strips are:
l1 = lo (1 + 1 T ) and
l2 = lo (1 + 2T ): Assuming that the radius of curvature is R, the subtending angle of the strip is , and the change of thickness is negligible, then l2 = (R + x=4) and l1 = (R , x=4): Therefore, l2 , l1 = x2 = x2 l12+Rl2 = 4xR lo (2 + ( 1 + 2) T ) :
But, we know that
l2 , l1 = lo T ( 2 , 1) :
Making the substitution, we have giving
lo T ( 2 , 1) = 4xR lo (2 + ( 1 + 2) T ) + ( 1 + 2 ) T ) : R = x (24 T ( , ) 2
1
1994-1995 Physics Olympiad Preparation Program { University of Toronto { Problem Set 4: Waves and Optics Due February
6th , 1995
3. You are conducting a feasibility study into the SDI (Strategic Defense Initiative) missile defense system. You are to consider the viability of a ground based laser defense against ICBM's, which will use powerful CO2 lasers to bore holes in the incoming missiles, thus destroying them. The lasers have an output power of 50 W in a beam diameter of 1 mm. The laser beam is red at the missile when it is 10 km away and the beam loses 3% of its intensity for every kilometer travelled. The outer skin of the missile is aluminum that is 3 cm thick. When the laser res the skin temperature of the missile is ,50 C and must be heated to its boiling point at 2500 C . The density of aluminum is 2:34 g=cm3 and heat capacity is 0:9 J=g C. (a) How long will it take the laser to burn through the outer skin of the missile, thus destroying it? Assume that all of the laser power that reaches the missile goes into heating the 1 mm diameter spot. (b) What angular accuracy must the aiming system of the laser have in order for the laser not to drift away from the target spot? (c) Suggest three ways that one could use to protect a missile from this laser defense system. 4. We have all seen those really cool holographic images of our favourite cartoon characters on the cereal boxes, not that I still watch cartoons anymore, well occasionally I see the odd one, actually on Saturday... anyway, back to the problem. Being a physics nerd, and proud of it by the way, I could never just enjoy those pictures without trying to gure out how they work. I will now in ict my geeky inquisitiveness on all of you, heh heh heh. (a) Holograms are 3-D images that change depending on the viewer's perspective. How is the dimensionality of the image captured on the holographic plate? How is this di erent than normal 2-D lm? (b) Sketch an experimental apparatus that could be used to produce a hologram. Be sure to explain what is going on. (c) When normal lm is exposed to light and developed, a negative is produced. It is called a negative because the dark and light areas are interchanged by the developing process. When light is shone through the negative onto photographic paper a positive image is produced, which then ends up in your photo album. In holography it makes no di erence whether the holographic plate ends up being a negative or positive. Why is this? 5. Now for a really dull question, for which I humbly apologize; it won't happen again. There are two thin lenses of focal lengths f1 and f2 separated by a distance d. In general, the focal length of a system of lenses will be di erent depending on which side the image is on. For systems with more than one lens we de ne a back focal length and a front focal length, which are just the two focal lengths for these two possibilities. (a) Find the back focal length and the front focal length of this system of lenses. (b) Something nifty happens when d = f1 + f2 . What is it? (c) Find the focal length of the system for d = 0. Does it matter which side of the system the image is on in this limit? 6. Now for a few small questions that will really get you wondering why you are doing these Olympics in the rst place. Just give qualitative answers. (a) I'm sure most of you have been on trips up into northern Ontario and seen the Northern Lights rst hand. What causes the Aurora? Why are they strongest in Northern Canada? (b) We were all kids at one time or another (except maybe Pekka) and probably sung Twinkle Twinkle Little Star. What makes stars twinkle? Do the planets and moon twinkle also?
(c) If you are walking on the beach one day and nd a seashell, and just happen to hold it up to your ear, you will be able to hear the roar of the ocean. What causes this? (d) If lightning strikes a tree, it can either be left unharmed or be completely shattered. Why will some trees be destroyed and others be ne? It turns out that Oak trees are preferentially destroyed over other species. Why?
1994-1995 Physics Olympiad Preparation Program { University of Toronto {
Solution Set 4: Waves and Optics
1. (a) To solve this problem we need the equation which describes how a curved surface bends light n1 n2 n2 , n1 So + Si = R where n1 and n2 are the indices of refraction of the two media, R is the radius of curvature of the surface, Si is the image distance and So is the object distance. Solving for the image distance o : Si = (n , nn2 RS 2 1 )So , n1R At the second surface we have n2 n1 ,(n2 , n1 ) 2R , Si + x = ,R where x is the image distance we are trying to nd and all of the other variables have the same meaning as above. Here, the second object distance So = 2R , Si because a virtual image is formed by the rst surface. Solving for x gives , Si ) x = (n , nn1 R(2R : 2 1 )(2R , Si ) , n2 R Putting n1 = 1:0, n2 = 1:5, So = 1:2m and R = 10cm gives Si = 36cm and x = 6:9cm. Therefore the image is 6:9cm from the crystal ball. (b) The image is real, inverted and mini ed. (c) For the gypsy lady to produce no image means that x ! 1. This corresponds to (n2 , n1 )(2R , Si ) = n2 R or Si = ,10cm and So = 5cm. In other words, that gypsy's mug is awfully close to the crystal ball. 2. (a) This problem is solved by looking at the interference that the two waves undergo at a particular point on the screen. The rst thing to do is work out the path length that each wave undergoes on its way to the screen (see gure #1). Consider rst r1
r
Using h << x gives and using D >> x gives Similarily
r1 = D2 + (x , h2 )2 :
r
2 r1 = D 1 + x D,2hx 2 , hx r1 = D(1 + x 2D 2 ): 2 + hx r2 = D(1 + x 2D 2 ):
(b)
(c)
3. (a)
(b)
(c)
4. (a)
The path di erence between the two waves will be r2 , r1 = hx=D and phase di erence = 2xh=D. The electric eld at the screen will be the superposition of the waves that went through the slits Etot = E2 ei + E2 exp i( + ) where the factors of two appear because the wave splits in wave in half when it hits the slits. Notice that one wave is phase shifted radians with respect to the other. To nd the intensity we multiply Etot by its complex conjugate, which gives 2 xh ): Iscreen = E2 (1 + cos 2 D If the laser is shifted so that the wave coming from one slit is out phase by radians with respect to the other then the electric eld becomes Etot = E2 exp i( + ) + E2 exp i( + ) Multiplying this by its complex conjugate gives 2 xh ): Iscreen = E2 (1 , cos 2 D This new formula has its maximum o centre with respect to the two slits, which is the major di erence between the interference patterns. The reason the experimental pattern fades is because the same amount of light doesn't hit all parts of the screen. Most of the light passes straight through the slits, with far less scattered at large angles. This means that the interference pattern is brightest opposite the two slits (ie at x ' 0). To solve this question we must nd out how much energy is needed to evaporate the aluminum, and then how fast the laser can supply this energy. The volume of aluminum to evaporate is V = (=4)d2h where d is the spot diameter and h is the thickness, which means that the mass of aluminun is M = V with . The heat needed to evaporate this amount of aluminum (neglecting the heat of vapourization) is H = CMT where C is the heat capacity and T is the temperature di erence. Plugging in the numbers gives H = 126:5 J. The laser beam is attenuated to P = 50(0:97)10 = 36:9 W by the time it arrives at the missle so it will take 3:4 sec for the laser beam to burn through the missle's skin. The angular accuracy needs to be enough to keep the laser aimed at the same spot on the missle for the entire 3:4sec. Assuming the beam can't move by more than its radius gives an accuracy of ,4 tan < 5 10104 or the drift in must be less than 8:4e , 7 degrees=sec. To defend against this sort of attack one could: rotate the missle, thicken the skin, make the skin re ecting, increase heat transfer over the skin, have dummy warheads, send up cha e, or as some of you suggested, not re the missles, use a cloaking device, blast laser system and nally, don't piss the guys at ICBM control o . The thing that makes holograms di erent from normal pictures is that holograms record phase information, in addition to amplitude information on the lm. A picture only records how intense the light is, which gives no information about depth. A hologram on the other hand is just a recorded interference pattern of the image. The depth information about an image is preserved on the plate in the form of an interference pattern.
(b) To create a hologram one needs to produce an interference pattern of the desired object. This is done by splitting the laser beam into a reference beam and an object beam (see gure #2). The probe beam is re ected o the object, and then interferes with the reference beam. Because the object has depth to it, the re ected beam will not have the same phase across it when it reaches the holographic plate. It will then interfere with the uniform reference beam, thus producing an interferomic record of the object. (c) It makes no di erence wether the holographic plate is a negative or a positive. Since the information is just an interference pattern, the di erence between positives and negatives will only e ect the overall phase of the pattern and will not e ect the nal image. 5. (a) I will rst work out the back focal length using the Lensmakers equation. At the rst lens we can write 1 = 1 + 1 f1 Si1 So1 or Si1 = SSo1,f1f : o1 1 This will be the position of the image after the rst lens. The same can be done for the second lens, which gives Si2 = SSo2,f2f o2
2
Since the lenses are located a distance d away from each other we can write So2 = d , Si1. Using these three equations to solve for Si2 while eliminating So2 and Si1 gives back focal length = Si2 = df2,(df ,,f1f) 1 2 Doing the same analysis but in the other direction gives front focal length = df1,(df ,,f2f) : 1 2
(b) If we set d = f1 + f2 then the system acts like it isn't there. Putting plane waves in one end produces plane waves at the other. Note that the magni cation of the system will not likely be unity. (c) If d = 0 in the above equations then back focal length = front focal length = f f1+f2f 1 2 or in other words it doesn't matter which side the image is placed on. We can write 1 1 1 feff = f1 + f2 : This is just a modi ed version of the thin lens equation. 6. (a) The Aurora are caused by low energy solar electrons that are caught by the Earth's magnetic eld. They are then sped up and enter the atmosphere at the magnetic North Pole. These electrons excite nitrogen and oxygen which produce the characteristic blue of the Aurora. The reason it is strongest over Canada is that the magnetic North Pole is centred over ??? Island in the Canadian Arctic.
(b) Stars twinkle because the atmosphere that the light passes through to get to our eyes is turbulent. This is due to uneven heating of the atmosphere by the sun. This turbulence causes random refracting of the light which we see as a twinkle. This twinkle also appears for larger bodies (eg. the moon) but is much less noticable. (c) The sound from the seashells is actually produced by air currents that pass by the shell and excite the shell's air volume at its natural resonances. Since these air currents are not constant, the resonances will be excited sporadically giving the impression of ocean waves. (d) When lightning strikes a tree, the current in the lighning stroke needs a path to ground. If the bark of the tree is wet, then the current will pass over the outside of the tree leaving it undamaged. If, on the other hand, the tree isn't wet then the current will pass into the tree and descend through the sap. When this happens the sap is superheated and expands, which ends up turning the tree into toothpicks. The Oak tree has especially rough bark which makes it dicult for the current to nd a path to ground along the bark. It is therefore not that Oak trees get struck more by lightning, just that they are more likely to be destroyed when hit.
1994-1995 Physics Olympiad Preparation Program { University of Toronto { Problem Set 5: Electricity and Magnetism Due March
6th , 1995
1. Okay, all you mental gymnasts { it's time for the parallel conducting bar problem, an Olympian event designed to exercise your upper bodies. Consider two long parallel conductors, each carrying currents in the same direction, as shown in Figure 1. Conductor A carries a current of 100 A and is held rmly in position. Conductor B carries 147 A and is allowed to slide freely up and down (parallel to A) between a set of non-conducting guides. If the equilibrium distance between the two conductors is 2 cm, what is the linear mass density of conductor B? 2. Pretend for a moment that you're an engineer working on the design of a resistor that is to have a temperature coecient of expansion of zero at 20C. The design is a composite of two bars having octagonal cross sections (each side of the octagon having a length d = 0:25 mm) as shown in Figure 2. The ratio of the resistivities of the two materials is 1 =2 = 3:2, and the ratio of the lengths is l1 =l2 = 2:6. The cross sectional geometry and dimensions are uniform throughout the resistor. Assuming that the temperature of the two sections remains equal, calculate 1= 2, the required ratio of the temperature coecients of resistivity of the two materials. 3. It is the year 2236, and the Federation has just banned all telecommunication optical bre in your sector for some unknown reason. The replacement con guration, consisting of two coaxial dielectrics, is depicted in Figure 3. The permittivities are 1 and 2 , respectively, the conductor has potential V0, and the outer shield is grounded. (a) Derive an expression for the capacitance per unit length for the cable. (b) Given that V0 = 1:2 kV, r1 = (4:5=36) 10,9 F/m, r2 = (3:0=36) 10,9 F/m, and r3 = 2r2 = 4r1 = 40 mm, determine the maximum electric eld intensity in each dielectric. Note that, for an in nite single dielectric coaxial cable, the capacitance per unit length is c = 2= ln(b=a) where is the permittivity of the dielectric with outer radius b, and a is the radius of the conductor (a < b). By Gauss' Law, the electric eld intensity for such a coaxial cable is given by E = q=2r, where a < r < b and q is the charge per unit length in the region enclosed by r. 4. Consider two conducting spheres, one of radius 6.0 cm and the other of radius 12 cm, where each has 3 10,8 C of charge and is positioned very far apart from the other. The spheres are then connected by a platinum wire maintained at a temperature of 312.5 K. Find: (a) the direction of motion and magnitude of the charge transferred, and (b) the nal charge on and electric potential of each sphere. 5. As an accelerator physicist, you have been asked to design a xed target machine to produce antiprotons, p, the antimatter partner of protons, p. The reaction is p + p ,! p + p + p + p, where one high energy proton hits another proton that is at rest (since it's part of the target). In addition to the original two reactant particles, a proton-antiproton pair is created. What is the minimum energy required by the incident proton for this reaction to occur? Express your answer in terms of the proton mass m and the speed of light c. Hint: consider the centre of mass reference frame. p 6. Imagine a long copper conductor with rhombic cross section having sides of length 5 mm and diagonal dimensions of 2 mm and 4 mm. The conductor is carrying a current of 10 A. Each second, what percentage of the conduction electrons must leave (to be replaced by others) a 100 mm length?
1994-1995 Physics Olympiad Preparation Program { University of Toronto { Solution Set 5: Electricity and Magnetism
1. All forces mentioned in the following are with respect to conductor B (refer to Figure 1). The magnetic force on a conductor in a uniform external magnetic eld is given by Fmag = IB~l B~ A ; where ~l is a displacement vector along the conductor B, and B~ A is the magnetic eld from conductor A. The above expression can be simpli ed to Fmag = IB lBA sin = IB lBA ; since B~ A is perpendicular to ~l ( = 90). >From the Biot-Savart Law, the magnetic eld B~ A at a distance r from a wire carrying a current IA is given by o IA ; BA = 2r
where o is the permeability of free space. The magnetic force per unit length on conductor B is therefore Fmag = o IA IB : l 2r The gravitational force per unit length on conductor B, Fg , neglecting the gravitational elds of conductor A and the non-conducting guides, is Fg = g; B l where B the linear density of conductor B and g is the acceleration due to gravity. At equilibrium, we set the forces to be equal, which gives: Fmag = Fg l l o IA IB = g B 2r o IA IB B = 2rg 10,7 N=A2 )(100 A)(147 A) = (12:566 2(0:02 m)(9:8 m=s2) , 2 = 1:5 10 kg=m = 0:15 g=cm:
2. Calculate the change in total resistance with respect to the change in temperature, Rt: R(T) = Ro (1 + tT) R(T) , Ro = R =) o t T t =) R T = Ro t : Given that t = 0 at 20C, Rt = 0: T For each of the two individual resistance regions: R1 = (R ) o1 1 T1 1 l1 ; (Ro )1 = A
where A is the cross sectional area of the octagonal bars. Therefore, R1 = 1 l1 1 : T A 1
Similarly,
R2 = 2 l2 2 : T2 A Since the two materials are in a series con guration: Rt = R1 + R2 = 0: T T1 T2 Since T1 = T2, R1 = ,R2 and 1 l1 1 = ,2 l2 2
,1 = ,0:12: =) 1 = , 2 ll2 = (3:2)(2:6) 2 1 1 3. (a) The capacitances per unit length due to each dielectric are given by: c1 = ln(r2=r1 ) 2 1 and
c2 = ln(r2=r2 ) : 3 2 The total capacitance per unit length is calculated by treating the above two capacitances in series. The total capacitance per unit length is therefore given by: c = c c1+c2c ; 1 2 1 2 : c = ln(r =r2 2 2 1) + 1 ln(r3=r2)
(b) 4:5 10,9 F=m c1 = 2 36 ln 2 =: 0:36 nF=m;
c2 =
2 363 10,9 F=m : = 0:24 nF=m; ln 2 V2 = c1 = 0:36 = 1:5; V1 c2 0:24 V1 + V2 = 1200 V:
Solving for V1 gives V1 = 480 V. Therefore q = c1V1 = 172:8 nC=m: The maximum electric eld intensity occurs at the inner surface of either dielectric. Hence, at r = r1 :
10,9 C=m : Emax (r1) = 2(0:010172:8 m)(4:5)(10,9=36 F=m) = 69:1 kV=m: Similarly for r = r2 :
10,9 C=m : Emax (r2) = 2(0:020172:8 m)(3:0)(10,9=36 F=m) = 51:8 kV=m: 4. First calculate the initial potentials on the sphere surfaces. For the smaller sphere: 1 q = (9 109 N m2=C2 )(3 10,8 C) = 4:5 103 V: V1 = 4 r (0:06 m) For the larger sphere:
9 m2=C2)(3 10,8 C) = 2:25 103 V: V2 = (9 10 N (0:12 m)
Since V1 > V2 , the direction of charge motion is towards the 12 cm radius ball. When the conducting wire is connected, the spheres seek to assume equal potentials. Therefore, 1 q1 = 1 q2 ; 4 r1 4 r2 q1 = q2 ; r1 r2 q1 = q2rr1 : 2
Since q1 + q2 = 6 10,8 C, q2
r1 + 1 = 6 10,8C; r2
and the nal charge on the larger sphere becomes q2 = 4 10,8 C; whereas the nal charge on the smaller sphere becomes q1 = 6 10,8 C , 4 10,8 C = 2 10,8 C: The magnitude of transferred charge is therefore 1 10,8 C. The nal potential of each sphere is given by: 1 q = (9 109 N m2=C2)(2 10,8 C) = 3 103 V Vsmaller = 4 r (0:06 m) and 9 m2=C2)(4 10,8 C) = 3 103 V: Vlarger = (9 10 N (0:12 m)
5. In the frame of the lab, the total energy is E + mc2 , where E is the energy of the incident proton and mc2 is that for the proton at rest. Let the incident proton have momentum p. The invariant mass, M, of this lab system is then given by 2 2 2 = Etotal , p : Mlab c4 c2
In the centre of mass (CM) frame, the 4 reaction products are at rest for this problem, since we are only calculating the minimum, or threshold energy, for this reaction. The total energy in this system is therefore 4mc2 and the invariant mass is ,
4mc2 2 = c4 : 2 = M 2 and E 2 = p2c2 + m2 c4, we get Since Mlab CM 2 MCM
(E + mc2 )2 , p2 = (4mc2 )2 ; c2 c2 (E + mc2 )2 , E 2 + m2 c4 = (4mc2 )2; E 2 + m2 c4 + 2Emc2 , E 2 + m2 c4 = 16m2 c4 ; 2Emc2 = 14m2c4 ; and E = 7mc2 : 6. First calculate the number of electrons per unit volume. Take Avogadro's number to be N = 6:02 1026 atoms/kmol, the density of copper to be 8:96 103 kg/m3, and the atomic weight of copper to be 63.54 g/mol. Assuming one conduction electron per atom, the number of electrons per unit volume is Ne =
6:02 1026
atoms 1 kmol 8:96 103 kg 1 electron kmol 63:54 kg m3 atom
Ne = 8:49 1028 electrons=m3: The rhombic cross sectional area is 21 (2 mm)(4 mm) = 4 10,6 m2. The number of electrons in a 100 mm length is N = (4 10,6 m2)(0:100 m)(8:49 1028 electrons=m3) = 3:40 1022 electrons: A 10 A current requires that (10 C=s)(1:6 10,19 C=electron),1 = 6:25 1019 electrons=s pass a xed point. The percentage leaving the 100 mm length per second is 6:25 1019 (100) = 0:184%: 3:40 1022
1994-1995 Physics Olympiad Preparation Program { University of Toronto { Problem Set 6: Electronics Due April 3rd , 1995
1. You just gotta love electronics. All of those little resistors and capacitors to play with. And just when you think there can be nothing as exciting as that, out come those digital circuits with all of the chips that can be used. It's even better than having... sorry about that, just got rambling...on to the business at hand. This problem concerns itself with a circuit called a tank circuit (see gure #1). I have no idea why it is called that, but whatta name. (a) Find the absolute value of the impedance of the tank circuit as a function of frequency. The impedance of this circuit is called the transfer function because it describes how signals are transferred across the circuit. At what frequency is the impedance of the circuit in nite? To do this pyou will need the resistance of both a capacitor Zc = 1=i!C and an inductor Zl = i!L, where i = ,1. The absolute value of a complex number is de ned as p Z = Z Z where Z is the complex conjugate of Z . (b) Sketch this impedance function as a function of frequency. (c) How will the transfer function change if the inductor has some resistance? Just determine the complex transfer function. 2. Often in electronics one uses lter circuits to remove unwanted frequencies in signals. Most of them work by using the frequency dependent resistance of capacitors and inductors. If a certain frequency signal is put into a circuit it will either be allowed to pass through or get blocked by the circuit depending on the frequency dependent resistance of the components. Filter circuits fall into the following categories: (a) High pass lters only allow high frequencies through them. (b) Low pass lters do the opposite. (c) Band rejection lters block an intermediate range of frequencies. (d) Not a lter at all. Classify the ve circuits given in gure #2. 3. You're out with your friends one day and you stumble across an ornately carved lamp lying on the ground. One of your friends says that it must be a magical lamp with a genie inside. You all tease him mercilessly until, sobbing uncontrollably, he rubs the lamp and out comes a genie. The ethereal entity speaks, and I quote:"For teasing my master so ruthlessly, I will ask you one question which you must get right to live. What is the resistance between opposite vertices of an octahedron, if each of the twelve edges has the same resistance." You begin to sweat and really wish you hadn't skipped your electronics class all last term...... 4. "Could the Romulin Warbird be hiding in that interstellar gas?" Captain Picard said from the bridge of the Enterprise. "Our sensors cannot penetrate the gaseous cloud, but I think it quite likely Captain," says Data in his usual monotone. Then Wesley piped up, "The homogeneous material of the cloud has known conductivity c , whereas the Warbird has a conductivity of w . By measuring the resistance of the cloud we could determine if the Warbird is in there." He sits in his chair with a stupid smile on his face.
Picard's face turns red as he shouts, "Will you shut your insolent little face. Worf, stun Wesley and throw him in the brig. After Wesley is unconscious, the captain remarks to Data, "Make it so." (a) Data assumes that the cloud is cylindrical with radius b and length L, and that the input and output currents are uniformly distributed across the ends of the cloud. Show that Data computes the resistance to be L : R= b2 c
(b) Data decides to model the Romulin Warbird as a metallic cylinder of radius a, length a and conductivity w in the centre of the cloud. Estimate the relative change in resistance of the cloud to rst order in c , w if c ' w . Assume that b >> a. Model the various parts of the cloud as resistors in series or parallel with each other. 5. This problem concerns itself with the recti cation of AC voltage. The circuit shown in gure #3 is called a full wave recti er and is used to turn an AC signal into a DC signal. The funny looking circuit elements are called diodes and only allow current to pass in the direction of the arrow. (a) If the voltage source produces a sinusoidal signal, what signal is expected across the resistor? What will be seen at the resistor for a square wave signal as input? Give reasons for your answer and sketch the output and input waveforms. Ignore the capacitor for this part. (b) If I add a large capacitor in parallel with the resistor, how will it a ect the voltage signal at the resistor? To be more speci c, why must the capacitor be large to produce an almost DC signal at the resistor? 6. This just wouldn't be an electronics problem set without at least one question on digital circuits. See gure #4 for a type of circuit that shows up when designing computers and such. (a) Find the truth table of the circuit. What is this circuit actually doing and how could it be useful? (b) Using the circuit from (a), design a full adder that can add three one digit binary numbers. Give the truth table for your circuit.
1995-1996 Physics Olympiad Preparation Program { University of Toronto {
Problem Set 1: General Physics Due October 16, 1995
1. Joe \build-em tall" Girders is designing a construction crane (did you know he learned his stu working with Tinker Toys?). He uses two beams with mass m1 = 9000 kg and m2 = 3000 kg with lengths l1 = 45 m and l2 = 15 m, attached to the vertical \trunk" of the crane and supported by cables, as shown in Fig 1. (a) How much mass, mc , should there be in the counterweight located at the end of the second beam to ensure that the crane is perfectly balanced when the crane is not carrying a load. (b) The cable support has a height h = 15 m. What are the tensions on the two cables supporting the beams? 2. In the original movie Batman, our masked avenger has a cool get-away planned when he and Vicky Vale, the hot-shot reporter, are trying to escape the Joker after the debacle in the museum (check out the movie if you folks forget this scene). He res up one of his awesome cables and attaches it to a girder above the street. With his super-cool winch located on his belt, he begins to lift the two of them out of harm's way. But wait! He only gets part way up. His winch appears to run out of juice. As he and Vicky twist in the wind, he ponders his potentially fatal mistake. (a) What is the minimum energy required to lift the two of them up 15 m, if we assume Batman tips the scale at a t 75 kg and Vicky really does weigh 47 kg (as she claims)? (b) They have to be rising at least 1 m/s in order to avoid getting plastered by Joker's henchmen. What is the power required to execute this daring move? (c) In the movie, after Batman hooks the winch to Vicky and lets go, she is winched up the rest of the way to the girder. Can you give a plausible explanation as to what really went wrong with the winch? Did it really run out of energy? 3. Sven is tooling down Highway 11/17 with a tractor-trailer load of pulp wood to be fed into the maw of the MacBlo mill in Fort Frances. He is doing 120 km/h, easy. He comes around a corner and nds { what else? { the icon of the northern Ontario bush, a moose, standing in the middle of the road. The old pro that he is, Sven hits the brakes. He is about 150 m from the moose when his size 13 work boot hammers the brakes. (a) How heavily does he have to brake to avoid bagging a moose out of season (and making a mess of his front end)? (b) He is still in the curve when he starts braking, so his trailer lled with cord wood makes a small angle with the tractor unit. He realises as he hits the brakes that he forgot to connect his trailer brakes, so that all braking is performed by the tractor unit. It has been snowing (what else can go wrong?) so the coecient of static friction between his wheels and the asphalt is s = 0:2. What is the maximum angle his trailer unit can make with the tractor unit before it starts to jacknife (the truly awesome maneuver when the trailer starts sliding and swings around)? 4. Rob Hanks was a wild and crazy man. By day he worked as an electrician for General Motors, but by night he assumed a secret identity, one that no soul knew a thing about { he became a physics problem solver. Now Rob often works with AWG #12 copper wire with an 80.8 mil diameter. He already knows that a 50 ft length can carry a current of 20 A, and that copper has a conductivity of = 5:8 107 ,1 m,1 and an electron mobility of = 0:0032 m2V,1s,1 at 20 C. Can you help Rob calculate the electric eld intensity E, the voltage drop and resistance across the 50 ft length, and the time it would take an electron to move a distance of 1 cm in the conductor? 1
h
mc l2
l1
Construction Crane
2
1995-1996 Physics Olympiad Preparation Program { University of Toronto {
Solution Set 1: General Physics
1. (a) We require that there is no net torque about the axis through the point where the two beams connect to the vertical support (recall that torque is ~ = ~r F~ where ~r is the vector separating the axis about which you are calculating the torque and the point where the force is applied, and F~ is the force). The torque due to gravity on beam 1 is centred through the midpoint of beam 1. Ditto for beam 2. The torque due to the counterweight is the cross product of the gravitational force and the vector between the axis and the end of beam 2. The force diagram for the crane is shown in Fig. 1a). Thus the net torque is X = ,m1 g l21 + m2 g l22 + mc gl2 = 0 (1) (2) ) mc = 12 m1 ll1 , m2 2 = 12; 000 kg; where g is the acceleration of gravity (g = 9:8 m/s2). Note that we can verify that the equation for mc makes sense. For example, if m1 = m2 and l1 = l2 , then we would expect mc = 0, which is the case. Can you think of other sanity checks? What about units? Note that we do not have to take into account the torque due to the support cable tension on each beam. Can you understand why not? (b) We require that the net torque on each arm to be equal. In this case, the tensions on the cable supporting the two beams are forces, T1 and T2 that act on the ends of the beams and on the vertical support. Let 1 be the opening angle between beam 1 and the cable (as shown in Fig. 1b)). Then the torque acting on beam 1 is X = 0 = l1 T1 sin 1 , l21 m1 g (3) 1g (4) ) T1 = 2 msin p1
2 2 = m1 g 2hl1 + h (5) = 139; 000 N; where h is the height of the vertical support above the beams. Similarily, for beam 2, the requirement that the net torque on the beam is zero gives X = 0 = ,T2 sin 2 + m2 g l22 + mc g (6) p 2 2 ) T2 = m22 g + mc g l2h+ h (7) = 187; 000 N: Its interesting that the tension is highest on the cable supporting the counterweight. Them cables are pretty strong. They have to be able to support almost ten tons, and that's without any load being carried. 2. (a) To lift Batman and Ms. Vale up h = 15 m, one has to at least expend the work equivalent to raising them that distance in a gravitational eld (it could be more{think about the work required
1
a)
b) Cable Tension
mcg
θ1
m1g m2g
m1g
Beam 1
Figure 1: The forces acting on the two beams around the point at shich they are connected to the vertical support are shown in a). Note that we do not consider the the force of tension on the two beams from the support cables. (Why not?) The forces acting on the end of beam 1 due to the cable and the weight of gravity are shown in b). to accelerate them into motion). This is equal to W = h (mB + mV ) g = 17; 931 Joules; (8) where g is our faithful acceleration due to gravity at Earth's surface. (b) Let's assume they accelerate to vh = 1 m/s and then rise steadily the rest of the way. The power expended (ignorning the power required for acceleration) is just the force velocity, which is P = (mB + mV ) g (9) = 1; 196 Watts = 1:6 hp: (10) Recall that 1 horsepower is equal to 746 Watts. That's a pretty hefty little motor he has there. (c) Let's consider why the winch stopped in the rst place. There are only a few likely scenarios (given that it wasn't really broken): i. motor overheated or otherwise failed, ii. the energy source couldn't continue to supply the needed power, or iii. the winch is based on some funky technology that I know nothing about that gives out at exactly the right moment in the script. The fact that it continued to operate and yank Vicky up when Batman let go and the tension dropped indicates that i) is unlikely; the drop in tension itself would not have been correlated with the moment when the motor became operative. Scenario ii) is plausible, as the power level required to haul Vicky up would be less than that required to haul both of them up. Of course, I would go for iii). Hollywood rarely cares about conforming to reality. 3. (a) We can safely assume that Sven and the pulpwood are going to decelerate uniformly, with deceleration a. This is an approximation, as brakes usually fade as they heat up, but we have to start from somewhere, and it isn't such a bad assumption. We can then employ our usual equations for an object of mass m decelerating uniformly over a distance d. If v0 = 120 km/h is Sven's initial speed, then the time required to decelerate is t = vo =a. Thus, 2 v2 o d = 21 a vao = 2a (11) v02 = 5:8 m=s2; ) a = 2d (12) 2
Fb θ Tractor Unit Trailer
Figure 2: The braking force acting on the trailer. which is over half a g. That's real heavy braking. I'm guessing Sven ended up with the meat. (b) The force diagram on the trailer is shown in Fig. 2. The braking force Fb is shown, along with the component of that force perpendicular to the axis of the trailer, Fp = Fb sin , where is the angle between the axis of the trailer and the braking force. There is one other force in the problem; the reaction force of the pavement acting on the trailer through its tires to keep the angle constant (note that it is not shown in Fig. 2). This reaction force would normally be equal to Fp , so that there is no increase in angular momentum of the trailer (note that the trailer is swinging in uniform motion as it turns the corner). The maximum frictional force is Ff = s mw g;
(13)
where mw is the weight of the trailer resting on the back axle, which we approximate as half the total trailer weight mt . Since the braking force is Fb = mt a, the maximum angle between the trailer and tractor unit is given by the requirement Ff = Fp s mt g ) 2 = mt a sin ) = sin,1 2as g = 9:7:
Not a big angle in this case. Note that it matters not a whit what the trailer weighs. 1 inch, the cross-sectional area of the conductor is: 4. Since a mil is 1000 A=
0:0808 in 2:54 10,2 m 2 = 3:31 10,6 m2: 2 1 in
The current density therefore is: J = AI = 3:3120 A10,6 = 6:04 106 A m,2: The electric eld intensity is: 6:04 106 A m,2 = 1:04 10,1 V m,1: E = J = 5:8 107 ,1 m,1 The voltage drop is given by: V = E` = (1:04 10,1 V m,1)(50 ft)(12 in ft,1)(0:0254 m in,1) = 1:59 V: 3
(14) (15) (16)
The resistance is given by:
V = 7:95 10,2 : R = VI = 1:59 20 A Since = , where is the charge density: 5:8 107 ,1 m,1 = 1:81 1010 C m,3: = = 0:0032 m2 V,1 s,1 Since J = U, where U is the drift velocity: 6:04 106 A m,2 = 3:34 10,4 m s,1 : U = J = 1:81 1010 C m,3 With the above drift velocity, an electron will require approximately 30 seconds to move a distance of 1 cm in the #12 copper conductor. 5. The image distance, v, can be calculated from the lens makers' equation: 1 + 1 = 1; v u f which yields a distance of: v = uuf ,f; where f is the focal length and u is the object distance. The Rayleigh criterion, = 1:D22 , where is the limiting angle of resolution, is the wavelength, and D is the diameter of the aperture, can be used to calculate the blur diameter: d = v = 1:22v D : The defocus blur is calculated by similar triangles as: d = (v ,ff)D : Equating the di raction blur and the defocussing blur gives: s p D = 1:22vf 1:22u: = (v , f) For = 550 nm and u = 15 m, we end up with D = 3:2 mm. 6. (a) The digital logic function is an OR expression encompassing the combinations that lead to a correct process, namely: X = A C S + A C S + C: The circuit is sketched in Fig. 3a). (b) Operating on the second term of the above expression with one of DeMorgan's laws of Boolean gives: algebra (A B = A + B) X = A C S + A + C + S + C: Applying the redundancy law of Boolean algebra (A + A B = A) yields: X = (C + A C S) + A + C + S = C + A + C + S: The simpli ed circuit is sketched in Fig. 3b). 4
a) A C S
b) A•C•S
A C S
x
A+C+S x
A•C•S
Figure 3: The circuit to handle the logic of the starship Compromise transporter control logic is shown in a). A simpli ed circuit is shown in b).
5
1995-1996 Physics Olympiad Preparation Program { University of Toronto { Problem Set 2: Mechanics Due November 13th , 1995
2
Figure 1: Figure for Problem 1.
3
Figure 2: Figure for Problem 2.
4
Figure 3: Figure for Problem 3.
5
1995-1996 Physics Olympiad Preparation Program { University of Toronto {
Solution Set 2: Mechanics
1. (a) Assume that the bungee cord obeys Hooke's law, F = Y l A l F = ( Y lA )l = kl; where A is the cross section of the elastic cord, Y is Young's Modulus, l is the length of the cord, l is the change in the length of the cord, and k is the elastic constant of the cord. Since Y and A are constants, kl is constant, i.e. kl = k0l0 . At h = 0 the velocity of the jumper has to be zero, i.e. v = 0, where h is the vertical distance of the jumper from the ground. Thus, using energy conservation mgh = 12 k0 (h , l0)2 = 12 k(l=l0 )(h , l0 )2 where h is the initial vertical distance. This can be rearranged as a quadratic equation in l0 . 0 2 l0 2 , 2h 1 + mg kl l + h = 0: Solving for l0 then, l 0 = h
r 2 ! mg mg 1 + kl 1 + kl , 1
Using g = 9:8 m/s2 , m = 55 kg, k = 77 kg/s2, h = 60 m, and l = 30 m, one gets the only physical solution, l0 = 21:4 m. (Note: In order for l0 to be physical, it must be 0 < l0 30 m.) (b) The force exerted on the agpole at h = 0 is: F = mg + k0 (h , l0 ) = mg + k(l=l0 )(h , l0 ): The torque exerted on the agpole around the pivot point P is = FL0 cos : But from Figure 1, it is obvious that L0 cos = L cos . Thus = FL cos : The maximum torque the bolt has to withstand is = bT; 1
L
F
θ L’
T
b
φ P
Figure 1: Figure for problem 1 where b is the distance between P and the bolt and T is the tensile strength times the cross section of the bolt. Thus equating the above two equations, T = FL cos =b = (55 9:8 + 77 (30=21:4) (60 , 21:4)) L=0:5 m kg/s2 = 1:16 104 L m kg/s: 2. (a) When rolling without slipping, the bottom point P of the yoyo has zero instantaneous velocity. Rotation around a xed axis passing throughP is determined by the torque about that axis, but only F has non vanishing torque. Thus the rotation is clockwise (Figure 2(a)) and the centre of mass moves to the direction of the pull. (b) In order for the yoyo to stand still, the net torque of the system should be zero about the contact point P (Figure 2 (b)). Thus mgb sin = F(b , a). Then F = (b ,b a) mg sin : When the yoyo is about to slip, the sum of the torques around the pivot point P 0, tot is tot = 0 = 1 + 2 = mga sin , mg(b , a) cos ; where 1 is the torque due to the centre of mass of the yoyo being pulled down by the gravitational force and 2 is the torque due to the friction between the yoyo and the inclined surface. 2
F P (a)
b
a
F P’ P
θ (b)
Figure 2: Figure for problem 2
3
Thus
sin = (b ,a a) cos a = (b , a) tan :
(Note: The moment of inertia of a cylinder whose rotational axis is perpendicular to the at surface of the cylinder and lies a distance a away from the centre is IR = IM + Ma2 . There was a typo in the problem set. Not that it would have made any di erence in solving this problem...) 3. (a) From energy conservation the energy at h = (R + a) cos is 2 mg(R + a)(1 , cos ) = 12 mv2 + 12 I!sph = 12 mv2 + 15 mv2 7 mv2 = 10
where we used Isph = (2=5)ma2 and v = a!sph . (Note: The trajectory of the centre of mass of the marble rolled to an angle from the top of the bowl is (a + R) (Figure 3 (a)). And R = a . Combining these equations, one gets (a + R) = a + R(a=R) = a( + ) = asph ; which means v = (a + R)! = a(! + ! ) = a!sph .) For a marble about to take o from the surface of the bowl the force perpendicular to the surface of the bowl has the same magnitude as the centripetal force on the marble. Thus, mg cos = mv2 =(a + R): From the above two equations (1 , cos ) = (7=10) cos Solving for cos gives cos = 10=17. Thus the marble leaves the surface of the bowl at the height of hc = R cos = (10=17)R (or hc = (10=17)(R + a), depending on what your de nition of hc is.) (b) For a \rolling pendulum" (due to the lack of a better name) the potential energy has to be distributed between the rotational motion of the marble and the translational motion, whereas the \simple pendulum" potential energy is converted only to the translational motion. Therefore, the simple pendulum has a higher translational speed than the rolling pendulum at the same angle, i.e. it takes more time for the rolling pendulum to complete an oscillation than the the simple pendulum. Brownie points: 4
a
θ
φ
mg mg to the centre of the bowl
R (a)
h (b)
Figure 3: Figure for problem 3 5
For a simple pendulum, the energy conservation equation at h looks like mg(h , h) = 21 mvs 2 . Then p vs = 2g(h , h) , where h is the release height of the pendulum. For a rolling pendulum, mg(h , h) = 12 mvr2 + 21 I!r2 7 mv2 = 10 r p
vr = (10=7)g(h , h)
. p p Thus vr =vs = 5=7 for any given height. Which means Ts =Tr = vr =vs = 5=7 4. (a) The surface of the liquid is perpendicular to the direction of the sum of the forces exerted, centripetal force and gravitational force (Figure 4). Thus
i.e.
2 tan = v g=r
. The liquid touches the rim of the the cup when tan sati es the following equation: tan = d2 (H , h ) . Using the above two equations, one can obtain r
2gr (H , h ) r d 30 (0:15 , 0:12) = 2 9:8 0:08 = 14:8 m/s: (b) When the inside wheels of the car are about to leave the ground (provided that the car does not slip): W F = hF l 2 d where W = the track width of the car (or the width of the car for our purpose); Fd = downward force; h = the height of the centre of mass of the car; and Fl = lateral force: v =
6
θ Fl θ
H ho mg
d
Figure 4: Figure for problem 4 The downward force is the sum of the gravitational force and the force generated by the aerodynamic device, i.e. Fd = mg+cv. Here c has units of N/(km/hr). Converting it into MKS units, it becomes N/(km/hr) = N/(1000 m/3600 s) = 3:6 N/(m/s). The lateral force is centripetal force, 2 i.e. Fl = mv =r. Rearrange it as a quadratic equation in v: Solving for v
c rW v2 , rW 2h m v , 2h g = 0: p
2 + 4ag v = ac=m + (ac=m) 2
, where a = rW=2h. Using r = 30 m, W = 2 m, h = 0:4 m, m = 1500 kg, and c = 150 3:6 N/(m/s) = 540 N/(m/s), one gets vr = 43:8 m/s: However, if (mg + cv) = mv2 =r, the car is about to slide. Rearrange it as a quadratic equation in v: v2 , rc m v , rg = 0: Solving for v p
Using = 0:23 one gets
2 + 4bg v = bc=m + (bc=m) : 2
7
vs = 9:6 m/s: Since vr > vs , the car will slide before it lifts its inside wheels. Thus the maximum speed you can go through the turn is at 9:6 m/s. 5. (a) For a satellite on a circular orbit of radius r, mv2 = GMm r r2 and p v = GM=r: The velocity of the satellite at the parking orbit (h = 200 km) is then vp = 7:772 103 m/s, where G = 6:673 10,11m3kg,1 s,2 and M = 5:974 1024kg are used. The angular velocity of the satellite on the geosynchronous orbit is ! = 2=(24 3600 s) = 7:272 10,5 s,1: The radius of the geosynchronous orbit, rg , can be obtained from the rst equation using v = r!, rg
1=3 GM = !2 = 4:224 107 m:
Thus the velocity of the satellite on the the geosynchronous orbit is vg = rg ! = 3:072 103 m/s. From energy conservation 1 mv , GMm = 1 mv , GMm ; 2 pt rp 2 gt rg where vpt = The velocity at the perigee of the transfer orbit; vgt = The velocity at the apogee of the transfer orbit; rp = The distance of the perigee from the centre of the earth = The radius of the parking orbit; and ra = The distance of the apogee from the centre of the earth = rg : From angular momentum conservation ~rp ~vpt = ~ra ~vat
, and at the perigee and the apogee, ~r and ~v are perpendicular to each other, so rp vpt = ra vat 8
or vat = (rp =ra )vpt : Thus the energy conservation equation now looks like 2!
1 mv2 1 , rp 2 pt ra giving vpt
rp = GMm ra 1 , ra
1=2 2GM = r (1 + r =r ) a p a = 1:022 105 m/s:
So vpt , vp = 2:248 103 m/s. The velocity at the apogee of the transfer orbit is then
rp v ra pt = 1:597 103 m/s: Thus the velocity change the satellite has to achieve in order to get into the geosynchronous orbit from the transfer orbit at the perigee of the transfer orbit is vg , vat = 1:475 103 m/s. (b) The transition takes half the time for the satellite to complete the elliptical orbit. So using the Kepler's law vat =
T1=2 = T2 2 3 1=2 4 ((r 1 p + ra )=2) = 2 GM
= 1:900 104 sec ' 5 hrs 17 min It takes 5 hours and 17 minutes to complete the transition. 6. (a) Of course not, silly. The possible physical e ects are i. Coriolis force. 2m~! ~v ii. Centripetal force. mv2 =r. iii. Decrease of gravitational force due to thepincrease in r. iv. Relativistic e ect (time dilation). t = = 1 , v2 =c2 , etc. International ight normally ies at an altitude of about 10 km. Thus the change in the gravia2 ,3 tional acceleration at an altitude p p h is g = g(1 p, (R=(R + h)) ) = 3:1 10 g. The change in , 4 T is T ' l=(g , g) , l=g ' 1:6 10 l=g, where l is the length of the pendulum. But it a ects the same way regardless of the direction of the ight. The speed of the east bound plane that Chris is on from the xed reference framepis v = =12 36006:4106 ' 460 m/s. Relativistic e ects are then on the order of T T(1= 1 , (v=c)2 , 1) 10,11T 10,7 sec: 9
(b) If we use the inertial system with the origin of the coordinates sitting at the centre of the earth as the reference system, the Coriolis force vanishes in this system. The e ective gravitational acceleration at the earth's surface for an object with angular velocity ~! against an inertial system is ^ ~ ~geff = , GM R2 R , ~! (~! R): Since Pat and Chris are moving along the equator, their angular momentum vectors ~! are per~ Thus j~! (~! R~ )j = j!2 Rj. Then the e ective gravitational acceleration looks pendicular to R. like
, and if we replace
2 geff = GM R2 , ! R GM R2
with g
geff = g , !2 R: The period of a simple pendulum with the length of the pendulum L is s
T = 2 Lg : Since Pat is moving westward, the angular velocity is zero. Thus s
TP at = 2 gL s
= 2 g +L!2 R s s = 2 Lg 1 + !12 R=g s = T 1 + !12 R=g : Whereas, Chris is moving eastward, so the angular velocity of Chris is 2!. Thus s
TChris = 2 g ,L4!2 R s = 2 g , L3!2R s s = 2 Lg 1 , 3!1 2 R=g s = T 1 , 3!1 2R=g : 10
TChris , TP at = T
s
s
!
2 1 1 2! R , = T 2 2 1 , 3! R=g 1 + ! R=g g
Using g = g , !2 = 9:8 m/s2, R = 6:4 103 km, T = 12 hrs, and ! = 2=T one gets TChris , TP at ' 298 seconds: Since TChris , TP at > 0, Pat's clock goes faster than Chris' clock. (Pat's clock would be 75 seconds faster and Chris' clock would be 224 seconds slower than the local clock at Y.)
11
1995-1996 Physics Olympiad Preparation Program { University of Toronto {
Problem Set 3: Thermodynamics Due December 18th , 1996
p
n
1
You know when water turns into ice, the volume it occupies increases. Now here is a heat engine to exploit that. Figure ?? shows the sequence of how the engine works. (a) Initial state: water is con ned in a cylinder with a piston and under 1 atm. (b) A weight is placed on the piston. (c) The cylinder is cooled and the water freezes. The expansion on freezing raises the weight, doing work. (d) The weight is moved laterally. No work is done in moving the weight laterally. (e) The temperature is returned to the initial value and the ice melts. The state of the system is now identical to the initial state. Since ice melts and water freezes at 0 C, which implies that this heat engine can operate at a single temperature, which in turn implies that this engine can run forever. Now you know that there is no such thing as \perpetual motion machine". What is wrong with this picture? Uncle Moe won't take answers like \yeah, you cannot violate the second law of thermodynamics. that's why." 5. You are a member of a team developing an autonomous volcano exploring vehicle. You are in charge of the cooling system design. The ambient temperature of the volcano is 530C. The electronics bay of the vehicle is insulated and should be maintained at 30 C. The electronics in the bay produces heat at a rate of 720 W-h and the the heat ow rate into the bay through insulation is 0 002 ( , ) W. :
Tout
Tin
(a) What's the maximum coeciency of performance of the cooling system? (b) What's the minimum power requirement for the cooling system? (c) The coecient of performance of the cooling system onboard the vehicle is rated at 0.2 at the conditions above. The typical mission lasts about 100 hours. What is the minimum energy requirement for the dedicated power source of the cooling system? 6. You work for CSA (Canadian Space Agency) and your team is involved with the Mercury Observer, a long waited Mercury probe spacecraft, project. Your supervisor asks you to do the following preliminary calculations. By the way, the funding for the project was approved on the condition that the spacecraft be painted black (bureaucracy, you see). Note: The average radius of the orbit of mercury is 5 79 106 km. The radius of the sun is 6 96 105 km and the temperature on the surface is 6000 K. :
:
(a) What would be the maximum radiation pressure that the spacecraft experiences from the sun's radiation when the spacecraft is on the mercury orbit. (b) What would be the surface temperature of the craft a long time after it arrives at the mercury orbit? The spacecraft is spherical in shape, has a 3 m diameter, and its cooling system expels 4 W from the interior to the exterior of the spacecraft.
2
Figure 1: Figure for Problem 4.
3
1995-1996 Physics Olympiad Preparation Program { University of Toronto { Problem Set 4: Wave and Optics Due January
22th , 1996
1. As a physics-illiterate rock concert organizer on Lilliput island (remember Gulliver's Travels), Donald Armen is still puzzled by the complaints from the rock fans. In fact, one of the two speakers was destroyed. He just cannot believe it. He bought a new speaker to increase the loudness of this mono sound system to the one he had. He placed them 8.00 m apart. The closest audience member, Christie, was 10.0 m away from each speaker, respectively (or 9.17 m from the midst of the both). Can you help him understand what was going on? Assume the average frequency is 690 Hz (being tiny, these Lilliputs have high pitch). The speed of sound is 344 m/s. (a) Could Christie hear the concert? (b) Albion, standing on Christie's left, was among the ones who got angry and destroyed the speaker. He just barely listen to the concert. Where did he stand? (c) This ction is far from reality. Explain! 2. A group of police cars moving at 80 km/hr are chasing O'Samson's white Bronco, which is travelling at the same speed. They are afraid that this former famous hockey player will commit suicide. It seems that 3000-Hz sirens from the police car do not bother him much. (a) What is the wavelengh of the sirens in the air between O'Samson and his chasers? (b) What frequency does he actually hear? The speed of sound is 344 m/s. 3. Learning from Donald Armen's story (Problem 1), you decide to study more about loudspeakers. You place two small loudspeakers facing each other at the two ends of a 5.0m-bench. The rst loudspeaker emits a sound of pressure amplitude twice as much as the second one. Both sounds are at 1334 Hz and are in phase at the speakers. (a) At what point on the line joining the two speakers is the intensity a minimum? (b) What is the intensity compared to one from the second speaker alone? (c) At what other points on the line joining the speakers will destructive interference occur? (d) At what point on the line joining the speakers is the intensity a maximum? (e) Express the maximum intensity terms of the intensity from the second speaker! 4. Your professor told you that di raction gratings are better than prisms for atomic and molecular spectroscopy. You don't believe it; you are sceptical .... So you perform experiments using both and you nally agree with him. Why is your professor right? To support your arguments compare your calculations for both prism and di raction grating for a source light which emits 434 nm (blue), 486 (green blue), 589 nm (yellow) and 656 nm (red). Take the indices of refraction of heavy int prisms n = 1.675, 1.664, 1.650, and 1.644, which correspond to the wavelengths above, respectively. The angle of the prism is A = 60o and the di raction grating has 8000 lines per centimeter. 5. Kids love to play with soap bubbles and are always fascinated by them. Ok, ... you are not kids anymore but you need to guess what colour a bubble of 400 nm thickness appears to be. Need some hints? Alright. (a) What colours do not appear in the re ected light?
(b) For what colours does constructive interference occur? (c) So the colour of the soap bubble is predominantly .... The index of refraction of the soap lm is 1.36. 6. Ok, let us have a break now with a few small questions. (a) Most large astronomical telescopes in use today are re ectors rather than refractors. Why is this so? (b) You are able to nd a very sharp focal length of a crystal ball. Right or wrong? Explain.
1995-1996 Physics Olympiad Preparation Program { University of Toronto { Problem Set 5: Electricity and Magnetism Due February
19th , 1996
(b) Instead of an electron, you can also use a muon which is heavier by a factor of 207. This heaviness makes the distance between the protons smaller by a factor of 207, too. What is the total kinetic energy of the system now? Of course, the story does not end here because the two protons will transform into a deuteron, and emit a positron and a neutrino. NOTE: This minimal distance between the protons is determined by the Heisenberg uncertainty principle of quantum mechanics. 6. Here are a few questions that do not require any calculations. (a) You have been told that wave requires a medium to propagate like sound requires air. However, light, which is electromagnetic wave, can travel in vacuum. How do you explain this phenomenon? (b) Electromagnetic wave, like other waves, can transmit energy from one place to another, say E . How much momentum does it transmit if it can? (c) Why do we all have an AC-, instead of DC-, power in the houses and buildings?
1995-1996 Physics Olympiad Preparation Program { University of Toronto { Problem Set 6: Electronics Due March 18th, 1996
1. The circuit in Figure 1 is one that you can't easily escape from. Whether you're taking in tunage from O spring, Metallica, or a Bach fugue for harpsichord, this little gem will most likely come into play (oops, no pun intended). It's a { wait for it { noninverting operational ampli er circuit. (a) Find an exact expression for the voltage gain ratio, Av = vv20 , in terms of R1, R2, Rd (the e ective resistance between the \+" and \-" terminals), and AOL vvd0 (the open loop voltage gain). (b) Assume that iin is zero, so that vd 0 and v1 v2 . Derive an expression for the voltage gain ratio for this simpli ed case. 2. Gill Bates is the billionaire founder of Sicromoft Corp., the creator of ScreenDoors 96 (a soon-to-be popular operating system). It is a little known fact that Gill spent her teen-age summer months designing power supplies for a small Ontario electronics rm. One year, she dreamed up the 5=2 kHz circuit shown in Figure 2. Her boss wanted it put on the market immediately, and asked her to write a speci cation sheet for it, including the magnitude and phase of (a) the open circuit voltage across the terminals AB and (b) the driving point impedance at AB (with the internal voltage source set to zero). Solve for these quantities and you too may be a billionaire someday. 3. Let me tell you about a nightmare I once had. It was a dark and stormy night and I had somehow become trapped in a dripping dungeon guarded by a band of simian subterranean gypsies. Their leader, Murd, handed me a circuit breadboard and a baggy containing half a dozen NOR gates. He told me that if I could realize the logic function f = A B + A B using the devices provided, then I would be free to leave his dungeon. How did I do it in my dream, using all the given gates? Of course, I then awoke in a cold sweat and realized how stupid I (and Murd's band for that matter) had been. Why? 4. Aren't CDs just too cool, with their glittering colours and cute size? In the early 80s they were heralded as the \perfect sound" medium, and thanks to the power of advertising many folks still think this to be true. The shortcomings of the CD format, which are rooted in the fact that the standard was established way back in the late 70s, could form the basis of many POPTOR problems. CDs are here to stay for a while, anyway, so let's be positive and examine the medium a little. First of all, the data words on a CD are 16 bits long. The digital sampling rate is 44.1 kHz, and the nominal frequency limits of human hearing are 20 Hz to 20 kHz. (a) How many di erent levels of sound pressure can be represented with this medium? (b) Discuss why the digital sampling rate was chosen to be 44.1 kHz. (c) What is the maximum possible dynamic range (in decibels)? (d) What is the data density (in bytes/m2) on a CD if the angular speed is 500 rpm when the laser pickup is at the innermost point and 200 rpm when it's at the outermost point. Assume the outer diameter to be 120 mm, that a CD plays for 74 minutes, and that there's no data redundancy or error-checking information. 5. The con guration in Figure 3 contains two AND gates and two NOR gates. The input marked CLK is connected to a square wave generator that oscillates between the \1" and \0" logic states. (a) Give a truth table for this logic circuit. (b) How do you think this con guration could be useful? 1
+
V +d
V2
+
+
R2
iin +
Vo i
V1 -
R1
-
-
Figure 1: The noninverting op-amp circuit in Problem 1.
2
21
50
5 kHz 2
A
12
B
30 4.8mH
Figure 2: Gill's circuit in Problem 2.
3
12mH
R
Q
CLK Q
S
Figure 3: The digital circuit in Problem 5.
4
(c) Can Figure 3 be replaced by a system of NAND gates? If so, how? If not, why not? 6. Your fairy godmother gives you a black box with four terminals on it (two inputs, A and B, and two outputs, C and D). She tells you that the resistance between terminals B and C is equal to the resistance between terminals C and D, and that there is negligible resistance between terminals B and D. She also informs you that, when a sinusoidal voltage with a frequency fo is applied across A and B, the output voltage across C and D is also sinusoidal, but has an amplitude that is 20% less than the input amplitude. If the input frequency, f , is varied either higher or lower, the output voltage amplitude is attenuated symmetrically about fo , roughly as a function of 1= jf , fo j. What does the circuit inside the black box look like and what can you say about the values of the devices inside? Note: You may assume that there's no active electronics inside { only passive devices are used.
5
1995-1996 Physics Olympiad Preparation Program { University of Toronto {
Solution Set 6: Electronics
1. (a) The feedback voltage, i.e. the voltage at the \-" terminal of the op amp, can be written as:
v1 = v0 , (i + iin )R2: Since i = v1 =R1 and iin = vd =Rd = v0 =AOL Rd: OL Rd v : v1 = 1 ,1 +R2R=A=R 0 2 1 But
vd = v1 , v2 = Av0 ; OL
or
v2 = v1 , Av0 : OL
Combining and rearranging the above expressions gives: R1 + R2 Av = vv0 = : 1 R2 , R1 +R2 R , 2 1 AROL Rd AOL (b) If iin = 0, then the currents through R2 and R1 are equivalent and: v0 , v1 = v1 ; R2 R1 and R2 : Av vv0 vv0 = 1 + R 2 1 1 2. Gill's circuit is a Bridge circuit and the problem is solved by calculating the Thevenin equivalent voltage and impedance. The impedance of a resistor and inductor in series is given by the complex expression Z = R + i!L, where R is the resistance, ! is the angular frequency, and L is the impedance. The quantity XL = !L is called the inductive reactance. The equivalent impedance at terminals AB with the source set equal to zero is:
Z0 =
,1 ,1 1 1 1 1 12 + i(5 kHz 4:8 mH) + 21 + 30 + i(5 kHz 12 mH) + 50 ;
which simpli es to:
Z 0 = 47:3 at 26:8: With the circuit open at AB , the current on the left side of the bridge is:
I1 = 21 + 12 +20i(5VkHz 4:8 mH) ; 1
and the current on the right side of the bridge is:
I2 = 50 + 30 +20i(5V kHz 12 mH) : If we assume that point A is at a higher potential than point B , then we have:
V 0 = VAB = I1 (12 + i(5 kHz 4:8 mH)) , I2 (30 + i(5 kHz 12 mH)): Substituting and simplifying gives:
V 0 = 329 mV at 170:5: 3. Just like in many other nightmares, things here didn't proceed optimally. The function, f = AB +AB , can be simpli ed to B , so no NOR gates are needed and the output can simply be wired to the input B . The question, however, insists that we design a logic circuit using NOR gates, so here goes. Since f = A B + A B , then:
f = A B + A B: Applying deMorgan's Laws (A + B = A B , and A B = A + B ) gives: f = (A B ) (A B ); and:
f = (A + B ) (A + B ); and:
f = (A + B ) + (A + B ): Therefore:
f = (A + B ) + (A + B ): The circuit of NOR gates is given in Figure 1. 4. (a) Since levels of sound pressure, or intensity, are related to signal (voltage) levels in the electronics, and the data word length for a single sample is 16 bits, the number of levels of sound pressure is: 216 = 65; 536 levels: (b) The Danish mathematician, Sven Nyquist, proved what is now called Nyquist's Theorem, which states that a sampling rate of at least double the highest recorded frequency is required for the reproduction of sine waves of the highest recorded frequency, and of any frequencies below that point. Since any waveform can be decomposed into a superposition of sinusoidal waves, the sampling rate for frequencies detectable by the human ear needs to be at least 40 kHz (2 20 kHz). 2
A
B f
Figure 1: The NOR gate con guration in Problem 3.
3
S 0 0 0 0 1 1 1 1
R CLK Q Q 0 0 Q Q 0 1 Q Q 1 0 Q Q 1 1 0 1 0 0 Q Q 0 1 1 0 1 0 Q Q 1 1 ? ?
Table 1: The truth table for the gated R-S ip- op in Problem 5. (c) The dynamic range of CDs is nominally said to be at least 90 dB. In this case, the decibel is a logarithmic measure of the relative highest and lowest power levels possible with the CD format. The decibel, when used to compare two power levels, is de ned as 10 log10 PP21 , where P represents power. We know that CDs o er 65,536 voltage levels, and thus, since P = V 2 =R (V is voltage and R is resistance), the maximum dynamic range is: 20 log 65; 536 =: 96:3 dB: 10
(d) For vinyl long playing records, the stylus spirals along a groove from the outer radius to the inner one while the turntable rotates at a xed angular velocity. For CDs, the laser pickup instead tracks a spiral from the inside radius to the outer one, and the disc is rotated with a variable angular velocity to provide a constant (linear) speed di erence between the pickup and the track for any given radius on the disc. The linear relative speed at the outer radius is:
120 mm (200 rpm) = 12 m=s: 2
Equating this with the linear relative speed where the magnitude of the angular velocity is 500 rpm gives an inner radius of 24 mm. Now that we have the inner and outer radii, we can calculate the active area of a CD as: ,
(60 mm)2 , (24 mm)2 = 9:5 10,3 m2: The storage capacity is approximated by using the maximum playtime and the digital sampling rate: (74 min)(60 sec=min)(44:1 103 words=sec)(2 bytes=word) = 3:92 108 bytes: The data density is therefore: 3:92 108 bytes = 4:1 1010 bytes=m2 : 9:5 10,3 m2 5. (a) The truth table for this sequential logic circuit, which is a gated R-S ip- op, is given in Table 1. (b) The principle of this R-S ip- op can be used to store information in a clocked computer memory register. (c) Refer to Figure 2 for an equivalent con guration made up of NAND gates. 4
R
Q
CLK S
Q
Figure 2: The NAND gate equivalent to Problem 5.
5
6. Combining the the given attenuation and interterminal resistance information suggests that there's a passive bandpass lter in the black box. Refer to Figure 3 to see the circuit. Since we're only concerned with amplitudes here (we can ignore the phase), we can deal with the equation V = I jZ j, where Z is the impedance. The ratio of the output over input voltage amplitudes is therefore given by:
Vout = r Vin
R : 2 1 (2fL)2 + r2 , 2fC + R2
The resonant frequency from the above equation, fo , is calculated to be: r
1: fo = 21 LC We know that the resistance r exists because of the nonzero attenuation (20%) at the resonance frequency fo . At resonance:
Vout = 80% = p R : Vin R2 + r2 The relationship between r and R is therefore:
r = 0:75R:
6
L
r
C
A
C R
B
D
Figure 3: The passive LRC circuit in the black box of Problem 6.
7
teX?.aux teX?.aux
1996-1997 Physics Olympiad Preparation Program – University of Toronto –
Problem Set 1: General Due November 1st , 1996 1. While playing a game of Laser-TagTM , Daniel is trying to target his pal Yiorgos with his laser beam. Daniel (from his sonar display) knows that Yiorgos is hiding in the next room separated from him by an opaque wall and a thin window. Daniel wants to fire at Yiorgos, but he can’t aim without exposing himself. Suddenly he gets a great idea! He pulls a Gas-Bomb from his belt and pulls the trigger. The invisible gas fills the room, slowing the speed of light to one half of its normal value in air.
Figure 1: For problem 1 (a) What is the smallest lateral distance (i.e. parallel to the window) Daniel must move to target Yiorgos? (Remember that for physicists, ’thin’, ’small’, ’minuscule’, and similar adjectives generally mean ’negligible’.) (b) Daniel, delighted with his ingenuity, moves this distance and fires his laser gun through the window. He is surprised to see the beam exit the window and continue straight. What went wrong? 2. An Ontario apple of mass M = 0.2 kg rests on fence post of height h = 3 m. A bullet with mass m = 10 g, traveling at v0 = 500 m/s, passes horizontally through the centre of the fruit. The apple reaches the ground at a distance of s = 10 m. Where does the bullet land? What part of the kinetic energy of the bullet was lost as heat when the arrow passes through the apple? 3. The prototype spaceship Stealth (mass: 250 000 kg) is in a solar orbit(r=1.5×108 km), testing out its new cloaking mechanism. The cloaking device makes the ship perfectly absorbing of all electromagnetic radiation. (a) Assuming the ship has a cross-sectional area facing the sun of 20 000 m2 and knowing that the sun produces 3.8×1026 W of power , what is the energy the Stealth absorbs after being cloaked for ten minutes? (b) Assume that the energy the ship radiates is negligible. Before the test period, the average temperature in the Stealth is 18 ◦ C. What will be the temperature after the ten minute test period? Assume that all the absorbed energy is going into heat, and the average specific heat of the Stealth and its crew is 2 J/g◦ C. 1
(c) In reality the ship does radiate energy. Taking this into account, what will be its final temperature if the ship stays cloaked? (you might want to use Stefan’s constant: σ =5.67×10−8 Wm−2 K−4 ). What is the peak wavelength of the spectrum that the ship emits? 4. Wesley (of Star Trek: Next Generations fame) is given a box with three electric terminals protruding from it labelled A, B, and C. He is told that the box contains three resistors and, using an ohm-meter, finds out that the resistances across AB, AC, and BC are exactly 8 Ω each. He is asked to describe the configuration and values of resistances in the box, knowing that if he gives the wrong answer, phasers on “kill” will automatically wipe him out. He’s under a little stress. Find the resistances. Is the answer unique? 5. In her laboratory, Marie-Eve finds an unknown component with two electrical leads. To determine what it is, she applies different voltages across the leads and measures the following current going through the device:
Figure 2: Device for problem 5 (a) What is the effective resistance of the device at 10V? at 4V? (b) Marie-Eve then uses the unknown device in the following simple circuit:
Figure 3: Circuit for problem 5 Using the I-V graph, circle the point (or points) that represents the voltage and current that could be passing through the unknown component in the above circuit. (c) Marie Eve replaces the 10V supply with a variable power supply and an ammeter (current measuring device with negligible resistance). Starting a 0 V, she increases the power supply voltage to 15 V and then decreases it slowly back to 0 V. Roughly sketch a graph of the current she would observe as a function of this voltage. 2
6. For some roll dispensers of toilet tissue, it is possible to drag the end of the roll over and flush the entire roll down the loo like spaghetti, with the tissue spinning freely off the roll. Presumably to interrupt such physics experiments, manufacturers often construct the holder to have a spring-loaded pressure plate at either end of the holder, which press flat against the ends of the cardboard core. The friction is enough to keep the paper from spinning off the roll, but as the roll becomes smaller, it soon becomes impossible to pull the paper directly off the roll without breaking.
Figure 4: Roll dispenser for problem 6 (a) Consider a cardboard core 4 cm in diameter and with walls 1mm thick, and take that the pressure plates push only on circular ends of this core with a force of 13 N on either end, and that the core has a static coefficient of friction µs = 0.27. Empirical studies (!) show that most toilet paper segments will burst at the perforations for a typical tension of T = 3.5N . If the toilet paper can be pulled off the roll without breaking at the perforations, what is the minimum diameter of the toilet paper for this to be possible? (b) The kinetic coefficient of friction µk is measured to be 0.18. If the paper is pulled off starting with a full roll (diameter 11.5 cm) and the motion never stops, what now will be the diameter of the roll of paper at the point where the paper tears? (c) Take that the paper on the roll wraps at 10 layers per millimetre. What length of paper remains on the roll for the critical diameter you found in (a) and (b) above? (d) Suppose that the pressure plates pushed on the whole side of the roll, cardboard core and rolled paper together, so that the contact area changed as the roll got smaller. How might the whole problem be changed? Assume you can use the same static coefficient of friction µs .
3
PROBLEM SET #1 Ð GENERAL 1.
LaserÐTag
This is a standard refraction-type question. Since Daniel is in a medium with a different (i.e. higher) refractive index than air, the laser beam will ÔbendÕ around the corner. Let x be the minimum distance Daniel must move to target Yiorgos. The only other variable to worry about is at what angle Daniel will aim (i.e. where will the beam hit the window). For minimum x, Daniel will aim to hit the very edge of the window. From SnellÕs law n i sinθ i = nr sin θ r We know that after the Gas-Bomb is used, ni = 2nr 4
sinθr =
√
42Ê+Ê12
=
4
√17
Also sinθi =
2n r
x x2Ê+Ê22 √
x x2Ê+Ê4 √
= nrÊ
=
x x2Ê+Ê4 √ 4
√17
17x2 = 4 (x2 + 4) 13x2 = 16 x = 1.1 m (Negative root extraneous). Daniel must move at least 1.1 m to the right to target Yiorgos. (Common error: In Snell's law, the incident and refracted angle are measured between the ray and the normal to the interface, not the ray and the interface.) b) Nothing went wrong, if Daniel was looking along the beam with the lasergun held high to aim. Daniel sees his laser beam due to light scattering back to him from dust or other particles in the air. This scattered light, along nearly the same path, but backwards, will also obey SnellÕs law, and as such the light that reaches his eye will follow a similar path as the laser beam. Therefore, to Daniel it looks straight. (Many of you came up with ingenious ways that would result in the laser beam not bending, and, if it was physically possible, got part-marks for your efforts. As you can see from the solution, none of these were necessary.)
2.
Apple shoot
The total momentum of the system remains constant in a collision. Thus, mv 0 = mv + MV
(2.1)
Here, v is the velocity of the bullet and V that of the apple after the collison. Both objects fall a distance h to the ground; their time of flight is t = √ 2h/g = 0.78 s. During time t, the apple covers a horizontal distance of s = 10 m, so its horizontal velocity is V = s/t = 12.8 m/s. Thus, the conservation of momentum equation yields v = 104 m/s for the horizontal velocity of the bullet immediately after the collision. Since the bulletÕs time of flight is also t, it travels a distance vt = 105 m horizontally from the column. Initially, the kinetic energy of the system is Ei = mv02/2 = 1250 J. After the collision, the kinetic energy is Ef = mv2/2 + MV2/2 = 93.2 J. Thus, 1250 Ð 93.2 = 1156.8 J
(2.2)
is lost as heat (92.5% of the original energy). Note that the collision is n o t completely inelastic. In a completely elastic collision, the kinetic energy is conserved. In a completely inelastic collision, the bullet would remain inside the ball. 3.
Stealth
In the world of science-fiction, a ÔcloakedÕ ship has to be perfectly transparent; or, next-best, absorbing all radiation that strikes it but emitting only the same radiation as the molecules it displaced would have. Our current knowledge of physics doesnÕt allow this, in steady state, but letÕs assume the second case. a)
We can assume that the sun radiates uniformly over a sphere.
What is the energy the Stealth absorbs per second? The sun radiates P = 3.8 × 1026 W over itÕs entire sphere. At r = 1.5 × 108 km, this corresponds to a flux of P/(4¹ r2) = 1.3 × 103 W mÐ2. ∴ Stealth absorbs a total power of 1.3 × 103 W mÐ2 ¥ 20 000 m2 = 2.6 × 107 W. Over 10 minutes, Stealth absorbs 2.6 × 107 J sÐ1 ¥ 10 min ¥ 60 s minÐ1 = 1.6 × 1010 J. b) With no radiation losses: ∆t = 1.6 × 1010 J / { (2.5 × 108 g)(2 J gÐ1 ¡CÐ1)} = 32¡ C, ∴ tfinal = (18 + 32) = 50¡C)
c) Unfortunately, the question does not give enough information to really answer this problem. We need the total surface area Asurf , not just the crosssectional area. We can make some guesses at this, though. Since the cross-sectional area =Ê20Ê000Êm2, the surface area must be at least 40 000 m2. There is no upper limit to this number. (Common error: many people just plugged in the only area value they were given, i.e. 20 000. A ship that has a cross-sectional area of 20 000 cannot have a surface area of 20 000, this answer is impossible.) At steady-state, power absorbed = power radiated. Using StefanÕs law: H = eσA surf T4 (T in Kelvin), and since ship is perfectly absorbing, emissivity e = 1, ship absorbs 1.3 × 103 W/m2, ∴ at steady-state 1.3 × 103 W mÐ2 ¥ 20 000 m2 = σAsurf T4 , 2.6×107 Ð1/4 Τ= σAsurf Since we do not know Asurf, we can determine only a rough range of T. Maximum for Asurf = 40 000 m2 (the minimum): 2.6×107 Ð1/4 Τ= = 389 K = 116¡ C σA surf Since Asurf can be any larger surface area, Tmin can approach zero K. A possible ship shape would be a sphere. The radius of such a sphere is πr2 = 20 000 m2 ⇒ r = 80 m Asurf = 4πr2 = 80 000 m2 giving T = 275 K → 2¡C No matter what your answer for T, you can determine the corresponding peak wavelength of the radiation distribution. WienÕs Displacement law is the quickest way to the answer. It states: λmaxT = 0.002898 m¥K Substitute for T (in Kelvin) and solve for wavelength in meters. 275K gives λ max = 20.5 µm. 4.
WesleyÕs test
A typical first guess at the contents of the black box is a triangular configuration of resistors of resistances Rab, Rbc and Rca shown in Fig. 4.1(a). Since Wesley is such a stereotypical thinker, he immediately grasps at this solution. When the resistance
across terminals ab is measured, the resistance of the parallel circuit formed by resistors Rab and Rbc + Rca is 1 1 [4.1] Rab-total = Ê+ÊR Ê+ÊR Ð1 R ca bc ab Similarly, the resistances measured across terminals bc and ca are 1 1 Rbc-total = Ê+ÊR Ê+ÊR Ð1 ca ab Rbc
[4.2]
Ð1 1 1 = Ê+ÊR Ê+ÊR ab bc R ca
[4.3]
and Rca-total respectively. Since all three measurements give the same result, Rab-total = Rbc-total = Rca-total . From the similar equations (4.1), (4.2) and (4.3) it can easily be shown that Rab = Rbc = Rca ≡ Rp
[4.4]
The total resistance for each pair of terminals, measured in terms of Rp, is 2 1 Ð1 1 = 3 Rp R total = Ê+Ê2R p R p
[4.5]
and so, if Rtotal is to be 8Ω, then Rp = 12 Ω.
Figure 4.1: Two black box circuits.
a
a
(a)
(b) Ra
Rab
b
Rca
Rb c
Rbc
b
Rc c
Is the answer unique? This is the tough part of the problem (assuming you got past the ÔguessingÕ part). Well, if the configuration above were the only solution the answer would certainly be yes. However, there is another possibility, which is shown in Fig. 4.1(b). The resistors are connected together and to each terminal is associated a single resistor named R a , Rb , and Rc . In this circuit, a measurement across terminals ab will give the resistance of the series circuit formed by Ra and Rb. As above, it can be shown that equal measurements across ab, bc, and ca must require Ra = Rb = Rc = Rs and for Rtotal = 8 Ω, Rs = 4 Ω. Wesley, that impetuous fellow, immediately thought that the answer was unique. Too bad. Shows that one should put oneÕs mind into gear before thinking. Would you have been able to confidently answer this question? 5.
Nonlinear electronics
a) For a linear device, R = V / I. But this is not a linear device, so this relationship is not useful. A more general definition is R effective = dV/dI, the local rate of change of voltage with current Ñ the slope of the tangent. When the device is linear, V vs. I is a straight line through the origin, so this definition comes back to R=V/I then. Even if you have no calculus experience, you can find an intuitive solution for VÊ=Ê10 V and V = 4 V. At both these points, an increase or decrease in voltage corresponds to no change in current, i.e., Reff is huge, approaches ° . b) We do not know the voltage drop across either component or the current. Start by listing the unknowns. VR : V? : iR : i?: Right away, we know
Voltage accross resistor Voltage across unknown device Current through resistor Current through unknown device
the current flows through one then the other i R = I? V R + V? = 10 Volts V R = iR ¥ R the resistor is a linear device I? = F (V?) where F() is the function shown in fig. 2, PS#1.
So we have 4 equations with 4 unknowns. Thus we can find a solution(s).
V R = iR R but VR = 10 Ð V? ∴ 10 Ð V ? = iR R but iR = I? ∴ 10 Ð V ? = I? R
i? = Ð V?/R + 10/R
Now we have 2 equations with 2 unknowns, but we must solve it graphically. The intersections between the two lines are possible operating points.
Current (mA) 25
possible operating points
Voltage (V) 10 The actual voltage/current across/through the unknown device depends on how Marie-Eve started the circuit (as shall be discussed in the section below). c)
At V = 0 V
Current (mA) 25
Voltage (V 10 i?=-v?/R+0/R This line moves as Marie-Eve increases the voltage. Since we assume there is no noise in the system, the operating point moves up the curve:
Current (mA) 25
Voltage (V 10 At this point (Vvar. P.s. = 13 V), the operating point jumps to
Current (mA) 25
new operating point
10 V var p.s. continues up to 15 V with I? = 0A As Marie-Eve decreases from 15 V:
Current (mA) 25
Voltage (V 10
At this point (Vvar p.s. = 9V), the operating point jumps to:
Current (mA) 25
10
Voltage (V
LetÕs put this all together.
i? (mA)
25
15
Vvar power supply (V)
This is an excellent case of hysteresis! 6.
TP Pull
Comment: see the POPTOR web pages, under Problem Set Extras for a photograph of such a roll-dispenser. Also a friction-lock, using stacks of many plates to increase the drag torque, in the quick-release adjustments of an adjustable office-chair. a) Knowing the normal force on each of two faces of the roll, we can find the total frictional (drag) force: FnÊ=Ê13N Ê ÊÊ⇒ÊÊFDÊ=ʵsÊFNÊ=Ê0.27Ê×Ê13NÊ µsÊ=Ê0.27 = 3.51N @ each end
The question is properly a matter of torques, since it is rotational forces Torque from drag:
= 2 ⋅ FD ⋅ ro = 2.351 N ⋅ 2 cm = 14.0 N⋅cm (0.14 N⋅m )
TD
Torque from pulling: Tpull = Fpull ⋅ b where Fpull ² 3.5 N. Tpull = Tdrag while pulling, and Tdrag is constant in the static case (i.e. testing from rest at different ÔbÕ) So, 14.0 N⋅cm = Fp u l l ⋅b unless it breaks b; maximized; the last b for not breaking is b=
b is minimized where Fp u l l is
14.0N⋅cm = 4 cm 3.5N
60 Tdrag-roll
Torque (N•cm)
50
Tpull
40 30 20 10 0
0
2
4
6
8
10
12
14
16
b
Figure 6.1: Comparing the constant drag-torque from the friction plates on the cardboard roll (red dashed line) to the maximum-deliverable pulling-torque, as the roll size increases (black solid line). The paper is pulled only with as much force as is necessary to overcome the drag-torque, but as the roll becomes smaller, even the maximum tension on the paper (burst-strength) is not enough, and the paper breaks at b = 4 cm, as the lines cross. b)
T D = 2 µ sF N ⋅ ro = 2 (0.18) 13 (2) = 9.4 N ⋅ cm (0.094 N ⋅ m)
T pull = Fpull ⋅ b b=
9.4N•cm = 2.7 cm 3.5N
TD = Tpull ⇒ 2.7 cm, i.e. 0.7 cm left on roll. c) Approximate, rather than finding the spiral length from an integral: consider separate layers, starting from the core (not a perfect spiral anyway ...) length layer = 2πr 2 cm ²r ² 5.75 cm (d/2) increment in r is ∆r = 1mm/(10 layers) = 0.1 mm layer-1 = 0.01 layer-1 Thus we have (4 cm Ð 2 cm) * 100 layers cm-1 = 200 layers 200
∑ n=0
2π(ro + n ∆r)
j 1 where ∆r = 0.01 cm. Using ∑ n = 2 j(j + 1) to simplify, we get n=1 200
200
= ∑ 2πo + 2π ∆r ∑ n n=0
n=0
= 200(2πro) + 2π ∆r (200(201)/2) = 200 2π 2 cm + 2π 0.01 (100) (201) cm = 3776 cm = 37.76 m Likewise for b = 2.7 cm
(2.7 - 2) * 100 = 70 layers length = 70 (2πro) + 2π ∆r (70(71)/2) = 70 (2π) 2 cm + 2π 0.01 ¥ 35¥71 cm = 1036 cm = 10.36 m
Compare this with the answer you can get by dividing the area of the side of the roll by the thickness of a single layer Ñ giving a length. A = (¹b2 Ð ¹ro 2 ), ∆r = 0.01 cm; for b = 4 cm, A = 37.70 cm2, so A/∆r = 37.70 m; for b = 2.7 cm, A = 10.34 cm2, so A/∆r = 10.34 m. This is probably just as good as the approximation above (thanks, Gordon Cook!) d) In this case the frictional force is the same (area increases but normal pressure decreases) BUT the torque changes. The torque changes because the frictional force is applied at a changing effective radius.
To best do the problem requires adding up all the torques contributed by all the similar regions (r = const). Plan ÔAÕ: If you can integrate, it is not hard: Consider a roll of diameter b, and core size ro . With a normal force FN evenly distributed over this area, the force per unit area is PN = FN/(πb2 - πro2). Then for an annulus (ring) very thin of width dr set at radius r, the force on the annulus is dFdrag = µsPNdA = µsPN2πrdr then the torque-contribution due to this annulus is dTdrag = r⋅dFdrag = µs PN 2πr2 dr The whole drag-torque on each side is then the sum of torques over all annuli 1 2 Tdrag =
T(b)
b d Tdrag =
T(ro)
µs PN
ro
2π2 dr b
= µs PN 2π ∫ r2 dr = µs PN ro
b 1 2π 3Êr3 r o 2π = µs PN 3 (b3 - ro3) So, since Fn = Pn π(b2 Ð r o2 ) = Pn π(b Ð ro )(b + ro ), then we can write πPn(b Ð ro) Fn = (bÊ+Êr ) and substitute for PN o (b3Ê-Êro3) 4 Tdrag = 3 µs FN 2 3 b Ê-Êr 4 = 3 µs FN
(b2Ê+Êbr
o 2 oÊ+Êro)
(bÊ+Êro)
now as you can see, the drag torque is not constant but scales with the outer diameter of the roll, b. To find out if the paper breaks Ñ ever, never, or always Ñ you have to compare this drag torque to the pulling torque you can produce from
the maximum (breaking) tension of the paper applied as a force at the radius b of the roll.
80 Tpull Tdrag-area Tdrag-area-kinetic
70
Torque (N•cm)
60 50 40 30 20 10 0
0
2
4
6
8
10
12
14
16
b
Figure 6.2: Comparing the changing drag-torque from the friction plates on the decreasing roll (dashed lines: redеstatic , greenе kinetic ) to the maximumdeliverable pulling-torque (solid black line), as the roll size increases. In the static case, the maximum pulling torque is never enough Ñ the paper always breaks; with a rolling start (the kinetic case), there can be barely a solution for a little while until the paper roll becomes too small. Plan ÔBÕ: If you do not yet integrate, you can also do it by series summation: ∆Fdrag (r) = µs PN ∆(Area) = µsPN2πr∆r the torque about the cylinder axis from just this ring is then ∆Tdrag (r) = r ⋅ ∆ Fdrag = µsPN 2πr2∆r we can take each ring of width ∆r to be an additional layer, then ∆r = 0.01 cm as in part ( c ) above. Then for a diameter b there are (b - ro)/∆r layers, each contributing a torque from the drag at its particular radius; adding all the torques up on one side gives us half the total drag torque, so
1 2ÊTdragÊ 2∆r) + ...
= ∆Tdrag (ro) + ∆Tdrag (ro + ∆r) + ∆Tdrag (ro + (b-ro)/∆r
=
∆Tdrag (ro + n∆r) ,
n=1
(b-ro)/∆r
=
µs PN 2π(ro + n∆r)2 ∆r,
n=1
= µs PN 2π∆r
(b-ro)/∆r
(ro + n∆r)2 ,
n=1
(b-r )/∆r
o = µs PN 2π∆r∑ n=1
(ro2 + 2ro ∆r •n + (∆r)2 n2) ,
(b-ro)/∆r
= µs PN 2π∆r
ro2 + ∑2ro ∆r •n + ∑(∆r)2 n2
n=1
= µs PN 2π∆r {
ro2
∑ 1 + 2ro ∆r ∑n + (∆r)2 ∑n2Ê }
Now N
∑ Ê1 = N n=1 N
∑ n=1 N
∑ n=1
n=
N(NÊ+Ê1) 2
n2 =
N(NÊ+Ê1)(2NÊ+Ê1) 6
where N is number of layers = 100 (b - ro), n is layer index-number. So N(NÊ+Ê1) 2 N(NÊ+Ê1)Ê(2NÊ+Ê1) } Tdrag = 4πµs PN∆r {ro2 N + 2 r o ∆ r + (∆r) 2 6 where N = (b - ro)/∆r ∆r = 0.01 cm b = diameter of roll (cm) ro diameter of core (cm) If now you let the layers become very thin, so that ∆r → 0 and N → °, this becomes TdragÊ = 4πµ P r2 (N∆r) + r (N∆r) 2 + 1 •2(N∆r)3 s N o o 6 with N∆r = (b Ð ro ),
1 = 4πµs P n r2o (b Ð ro) + ro(b Ð ro)2 + 3 (b Ð ro)3 4 = 3 πµs PN(b Ð ro)3 + 3ro(b Ð ro) 2 + 3r2o(b Ð ro) 4 2 = 3 πµs Pn (b Ð ro) (b Ð ro)2 + 3ro(b Ð ro) + 3r o Fn Then with πPn(b Ð ro) = (bÊ+Êr ) o
as above,
(bÊÐÊro)2Ê+Ê3ro(bÊÐÊro)Ê+Ê3r2o 4 TdragÊ=Ê3ʵsÊFnÊ (bÊ+Êro) (b2Ê+ÊbroÊ+Êr2o) 4 =Ê3ʵsÊFnÊ (bÊ+Êr ) o The same answer as the integral, because this is in fact doing the integral from first principles.
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Problem Set 2: Mechanics Due December 13, 1996 1) Pinball pastimes A pinball machine is one of the ancient ÔarcadeÕ games (see figure). As a good pinballer, you do not need to pull the spring knob too hard. In this machine, the table is tilted at 15¡, and the ball has a 0.60 m diameter of 3.0 cm and a mass of 100 g. The pinball-plunger spring constant is 200 N¥mÐ1 . C The ball is assumed not to F B D roll and the contact A E plunger between the ball with the 1.40 m 20 cm table is frictionless. ball Assume that all thicknesses not 15 deg specifically indicated are knob side view negligibly small. 3.0 cm a) What is the minimum d is ta n c e fro m the top view equilibrium point of the spring that one has to pull the knob so that the ball could reach the top of the semi-circle? b) After it reaches the top, where will the ball go: bumper A, B, C, D, or E? Explain. c) Pinball semi-wizard Elton does not pull the spring hard enough, and the ball does not even get into the semi-circle (it barely touches F) after he releases it. How far did he pull the knob? [Chairul] 2) Net result Martina is 6 feet tall and is a very good tennis player. Amanda is also a good tennis player, at 5Õ 3Ó tall. In tennis, a good serve can make the difference between a mediocre player and a very good one. For a serve to be a legal one, it must bounce in the service area after clearing the net (see figure). We model the service stroke of both players as a circular arc with the arm and racquet forming a straight line from the shoulder to the tip of the racquet. 1
Assume that they each can generate the same racquet-head speed, and further that the angular velocity remains constant throughout the stroke. Take it that the ball speed after leaving the racquet is 1.8× the racquet-head speed. Here are Martina and AmandaÕs vital statistics: Statistic
Martina
Amanda
Height
1.82m
1.60m
Height from ground to shoulder
1.57m
1.35m
Arm length
0.71m
0.56m
length of racquet from handle to middle of racquet head: 0.46m. Height of the net 0.91 m. a) Serves must land in the service area, which includes the lines. The fastest way to get the ball past your opponent is to maximize the itÕs velocity component parallel to the ground, vp. At what point in the stroke should each player hit the ball to maximize vp? Now assume both players hit the ball at this point. If they want to hit a power serve touching the centre service line, what is the maximum v p with which each player can hit a legal serve?
Point of service
23.7 m
Service Area
6.40 m
8.23 m
b) Contacting the ball at the top of their stroke, what is the minimum v p with which each player can hit the ball and still clear the net? (Ignore air resistance.) c) What is the maximum speed with which each player can hit the ball at the top of their stroke and still have it land in the service area? (Ignore air resistance.)
d) In a match, Amanda tried a trick serve in which she struck the ball early Ñ she hit the ball with her arm at an angle of 50¡ to the ground. What would the angular velocity of her arm need to be in order that the ball would make it over the net into the service area? How high in the air would the ball go? (Ignore air resistance.)
2
e) If Amanda wants to increase the speed v p with which she can serve the ball into the service area, explain qualitatively how she could apply spin to the ball and use it to her advantage. [Nipun] 3) Mind for rocks, or rocks for brains? Some of us students here at the University of Toronto have taken up gym climbing (indoor rock climbing). For safety reasons, a rope is attached to the harness of the climber. It runs through a metal loop bolted to the top of the wall, and down to a device attached to the harness of a partner (or ÔbelayerÕ) standing on the floor. A good belayer never allows any slack on the rope. For the sake of this problem assume that climbing ropes donÕt stretch. a) Assume no friction in the system, what is the maximum mass of climber with whom you would want to be partnered? Why? b) When we started climbing, we were told that the general rule for relative masses between partners was ÒMore than double, youÕre in troubleÓ. If this is the case, what would be the coefficient of friction between the rope and the metal loop if this is the only friction in the system? c) There is also friction between the floor and the belayer. For simplicity ignore the friction between the metal loop and the rope for this part of the question. Does standing farther away from the wall allow you to support more mass than in part (a)? Is there an optimum angle? Assume the floor friction coefficient is 1.0. [James] 4) Dense and denser Diane walked into her physics lab to find her demonstrator Erica standing in front of a cylinder floating upright in a tank of a clear fluid. Diane asked her what she was doing. ÒIÕm measuring the density of the fluid. I know that the 10 cm long cylinder has a density of 0.63 g¥cmÐ3 Ó. 5 cm a) Diane, trying to impress her demonstrator, remarked that since the cylinder was half submerged, she knew what the fluid density was. What was her answer? b) ÒGoodÓ, responded Erica,Ó now try something more challenging. Cause the cylinder to sink farther into the fluid without touching it.Ó Diane looked around the lab and spotted a compression chamber. ÒI need a force, and pressure is force per unit area, so IÕll let air pressure do my work for me.Ó She put the tank with the cylinder in the chamber and increased the air pressure to 100 atm while keeping the temperature constant. What happened to the floating cylinder? Assume that the fluid is incompressible, and that air is an ideal gas (density= 1.29 kg¥mÐ3 at 1 atm).
3
c) What would Diane observe if she could increase the air pressure in the chamber to 1000 atm? (Keep the totally silly assumption that the gas remains ideal.) [James] 5) ÔI crush your moon Ñ crush crush!Õ a) The moon takes about 29 1/2 days to orbit the earth, as related to the earth (the synodic period) or about 27 1/3 days as related to the stars (the sidereal period). Knowing the constant of acceleration on the surface of the earth, and the earthÕs radius of 6.38 × 10 6 m, show how to estimate how far the moon is from the earth. b) Knowing the result of part (a), the neighbourhood tyke, O. Bray, takes a ruler out at night and uses it to figure out the approximate diameter of the moon. Explain how he might best have made his measurement. [Robin] ¥ For Hubble Space Telescope info & pictures, try http://www.stsci.edu/public.html ¥ For NASA info, pictures and teaching materials, try: http://spacelink.msfc.nasa.gov:80/Instructional.Materials/Curriculum.Materials/Sciences/Astronomy/Our.Solar.System/
6) WhatÕs the buzz? P ART I: Sepsis is a housefly. Interestingly, when houseflies land on a ceiling, it goes something like this: they fly up close to the surface while still upright, put up their forelegs up above their head, and stick Ôem to the ceiling Ñ then they swing around that way, doing a sort of flip, until they smack all their legs onto the ceiling and they are upside down.
a little like... Take it that Sepsis weighs 0.3 gm, and that he is about 15 mm long from ÔtailÕ to compound eyes. We need a simplified model of a fly, so take Sepsis to have legs that reach 1 cm from his centre-of-mass, and take that his main legs let him bounce about 1.5 mm on landing. Say that Sepsis is cruising along at a good 1 m sÐ1, when he decides to alight on the spot of the last ceiling spaghetti-test. He sticks his forelegs to the ceiling when his centre-of-mass is 7 mm below the ceiling, and pivots rigidly around that sticking-spot to a landing. a) If you ignore any slowing-down by drag in the air, how quickly does he land Ñ how much time between sticking his forelegs and touching down all his legs? b) How fast is Sepsis going toward the ceiling when he finally ÔsticksÕ his landing? c) If he has only 1.5 mm to bounce, then he has only 1.5 mm to stop Ñ what is the minimum acceleration Sepsis suffers at the end? 4
5
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Solution Set 2: Mechanics
1.
Pinball pastimes
We can consider the ball as a point concentrated in its centre of mass. This assumption is valid since the ball is not rolling and the contact between the ball and the table is frictionless. a) When the ball reaches the top of the semi–circle, it already has speed v; thus, it also experiences a centripetal force: mv2/(0.285 m) = mg sin 15°
[1.1]
(0.285)(9.79) sin 15° = 0.850 ms–1 v=√
[1.2]
or
Let x be the distance between the ball with the equilibrium point of the spring. Using the conservation of energy (with the equilibrium of the spring as the reference point, x = 0): 1 2
kx2 – mgx sin 15° = 21 mv2 + mg(1.170) sin 15°
[1.3]
or 100x2 – 0.253x – 0.333 = 0
[1.4]
Solving this equation, one obtains x = 0.0590 m or 5.90 cm. (The other root is not used for obvious reason). b) After it reaches the top, the ball continues to pass A since it has momentum. Due to the inertia, the ball tends to go straight but the curve bends the path. One can also get the same answer using the symmetry argument. c) Since the ball barely touches F, its speed v = 0 and its position from the equilibrium is s = 0.870 m. 1 2
kx2 – mgx sin 15° = mg(0.870) sin 15°
[1.5]
or 100x2 – 0.253x – 0.220 = 0 Solving this equation, one obtains x = 0.0482 m or 4.82 cm.
[1.6]
2.
Net Result
To maximize vp, want to maximize the velocity component of the ball parallel to the ground. Therefore, one should hit the ball at the top of the stroke, i.e., when the racquet head is perpendicular to the ground. a) The "power serve" hitting the centre service line with the maximum vp (and still being a legal serve) would land at the back of the service area. Horizontal distance travelled by ball: ball’s trajectory
⇒ x = 18.7 m
x2 = a2 + b2 = (4.12)2 + (18.25)2
40
vertical distance travelled by ball: Martina: 1.57 m + 0.71 m + 0.46 m = 2.74 m Amanda: 1.35 m + 0.56 m + 0.46 m = 2.37 m 50
1 2 gt ⇒ t = 2 = 0.75 s = 0.70 s
time ball is in the air: d = Martina: tM Amanda: tA
2d g
∴ maximum vp for: Martina: v = x/ tM = 18.7m / 0.75s = 25m/s or 90 Km/h Amanda: v = x/tA = 18.7m / 0.70s = 26.9m/s or 97 Km/h We, of course, have to make sure that balls with these velocities will clear the net! (see part b) b) (Note that a more precise solution to this problem can be obtained by the methods of calculus. We sketch here an approximate solution.) Using the same trajectory for the ball as in part (a), we need to find the distance from the server to the net. By similar triangles: 4.12 x 18.25 = 6.40 2
net
⇒ x = 1.44m
∴ y2 = (6.40)2 + (1.44)2
y1
⇒ y2 = 6.56m
∴ y1 = 18.7 m – 6.56 m = 12.14 m To find the minimum vp for each player, time for ball to fall from point of stroke to height of the net:
4.12 m y2 x centre line 18.25 m
tm′ =
ta′ =
√ √ 2(d m – h net ) = g
2 (274 – 0.91) = 0.61 s g
√ √ 2(d a – h net ) = g
2•(2.73 – 0.91) = 0.55 s g
vmin: Martina: Amanda:
12.14m / 0.61 s = 19.9 m/s or 71.5 km/h 12.14m / 0.55 s = 22.1 m/s or 79.5 km/h
So, the serves in part (A) are fast enough to clear the net. c) maximum speed is for longest-possible horizontal trajectory, as horizontal distance vp = time for ball to fall to ground so we want the ball to land in for rear corner of service area: Horizontal distance travelled by ball = ((8.23)2+ (18.25)2)1/2 = 20.0m Maximum speed: Martina: Amanda:
20.0m / 0.75s = 26.7 m/s, or 96.0 km/h 20.0m / 0.70s = 28.6 m/s, or 103 km/h
d) We need to know: 1) how high above the ground the ball is when Amanda hits it 2) how much vertical distance between the release point of the ball and the top of the net — gives time available to clear net 3) the horizontal velocity component of the ball needed to reach the net at the closest distance to the point of service such that the ball will land in the service area. (See part b) 1) ball height at release point: 1.35 + (0.56 + 0.46) sin 50° = 2.13 m 2) d = (2.13 m – hnet) = 1.22 m ∴ Time available to clear net = (2d/g)1/2 = 0.50 s 3) From part (b): distance = 12.14 m vhorizontal = 12.14m / 0.50s = 24.3 m/s vtotal (tangential to the arc of the racquet): vhorizontal = vtotal cos 40° vtotal = 31.7 m Racquet head speed × 1.8 = ball speed vracquet = 17.6 m/s
angular velocity of arm = ω = v / r = 17.6 / (0.56+0.46) = 17.3 rad/s vertical velocity of ball: vy = vtotal sin 40° = 20.37 m/s So, the maximum height above the ball’s release point is 2gy = vy2 ⇒ y = 21.1 m Max. height of ball = 1.22 m + 21.1 m = 22.4 m
3.
Mind for rocks, or rocks for brains?
The system has been simplified since ropes don't stretch and so no slack is allowed on the line. This of course is not physically reasonable (climber would almost always be supported by belayer and so could never actually ‘climb’!) but is a good first approximation. a)
Since belayer does not want to accelerate off the ground: Σ Fb = 0
Tc
Since there is no friction Tc = Tb
mcg
Tb
Let 'c' indicate climber Let 'b' indicate belayer Let R be force by floor on belayer
∴ Τb + R = mbg Tc + R = mbg
R
Climber is also not accelerating. ∴ Τc = mbg
mbg
Maximum mc for a given mb occurs for R = 0, i.e., belayer barely touching floor. ∴ as expected, max. mass of climber is your mass, as belayer (Of course in real life, you would not want to be right on the edge of stability, but within the assumptions given the correct answer is ‘maximum mass = your mass’, not ‘< your mass’ which some people stated. This is a minor point.) b) Same pix as before except for friction on metal loop. Friction force opposes motion. Again we are trying to find a max. mc for a given mb. Start by setting R = 0. Note that forces on rope must balance ∴ mcg = f + mbg (direction of f choosen since we are opposing motion of climber dropping, not belayer dropping).
f
Frictional force is standard static friction, i.e. f ≤ fmax ≤ µs (mcg + mbg) For maximum mc, set f = µs (mcg + mbg) ∴ mcg = µs (mcg + mbg) + mbg mcg (1 – µs) = mbg (1 + µs) ∴ mc = mb (1+ µs) / (1− µs)
Tc mcg
but question states that mc = 2mb for maximum mc ∴ 2 = (1+ µs) / (1− µs) ⇒ µs = 1/3 <— cofficient of friction between loop and rope.
Tb
c) Since there is no friction in the rope, Tc = Tb. Since climber does not fall Tc = mcg, ∴ the forces on the belayer look like: f
≤ ≤
µs R 1.0 R
For maximum mc, f = 1.0 R Since there is no acceleration, Σ F = 0. Standard way would be to break this up into horizontal and vertical forces but more elegant way is to consider vector sum of forces: (since Σ F = 0, can draw triangle)
but we know:
mcg mbg-R
θ f=R
4.
mbg
θ mcg
(mbg – R)2 + (1.0 R)2 = (mcg)2
And write:
R
R f m bg
(a + b)2 ≥ a2 + b2 ∴ (mbg – R)2 + R2 ≤ (mbg – R + R)2 ≤ (mbg)2 ∴ (mbg)2 ≥ (mcg)2 ∴ max. mc is mb ∴ friction of floor does not help!
Dense and denser
a) Effective buoyancy force = g (mdisplaced air + mdisplaced fluid), but the masss of the displaced air is very small, so can be ignored. Net force = 0, ∴ g Vc /2ρfluid = g Vc ρc ∴ ρfluid = 2 ρc = 1.26 g/cm3
b) This problem explores what happens when you take a model and go to a silly extreme. You are required to make one simplification. The question states only the pressure for one point in the chamber. Of course, as the gas becomes more dense the pressure increase due to gravitational forces (ρgh) becomes more important. It is this very pressure gradient that causes the cylinder to rise. For this part of the question, since the pressure is increased only a small amount one can assume that the gas it displaces would have a minor pressure gradient across the length of the cylinder. Thus one can assume a constant density and that Diane’s measurement was taken at a height similar to the height of the cylinder. Let x be the distance that the cylinder is submerged. Net force = 0 ∴ g Vc ρc = g (x Vc ρfluid +(1 – x)Vc ρair) You cannot ignore the displaced air. ρc = (l – x) ρair + x ρfluid ρair is calculated from PV=nRT . ∴ x = 4.4 cm Cylinder moves up 6 mm. Note: cylinder floats higher in fluid. Diane should have reduced air pressure, not increased it. c)
Cylinder + fluid 'float' to top of chamber.
5.
‘I crush your moon – crush crush!’
a) Knowing the period of the moon’s orbit, we can figure out its distance from the earth, because the period depends on the orbit radius r and orbital speed v: Torbit =
distance 2π r M,m = speed v
rM, m = earth-moon separation, v = speed of moon
[5.1]
The orbital speed and radius are also related by the centripetal-force formula: FMm
m v2 = r M,m
m = mass of moon
[5.2]
Where the centripetal force is gravitational attraction: FMm =
GMm 2 r M,m
M = mass of earth, m = mass of moon, G = gravitational const.
[5.3]
Equations [5.2] and [5.3] both give the force, so with right-hand-sides equal, we can eliminate the moon mass m, to get: 2
2π r M,m GM = v2 = , where the last equality uses [5.1]. r M,m Torbit Then we have r M,m
2 GM Torbit = 2 4π
1/ 3
For G = 6.67 × 10–11 N m2 kg–2, M = 5.98 × 1024 kg, Torbital = 27.333 days * ( 8.64 × 104 seconds/day) = 2.36 × 106 s, this gives rM,m = 3.83 × 108 m (cf. 3.844 × 108 m). b) Ah, now that we have the distance rM,m we can figure out the diameter of the moon by constructing similar triangles: r earth-moon r-measure
disk, or ruler moon By holding a small paper disk, and moving it toward and away from the eye until it exactly covers the moon, then recording both the disk size ddisk and the distance held away from the eye rmeasure, the earth-moon distance from (a) can be used to figure out the moon’s size: dmoon d = disk r M,m rmeasure dmoon = ddisk
or finally
r M,m rmeasure
When I did this, I found roughly ddisk = 5 mm at rmeasure = 50 cm, and so dmoon = (0.5/50)3.83 × 108 m = 3.83 × 106 m. Compare this with a reference-table value of 3.476 × 106 m. ------------------------------------------------------------------------------------------------------Small Shortcut: Instead of using the values of G and of M, you might note that at the earth’s surface (rearth,m = rearth) we must have FMm = m g, where g = 9.8 kg m–1 s–2, so:
FMm
2 GMm rearth 2 = 2 = m g 2 , i.e., GM = g rearth , constants easier to remember r M,m r M,m
[5.4]
------------------------------------------------------------------------------------------------------Synodic vs. sideral: How should you measure a complete rotation of something in space? Suppose the earth did not rotate at all. If it made one orbit of the sun, every spot on earth would have daytime and night-time, both, so the earth would see one revolution or one day in terms of the sun, though in terms of the ‘fixed’ stars it had not rotated (no sidereal days at all). So the synodic and sidereal periods for the earth are not the same. The same holds for the moon and its orbit of the earth. As the moon orbits the earth, the earth orbits the sun. If the moon ‘start-finish line’ for an orbit is an imaginary line between the earth and some very distant star, then it will orbit and come back to this line again in a sideral period. If the start-finish line is a line between the earth and the sun, then while the moon is orbiting, this line will change along with the earth’s orbiting, so the finishing-line also will move through some angle in space. With a retreating finish-line, a complete revolution in these terms will take longer. (Of course, during the time it takes to catch up, the line will have moved a little bit again, making a bit more to catch up, and so on, and so on. The problem ends up like figuring out from 8:00 pm exactly when the minute hand ends up opposite the hour hand: in 10 minutes it is opposite where it used to be, but the hour hand has moved an angle 1/12th as much as the minute hand just moved. So the minute hand reaches the corrected place 1/12 of 10 minutes later, but then the hour hand in that time moved 1/12th of 1/12th of 10 minutes... The answer can be found using the sum of the series (1/12)n, which has a well-known formula.) 6.
What’s the buzz? a)
Need some approximation 0.7cm angle θ: 1.0cm = sinθ = 0.7 ⇒θ= 0.775 radians = 44.4 °
Momentum: 0.3 g = 3 × 10–4 kg; mv = 2 × 10–4 kg · 1 m s–1 = 3 × 10–4 kg · m s–1 . The momentum resolves along the rigid rod and along the ⊥ to the rigid rod. p⊥ 0.7cm p = 1cm = p⊥ = 0.7 p Angular momentum is conserved; with the free-pivot action, it sees no torque and so is conserved without contribution from the ceiling, earth, etc.. L = P⊥ • r = 0.7 p •10–2 m, = 2.1 × 10–6 kg m2 s–1
It’s undefined what the final angle is when he reaches the other side — he could swing around until he is parallel to the ceiling, or maybe until his 1 cm-long legs, halfway down his body, hit the roof. Either is good for the sort of rough estimates we are making on behalf of flies! You figure. Say, take 1st case: the final angle is π radians. Then he swings through (π – 0.775) radians = 2.37 radians or 135.6° Arc length: ∆θ · r = 2.37 rads · 1 cm = 2.37 cm = 2.37 × 10–2 m Speed: v⊥ = 0.7 · v = 0.7 × 1 m s–1 = 0.7 m s–1 d 2.37 × 10 –2 m t=v= = 3.39 × 10–2 s = 33.9 ms –1 0.7 ms we’ll accept 15 – 35 ms, about one frame of a video camera. b) Again depends on final angle, leg length, etc., but approx v⊥ = 0.7 m s–1; we’ll accept answers 0.35 – 0.7 depending on final angle u2 v = 0 ⇒ –2d = a c)
v2 = u2 + 2ad a=–
d)
0.49 (0.7)2 = = 0.16 × 103 = 1.63 × 102 = 163 m s–2 – 3 – 3 21.5 × 10 3 × 10
F = Mfly • a = 0.3 × 10–3 kg (–163 m s–2) = –49 × 10–3 kg m s–2 = –4.9 × 10–2 kg m s–2 = –4.9 × 10–2 N
F = Mhuman • a = 70 kg (–163 m s–2) = –1.14 × 104 kg m s–2 = –1.14 × 104 N
PART II 30 km/hr = 30 × 103 m / 3600 s = 8.3 ms–1 e)
v2 = u2 + 2ad v=0⇒ (8.3ms–1)2 –2 a=– 21.5m = –23 m s , much less than for the fly F = m • a = (1,000 kg) (–23 m s–2) = –2.3 × 104 kg m s–2
f) not very significant that belts stretch a little, if assume constant acceleration force (unless you let the passenger bounce)
total distance is 1.75 m ∆x = 1.75 m 50 km/h = 50 × 103 m / 3600 s = 13.9 m s–1 u2 (13.9 m s –1 )2 a = – 2d = – 2(1.75 m) = – 55.2 m s–2 F = m • a = 80 kg • (–55.2 m s–2) = – 4.416 kN Then 0.35 m • 0.05 m = 0.0175 m2 area for belt. Pressure is (4.416 kN / 0.0175 m2) = 2.52 × 105 Pa [1 ATM = 1.01325 × 105 Pa] – about 2.5× atmospheric pressure; can bruise heavily where the belt cuts in – or, say my fist is 7 cm × 9 cm = 63 cm2 = 0.0063 m2, so to produce the same pressure I would need a force of about 1.6 kN, the weight of about 162 kg — much more than I weigh, and probably a really decent heavyweight prizefighter punch if the followthrough is strong. Or, to compare another way, the whole force on the person is F = 4.416 kN corresponds to the weight of ~450 kg – almost half the weight of a small car, so this is a serious force. How long does it last? 13.9 m s – 1 t = – = 0.126 s = 126 ms 2(–55.2) m s –2 Perhaps this is something like being directly hit by a motorcycle? You can hit the belt, or you can hit the windshield, but much better the belt! Buckle up — it saves lives.
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Problem Set 3: Thermodynamics Due January 13, 1997 1) Great balls oÕ fire! The sun may be modelled as a ball of hydrogen gas. Assume its main energy source is the conversion of hydrogen (atomic weight = 1.0078 amu) to helium (atomic weight = 4.0026 amu) roughly by the process 4H → He (the real process is much more intricate than this). The energy emitted is given by the famous Einstein equation, E = ∆m c 2 , where c is the speed of light. a) How much energy is given off during the life of the sun if only 10% of its hydrogen is converted to helium? b) The power radiated from the sun and arriving at the earth has a density today equal to 1400 W¥mÐ2 . If we assume that this rate is constant throughout the sunÕs life, estimate the life expectancy of the sun. Is this value plausible? [Nipun]
EIT image in the He II emission line at 304•, upper chromosphere/lower transition region, ~60,000 K. http://sohowww.nascom.nasa.gov /gallery/EIT/index.html
2) Running Amok The Amok IV space probeÕs initial design included no insulation or shielding. The electronics require a minimum operating temperature of 200K. Since it is a deep-space probe the engineers had to provide it with a heat source. a) What would be the minimum power requirements for the heat source for a black spherical satellite of radius 2 m? Assume that outer space is really,really cold. b) To reduce heating bills, the engineers decided to put a nice flat black outer layer around the satellite and stuff it with Fibreglass Fuchsia insulation. This insulation has an R value in vacuum of 18.8 ft2¥F¡¥h¥BTUÐ1. (Believe it or not, this is the unit often used for insulation, even in our SI-enlightened society.) What is the new power requirement to keep the electronics working properly? The radius of the shield is only slightly larger than the radius of the probe. (1 BTU=1.054 kJ, 1ft = 30.48 cm, 1F¡ = 5/9 K) c) Unfortunately, Alain the co-op student forgot to pick up the insulation before the launch, and somehow also the outer layer never got painted. Assuming that the emissivity 1
of the silver shield was 0.2, what was the steady-state temperature of the interior of the satellite? [James] 3) Too blue to be true? Light from a star has the broad spectrum pictured below, and it fits a blackbody distribution pretty well over much of the frequency range recorded. 6 fitted blackbody spectrum star's emission spectrum
Relative Brightness
5 4 3 2 1 0
0
300
600
900
Wavelength (nm)
1200
a) From the distribution, what is the temperature of this star? b) Travelling in a spacecraft toward this star at 0.1 c, all the frequencies of the light from the star will be Doppler shifted, and so the peak of the spectrum shifts. By WienÕs displacement law, λmax [nm] = 2.8978 × 106 / T [K], what is the new recorded temperature of the star? Present your arguments about whether or not this is really a new temperature for the star, as measured in the frame of reference of the traveller. [Robin]
4) Huddle up! Edna Hillary and her team have set up a research station in the antarctic. Their base is a structure of dimensions, 20×20×3 metres. While they have brought along with them many of the conveniences of home, they have overlooked one very important detail Ñ they forgot to insulate the walls of their cabin from the cold! As soon as they set up, they powered up their generator to heat the room to a cozy 20¡ C and then shut the generator off, only to find that the temperature then dropped at a rate of 1¡ C every 5 minutes! a) Assume that the research team is well-insulated (they're all wearing parkas), so that the heat capacity inside the cabin is due only to the air. Furthermore, treat the air as a diatomic ideal gas. If the air pressure is 1 atmosphere, how much power would the generator have to emit to stabilize the temperature? b) Bicycle pumps get hot by adiabatic compression of air: if they wanted to try and stabilize the temperature by compressing the volume of the cabin, at what rate would they have to ÔshrinkÕ the cabin. Express your answer in m3¥sÐ1. [Nipun] 5) They chilled out and got it together The first realization of Bose-Einstein condensation was achieved last year by researchers in Colorado. By cooling rubidium atoms to extremely low temperatures, they underwent a 2
phase change resulting in the group of atoms condensing into one single state Ñ a kind of ÔsuperatomÕ. Much work has been done in the quest to create this new form of matter since it was originally proposed by Albert Einstein and Satyendra Nath Bose. A rough understanding of this new state can be understood in the following way: de Broglie proposed that all matter can be viewed as waves, with wavelength: λ = h / mv. BoseEinstein condensation occurs when the de Broglie wavelength exceeds the interatomic spacing. Effectively, the atoms overlap in space. a) To begin, estimate the wavelength for a car speeding down a highway. What about a nitrogen molecule at room temperature and pressure (vavg = 400 m¥sÐ1)? While you are sitting in your chair doing this problem set, is your wavelength infinitely long? b) By recalling the equipartition theory (each degree of freedom contributes on the average (1/2)kT of energy), estimate the temperature one must cool a group of rubidium atoms (ideal gas) down to achieve Bose-Einstein condensation. Assume that you start at room temperature with a pressure of 1 atm and that you maintain constant volume. You should realize that the equipartition theory does not apply well to these low energy atoms, but it is adequate to get a rough idea of the required temperature. c) If you could reduce the atomic spacing, you wouldnÕt have to go to as low temperatures. Why do you think the first experiments didnÕt use very high pressures? d) Obviously this new state of matter is much more ordered than your original state. Have you broken the second law of thermodynamics, which says that entropy (and randomness) must always increase? [James] 6) The curious case of the cold calculating counterfeiter Svend just arrived in Toronto. When he got a ÔtooneyÕ in his change from the newstand at the Pearson airport, he got an idea, ÒIt is a real challenge for me, I will duplicate it.Ó Svend was very good at counterfeiting coins. So he bought two plates of metal: brass and nickel with the same thickness, 2.00 mm. Cooling them to liquid nitrogen temperature (77 K), he drilled a hole in the nickel plate in and fitted it with a brass centre-plug. The diameter of the hole was 12.00 mm and the outer diameter of the nickel is 28.00 mm. He made sure that the brass could barely fit into the nickel while cold. a) What is the approximate pressure between the two metals at room temperature (293 K)? Assume the hole expands in the same proportion as the rest of the plate. b) Would police catch Svend if he used this ÔtooneyÕ based on its weight? Compare with an actual one. (Measure one yourself!) (Note: the thermal coefficients of linear expansion brass and nickel are 19 × 10Ð6 KÐ1 and 11 × 10Ð6 KÐ1; their densities at room temperature 8.87 g¥cmÐ3 and 8.85 g¥cmÐ3 ; their YoungÕs moduli 9.1 × 1010 N ¥ mÐ2 and 19.0 × 1010 N ¥ mÐ2). [Chairul]
3
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Solution Set 3: Thermodynamics 1.
Great balls o’ fire!
a) Total mass of the sun = 1.99 x 1030 kg If sun is made of hydrogen, then amount of hydrogen converted to helium = 10% of (1.99 x 1030 kg) = 1.99 x 1029 kg For reaction 4H → He, mass loss is: ∆m = 4 x mass of H – mass He = 4(1.0078 amu) – 4.0026 amu = 0.0286 amu = 4.75 x 10–29 kg Energy given off by 4H → He: E = ∆mc2 = (4.75 x 10–29) x (3.00 x 108 m/s)2 = 4.27 x 10–12 J Total # of hydrogen atoms in reaction: 199 . × 10 29 kg 1.0078 amu / atom × 1.6606 × 10 − 27 kg / amu
. × 10 56 atoms = 119
4 hydrogen per reaction = 2.97 x 1055 groups of 4 H ∴ total energy: (2.97 x 1055) x (4.27 x 1012 J per reaction) = 1.27 x 1044 J b) mean radius of earth’s orbit = 1.49597 x 1011 m = re ∴ "surface area" of orbital sphere = 4Bre2= 2.8 x 1023 m2 ∴ energy flux from sun: 1400 W/m2 (at earth’s distance) ∴ total power = 1400 x 2.81 x 1023 = 3.94 x 1026 W 127 . × 10 44 J = 3.23 × 1017 s = 1.02 × 10 10 years ∴ life of sun = 26 3.94 × 10 W The life expectancy of the sun is ~10 billion years, so the calculated value is pretty good! 2.
Running Amok a) The probe will lose heat due to radiation. The net rate of heat loss depends on its temperature, emissivity (e.g., black, white or grey), surface area, and the background temperature.
1
Power radiated from a surface follows Stefan’s Law: P = σ A e T4
where
σ = Stefan-Boltzmann constant A = area e = emissivity (between 0 (white) and 1 (black) T = temperature (Kelvin)
(we are neglecting background temperature since question states “outer space is really, really cold,” i.e., To.s.= 0). A = 4πr2 = 50.3 m2 Since it is black, e = 1. Minimum power radiated corresponds to minimum temperature, i.e. Tmin = 200K ∴ Pmin = 4.6 kW b) We must find the temperature of the shield. It is radiating power as in part (a). Energy is being transferred from probe to shield at a rate determined by R value, Tprobe, Tshield, area
P1 TPROBE
P2
At steady–state P1 = P2 P1 = σ Ashield εshield Τ4 (T - Tshield ) ∆T = A probe P2 = A R R Thus σAshield e shield Ts4 = Question states
A(Tprobe − Tshield ) R
Ashield = Aprobe eshield = 1
ft 2 o F h Thus, R = 18.8 BTU m2 ⋅ oC ⋅ s = 3.31 J We want minimum power requirement, thus Tprobe = 200K Need to solve: Tshield = Tprobe – R σ Tshield4 — cannot isolate Tshield easily... Many ways to solve this: 1) computer or fancy calculator (HP28s or better) 2) graphically 3) by iteration, substituting each successive approximation back in I will demonstrate method (3): numbers [x] indicate the iteration being made 1) Start by assuming Tshield << Tprobe, thus take Tprobe = R σ Tshield4 and solve for Tshield: 1 /4 Tprobe Tshield = = 181 K Rσ
2
Plug this in and find new Tshield, then repeat as it converges and a self-consistent value emerges: 2)
200 − 181 1/4 = 100K Tshield new = Rσ 200 − 100 Tshield new = Rσ
1 /4
3)
200 − 152 Tshield new = Rσ
1 /4
4)
Tshield
200 − 126 = Rσ
1 /4
new
Tshield
200 − 141 = Rσ
1 /4
new
7)
Tshield
new
200 − 133 = Rσ
8)
200 − 137 Tshield new = Rσ
9)
200 − 135 Tshield new = Rσ
5) 6)
= 152K = 126K = 141K = 133K 1/ 4
= 137K
1/ 4
= 135K 1/ 4
= 136K
200 − 136 = Tshield = 136K Rσ ∴ Tshield = 136 K ∴ P = σ A e T4 = σ (4π (2)2) (1) (136)4 = 0.97 kW 1 /4
new
10)
This is larger than part (a) by more than a factor of 4. c) All heat transfer here is radiative. Consider all possible radiated power processes:
P3
P1 P4 P5 P2
P1: P2: P3: P4: P5:
from outer shield to outer environment from inner shield to inner environment from inner environment to shield from probe to inner environment from inner environment to probe
Right away we can write (‘p’ = ‘probe’, ‘s’ = ‘shield’): P1 = σ Aes Ts4 P2 = σ Aes Ts4 = P1 P3 = e s P4 P4 = σ Ae p Tp
4
P5 = P2 + P4 − P3 (since we are at steady state and eprobe = 1)
3
For shell, at steady–state: P 1 + P2 = P3 2σAeshieldTshield4 = eshieldσAeprobeTprobe4 ∴ 2Tshield4 = Tprobe4 Power radiated out by cabin = P4 – P5 = P 4 – P2 – P4 + P 3 = P 3 – P2 = eshield σATprobe4 – σAeshieldTshield4 = eshield σA(Tprobe4 – Tshield4) = eshield σA(Tprobe4 – 1/2Tprobe4) e σA 4 ∴ power radiated by cabin: s Tp 2 Power source is still the same e shield σ A Tprobe4 ∴ 0.97 kW = 2 ∴ Tprobe = 242 K 3.
Too blue to be true?
a) For a black–body distribution, Wien’s Law gives the peak of the distribution, according to: λmax T = 2.9 x 10–3 m · K In the figure the peak is around λmax = 460 nm, so T=
2.9 × 10 −3 m ⋅ K = 6,300 K 460 × 10 −9 m
b) The relativistic Doppler–shift means a change in frequencies: 1 v (1 − cos φ) 2 2 c 1− v / c where N is the angle between the direction of the photon and the spacecraft motion in the star’s frame.
ν′ = ν
Here N = 0 so ν ′ =ν =ν
1− v / c
1−v / c 2
2
=v
1− v / c (1 − v / c)(1 + v / c )
1− v / c 1+ v / c
with v = –0.1 c this gives ν ′ = 1.11 ν
4
c ν c c c so λ’ = = . λ = 0.90 = 090 ν ′ 111 . λ
νλ=c
⇒ λ=
This means a shift in the peak to shorter wavelengths, and so, nonimally, from Wien’s law, a higher temperature T’ = 1.11 T = 6965 K Is this a ‘real’ new temperature? Well, does the blackbody distribution still fit? 8π hν 3 uν = 3 c [exp( hν / kT − 1)] hν ′ hν = if ν ′ = 1.11 ν and we take T’ = 1.11 then , and (ν’)3 = 1.37 ν’, so a distribution of this kT ′ kT type still fits, with an overall factor of 1.37. Since the overall amplitude of the distribution uν changes, it is no longer normalized properly. However, this is not really an issue in a typical real measurement — it may simply seem like a larger star radiating more power. If it were not a perfect blackbody but included some spectral lines, the patterns of these lines would be shifted from their usual positions, helping you guess what had happened. In thermal terms, fitting a Maxwell-Boltzmann distribution, adding a constant velocity to all particles does not change its kinetic temperature.
4.
Huddle Up!
a) Temperature drop: 1°C / 5 mins Cv = 5/2 R for a diatomic ideal gas (J • K–1 • mole–1) In finding total # of moles, let’s assume we deal with the gas only, and that the volume of the researchers is negligible (probably not true) 1200m 3 = 12 . × 10 6 l ∴ volume = 20 × 20 × 3 = 1 × 10 −3 (1l = 1000 cm3 = 1 x 10–3 m3) at S.T.P., 1 mole of ideal gas occupies 22.4 l 293K ∴ at 20 °C: 22.4l = 24.04 l • mole–1 273K Or use PV = NRT which is right for an ideal gas and often OK for a diatomic gas too. (Diatomic nature of gas doesn’t change the equation of state, as diatomic molecules add to the molecules’ internal energy but don’t affect the entropy so the pressure isn’t affected. Therefore we have the same equation of state. 12 . × 10 6 l = 4.99 × 104 moles ∴ Total # of moles: 24.04l / mole
5
5 5 R = k B = 20.7 J ⋅ mole −1 ⋅ K −1 2 2 1°C / 300 s = 3.33 x 10–3 °C • s–1 ∴ Power = (3.33 x 10–3 °C • s–1 ) × (4.99 x 104 mole) × 20.7 J • mole • °C = 3.46 x 103 J • s–1 = 3.46 x 103 W For a diatomic ideal gas, CV =
b)
Adiabatic compression: For an ideal gas pV = nRT, ∴ at constant temperature and particle number, pV = constant Instead, assume the gas is thermally insulated from its surroundings (adiabatic conditions) Changing the volume of the gas will do work on the gas and the internal energy of gas will change, so the temperature will change. We need the relationship between volume and temperature. From 1st law of thermodynamics, heat-energy transfer is ∆Q = ∆E + p∆V (1) In terms of heat capacity at constant volume, ∆E = n CV ∆T, where n = # of moles. For an adiabatic process, there is no heat transfer, so ∆Q = 0, and thus from (1) (2) 0 = n CV ∆T + p∆V We can write the differential of pV: ∆(pV) = p(∆V) + (∆p)V p∆V + V∆ p so, ∆T = nR into (2): (CV + R) p∆V + CVV∆p = 0
(3)
Dividing by CVpV: ( CV + R ) ∆ V ∆p + =0 CV V p ∆V ∆p ⇒ γ + =0 V p where
(4) (5) C C +R γ= V = p CV CV
7 Cp 2 R 7 For diatomic ideal gas, = = CV 5 R 5 2 Integrating equation (5), with ∆V as a differential dV: γ ln(V) + ln(p) = constant γ ⇒ pV = constant γ–1 ⇒ V T = constant ⇒ V2/5 T = constant (γ = 7/5 for diatomic ideal gas)
6
For a change in temperature of 3.33 x 10–3 °C from 20 °C V12/5 T1 = V22/5 T2 ∴
T1 V2 2/5 20 o C V2 2 / 5 = ⇒ = T2 V1 20 − 3.33 × 10 −3 (1200m 3 ) 2 / 5
Take the difference: |V2 – 1200 m3| ∴ Need to compress at a rate of 0.5 m3 • s–1 [0.034?] 5.
They chilled out and got it together a) DeBroglie wavelength, λ = h/p = h/mv assume car v = 100 km • h–1 = 28 m • s–1 assume car mass = 1000 kg ∴ λ = 2.4 × 10–38 m N2 mass is 14 g/mol, ∴ for one molecule, m = 2.3 × 10–26 kg v = 400 m • s–1 ∴ λ = 7.1 × 10–11 m My net velocity over time might be zero, but my instantaneous velocity is not ∴ my λ does not approach ∞.
b) Since these are atoms, they have 3 degrees of freedom (corresponding to 3 directions of motion), ignoring internal degrees of freedom for now. 3 ∴ Ekin = kT 2 But this also equals: 1 E kin = mv 2 2 3 = kT 2 3kT ∴v = m Mass of a rubidium atom: 85.5 g • mole–1 ⇒ 1.42 × 10–25 kg atom–1 λ= =
h h m = mv m 3kT h 3 kT m
7
What is the interatomic spacing as a function of T? Since question states that we start at room temperature, and maintain a constant volume as we cool, spacing does not change. From ideal gas law:
PV = nRT n P = RT V n P N P = ⇒ = V RT V kT
where now N = number of atoms and k = Boltzmann’s constant. Assuming atoms are evenly distributed in space, x distance apart from each other: density is 1 atom/x3 1 N P ∴ 3 = = x V kT
atoms
x
with P = 1 atm = 1.01 × 105 N • m–2 k = 1.38 × 10–23 J • K–1 T = 300 K ∴ x = 3.4 × 10–9 m, independent of T for this question
For condensation, x = λ (spacing between particles is about one de Broglie wavelength) h ∴ x= 3kTm h2 T= 2 x 3k m T = 6.5 mK This is actually much warmer than what is really required. (Our problem is that the equipartition theory breaks down for low temperatures.) c) Relatively vague question, possible answers: 1) increasing P often increases T1 which is undesirable 2) methods to cool atoms lose the warmest atoms, thus decreasing pressure d) Overall entropy of system does go up. 6.
The curious case of the cold calculating counterfeiter The assumption that the hole expands in the same proportion as the metal is valid. From the microscopic point of view, when one considers that the plate is composed of atoms, the inter–atomic distance increases in all direction for each atom. (It is similar to scaling up the grid points). One can also look at it from macroscopic view point.
8
Mathematically, it is a scale transformation, like a photocopier set to enlarge to, say, 110%. Suppose we have a situation that the brass is NOT inside the nickel. The brass will expand according to thermal expansion: r o → ro + ∆ B where ∆B = ro αB (T – To) Similarly the hole (of the nickel) will expand r o → ro + ∆ N where ∆N = ro ∆N (T – To) a) Now, the brass is inside the nickel. Since ∆B > ∆N, there is a contact force (consequently a pressure) between the two, as action = reaction forces, so that both the brass and the hole expand with the same expansion: r o → ro + ∆ Of course ∆N < ∆ < ∆B, and thus using different materials creates pressure. Since the pressure forces act perpendicular to the contact surface (we are dealing with the geometry of annulus), the pressure (Jpress) is primarily determined by Young moduli YB, YN. The pressure which inhibits the thermal expansion of the brass: ∆ −∆ pB = YB B Ι ro + ∆ where ro + ∆I is the radius of the brass after an equilibrium temperature T (room temperature). Similarly for the nickel, p N = YN
∆ − ∆N r + ∆ II
where ro + ∆II is the radius of the hole. Both ∆I and ∆II are negligible compared to ro and pB = pN = p due to action = reaction. So we have two unknowns and two equations: (∆B – ∆)/ro = p/YB (∆ – ∆N)/ro = p/YN 1 1 Y (∆B – ∆N)/ro = p + YB YN which we solve for p: p=
Thus
p=
(∆ B − ∆ N ) / ro (α B − α N )(T − To ) = 1 1 1 1 + + YB YN YB YN (19 − 11) × 10 −6 (293 − 77) = 1.06 × 10 8 Pa 1 + 1 −10 × 10 9.1 19.0 = 1050 atm
9
b) The expansion of the brass at T = 293 K p ∆ = ∆B − ro YB = [α B (T − To ) − p / YB ]ro 1.06 × 10 8 = 19 × 10 −6 ( 293 − 77) − × 6.00 mm 91 . × 1010 = 17.65 µm Thus the radius of the brass r = 6.0177 mm, and so is the inner radius of the nickel (the hole). The volume of the brass, where r is radius, t is thickness: VB = π r2 t = π (6.0177)2 2 = 227.53 mm3 = 0.228 cm3 The outer radius of the nickel also expands R = Ro [1 + αNi (T – To)] = 14.00 mm [1 + 11 × 10–6 (293 – 77)] = 14.0333 mm The volume of the nickel annulus VN = π (R2 – r2)t = π [(14.0333)2 – (6.0177)2] 2 = 1009.84 mm3 = 1.010 cm3 Hence the mass of tooney: m = (VB • 8.87) + (VN • 8.85) = 10.95 g compared to the one I measured: 7.31 g. This is about a 30% difference in weight — easy enough to tell the coin. In fact, Svend would NOT get away with his coin even by its rough appearance. The brass centre piece is actually 16 mm wide in a real tooney, not 12 mm!
10
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Problem Set 4: Optics and Waves Due February 14♥, 1997 1) JesseÕs Ôanti-musicÕ Jesse has a problem with his roommate Bongo. BongoÕs stereo is way TOO LOUD! In fact, Bongo has already blown his right speaker, and is pumping his Too-Much Music over the left one. So Jesse decides to fight sound with sound. He places an identical speaker of his own, on the left channel of the same station, face-to-face with BongoÕs and 6.0 m away. Jesse then sits with his one (remaining) good ear 1.5 m from his own speaker, on a direct line between the two. For the following, you can assume that Jesse and Bongo are pumping out exactly the same signal, in amplitude and in phase, and that JesseÕs head doesnÕt interfere with the sound. The speed of sound in his apartment is 331.45 m s Ð1. a) At what frequency(ies) in the music would Jesse hear minimum sound at his position? b) Are there points along the line joining the speakers at which Jesse cannot hear sound of any frequency? c) What ways might Jesse set up his speaker, better to cancel BongoÕs racket? [Chairul]
θ
2) A new twist on light Like any vector, the electric field of linearly polarized light can be considered as the sum of two components. A linear polarizer allows through only the component that is aligned to its ÔaxisÕ, and absorbs the other component. This results in MalusÕ law, which states that the light transmitted by a polarizer is: I = Io cos2θ, where I is the transmitted light intensity, Io is the incident light intensity, and θ is the angle between the Êv axis of the polarizer and the input E field direction. a) Why does MalusÕ law involve cos2θ, not cosθ?
b) Kimberly wanted to verify MalusÕ Law. She shone a flashlight on a photodetector and measured 1.0 mA of current generated by the detector due to the light intensity. She inserted a linear polarizer between the flashlight and the detector and measured the current. What was this value?
1
c) She inserted a second polarizer and rotated it to minimize the current generated by the photodetector. What was this current, and what was the angle between the polarizersÕ axes? d) On a whim, she placed a third polarizer after the first two and aligned it with its axis 45¡ to the first one. Without changing the orientation of the first two polarizers, she measured the current. What current did she measure? She exchanged positions between the second and third polarizers. What was the new current generated? e) As a result of her findings in section (d), she decided to extend her studies. With the first and last polarizers aligned perpendicularly, she inserted two polarizers aligned at 30¡ and 60¡ to the first one. What current did she measure? Write down the general expression for the amount of current generated for a total of N polarizers, all aligned at 90¡/(N-1) to each other. For really big N, what do you expect this value to be? f) For large N, Kimberly effectively rotated the polarization of the beam by 90¡. What problems would you face if you attempted part e) in real life? What is a much easier way to produce vertically polarized light from a horizontally polarized beam, using only a few standard mirrors? [James] 3) People see through Claire Claire (not her real name) plans to smuggle a tiny glowing radioactive pellet out of Ukraine. She safely implants the pellet in the centre of a solid transparent cube of index of refraction n = 1.75, but realizes the glow of the pellet will give her away. She decides to paint the cube black, but knows that Customs officers will be suspicious of such an obvious concealment.. Therefore she wants to paint only some parts of the surface. a) What is the least surface she must paint Ñ and in what shape? b) Unfortunately, at Customs she accidentally drops the cube in a glass of water and the inspectors catch her. What is the smallest surface she should have painted? (The index of refraction of water is 1.33.) [Chairul] 4) NothinÕ but blue sky... A plane wave of light has a wavelength λ. It scatters off a small sphere of radius a << λ. The light scattering off of the sphere causes the charges in it to oscillate and thus induces an electric dipole moment p in the sphere. p increases as a function of the volume of the sphere, p ∝ a3. Oscillating dipoles radiate electromagnetic waves, so that the intensity of scattered radiation Iscatt from such a sphere scales as the square of p: Iscatt ∝ p2. If we know the intensity of the incident wave, Io , we should be able to calculate the intensity of the scattered wave Iscatt as a function of the distance R from the sphere. In particular, we want to know how the scattered intensity varies with the wavelength of the 2
incident radiation. Note that the three length-dependent variables in the problem are λ, R, and a. a) By conservation of energy, the amount of radiation flowing through any spherical shell at any distance R from the sphere must be a constant. How does Iscatt scale with the distance R from the scattering sphere? b) The ratio Iscatt /Io is dimensionless. As a result, how must Iscatt depend on λ? Use the dependence of Iscatt on the other length-dependent variables (which you know from above) to obtain your answer. c) The situation above can be used to model the sky, where visible sunlight of all wavelengths scatters off of water molecules in the air. The human eye has evolved so that it is most sensitive to that part of the spectrum for which the intensity of light given off by the sun is maximal. As a result, we are most sensitive to green and this sensitivity falls off as we move to either the red or violet ends of the spectrum. Use this knowledge and the result of part (b) to explain why the sky appears blue, rather than, say, green. [Nipun] 5) Just tweezing... Focussed laser light can be used to push tiny spheres around. It is possible to coat these tiny spheres with biological antigens and make them glue themselves to particular polymer molecules, even the end of DNA strands. Then in moving the spheres with a laser, the polymer molecules can be manipulated, stretched, moved. Sometimes this is referred to as optical tweezers The force doesnÕt come from light pressure, exactly, but through refraction. Consider a plastic sphere 25 µm in diameter and with an 6 µm index of refraction n = 1.4. Visible laser light can be focussed to a spot a few micrometers across, through a microscope, so letÕs approximate the light by a single ray as illustrated, incident on the sphere at 6 µm off the normal-incidence axis. a) Find the approximate path of refraction through the sphere, and analyze the change in momentum for a photon of wavelength λ = 530Ênm. b) For a laser with power 1 W, what is the net force exerted on the sphere as it changes the momentum of all the photons in the beam? You can assume the sphere doesnÕt move [Robin] Web reference: http://www-leland.stanford.edu/dept/news/relaged/940509Arc4280.html (a sort of press-release on related work at Stanford University); perhaps also see ÔOptical Levitation by Radiation Pressure,Õ A.ÊAshkin and J.M. Dziedzic, Appl. Phys. Lett. 19, p.182 (1971)
3
6) Photons get the bends In one kind of description, lenses work by changing the curvature of the wavefronts of an incoming beam. When a plane wave passes through a positive lens, the flat wavefronts come out curved concave-forward. Then by HuygenÕs construction, it is not hard to see that the beam must collapse to a focus. The speed of the wavefronts is the phase velocity vφ = c/n, where n is the index of refraction. a) Consider a simple lens, flat on one side and with a spherical surface on the other, 50 mm in diameter, and 5 mm thick in the middle (see figure). If the glass has an index of refraction n = 1.66, find out how much the wavefronts in the middle of the beam are delayed in time relative to the part of the same wavefront that passes through the very edge of the lens. b) What is the radius of curvature of the new wavefronts, and where do they come to a focus? Something similar can happen in an ordinary flat block of glass when very intense light is incident on it. It turns out that the index of refraction depends also on intensity I, as: n = no + n2 I where n is the actual index of refraction, n o is the ordinary index of refraction for lowintensity light, I is the intensity of light in W cmÐ2, and n2 is a constant equal to 5 × 10Ð15 W cmÐ2 for glass. Where light is very intense, it increases the index of refraction it encounters. c) Consider light which has flat wavefronts as before, but is more intense in the middle than at the edges, with the distribution:
(
2 × 1013 −b + 10 6 – x 2 I [ W cm ] = 0 –2
where b ≡
)
for x ≤ 2.5 else
10 6 – (2.5)2 = 999.996875
Because the light is more intense in the middle, the wavefronts are delayed in the middle of the beam relative to the edge, and again the beam will come to a focus. For this intensity and for a block of glass 10 cm thick, find the approximate focal point of the beam. [Robin]
4
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Solution Set 4: Optics and Waves 1.
Jesse’s anti-music
This problem here is a simple standing-wave problem. Each speaker is at the antinode position. To have minimum sound, Jesse has to be at one of the nodes of the standing wave pattern. Wave patterns
Mode
n
fundamental
1
L = 1/2 λ
2
L= λ
3
L =3/2 λ
4
L=2λ
5
L = 5/2 λ
6
6
L=3λ
7
7
L = 7/2 λ
8
8
L=4λ
9
9
L = 9/2 λ
1
2
3
4
5
0.0
1.5
3.0
4.5
6.0 m
where L=6.0 m is the distance between the two speakers and λ the wavelength.
1
a) Analytically, the appropriate wavelengths can be obtained by imposing the antinode boundary conditions: cos ([2π/λ] L ) = ± 1, thus
[2π/λ] L = n π for n = 1, 2, 3, …
At position x =1.5 m, the intensity must be a minimum, meaning cos([2π/λ] x ) = 0 ⇒ cos([2π/λ] Lx / L) = 0 ⇒ cos([2π/λ] L [x / L ]) = 0 or 0 = cos ([n π]x /L ) = cos (0.25 n π) for the acceptable n. Thus n = 2, 6, …, and f = v / λ = v (2 m – 1) / L
for
m =1, 2, 3, ….
So the frequencies would be minimum for f = 55.24 Hz, 165.73 Hz, 276.21 Hz, …. b) No. He, in fact, would hear some frequencies louder (at maximum with intensity four times larger than its original without his own speaker). c) If Jesse has only one speaker, he could not do any better no matter what direction his speaker faces. If he has more speakers he could arrange them as such that more frequencies can be cancelled out. Nevertheless, there is some complication due to the interference between his own speakers. 2.
A new twist on light
This problem can be answered with different levels of rigour. To try and be as ‘correct’ as possible, while avoiding integration. I will define intensity as the time average of the square of the electric field: r I = < | E (t)|2> This is necessary since for visible light fields, the electric field is oscillating very fast (≈ 6 × 10 Hz), much faster than a detector can follow. This method can also be used for a slowly-varying –12 field by taking the time average over a length of time corresponding to detector response (≈ 10 – –6 10 s). We do not require ‘I’ as a function of time for this question. 14
PLEASE NOTE: Many of you may have taken a simpler approach (more likely to be found in an OAC-level course) by writing: r r I = | E |2 (ignore oscillating nature of E (t) ) This is o.k., but not as rigourous as my solution. a) Consider with a set axis and a general r (instantaneous) electric field vector. r a polarizer r where | E || | = | Eo |. Transmitted electric field is E ||. Thus transmitted intensity is: r IT = < | E || |2 >
2
r 2 = < Eo cos2 θ > r 2 = < Eο > cos2 θ r 2 but Ι ο = < Eo > ∴ IT = Io cos2 θ b) Required assumption: current on photodetector ∝ intensity of light. A flashlight is an unpolarized source. This means the polarization of the EM (light) field does not have a preferred direction. A really rough answer to this would be that since there is no preferred direction, 1/2 of the intensity is polarized parallel to the polarizer, and 1/2 is orthogonal. ∴ she measured 0.5 mA.
E polarizer axis
E||
θ
E
Let’s do this a little more rigourously. We can model the flashlight as a very large number of linearly polarized sources, all with the same amplitude, and with their polarizations pointing in all directions. What does Kimberley measure without the polarizer? N r 2 ITOT = <| ∑ En | > n=0
where N is very large. This can be written as: N r N v ITOT = <| ∑ E n|| + ∑ En⊥ |2 >
r where E n|| : N:
n=0
n=0
(1)
is a parallel component of the nth source. a very very big number (I’m working hard to avoid calculus here. If using calculus, N → ∞ and all the sums become integrals).
3
With the polarizer she oberves N r 2 Ipol = < | ∑ E n|| | >
(2)
n =0
We need
I pol ITOT
Take (1) ⇒ N r N r N r N r 2 2 ITOT = <| ∑ E n|| | + | ∑ En⊥ | + 2|( ∑ E n || )⋅( ∑ En⊥ )|> n =0
n= 0
n =0
n=0
Due to the absolute value brackets, we can write. N r N r N r N r 2 2 = < | ∑ E n|| | > + <| ∑ En⊥ | > + 2 < |( ∑ E n || ) ⋅ ( ∑ En⊥ )|> n =0
n =0
n =0
n =0
We know that the flashlight polarization had no preferred direction. Therefore no matter how Kimberly placed the polarizer, she would get the same result. Thus ‘parallel’ and ‘orthogonal’ are arbitrary and the first two terms must be equal. r r To simplify the third term, consider a case with only two sources. A, B The third term looks like r r r r <|( A || + B || )⋅( A⊥ + B⊥ )|> r r =0 since A || ⋅ A⊥ = 0 r A || ⋅ B⊥ = 0 r r B || ⋅ A⊥ = 0 r B || ⋅ B⊥ = 0 Therefore (1) ⇒
N r ITOT = 2 < | ∑ E n|| |2 > (since first two terms are equal). n =0
I ∴ Ipol = TOT 2 She observes 0.5 mA of current. c) Only linearly polarized light is transmitted by the first polarizer. Therefore the second polarizer transmits 2 I2 = I1 cos θ For minimum I 2 , chose θ =
π , ∴ I 2 = 0 and current = 0.0 mA. The angle between the axes is 2
90°.
4
d) Let I1 be intensity of light passing through first polarizer. Let I2 be intensity of light passing through second polarizer. I2 = I1 cos2 90° = 0 And the final polarizer is aligned at 90° – 45° = 45° to the second one: I3 = I2 cos2 45° = 0 ∴ she measured 0 mA. After exchanging: I1 2 I 2 I3 = I2 cos (90° – 45°) = 1 4 ∴ she measured 0.13 mA. I2 = I1 cos2 45° =
e) Output intensity is
2 2 2 Ifinal = I1 cos (30°) cos (30°) cos (30°)
3 = I1 2
6
≈ 0.42 I1 Therefore she measures 0.21 mA. By adding more polarizers, she is outputting more light! For N polarizers: N 2 90o I FINAL = I1 cos ( N − 1) By ‘subbing’ in increasing values for N, we see this coefficient is approaching 1. Slightly more rigourously 90 =0 lim N →∞ N − 1 2N lim cos 0 = 1
N →∞
Therefore she measures 0.5 mA. (Recall I1 is intensity of light passed by one polarizer). f) Real polarizers are not perfect. They still absorb some of the EM wave polarized parallel to their axes. Thus very little light would make it through N polarizers, where N is a large number. To change the polarization direction using two standard mirrors, recall what happens to a polarized beam in reflection:
5
polarization parallel to surface of paper
polarization orthogonal to surface of paper
So arrange two mirrors as follows:
3.
People see through Claire
A ray with incident angle less than θc
θi θc
θr ray with incident angle θc ray with incident angle θi > θc
B
The light-ray from the pellet will be refracted out if the angle of incidence (θ I), measured from the normal, is less than the critical angle (θ c). Due the geometry of the cube (shown as a square in the figure at left), θc has to be less than 45°. Otherwise the whole surface has to be covered. Any direct rays fall with θ i > 45° in one side will fall on the adjacent one.
If θc < θi < 45°, the ray will be totally reflected and it will be reflected forever! To see this effect refer to the figure. Coming from the source, the light will be reflected by side A with θ i . The second reflection will be θr = 90° – θ i > 45° which is certainly 6
greater than θ c , and the third will be θ i , and so on alternately. In this way a cube is a good container for hiding an object. The light rays can also run in a direction not parallel to the edges of the faces of the cube: the extreme case is one in which the rays are directed along the planar diagonal of the cube. In this case θi can be as large as 54.73° (i.e., tan–1 ( 2) ), with θr = 35.26°. To contain this ray also, we need θc < θr, i.e., θc < 35.26°. a) A point source will emit light spherically. So Claire must paint in a circular shape on each side. The area she covers must be such that θ i < θ c cannot penetrate. The radius of each circle will be determined from θc. θc = sin–1 (nfluid / ncube) For air nfluid = 1.00, thus θc = 34.85° and the radius of the circle r = d is the vertex of the cube. So the area she must cover is
1
2
d tan θc = 0.349 d, where
R = π r2 / d2 = 1 4 π tan2 θc = 0.3808 i.e., 38.08% of the total surface. b) First we have to check whether in water, θc < 35.26°. Using the same formula, θ c = 49.46° so Claire has to paint it all! 4.
Nothin’ but blue sky ...
a) Assume that the radiation is emitted symmetrically for 2 radii, R and R . We must have same 1 2 radiation flux through surfaces 4π R12 and 4π R22, so 1 I∝ 2 R Iscatt ∝ p 2 ∝ ( a 3 )2 Io
b) and
0.014
Iscatt 1 ∝ Io R Iscatt Io
to be dimensionless
1 dependence. As (length)4 ∞ is the only remaining length– dependent variable, we have requires a
Iscatt 1 ∝ 4. Io λ
Power (arb. units)
So for
scattered power (arbitrary units)
0.012 0.01
much more blue is scattered out of the beam than red is
0.008 0.006 0.004 0.002 0 200
300
400
500
600
700
800
900
Wavelength (nm)
7
This 1 dependence is known as ‘Rayleigh λ4 scattering’.
10-1 Relative Sensitivity
c) From part (b), longer wavelengths experience smaller scattering. Thus especially at sunset (when the sunlight passes through the longest stretch of atmosphere), looking at the sun, the blue end of spectrum is preferentially scattered away by water molecules and the sun appears reddish or red-orange. The shorterwavelength, bluer light — sent sideways from the transmitted beam of sunset light — has showered sideways to become someone else’s blue sky.
100
10-2
rods (scotopic)
cones (photopic)
10-3
10-4
Our eyes’ sensitivity peaks near green and sensitivity falls off towards red and violet. So, superposing the eye’s sensitivity and the Rayleigh scattering law, we see a peak near the blues.
5.
10-5
Wavelength (nm)
Just tweezing ...
As the beam enters the sphere, its direction deviates, due to refraction at the interface sinθ1 = n sinθ2
(Snell’s Law)
The amount the beam deviates is then ∆θ ⇒
400 450 500 550 600 650 700
α = θ 2 + ∆θ = θ 1 ∆θ = θ 1 – θ 2
θdev θ1
(opp. ∠ th’m)
when the beam leaves the sphere the angle of incidence at the exit is again θ 2 and then θ 1 at the exit, and again the angular deviation is ∆θ. Thus the overall deviation in angle is θdev = 2∆θ = 2(θ1 – θ2) b b sin θ1 = sin α = → θ1 = sin −1 r r 1 b b → θ 2 = sin −1 sin θ 2 = sin θ1 = nr n nr b b Thus θ dev = 2 sin −1 − sin −1 r nr
θ2 r
r θ2 ∆θ α
θ1 b 8
b = 6 µm n = 1.4
6 µm 6 µm θ dev = 2 sin −1 − sin 25 µm 1.4 25 µm = 2 (13.88° – 9.87°) = 8.03°
This new direction means photons have a new x-component of momentum. momentum is pγ originally, then px
If the photon
= pγ sin(θ dev ) b b = pγ sin 2 sin −1 − sin −1 r nr
here px = pγ sin(8.03°) = pγ (0.140) For a photon: p =
so
hν h –34 –7 = , where h = 6.62 × 10 J • s. Here λ = 530 nm = 5.30 × 10 m: c λ 6.62 × 10 −34 J s [but J: kg•m2•s–2 ] p= −7 5.30 × 10 m = 1.25 × 10–27 kg• m•s –1 ∆px = 1.75 × 10–28 kg•m•s–1
b) Laser 1W = 1 J/s 1 photon has energy hν =
hc (6.62 × 10 −34 J s) 3 × 108 ms −1 = λ 5.3 × 10 −7 m = 3.75 × 10 −19 J
so
1 J = 2.67 × 1018 photons per second dp 18 –28 –1 FTOT = x = Nx • ∆px = 2.67 × 10 ⋅1.75 × 10 kg• m•s dt = 4.67 × 10–10 kg•m•s –2 = 4.67 × 10–10 N
6. a)
Photons get the bends air only glass only
vφ = c / n
phase fronts move slower in glass where n = 1.66 9
nd 1.66 5 × 10 −3 m = • time through 5 mm glass t = d / vφ = c 3 × 108 ms −1 = 2.77 × 1011 s = 27.7 ps (picoseconds) • time through 5 mm air t =
d nd 1.00 5 × 10 −3 mm = = vφ c 3 × 108 ms −1 = 1.67 × 10 = 16.7 ps
–11
s
⇒ time difference is (27.7 – 16.7) ps = 11 ps b) In one way of describing it, lenses focus by putting a spherical curvature on phase fronts, then Huyghen’s Principle shows the wavefronts converge to a point — the focus 8 –1 –12 Right after lens, the 11 ps relative delay means distance ct = 3 × 10 ms • 11 × 10 s = 3.3 mm behind edges:
b 3.3 mm
50 mm
R a R
This is a chord of a circle. What is the radius of the circle? b = 25 mm 2 2 2 a + b = R a = R − 3.3 mm 2 2 2 so R = (R – 3.3 mm) + (25 mm) 2 2 R = R – 6.6 mm⋅R + 625 mm2 625 mm 2 R = = 94.7 mm 6.6 mm = 9.47 cm So the radius of curvature is 9.47 cm, and this is where the beam comes to a focus 9.47 cm after the lens!
9.47 1 3/8" cm 10
c) The form of I looks complicated. That is because it is pre-cooked to give an easier answer! n = no + n 2 I ∆n = n2I across the beam, making wavefronts curved following intensity changes across beam. Let’s find wavefront delay according to x ∆t =
n( x )d c
n(x) = no + n2 I(x) d = 10 cm = 0.1 m c = 3 × 108 ms–1
no d n2 I ( x )d + c c constant for all ↑ ↑ causes curvatures in wavefronts ∆t =
and the physical displacement of the wavefronts is: c∆t = no d + n2 I(x)d constant (0.17 m) ↑ ↑ depends on x So all the curvature in the beam comes from the second term: ∆z = n2 I(x) d = 5 × 10–15 • 2 × 1013 W • cm–2 • (–b + 10 6 − x 2 ) = 0.01 (–b + 10 6 − x 2 ) = 0.01 ( 10 6 − x 2 – b) 2 –1 Note the units of n2 : cm W — there was an error in the question! (eek!)
Then
(100 ∆z + b) = 10 6 − x 2 (100 ∆z + b)2 = 106 – x2
Take y = 100 ∆z, then x2 + (y + b)2 = 106 cm2 the equation of a circle, offset by –b along y, with R = 103 What is the y-displacement? y = yo where x = 0 2 6 (y + b) = 10 3 y + b = 10 y = 103 – b = 1000 – 999.996875 –3 y = 3.13 × 10 so
∆z = y/100 = 3.13 × 10
–5
m
11
Use the same method as in (a) to get the curvature a 2 + b2 = R2 2 6 2 R = 10 cm 3 R = 10 cm = 10 m
a = 2.5 cm b = 999.996875 cm
The beam focuses in 10 m.
12
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Problem Set 5: Electricity and Magnetism Due March 7, 1997 1) Gauss visits Planesville... The (very obscure) town of Planesville, Manitoba is much like any small Canadian town, except that the town is two-dimensional Ñ it is lacking the third dimension that we experience everywhere else on earth. We want to see how GaussÕs law might be different in Planesville. For an arbitrary charge distribution, GaussÕs law takes the form of an integral, but recall that for a point charge Q, we can draw a gaussian sphere at radius R from the charge and use the radial symmetry to write GaussÕs law as: Esphere ¥ (area of sphere) = Q/εo a) If CoulombÕs law applies in Planesville in the same form it does everywhere else, would GaussÕs law still apply in its usual form? (Hint: Derive the form of GaussÕ law in a 2-D world for a point charge.) b) How could we ÔfixÕ CoulombÕs law so that GaussÕs law would apply in Planesville? Write the expression for the new force law. c) For the force law of part (b), consider a solid disc of charge of radius 10 cm and charge density 0.5 mC mÐ2. What is the electric field of such a disc in Planesville as a function of the radial distance from the centre of the disc? Sketch a graph of this field as a function of the radial distance from the centre of the disc. [Nipun] 2) ItÕs not just a good idea... itÕs OhmÕs law!
Êv Êv Êv OhmÕs law for conductors can be written in the form V = IR or as J = σ E , where J is the current per unit cross-sectional area flowing in a wire and σ is the electrical conductivity of the material. Ê For our purposes, the important thing is that the current is proportional to the v electric field E , so a constant electric field produces a constant current. Note also that the current in a wire is proportional to the average velocity with which the charges are moving. a) Êv A single point charge Q is moving freely under the influence of a constant electric field E . Is such a charge moving with a constant velocity, or is it accelerating? Does this charge obey OhmÕs law?
1
b) Now, instead of free point charges, we consider electrons flowing through a wire. Such electrons undergo occasional collisions with ions making up the metal. On average, an electron travels a distance d before colliding with an ion and stopping altogether. For a Êv constant applied field E , what is the average time between collisions? What is the average velocity of an electron? Does such a result agree with OhmÕs law? c) In part (b), we neglected the fact that the electrons (at room temperature) have a lot of thermal energy. Therefore, they are constantly jiggling about in random directions with a thermal velocity v t. If this thermal velocity determines the time between electron-ion collisions (because vt is much greater than the electron velocity due to any applied field), Êv what is the new time between collisions? If we then apply an electric field E to the wire, what is the average electron velocity? Does this result agree with OhmÕs law? d) If the model of conduction in part (c) was accurate, what would Êvhappen to the current at very high and very low temperatures for a constant applied field E ? [Nipun] 3) Total Recoil After seeing a recent Schwarzenegger film ERASER, Noah was impressed by Arnie's nasty rail-gun type weapon. The hand-held gun fired metal projectiles at near-light speed with rather destructive results. Noah decided to try to build one himself. He recalled a physics demonstration, in which a metal ring is placed on a metal post, attached to the top of a solenoid. The solenoid consists of a wire wrapped many times around a cylindrical metal core. When a sudden voltage was v applied across the ends of the B wire, the metal ring was launched away from the solenoid. Noah decided to make use of this effect ring post to build his own rail gun. switch Noah's first design contained the following components: solenoid (core wrapped with 2000 turns of wire, length = 50 cm, µ = 500 µo), metal post (length = 5 cm, µ = 500 µo), metal ring (inner diameter = 1 cm, outer diameter = 1.1 cm, thickness = 0.5 cm, resistance = 10Ð3 Ω , mass = 0.5 g ). a) With the metal ring anchored to the post so it could not slide, Noah turned on the voltage source and found that the current did not reach its maximum value immediately. He measured the current (in amperes) as a function of time and determined that it satisfied the following equation: I = 1.0 (1 Ð exp(Ðt /T)) with T = 0.04 s. What caused this time lag?
2
b) From a physics text, Noah determined that the magnitude of the magnetic field at the end of a solenoid is: B = µ N I /2l where N is the number of turns of the solenoid, l is the length of the solenoid, and I is the current in the wire. What value of magnetic field does Noah generate, as a function of time? c) Recalling Faraday's law of induction, what is the magnitude and direction of the current induced in the anchored metal ring as a function of time? You can assume that the current response of the ring is instantaneous. d) If the solenoid creates a uniform Êmagnetic field, pointing exactly parallel to the metal v Ê v Êv Êv ringÕs axis, what is the Lorentz force (F = q v × B , where v = velocity of particle of charge q) the ring feels as a function of time and in what direction? If the ring weren't anchored to the post, what would be its resulting motion? e) The magnetic field is of course not uniformly pointing parallel to the solenoid axis but starts to diverge as it exits the solenoid (as shown in the diagram). For simplicity, assume that at the fixed position of the ring, the field lines leave the post at 15¡ to its surface. What is the new net force on the anchored metal ring, as a function of time? f) Noah turned off the voltage supply and removed the anchors so that the metal ring could slide freely on the post. He closed the switch again and the ring was projected forward. To get an approximate value for the resultant speed, approximate the force applied to the metal ring for arbitrary time t to be the same as calculated in part e) at time t = T. Use this constant force to get a rough idea of the velocity of the ring as it left the metal post. g) Noah decided to improve his design. In the movie, Arnie's gun was supposed to fire at near-light speed, but Noah didn't want to be greedy. He redesigned his device to launch the projectile with a velocity of 0.01 c (c = speed of light, 3 × 108 m¥sÐ1). Somehow Noah succeeded in building it and though it worked perfectly, Noah was seriously injured. What had he forgotten to take into consideration? [James] Êv 4) The Lambda particle: outstanding in its (B ) field. In the dawn of the field of particle physics, the discovery of the Λ Ð particle played a significant role. It is relatively long-lived. One of the most important decay modes of this particle is the decay to proton and pion: Λ → p+ + ¹ Ð The minus sign for ¹ denotes that the charge of this pion (¹) is Ðe. Suppose the free kinematics of this decay is such that all particles before and after the decay move in the
3
same direction (see figure). Now a magnetic field of 1 T, perpendicular to the motion of the particles, is applied to separate the moving proton and the pion. If the radius of the proton path is 3Êcm, what is that of the pion? (rest-masses: MΛ = 1115 MeV ¥ cÐ2, Mp = 938 MeV ¥ cÐ2, M¹ = 140 MeV ¥ cÐ2, where c is the speed of light) This problem can be solved in a non-relativistic way (even though it will not be accurate), however, one should take into consideration the mass missing after decay, which will contribute to the kinetic energies of the proton and the pion. [Chairul] 5) Bohring after the truth
p
Λ
π decay (no magnetic field)
p Λ π decay (under magnetic field)
In the Bohr theory of the hydrogen atom an electron circles the proton in an orbit of radius 0.053 nm. The electrostatic attraction of the proton for the electron furnishes the centripetal force needed to hold the lectron in its circular orbit. Find, in this classical model: (a) The force of electrical attraction between the proton and the electron. (b) The speed of electron. What is this speed as a fraction of the speed of light c? [Chairul] 6) Plasma, plasma, on the wall Practically all of the mass of the universe is not solid, not liquid, and not quite gaseous, but in the form of plasma. Plasma is any ionized matter, whether ionized by heating (as in the sun), by electric discharge (as inside a fluorescent light tube), or by dissociation in solution (as in blood). When spacecraft re-enter the earthÕs atmosphere, the heat of re-entry produces an envelope or sheath of ionized air and ablating heat-shield surrounding the craft. The density of this plasma increases as the craft enters thicker atmosphere, and later dissipates as the vehicle slows down and cools. This plasma sheath can actually act like a mirror for radio-wave communications between vehicle and ground-control, reflecting the signals and blocking communication for a period during re-entry. A similar kind of reflection takes place for short-wave radio waves reflecting off the right part of the ionisphere. This happens because plasma is a good conductor and because plasma of a certain density has a characteristic resonant frequency, the plasma frequency. Consider a hydrogen plasma, made of a mix of electrons and protons only, with equal density of electrons and protons. If in a region of plasma (as illustrated) we pull the electron component to the left, away from the proton component, it leaves two slabs of excess charge, like a parallel-plate capacitor.
4
b) Find the force on the electrons and ions within this parallel-plate capacitor.
+ + – – –++ + + ++++++ +++ ++ + +++ – – – ++++++ + + + ++ ++++ + + + ++ + + – ++ + + + + + + + – + + –– –– ++ ++– –+– + + + +++++ ++ –+–++++++ ––––– +–++ – –– ––––– –++ + + –+––+++ + ++++ ++++ +++ –+––+ + ++ +++ ––– ––+–+– + + + + – +– + – –+ – + – – – – –––––– ––– –––– + –– –– – + – – ––– ––––– – + – – + –––– –––– ++ + – – +++ ––– ––––––– ––– – – – – + – – –– – ++++ ––––– ––– –– – – – + + ++ – – ––– – ––
a) Find the amount of charge in each charged slab, assuming a displacement x of the electrons relative to the protons, and a particle density of N cmÐ3 for the protons and the the electrons, each.
c) Show that this is a restoring force F = – kx, like a mass on a spring. Under the assumption that the protons have so much more mass that they hardly move, show that the force constant k leads to an oscillation frequency for the electrons sloshing back and forth given by:
νp =
1 1 ωp = 2π 2π
N e2 ε o me
where N is the particle density (electrons or ions), e is the electron charge, and m the electron mass. This is the frequency of best reflection of electromagnetic waves by the plasma Ñ any frequency lower than this is reflected. d) Find the minimum density N of plasma needed to reflect shortwave radio at λ = 30 m. [Robin]
5
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Solution Set 5: Electricity and Magnetism 1.
Gauss visits Planesville ...
a) for a pt. charge in 2-D: GAUSS’ LAW: E • (circumference of circle) =
Q 4 πε o R
2
2 πR =
Q 1 Q ⋅ ≠ 2ε o R εo
which is what one would expect from Gauss’s law b) to ‘fix’ this, make Coulomb's law ∝
1 R
, i.e., E= then
Q 1 ⋅ 2 πε o R R
(point charge)
E ⋅ (circumfere nce of circle ) =
Q 1 Q ⋅ ⋅ 2π R = 2πε o R εo
(satisfies Gauss’s law)
c) inside disc: 2 E •2π•R = σ • πR –2 where σ = 0.5 mC•m σ R, R < R o , 2
(Ro = 10 cm)
outside disc: 2 E • 2πR = σ • πRo –2 where σ = 0.5 mC•m ⇒ E=
E(R)
⇒E =
Field needed to satisfy Gauss's Law in 2-D
σ Ro2 2 R disc radius Ro
2.
It's not just a good idea ... it's Ohm's law!
a)
F = ma = QE, J ∝ Vavg =
∴a=
qE m
R
∴ charge experiences constant acceleration
qEt ∴ current should increase linearly with time! (‘non-Ohmic’) m
1
b)
d=
1 2 at ⇒ t = 2 ∴ vavg =
2d 2d ∝ a E 1 1 at = a 2 2
2d ∝ a∝ E a
∴ it is non-ohmic c)
if v
thermal
>> v
due to field
, t=
d vthermal
1 1 d then, v avg = at = a ∝E 2 2 vthermal ∴ it is ohmic d)
σ∝
1 v thermal
from part (c)
so, as T increases, vthermal increases so the resistance is greater. If T decreases, vthermal decreases, so the resistance decreases. 3.
Total Recoil
a) The voltage increase causes an increasing current in the wire of the solenoid. This increasing current creates an increasing magnetic field through the solenoid. This increasing magnetic field acts back on the solenoid wires to create an ‘induced’ voltage which opposes the original voltage change, and will continue until the system reaches steady-state. In brief, the effect is the ‘induction’ of a ‘reverse’ electric field (and current) in the solenoid. µNI µN (1 − e − t / T ) = 2l 2l 500µ o (2000 )(1 − e − t / 0.04 ) = 2(0.5) = 1.3(1 − e − t / 0.04 ) Tesla
B= b)
c) Faraday’s law of induction dΦ E =− dt where ε is the induced electric field r r M is the flux = B ⋅ A r B is the magnetic field r A is orthogonal to the loop area r A = area of loop r Since the B direction is effectively orthogonal to the loop 2
r r Φ = B⋅ A = A B so dB dt µN d 0.01 2 = –A (1 − e − t / T ), a = π m2 2 2l dt –3 –t/T V = 2.5 × 10 e ε ..i= R –t/–0.4 = 2.5 e •A
ε = –A
(in opposite direction to current in solenoid.) (We can ignore back-induction from the ring since it is very small.) r r d) F = qv × B Let n be the charged particle density per unit length. Thus q = –neR r r F = − nelv × B r F r r = − nev × B l r r But within this model B is orthogonal to v and –ne |v| = i (current). r F r =i B v l µN = 2.5 e − t / 0.04 (1 − e − t / 0.04 ) l 2 = 3.3 e − t / 0.04 (1 − e − t / 0.04 )
F
r r r r Direction of F is parallel to v × B; v r is parallel to the ring; B is orthogonal r to the ring. Thus F is on the surface enclosed by the ring, pointing towards the centre of the ring.
out of page
B
ring, looking down solenoid
This results in NO motion (unless the ring implodes).
3
r
e) Even in this case v is still r r orthogonal to B , but now F is not on the surface enclosed by the ring. r The vertical components of F cancel, and we are left with a horizontal component facing to the right. r F FNET = l sin l –t/.04
v F
post
solenoid v
–t/.04
F(t = 0.04 s) = 2.7 × 10 –3 = 6.3 × 10 N but
out of page into page
F
(1 – e ) (.01) π sin 15 = 3.3 -2 –t/.04 –t/.04 = 2.7 × 10 e (1 – e )N f)
B
–2 –1
e
B
°
–1
(1 – e )
F = ma F ∴a= m 2 = 13 m/s
For a constant acceleration: 2
2
but v1 = 0
The ring would leave the post going 1.1 m/s. g) Noah forgot about the conservation of linear momentum. Consider
m1 to be the mass of the ring m2 to be the mass of the gun and Noah v1 to be the final velocity of the ring v2 to be the final velocity of the gun and Noah
Since the system starts at rest: m 1 v 1 + m2 v 2 = 0 v1 = –v2 m 2 m1 Guess that Noah and gun mass is on the order of = 100 kg. Their final velocity is
4
v1 = − 3 × 10 6
0.0005 100
= 15 m / s = 54 km / hr Being accelerated to > 50 km/hr in such a short time could hurt quite a bit. Even if Noah wasn’t hurt in the acceleration, the deceleration would probably be quite disastrous. 4.
The Lambda particle: outstanding in its (B) field
We can determine the speed of proton from the radius of its circular path Rp = pp/e B = Mp vp / e B
(for non-relativistic case).
So, its speed –2
6
–2
vp = Rp e B / Mp = (3 × 10 m) e (1 T) / (938 • 10 eV • c ) 6 –1 = 2.88 × 10 m • s = 0.00959 c.
(4.1)
The assumption of non-relativistic proton is applicable here. We apply the conservation law of energy and momentum: EΛ = Ep + Eπ p Λ = pp + pπ which can be written non-relativistically as 2
2
2
2
2
2
MΛ c + 1/2 MΛ vΛ = Mp c + 1/2 Mp vp + Mπ c + 1/2 Mπ vπ ,
(4.2)
MΛ vΛ = Mp vp + Mπ vπ.
(4.3)
vΛ can be eliminated by taking the square of (4.3) and substitute the term in (4.2) 2
2
2
2
2
2
MΛ c + 1/2 (MΛ) (Mp vp + Mπ vπ) = Mp c + 1/2 Mp vp + Mπ c + 1/2 Mπ vπ 2
(1115 − 938 − 140) MeV + (938 × 0.00959 + 140 vπ/c) /(2 × 1115) MeV − 2 2 938/2 MeV (0.00959) − 140/2 MeV (vπ/c) = 0 2
61.21 (vπ / c) − 1.13 (vπ / c) − 36.99 = 0 6
–1
So we can obtain vπ = 0.787 c = 2.36 × 10 m • s . This is a relativistic problem and the assumption of non-relativistic pion is incorrect. Nevertheless, let us find out the radius of the pion path. 6
–1
Rπ = pπ / e B = Mπ vπ / e B = (140 × 10 eV • c • 0.787)/ e (1 T) = 0.367 m = 36.7 cm The full relativistic treatment (especially for pion) gives the radius of 40.0 cm.
5
Use the following formulae: 2
2
2
2
MΛ c + 1/2 MΛ vΛ = Mp c + 1/2 Mp vp + Eπ, MΛ vΛ = Mp vp + pπ, 2
4
2
E π = Mπ c + p π c , to solve for pπ and Rπ = pπ / e B.] 5.
Bohring after the truth
a)
Applying Coulomb’s law: 2 2
9
F = k e /r = (9.00 ×10 ) • (1.602 × 10 –8 = 8.22 ×10 N b)
–19 2
–9 2
) / (0.053 × 10 )
2
The centripetal force F= m v / r. Hence the speed of the electron is –8
–9
v = √ (F r / m) = {(8.22 ×10 ) • (0.053 × 10 ) / (9.31 × 10 6 –1 = 2.19 × 10 m • s = 1/137 c 6.
–31
)}
1/2
Plasma, plasma, on the wall
a) Consider a slab of plasma, as in the question, which measures x × y × z. If the negative electrons are pushed off the positive protons to one side, by a tiny amount ∆x, then there will be a thin slab of excess charge on either side — one positively charged and one negatively charged. The volume of each little excess slab will be: (∆x) × y × z
[1]
If the density of electrons is N [cm–3], then the amount of excess charge will be N e (∆x) × y × z = N e (∆x) × A
[2] where A is the area of the side of the slab
In other words, what we have is something like a parallel-plate capacitor, with the two thin excesscharge slabs as the two charged plates of the capacitor. b) For this, we can find the field between the two plates of a parallel-plate capacitor. In the middle, where there are both electrons and ions, the charges of each cancel each other out (unless you give them time to move around and redistribute themselves), so the net field in the middle is just what is produced by the thin slabs of excess (unbalanced) charge. E = 4 π kc σ
where kc = 9 × 10–9 N m2 C–2 σ = charge per unit area on each plate
[3]
6
[The easiest way to see this is to use Gauss’s Law (if you know it), drawing little rectangular boxes with sides parallel to the plates, and having one side between the plates and the opposing side outside the plates. The field is perpendicular to the plates, by the symmetry of the situation, and it is quick to find the contribution each parallel plate makes to the overall field.] Then from the second part of [2], we can find the charge per unit area of the two slabs of excess: σ = (N e (∆x) × A) / A = N e (∆x)
[4]
and the field between the plates is: E = 4 π kc N e (∆x)
[5]
Now the charges in the middle see the field produced by the excess charges, and experience a force: F=qE
which depends on q, +ve or –ve
[6]
This then gives us the force on each charge within the thin slab: F = q E = q 4 π kc σ = e 4 π kc σ = e 4 π kc (N e (∆x) )
[9]
= 4π kc N e2 (∆x) c) The whole block of electrons in the diagram of the question is free to move (the ions, being more massive, have a certain ‘right of weigh’) in the electric field they see. Most of them see the electric field between the two plates, and this pulls them back onto the ion background. So the force on an electron between the charge-excess thin slabs is a restoring force against the separation of charge. Fr = – k (∆x)
where k = 4π kc N e2
[10]
This is the restoring force of a simple harmonic oscillator (SHO) — it means the electrons mostly will slosh back and forth past the ions, barring collisions, once they are released from their initial displacement. Since the ions are relatively massive, they hardly move, but the electrons oscillate sinusoidally, as does a SHO, with a frequency which depends on the electron mass: ωp =
where ε o =
1 4π kc
[11]
Then νp = ωp / 2π to get the result shown. This is called an electron plasma wave. d) When an electromagnetic wave is incident on a plasma, the E-field can begin to drive electrons back and forth, causing something like the excess charge in the model above. The electrons then oscillate at their own frequency, given just above in [11]. If the EM wave is at exactly the same frequency, it is in resonance with the electron plasma wave, 7
and oscillations can grow very large. This means that a large AC current flows back and forth in the plasma, and that current can radiate EM waves itself, cancelling the light that comes in and sending it back as a reflection. At what frequency does this happen for shortwave radio at λ = 30m? c = λ ν ⇒ ν = c / λ = 3 × 108 cm•s–1 / 30 m = 1 × 10 7 s–1 v = 1 × 10 7 = ⇒ N =
1 2π
Ne 2 εo m e
vp =
1 1 ωp 2π 2π
[12]
Ne 2 εo me
(2 πv) 2 ε o m e = 124 . × 1012 m −3 = 1.24 × 106 cm − 3 2 e
Compare this to gases at sea-level: 1 mole of ideal gas occupies 22.4 l @ STP, 22.4 l = 22.4 × 103 cm3 6.022 × 10 23 cm − 3 = 2.69 × 1019 cm − 3 3 22.4 × 10 Much less dense than this — shortwave radio is usually reflected not from fully ionized gas at atmospheric pressure, but from a layer of partially ionized highly rarefied gas high up, in the ionosphere.
8
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Problem Set 6: AC Circuits and Electronics Due March 28, 1997 1) WhatÕs a gigawatt? Thirty of Ôem?? The province of Ontario requires something like 3 × 1010 watts of electrical power (aÊplausible guess?: a kilowatt or so for every person, and then twice as much again for all of industry), at 110 volts AC (r.m.s.). A heavy power cable might be an inch in diameter. Say the whole province was supplied by a single cable a metre in diameter Ñ what would happen? Take it that the cable has a resistance of 0.15 µΩ mÐ1. Calculate: a) the power lost per metre from ÔI2R losses,Õ b) the length of cable over which the whole 3 × 1010 watts would be lost, and c) how hot the cable will get, if it loses all this power by blackbody radiation. Your answer ought to be preposterous Ñ why is our electrical power system not ridiculous in this way? [Robin] 2) Waffling on the issues... the issue of waffles A diode is an electronic device that allows current to flow only one way. An ideal diode has zero resistance for positive voltage and huge resistance for negative voltage. A slightly more realistic I(A) diode has a turn-on voltage, as shown in the figure at right. One typical use of a diode is as a light source. These light-emitting diodes (LED) produce visible light 0.1 when a voltage greater than the turn-on voltage is applied across the device. -2 2 Voltage (V) a) After many serious burns, Anton (the Waffle Czar) Duzzleeater pledged to improve current waffle-maker technology. He decided to incorporate an LED as an indicator that would light up whenever the waffle iron is plugged in. To begin, Anton decided to characterize his voltage source Ñ normal household AC Ñ so he measured the voltage as a function of time. Graph what Anton measured (2 cycles is sufficient). DonÕt forget to indicate your axis and scales. 1
b) Without bothering to think about it, Anton wired the LED in series with the waffle heating element (schematic at right). If the diode had a similar I–V curve to the one shown in figure 1, what would be the voltage drop across it as a function of time? Does the LED emit light? When?
LED
heating coils (R=10 Ω)
c) Unfortunately for Anton as soon as he plugged the circuit in, the diode flashed a brief burst of light and began smoking. The diode couldnÕt take the excessive current. Draw a more intelligent circuit that would avoid this problem. [James] 3) 2B ∨ (~2B) ← Q! [...to be or not to be, that is the question!] Wendy, a Physics Olympiad student, is given a bunch of digital gates. Unfortunately, she only has three types of gates: AND, OR, and NOT.
OR
Her supervisor said to her that she can make any combination. Lo and behold, she finds AND ((NOT A) AND B) OR ((NOT B) AND A) circuit is a very useful combination. Draw the circuit for that combination. Make a Boolean table for that circuit. Can you name that NOT simple circuit? [Chairul] 4) Connect the dots! Who said that physics wasnÕt all fun and games? See the figure on a separate page: the challenge is to connect the dots using the components given, to simultaneously achieve the desired voltage and current. Note that you may not need to use all the dots (or components) and you may want to attach more than two components to the same dot. The symbols ÔAÕ and ÔVÕ correspond to ideal ammeters and voltmeters. [James] 5) Impeding oneÕs reactance In a DC circuit, the total resistance is a sum of the individual resistances in the circuit, the nature of the sum depending on whether the resistors are in series or in parallel. In an AC circuit, the impedance, Z, plays the role of resistance and sums in the same way as resistance, however it has both real and imaginary parts. For ideal resistors, inductors and capacitors, the contributions to the total impedance are as follows: resistors:
ZR = R
inductive reactance:
Z L = iω L
capacitive reactance: ZC = (iωC)–1
2
Here, ω = 2¹ f, where f is the frequency of the input AC signal. a) What is the total impedance of the circuit in the figure below? b) We are interested in only the amplitude of V out, which is given by |V out| = (Vout ¥ V out*)1/2 where Vout* is the complex conjugate of Vout. What is |V out| for this circuit? Sketch |V out|/|Vin| as a function of log10 ω. R
c) How would such a circuit affect a highfrequency input signal? A low-frequency signal? What do you conclude it might be used for? [Nipun]
V in
V out
C
6) Say ÔcheeseÕ! (...Zap !!) In high-power lasers pumped by a flashlamp discharge, large capacitors are charged to several kilovolts and then a fast, highvoltage switch closes to discharge them through the flashlamp. It is very much like an electronic camera flash, but hugely larger. These are the facts about the flashlamp circuit: ¥ the electric current pulse through the flashlamp must last for a certain amount of time τ in order best to ÔpumpÕ the laser ¥ if only a flashlamp and capacitor are connected in series, the current will oscillate back and forth through the lamp before dying away, and this can damage the lamp or cause it to explode ¥ the best shape results when the capacitor voltage is a simple decaying exponential
L
¥ the exponential should be the fastest-decaying one possible To make these things possible, an inductor like a coil of heavy wire is added to the circuit, which can then be represented as shown at right: C Ð capacitor; L Ð inductor; R Ð resistance of flashlamp; i(t) Ð time-dependent current through flashlamp.
C
R
i(t)
a) What is the equation for the charge q on the capacitor? What is the voltage across the capacitor in terms of the charge q on the capacitor? Show your approach. b) Assuming the voltage is the fastest-possible decaying simple exponential, and that C and R are fixed, what must be the value of L? What is the exponential decay time, given RÊ= 1Ω, C = 200 µF? c) Find the current i(t) through the resistor R. If the capacitor is charged to 2.5 kV, how much energy is discharged through the lamp? [Robin]
3
Figure for ÔConnect the Dots!Õ
e.g.) Set voltage=5V, I=5A
A
V
A 5V
wires
1Ω
10Ω
1Ω
5V
V
a) Set voltage=4V, I=.2A
A
6V
10Ω
V wires
5Ω
b) Set voltage=6V, I=.2A
A 4V 6V
6V
6V
V 10Ω
c) Set current I=0A (note: short-circuiting batteries is baaaaaad!)
10Ω
5V
A 10Ω
50V
90Ω wires
4
1996-1997 Physics Olympiad Preparation Program — University of Toronto — Solution Set 6: AC Circuits and Electronics 1.
What's a gigawatt? Thirty of 'em??
a)
Power lost per meter, with P = IV (i.e., power and voltage are pre-set) 2 P loss = I2 R = (P/Vrms)2 R = (3 × 1010 W / 110 V ) • (1.5 × 10–7 ohm • m–1) = 1.1 × 1010 W • m–1
b)
length for loss: 3 × 1010 W / 1.1 × 1010 W • m–1 = 1.1 × 1010 W • m = 2.7 m (a pretty pathetic distance!)
c)
power loss by Stefan-Boltzmann law: P= σ A T4 trivial power gain from 20°C room temperature, i.e., likewise: P= σ A (293 K)4 so that net: P= σ A [(T4 – (293 K)4] ≈ σ A T4 Area of surface of 2.7 m of cable is A = 2π r L = 2 • 3.14 (0.5 m) 2.7 m = 8.48 m2 T = (P/(σA))1/4 = (3 × 1010 W / (8.48 m2 • 5.670 × 10–8 W m–2 K–4)) 1/4 =15,800 K
This doesn’t happen, basically because of Nikolai Tesla: with AC current we can easily manage step-up transformers, so the power is delivered at much higher voltages (like 250,000 V, I recall) with smaller losses, and then stepped-down several times, the last time quite near the delivery site. As well, the total diameter of cables is far in excess of 1m. 2.
Waffling on the issues ... the issue of waffles
a) Normal household voltage in about 120 V (rms) which corresponds to 170 V peak-to-peak 170
voltage (V) 1/60
time (s) 2/60
-170
1
b) The LED emits light only when it is forward-biased, as indicated by the asterisks in the graph below. Thus it flashes at 60 Hz. 170 voltage across 2 L.E.D. (V)
*
* time (s) 2/60
1/60 -170
c) One possible circuit is shown at right, where R b is a large resistor. This reduces the amount of current that flows through the LED.
3.
2B ∨ (~2B) → Q!
Rb AC
heating coils
[... to be or not to be, that is the question!]
The circuit of this combination is A
C
E
G
B
D
F
Boolean table Input A T T F F
B T F T F
C F F T T
intermediate nodes D E F F T T F F T F
F F F T F
Output G F T T F
Clearly, this circuit is an exclusive OR gate (XOR).
2
4.
Connect the dots! a) Set voltage=4V, I=.2A
A
All R=10Ω
V wires
5Ω
6V
A
b) Set voltage=6V, I=.2A
V
6V
6V
6V
10Ω
4V
10Ω
c) Set current I=0A (note: short-circuiting batteries is baaaaaad!)
50V
5V
10Ω
90Ω
A 10Ω
wires
5.
Impeding one's reactance
a)
As they are seen by VIN, the impedances are in series. Therefore, the total impedance is ZT = R + (iω C)
–1
Similarily, as seen by VOUT Z= b)
1 iω C
The ratio of VOUT to VIN is given by the ratio of the impedances.
3
VOUT ZT 1 1 = = R+ VIN Z iω C iω C
−1
=
1 1 + iω CR
1
VOUT VIN
1
1
* V VOUT 2 2 2 1 1 1 OUT = = × × = 2 2 2 VIN VIN 1 + iω CR 1 − iω CR 1 + ω C R 1
so, VOUT = VIN
2 1 2 2 2 1 + ω C R
1
c) for ω << | Vout | / |Vi n|
0.8
for ω >>
0.6
0.2
–log(RC) 1
ω (arb. units)
1 , all voltage is dropped RC
So this is a filter which allows only low frequency AC signals to pass. The cutoff frequency can be changed by adjusting R and C
0.4
0 0.1
1 , all signal is passed. RC
10
100
6.
Say 'cheese'! (... Zap !!)
a)
Each element has a charge-dependent voltage associated across it: resistor: ∆V R = IR capacitor: ∆VC = q/c dI inductance: ∆VR = –L dt
dq , and since the potential drops around the circuit must sum to zero, we can establish a dt relation in terms of change of charge. Be careful about the signs: Since I =
• choose allocation of +q/–q on capacitor • +q plate is at higher potential, so sketch current direction (flow of +ve charge) consistent with dropping potential Then the current carries charge away from +q dq so I= dt
4
d2q q/c + L + dt 2
dq =0 R
or L Then
b)
d2q dq 1 + R + q =0 dt C dt 2
a ‘second-order ordinary differential equation’
∆Vc (rise) + ∆VL (drop)+ ∆VR (drop)= 0 dI q/c – L – IR = 0 dt assume the form V(t) = Voe so q(t) = CVoe
–t/τ
–t/τ
and substitute into differential equation 2 –t/τ 1 + RCV − 1 + Vo}e = 0 {LCV o o τ τ e
–t/τ
0 must mean
2 1 – RCV LCV o o τ
1 + Vo = 0 τ
2
or multiplying by τ both sides 2
τ – RC τ + LC = 0 then −b ± b 2 − 4ac τ= 2ac
+ RC ± R2 C 2 − 4 LC = 2 4L 1 1 ± − R2 C = RC 2 The solution should be a simple exponential, meaning that the discrimant (root here) should be non-negative (else get imaginary numbers, which lead to cosine/sine terms — oscillations) 1−
4L >0 R 2C
or
2
4L < R C
5
R2 C L> 4 Then there are two solutions because of ± q(t) = CVo (a1e
–t/τ1
–t/τ2
+ a2 e ) a 1 + a2 = 1
4L 1 + 1 − R2 C τ1 = RC 2 4L 1− 1− 2 R C τ 2 = RC 2 One of these τ1, τ2 is always the longer time — the fastest decay is as τ1 → τ2, i.e. as 1−
4L → 0+ 2 R C
RC then τ1 , τ 2 → 2 R = 1Ω, C = 200µF
i.e., from above
⇒ T = 100µs
c) This was rather (too?) tricky: one way to see the solution is to start with τ 1, τ2 and the condition 0 = I (t ) = −
so
but
1 dq(t ) 1 −t / τ 2 = − CVo − a1 e − t / τ1 − a2 e dt τ2 τ1
I(t = 0) = –CVo − a1 − a 2 τ1 τ 2 a1τ2 = –a2τ1 a1 + a2 = 1, τ1 ⇒ a1 = τ1 − τ 2 − τ2 a2 = τ1 − τ 2
6
1 1 e − t / τ1 − e −t / τ 2 τ −τ τ1 − τ 2 1 2
I (t ) = + CVo = as L →
CVo − t / τ1 −t / τ 2 (e −e ) (τ1 − τ 2 )
R2 C 0 and τ1 → τ2 this gives a indeterminacy, which we can work with l’Hôpital’s 4 0
rule: numerator demominator
1 2 d (e − t / τ1 − e − t / τ 2 ) = e − t / τ1 −t − dτ1 τ1 d ( τ1 − τ 2 ) = 1 dτ1
So the answer I (t ) = CVo
t −t / τ e τ2
CVo −t / τ 2 te ( RC) t2 Vo – I (t ) = 2 t e RC R C =
(close enough)
1 2 CV 2 1 3 2 = 200µF (2.5 × 10 V) 2 = 625 J of energy.
E=
7
1997-1998 Physics Olympiad Preparation Program — University of Toronto — Problem Set 1: General Due November 3, 1997 1) Connect the dots! Who said that physics wasnÕt all fun and games? See the figure on a separate page at the end of this set: your mission, should you decide to accept it, is to connect the dots using the components given, to simultaneously achieve the desired voltage and current. Note that you may not need to use all the dots (or components) and you may want to attach more than two components to the same dot. The symbols ÔAÕ and ÔVÕ correspond to ideal ammeters and voltmeters. [James] 2) Out with the bad air, in with the good... In homes which meet the R-2000 standard of energy conservation, there are virtually no drafts which let in (cold) fresh air, and there might arise a risk of pollution-buildup from building materials, from cooking and cleaning, or from the occupants themselves. So for such houses, designers may include a heat-recovery ventilation system Ñ basically a kind of heat-exchanger in which stale air exits the house but first passes much of its heat on to the incoming cooler air. Because the outgoing and incoming air streams are kept separate as they pass each other, but still allowed to exchange heat, fresh air enters the house fairly warm and stale air exits the house fairly cool. In fact, the efficiency is surprising. It is a bit too complicated for POPTOR to calculate efficiency for a heat-recovery ventilation system, but consider this problem: Say you have 1L of hot water (say, dyed red), of temperature 100oC, and 1L of water (say, dyed blue), of temperature 0¡ C, presenting the warm and cool air above. Say also that you have many specially-insulated containers, any sizes you wish. These containers prevent heat loss to the air, but exchange heat perfectly when two (or more) touch each other. The question is like warming the incoming air in the house: without mixing the two together, to what temperature can you heat the initially cold (blue) water, and to what temperature can you cool the initially hot (red) water, by arrangements of letting them swap heat? [Peter] [Canada Mortgage and Housing Corporation (CMHC) is a remarkable Canadian gold-mine of housing research and public information of every imaginable sort, including energy-efficient R-2000 houses, heat recovery ventilators, indoor pollution, etc. To order publications, check out: www.cmhc-schl.gc.ca/InfoCMHC/contact.html#Publications] [Wonder what a heat-exchanger looks like? There are some photographs of heat-exchangers for industrial processes at: www.souheat.com/gallery.htm]
1
3) Waves ˆ la mode If you have ever sung in the shower, or maybe a marble washroom stall, you may have noticed that certain frequencies really pick up: they resonate within the space. For sound, the air molecules moving cannot move into the wall, so the walls are ÔnodesÕ or zero-points for the molecular motion. Only certain-frequency waves naturally meet this condition, so they can be resonant within the space, and their reinforced amplitudes can be larger than that for other frequencies. These are Ôstanding waves,Õ and sometimes called Ômodes.Õ In making a laser, several items are brought together: a laser cavity (consisting of two mirrors facing one another) and an optical amplifying laser medium are the bare minimum. The laser cavity provides ÔfeedbackÕ Ñ a way to recirculate the light many times through the amplifying medium to make it more intense. Metallic mirrors also set a condition on the electric field in light: the transverse electric field must vanish at the surface of a conductor (see question below). Therefore, standing electromagnetic waves are formed in the laser cavity, which is for that reason sometimes called a Ôlaser resonatorÕ. a) Luciano Pavarotten is an operatic shower-singer, with a three-octave range starting at C below middle C (middle C is 256 Hz). You may not know that a note an octave above another has twice the frequency. If his shower stall is 1.5 m × 1 m × 2 m, how many standing wave modes in the shower can his voice excite? (Alas, only the standing waves are excited by his voice) How many different frequencies is this? You should ignore the issue of how his (large) body interferes with the sound in the shower. b) Krystal has a Nd:glass (Ôneodymium glassÕ) laser, which typically operates at about 1 Ð6 micrometer (1 µm = 10 m) wavelength. It can amplify any wavelength within ±5 Ð9 nanometers (5 nm = 5 × 10 m) of this wavelength. How many standing-wave modes of light are amplified in a laser resonator which is 80 cm long? [Robin] 4) Opposites attract, but a narcissist always loves himself... Consider a grounded cylindrical conductor, of great length compared to its radius r. In orbit around this fat wire is a small charged particle of mass m and charge q. The distance from the point to the surface of the cylinder is d << r. Why should a charged particle orbit a grounded conductor? Explain the phenomenon, and find the period of the orbit. Neglect gravity, ignore air particles, and pay no attention to the man behind the curtain. [Peter] HINT: For this quite-neat question, you may need to know these two things that you may not already know: a) A perfect electrical conductor has no electrical field inside. Not obvious? Say that there were a field inside: then the free charges in the conductor would see that field (and so a force), and would begin to move Ñ any positive charges in one direction and any negative ones in the exactly opposite direction. If the conductor is of finite size, the charges canÕt keep on forever, but when
2
would this whole thing stop?
It would stop when the conductor ran out of free charges to move
(phenomenally unlikely, for real-sized fields) or until the separated + and Ð charges made their own field which exactly cancelled the original field. The net result is zero DC field.
b) Therefore the surface of a perfect electrical conductor is a surface of constant electric potential. If the field everywhere in the conductor is zero, then a small ÔtestÕ charge could be moved anywhere inside without seeing a force. Therefore the potential energy of the test charge is the same anywhere in the conductorÑ no force, no work, no potential energy change. So a charge a tiny distance inside the surface of the conductor is everywhere at the same potential energy: this means, then, that the surface of a conductor is an electrostatic equipotential.
5) Fermi Questions The famous and fabulous Italian physicist Enrico Fermi was known for a curious type of question he sometimes posed his students. One of them was Ôhow many piano tuners are there in Chicago?Õ. The students were expected to work out the answer without resorting to any reference materials. The questions were an exercise in a very important problemsolving talent: developing the judgement that lets one make reasonable assumptions. With some notion of how many people are in Chicago, one might estimate roughly what percentage would have pianos, how much a piano-tuning costs, or how long it takes, etc. Here are some ÔFermi questionsÕ to try. The idea is to use only what is already in your head, and to make plausible guesses for the rest. For each, give in brief point form the steps you followed in your reasoning. Estimates within a factor of 3 might be considered good, for some of these. a) about how many piano tuners are there in Toronto? (DonÕt just check the Yellow Pages) b) Claudio ChiapucciÕs bicycle has reflectors attached to the spokes of his wheels. He is fussy, and starts out each ride with the wheels set so each reflector is in the 12 oÕclock position, at the top of the wheel, but they never stay that way. Roughly how long does he ride before they end up opposite each other, e.g., 10 oÕclock and 4 oÕclock? c) Jacques Villeneuve went shopping with his car one recent Saturday, but found his favorite small lot at EatonÕs was full. He parked at the top of the lot, and from there he could see, and quickly get to, about fifty parking spaces. On average, about how long would he have to wait for someone to leave? d) About how many pounds of cigarette-butts are dropped on sidewalks in Canada each year? e) Half-time during an exciting Grey Cup football game: trips to the kitchen and washroom. How much, roughly, are the sudden increases in electrical power and water demand? [Robin] 3
6) ÔStacking the Deck Ñ Friction in the House of Cards,Õ by Kitty Kelly PART I Ð Consider the problem of making a structure out of cards. It's pretty difficult. Step One: consider the simplest problem of leaning two cards against each other on a flat surface. Let θ be the angle between the cards (we're assuming symmetrical stacking), µ1 be the static coefficient of friction between a card and the smooth table-top, and µ2 the coefficient between the two cards. I'm using a brand new Bicycle deck, so the cards are slippery and µ2 < µ1 .
θ
When two cards are stacked together at some angle θ, there is some minimum force F min required in order to collapse them. Find out whether there is some optimal angle, θbest Ñ an angle at which the cards will be most stable (i.e., the largest Fmin) Ñ and determine that angle. (θ = 180¡, with the cards lying flat on the ground is stable, alright, but useless Ñ that special case does NOT count!)
PART II Ð Suppose that you want actually to predict the value of the angle θbest, using results from PART I. How do you get µ1 and µ2 Ñ just look them up in the CRC Handbook of Playing Card Constants? Doh! You could measure it, you know. a) Find a deck of playing cards Ñ a couple of baseball or hockey cards (or perhaps even computer diskettes) will do. For your cards, and your smooth table-top, measure µ 1 and µ2 . Describe what you do, and why, and figure out estimates of uncertainties about your measurement process. Give an error-range for your final answer; this is essential because it represents to others how closely they can trust your exact answer. Note that everyone will (most likely) get a different answer Ñ it's how you do it that matters for POPTOR. b) Now knowing µ1 and µ2 , predict θbest. Does it seem reasonable? Explain why. [Peter]
4
For Question #1: e.g.) Set the voltage: V=5V, current: I=5A using only the given components.
A
V 5V
5V
wires
1Ω
10Ω
1Ω
A
V
a) Set the current measured by A1 to be I=2A, using only the given components. What is the current measured by A2, A3 and A4?
A2
A3
1
A4
A
2 wires 15V
All R=5Ω
A
b) Set the current: I=3A using only the given components.
2 wires
13V each R=1Ω
3
V
2
V
V
1
c) Set voltages: V1=11V, V2=7V, and V3=5V using only the given components.
17V 6V 3 unknown R 4V 2V
7 wires
5
1997-1998 Physics Olympiad Preparation Program Ñ University of Toronto Ñ Solution Set 2: Mechanics 1) The Pluto Problem KeplerÕs Second Law results from the conservation of angular momentum of the planet about the sun. According to this law, the straight line joining the sun and a given planet sweeps out equal areas in equal intervals of time. Thus, the ratio of the time interval t, during which the PlutoÕs elliptical orbit is situated inside the NeptuneÕs circular orbit, to the PlutoÕs orbital period T is equal to t/T = S/(πab) The denominator of the right side is the area of the ellipse representing the PlutoÕs orbit, and the numerator S is the area of the sector of subtended angle 50 degrees. We can estimate the value of S as the area of a circular sector (50/360) πR2, where for more accuracy we can substitute R by (rmin + R)/2. Therefore, t = (50/360) x (((rmin + R)/2)2 x T)/(ab) = (5/36) x ((4.4 + 4.25)/2)2 x (248)/(5.9 x 5.73) = (5/36) x (18.71 x 248)/(5.9 x 5.73) = approx. 20 years. It turns out that the question was in error, and Pluto became closer to the sun i n 1979, not 1969. Therefore, Pluto will be again the ninth planet from the sun in 1999. [http://seds.lpl.arizona.edu/nineplanets/nineplanets/pluto.html] 2) Free falling, at \$4.50 a throw a) Let l be the length of a non-stretched rope (l + h < H), k is its elasticity constant, m the mass of a person who is jumping down and v is a value we are looking for. It is obvious that at the equilibrium height h your velocity had the maximum value. We can write the following three equations: 1) the equilibrium condition at the height h : 1
mg = k (H Ð l Ð h)
[2.1]
2) the energy conservation law at the height h : mgH = mgh + (k/2)(H Ð l Ð h)2 + (1/2)mv 2
[2.2]
3) the energy conservation law on the ground level : mgH = (1/2) k (H Ð l)2
[2.3]
Divide [3] by [1] and obtain l = √H (H Ð 2h)
[2.4]
Then rewrite [2] using [1] in the following form : mg (H Ð h) = (mg/2) (H Ð l Ð h) + (mv 2/2) and get v = √g (H Ð h + l)
[2.5]
or using [4], v = √g (H Ð h + √H (H Ð 2h))
[2.6]
If H = 25 m, from [6] we have v = 16.2 m/sec or 58 km/hour. If H = 50 m, then v = 28 m/sec or 101 km/hour. b) Only the third equation will be different compared to case a) : mg (H Ð h 1) = (1/2) k (H Ð l Ð h 1)2
[2.3Õ]
Hence, l = √H (H Ð 2h) + h 1 (2h Ð h 1 )
[2.4Õ]
Use [4Õ] in [5] and get the new value for v. If H = 25 m, then v = 17.1 m/sec or 61.4 km/hour. If H = 50 m, then v = 28.2 m/sec of 101.6 km/hour. c) Elasticity constant can be found from [1] : k = (mg)/(H Ð l Ð h). For H = 25 m, h = 10 m, m = 100 kg and l = 11.18 m (from [4]), we have k = 261.8 H/m. For H = 50 m, h = 10 m, m = 100 kg and l = 38.73 m (from [4]), we have k = 787.4 H/m (the rope must be 3 times stronger for the height H = 50 m compared to the one for the height H = 25 m). 3) ÔBobÕs your uncleÕ, or sometimes heÕs a simple harmonic oscillator (SHO) a) submerged length of the cylinder force of gravity on the cylinder: volume of water displaced by the cylinder
Y F=mcylg Ya 2
So the force of gravity on the cylinder = Yaρwaterg At equilibrium, the two forces are equal, so: a m cyl g = Yaρ water g
Y=
mc yl
Bob
ρ waterπa
h 1'-1"
y -8" b) For a small displacement from equilibrium, y, there is a small difference between the buoyant force and gravity. Assume we push the cylinder slightly deeper into the water than the equilibrium point. Then the increase in the buoyant force is an excess force, in the amount: F = − mg = − ayρ water g
and likewise if we pull the cylinder up, the excess is in the opposite direction. c) (Note: in this section, A is acceleration, while a is the cross sectional area of Bob) F = m cyl A = mcy l
2
d y = − ρ water ayg 2 dt
[3.1]
so, d 2y ρ ag = − water y. 2 dt m cyl
[3.2]
d) Just by substituting, it is easy to show that y(t) = y0 cos(ωt+φ) is a solution of any equation of the form m(d2y/dt2) = Ðky, as in part (c). We will use this same form in future POPTOR problems! The maximum/minimum value that y can take occurs when cos(ωt+φ) = ±1. So, y0 is the maximum amplitude of Bob, measured from the equilibrium point. If we let go of Bob at an amplitude y0, at time t=0, then this should be his maximum amplitude. Then y (t = 0) = y 0 cos φ = y 0
[3.3]
holds, whether we start off by lifting Bob or by pushing him down a little. This requires cos(φ) = 1, which is true as long as we make φ = n 2π, where n is any integer. 3
For convenience, we simply set φ = 0. (If however t=0 is chosen at some other point in BobÕs oscillation, we will need φ to take on some other value.) dy = −ωy 0 sin(ωt + φ ) dt
[3.4]
To determine ω for Bob, note that, 2
d y k = −ω 2 y0 cos(ωt + φ ) = −ω 2 y = − y 2 dt m
[3.5]
so, ω = (k/m)1/2. From part (c), k ρ ag = water mcyl mcyl
[3.6]
therefore,
ω=
ρ water ag mcyl
[3.7]
e) F = buoyancy + gravity
a
= ρwga(yw Ð yB) Ð mBg at equilibrium F = 0, so
Bob
0 = ρwga(ywo Ð yBo) Ð mBg mBg = ρwga(ywo Ð yBo)
y[3.8] w 2'
yb
1'-5"
where ywo and yBo are equilibrium (rest) values. In general,
A
mB¥ accelÕn = F, so, m BØ B = ρwga(yw Ð yB) Ð m Bg, equilibrium:
but we substitute for m Bg the value from
mBØB = ρwga(yw Ð yB) Ð ρwga(ywo Ð yBo) = ρwga((yw Ð ywo) Ð (yB Ð yBo)) = ρwga(∆yw Ð ∆yB);
∆yw ≡ yw Ð ywo 4
since (ØBo) = 0, we have ØB = (∆ØB), (where by (∆ØB) I mean the second derivative in time of (∆yB) ). Thus, mB(∆ØB) = ρwga(∆yw Ð ∆yB)
[3.9]
The overall volume of water in the tank is constant, and this is the same value at equilibrium: V = A⋅yB + (A Ð a) (yw Ð yB) = const = A yBo + (A Ð a) (ywo Ð yBo)
[3.10]
thus A(∆yB) + (A Ð a) (∆yw Ð ∆yB) = 0 or (A Ð a) (∆yw) + a(∆yB) = 0
[3.11]
Substituting this into (A Ð a) × I we get mB(A Ð a) (∆ØB) = ρwag[(A Ð a) ∆yw Ð (A Ð a) ∆yB] = ρwag[Ða∆yB Ð (A Ð a) ∆yB] = ρwag[ÐA∆yB ] So m B(∆y B)+
aA gρ ∆y = 0 (A − a ) w B
[3.12]
Thus ωo2 =
aA gρw (A − a ) m B
Note: as A → ° , i.e., as the cylinder becomes huge, ωo2 → T = 2π
[3.13] a g ρw as above! mB
( A − a) m B aA g ρw
[3.14]
i.e., the period is smaller if the water level also rises and falls.
5
4) Full of fury, and signifying nothing... a) The system will annihilate. There is a net force of 3kee [towards centre of square] 2a2
(k =
1 ). 4πε o
on each particle To see this, consider any particle Ñ the net force is, by symmetry, directed towards the centre of the square. It has a magnitude of: FNET =
kee kee kee 3kee + 2 − 2 = 2a 2 a2 a2 a
Since this result holds for any a, and the masses of each particle are equal, and they start from rest, they will accelerate toward the centre of the square at the same rate. Since they all start from rest, their velocities along the diagonals will be equal, and hence their positions will be equal. So the formation will be preserved, and they will keep accelerating inwards. Their final velocities [same number] could be found from: potential =
1 kqi q j 4kee 2kee = + ∑ a 2 i ≠ j rij 2a
Hence, m V2 4kee 2kee 4kee 2kee + − − = 4 e b a 2b 2a 2 where b is the final size of the box. Note that this gives infinity for b = 0. b) This is not as easy as it looks. The particles will obviously be repelled to some large distance, where their interactions will be virtually zero. By symmetry, the final velocity of each of the positrons will be equal and opposite; the same is true for the protons. Conserving energy E=
kqq kqq kqq mV 2 MU 2 + +4 =2 +2 a 2 2 2a 2a
where V is the final speed of the positrons, U is the final speed of the protons and q is the magnitude of the charge of an electron (k = 1/(4πεo)). The terms on the left are due to: the energy of the positron-positron pair, the proton-proton pair, and the positronsÕ with their neighbouring protons (respectively). 6
Conserving momentum yields no new information Ñ it is conserved because of the symmetry of the system. Note, however, that because M >> m, the accelerations of the protons = 1/2000 of accelerations of positrons Ñ thus, the positrons will escape before the protons will have moved. I.e., the protons are virtually stationary. This gives E=
kqq kqq kqq mV 2 kqq + +4 =2 + a 2 2a 2a 2a
Solving,
mV 2 =
kqq kqq +4 a 2a
[1]
Now, that the positrons have escaped, we can consider the protons. We get:
MU 2 =
kqq 2a
[2]
Dividing [1] by [2] gives mV 2 +1= 4 2 + 2 MU 2 V M = 4 2 +1 U m
(
)
Or, 115 plugging in M/m = 2000. 5) Sikorsky meets Newton From Newton's 2nd law, the force is dp/dt. Now, we'll put ourselves in a frame rotating with the blades. At a distance r from the origin, an oncoming air particle has a horizontal velocity of r*ω m/s. We'll assume that the collision is partially elastic with a parameter k. I.e., after the collision the air particle has a velocity k*r*ω (for perfect elasticity, k = 1). After a collision, the particles will move downward with some speed U and backward with some speed V. Note that the magnitude of their velocity is krϖ = U 2 + V 2 By NewtonÕs second law there must be equal an opposite forces acting on the blade Ñ thus there is a vertical component (lift) mU, and a horizontal one mV, slowing 7
the rotation of the blade. This would make the problem complicated, but luckily we assumed that the rotation rate is constant (the engine compensates). (m = mass of one air particle) Now, after rotating through an angle dα, the area swept out by a blade is: area = 1 / 2 * dα * a 2 The volume thus swept out is: volume = 1 / 2 * dα * a 2 * b sin(θ ) And the mass hit is: mass = ρ *1 / 2 * dα * a 2 * b sin(θ )
θ
The downward component of k*r*ω (U) is: krϖ cos(θ )sin(θ ) Note that this varies this r Ñ technically speaking, we should integrate here, but weÕll make life easy and use the average r Ñ that is, R/2. Thus, for 1 blade:
(
)
F = ρ *1 / 2 * dα / dt * a 2 * b sin(θ ) * (kR / 2ϖ * cos(θ ) *sin(θ )) For N blades:
(
)
F = N ρ *1 / 2 * ϖ * a 2 * b sin(θ ) * (kR / 2ϖ *cos(θ )* sin(θ )) BONUS: Well, tiliting the rotor axis seems like a good idea (so part of the lift is sideways) but I suppose too hard to build. What is done, instead, is something much more interesting Ñ during a part of the cycle (say, when passing over the front) each bladeÕs pitch (θ) is changed Ñ this adds an extra force, as necessary. Electronic circuits control that ω is steady and the helicopter does not spin around.
8
6) Pulse-height analysis a) The data in the table can be plotted as a histogram Ñ a range of values of the output, and then a count of how many times the values fell into that range. Here is such a table, and two different ways of plotting it: 14
N
137
Cs Pulse Height Distribution
12 10
Count
0 2 2 3 2 4 2 3 3 2 1 2 1 1 1 1 0 1 0 0 0 4 12 3 0
8 6 4 2 0 0
7
14
21
28
35
42
49
56
63
70
77
84
Range 14 137
Cs Pulse Height Distribution
12 10
Counts
Xo 1.80 5.40 9.00 12.6 16.2 19.8 23.4 27.0 30.6 34.2 37.8 41.4 45.0 48.6 52.2 55.8 59.4 63.0 66.6 70.2 73.8 77.4 81.0 84.6 88.2
8 6 4 2 0 0
20
40
60
80
Pulse height range
The first plot better shows how there are bins, and that the columns marked indicate how many count data-points fell into the range of each box of pulse-heights.
9
For comparison, here is how a Ôpulse-height analyzerÕ package does the job: In the foreground is the Cs gammaradioactive source, and the package which holds the NaI scintillator and the photomultiplier tube. In the background is the electronic apparatus which sorts pulses as they happen, and adds them to bins much as you did by hand. b) The raindrops will have a terminal velocity, i.e., some maximum constant velocity. If the velocity is terminal (constant) there must be no net force on the droplet. So we can find the terminal velocity by figuring out the balance of forces: gravity and wind resistance. Working from: g acceleration due to gravity: ρ w density of water: ρf density of air: r droplet radius
Ð1
980 cm s Ð3 1 g cm Ð3 Ð3 1.2928 × 10 g cm at NTP
we need to subtract the buoyant force of the air Ñ a small correction given the density of water compared to air. Fg = mdrop g − mair g = (ρw − ρ f )V g = (ρw − ρ f )
4π 3 r g 3
[6.1]
For wind resistance, with density of air ρf, drag coefficient CD, speed v and crosssectional area S: Fd =
1 2
ρ f CD v 2 S =
1 2
ρ f CD vterm2 πr 2
[6.2]
10
1 2
ρ f CD vterm2 πr 2 = (ρw − ρ f )
1 2
ρ f CD vterm
2
4π 3 r g 3
or, cancelling,
4π = (ρw − ρ f ) gr 3
[6.3]
If we assume that we have a sphere, we can consider our two limiting cases for CD: we assume each case to start, and then see from the results what it will take to justify the assumption. Case I: Re < 80; CD ~ 24/Re. First we find vterm: 1 2
ρf
24 4 vterm2 = (ρw − ρ f ) g r 3 Re
6 η vterm 4 = (ρw − ρ f ) g r 3 r 2 (ρw − ρ f )g 2 vterm = r 9η
and substituting for Re thus, solving for vterm ,
[6.4]
Then this goes into the formula for the Reynolds number: Re =
ρ f vterm 2r η (ρw − ρ f )2 g r 2 2 r ⋅ = ρf 9η η =
[6.5]
ρ f (ρw − ρ f )4 g r 3 9 η2
Now, using this we require that Re < 80, letting us find the corresponding condition on r: 80 > Re =
ρ f (ρw − ρ f )4 g r 3 9η2
then solving for r ,
180 η2 = r < 80 • ρ f (ρw − ρ f )4 g r 3 ρ f (ρw − ρ f ) g 9η2
3
[6.6]
180 η2 r< ρ f (ρw − ρ f ) g 3
⇒ r < 0.0166 cm ; vterm = 333.8 cm s ±1 11
So this approach works for droplets smaller than 0.17 mm, for which the terminal Ð1 velocity will be around 3.3 m s . Case II: Re > 1000; CD ≅ 0.4 First we find vterm: 1 2
4 ρ f 0.4 vterm2 = (ρw − ρ f ) g r 3
vterm =
4 (ρw − ρ f ) g r 3 = 1 ρ f 0.4
thus, solving directly for vterm , [6.7]
(ρw − ρ f ) 20 gr ρf 3
2
Then this goes into the formula for the Reynolds number: Re =
ρ f vterm 2r η 2 ρ f vterm 2r (ρw − ρ f ) 20 = gr 3 η ρf
=
ρ f (ρw − ρ f ) 80 2
η
3
[6.8]
g r3
Now from this we require that Re > 1000, letting us find the corresponding r: Re2 > 106 ρ f (ρw − ρ f ) 80 η
2
3
g r 3 > 106
r3 >
3η2106 80 g ρ f (ρw − ρ f )
r>3
3η2106 80 g ρ f (ρw − ρ f )
then solving for r , [6.9]
⇒ r > 0.098 cm ; vterm = 703.4 cm s ±1 So this approach works for droplets larger than 0.98 mm, for which the terminal Ð1 velocity will be around 7 m s .
12
Momentum The microphone will approximately record the impulse, or momentum deposited by the droplets, so it is the scaling between terminal m o m e n t u m mdrop¥ vterm and droplet radius which is needed: ∆p = mdrop vterm = ρw V vterm = ρw ρw = ρ w
4π 3 r vterm 3
4 π 3 2 (ρw − ρ f )g 2 8π ρw (ρw − ρ f )g r 5 r • r = 3 9η 27η = 5.07 × 106 r 5 4π 3 r • 3
(ρw − ρ f ) 20 4π • gr = 3 3 ρf
(r < 0.016 cm )
ρw (ρw − ρ f ) 20 g ρf 3
= 9.39 × 103 r 3.5
[6.10]
7
r 2
(r > 0.98 cm )
We can plot this; for points in-between, there is a plausible smooth curve to sketch the transition between the two regimes. 101 9 14
momentum (g cm s– 1)
10
R e <80 R e >1000
9
10
104 10-1 10-6 10-11 10-16 10-21 10-26 0.001
0.01
0.1
1
radius (cm) In order to find radius from experimental m o m e n t u m data, we start on the y-axis (ordinate) and look up the radius on the x-axis (abscissa).
13
For your interest, a fuller set of data for CDs for spheres Ñ found by experiments Ñ looks like this:
1 02
CD
1 01
1 00
1 0-1 1 0-1
1 00
1 01
1 02
1 03
1 04
1 05
1 06
R adapted from: A PhysicistÕs Desk Reference, H.L. Anderson, ed., American Institute of Physics, New York (1989).
14
1997-1998 Physics Olympiad Preparation Program Ð University of Toronto Ð Solution Set 3: Thermodynamics NB: in thermodynamics, some of these questions draw on the relatively advanced international syllabus of physics used in the International Physics Olympiad (IphO) Ñ not all of which is taught in North America. 1) The heat is on at the Pentagon Ð8 a)There was an error in one of the given constants. StefanÕs constant is 5.6696E 2 4 W/(m K ). All I can say is that you didnÕt notice the sign error in the constant during the question, you would have found that your face would be radiating more power than Pickering Nuclear Power Station can generate. Even though the numbers were wrong, the given constants were meant to help you if you have never seen this type of question before. To find the peak in the blackbody radiation curve, you could actual plot the black body radiation curve as a function of wavelength for each temperature, but this is too much work. A little calculus provides WienÕs displacement law, which you can use directly:
λmax = 2898/ T where T is temperature measured in Kelvin, and λmax is the peak wavelength in the black body emission curve, measured in micrometers. 37°C human emits at 9.3um. 75°C car peak emission at 8.3um. 5°C terrain peak emission at 10.4um. What is the power/area emitted? We look to StefanÕs law for that: P/A= σ eT
4 Ð8
2
4
where σ is StefanÕs constant = 5.6696E W/(m K ) e is the emissivity (for a black object, e=1) and T is measured in Kelvin 2
37°C human emits 520 W/m . 2 75°C car radiates 830 W/m . 2 5°C terrain radiates 340 W/m . How about a flashlight? Assume the flashlight has a an excellent lens system that provides a uniform beam of radius 5cm. A guess of a bulb wattage of 0.1W is not 2 bad, thus the power per unit area is 13 W/m . Surprising eh? Remember also that the vast majority of this power is also emitted in the infrared (i.e. as heat).
1
b) Missile time! In a very short time the missile surface heats up and starts emitting. How much power is it radiating at equilibrium? The same as it is absorbing: 5MW (ow!). LetÕs use the same expression as used above, assuming that the missile is painted a nice dull black so that emissivity =1: P/A= σ e T
4
For spot size of diameter 1m: T=3300K or 3000°C. 5 For a spot size of diameter 1mm: T=10 K ! Kappoey!! 2. Planet X a) Let us find first the value of the gravitational field constant for the new planet. From the universal gravitation law we have for any mass m 2
mg = G (mM/r ) 2
g = GM/r = ((6.7)(10
(1) Ð11
25
14
)(10 ))/((9)(10 ))=0.74 N/kg
The atmospheric pressure on the surface of the planet is equal to p = ρ gh
(2)
where ρ is the atmospheric density. Assume that g is constant since h << r. From the equation of state for ideal gas pV = nRT = (M/(µ ))RT
(3)
ρ = M/V=(p(µ ))/(RT)
(4)
we find
and from (4) and (2) T = (µgh)/R = (10)(0.74)(5)/(8.3) = 4.5 K = Ð268°C
(5)
Unfortunately, it is too cold to live on this planet !!! b) For the molar mass of our atmosphere (78% of nitrogen and 21% of oxygen), we have µ = (0.78)(28) + (0.21)(32) = 28.56 g/mol. The EarthÕs atmospheric density decreases at the same rate as the atmospheric pressure (1/2 per each h 0 = 5.6 km). Ðx/ho We can express this dependence as the function ρ(x) = ρ0Ê2 =Êρ0Êexp{(Ðln2)(x)/h0}, where x is the distance from the surface of the Earth. The good approximation for the height of the atmosphere equivalent to ours but of constant density can be found from the following condition: ∞
∫ ρ(x) dx = r0 h
0
From this condition we obtain the value of h = (h0)/(ln2)=(5.6)/(0.693)=8.1 km. Now we can use the formula (5) to calculate the temperature on the EarthÕs surface: T = (µ)(g)(h)/(R) = (28.56)(8.1)(9.8)/(8.3) = 273 K = 0°C. 2
3. Blow ye winds and crack your, er, skin ... a) From the equation of state for ideal gas pV = nRT = (m/µ )RT, where n is the number of moles and µ is the molar mass, we obtain
ρ = m/V = (µ p)/(RT) = (µα p s)/(RT),
(1)
where ps is the saturation vapor pressure for given temperature T and α is the relative humidity. The molar mass of water µ = 18 g/mol. In November, 3
ρ1 = ((µ)(α1)(ps1))/((R)(T1)) = ((18)(0.95)(600))/((8.3)(273)) = 4.5 g/m . In July, 3
ρ2 = ((µ)(α2)(ps2))/(R)(T2)) = ((18)(0.4)(5500))/((8.3)(308)) = 15.5 g/m . Thus, in dry July (α 2 = 40 %), the air contains 3.4 times more vapor than in humid November (α 1 = 95 %). 3
b) The mass of water vapor in 1 m of outside air is equal to (0.25)(3.48) = 0.87 g. When the outside air is heated from Ð4°C to 20°C, its volume increases. Therefore, pV 1 /T 1 = pV2 /T 2 , 3
V 2 = V1 (T2 /T 1 ) = 1.089 m . 3
3
The density of water vapor in the heated air is equal to 0.87 g / 1.089 m = 0.8 g/m , 3 3 so the relative humidity will be (0.8 g/m ) / (17.3 g/m ) = 0.046 = 4.6 %. The air will be uncomfortably dry. When the Santa Ana winds blow about this dry into Los Angeles, some people get nosebleeds. 4. Jessica cycles heat a) The gas presses on the pump handle with a force = P A where A is the crosssectional area of the pump and P is normal atmospheric pressure. The distance the handle moves is δx. The definition of work is the force vector dot product with the displacement. In this situation, this gives that the work done by the gas is P A (δx). You should be careful about signs here: I have implicitly defined that compression yields negative δx and expansion corresponds to positive δx. The work done on the gas is the opposite of this, thus: W= ÐP δ V, where δ V=( δ x
A) ,
where A is the cross-sectional area of the bike pump. b) Given :
P′ = 1.0 atm (at steady-state with the room) V′ = 1 l V″ = 0.5 l
3
Using adiabatic compression expression we find that: P ′/P ″ = (V″ /V ′)
1.4
so
P 2.6
P″ = 2.6 atm Using the ideal gas law: PV=nRT or for a constant number of molecules: PV/T = constant. Thus we can say:
1.0
P ′V ′/(P ″ V ″ ) = T′/T ″ so that the final temperature is: T″ = 381K or about 110 ¡C . This assumes that the system started at room temperature (293K).
0.5
1.0 V
c)Using part (a), the work corresponds to the area under the graph give in part (b). Integration of this function will provide an accurate value, but you can approximate the curve as a straight line connecting the beginning and final states. Thus the area is a square + a triangle: Work = ÐArea = Ð(Ð0.5 l*(1 atm)+(1/2)*(Ð0.5 l)*(2.6 atm Ð 1 atm)) = 0.9 atm l or in the more usual units for work: Work = 91 J
P d) Using the ideal gas law again: P ′V ′/(P ′″V ′″) = TÕ/T′″ but here TÕ = T′″ so
2.6 2.0 1.0
P′V′ = P′″V′″
giving
0.5
P ′″ = 2 atm There is no work done on the gas, as can be seen in the PV graph (area under the curve is 0).
1.0 V
P
e) Given : P″′ = 2.0 atm (at steady-state with the room) V′″ = 0.5 l V″″ = 1.0 l
2.0
Using adiabatic compression expression again we find that:
1.0 .76
P″″ = 0.76 atm
0.5 4
1.0 V
Using the ideal gas law again: PV/(P ″″ V ″″ ) = T/T″″ so T″″ = 223K or Ð50°C Wow eh! Using a linear approximation: Work= ÐArea= Ð.69 atm l = Ð70 J Thus the gas does work of 70 J (the minus sign indicates that the gas is doing work on something, not that work is being done on the gas). P f) Easy one: final pressure: final T:
1 atm room temperature (293K)
No work done on the gas, since the area under the curve o n the PV diagram at right is again 0 (as in section d).
1.0 .76
g) Net work on the gas is 91 J + 0 J Ð 70 J + 0 J = 21 J 0.5 h) No heat is transferred in the adiabatic compression and expansion. It is only transferred when the gas is held at constant volume. δT for the cooling stage (stage 2):
381KÐ293K=88K
δT for the warming stage (stage 4):
223KÐ293K=Ð70K
1.0 V
The total number of moles in the gas is: n=PV/(RT)=0.041 mol Thus the total heat transferred from the gas to the surroundings is: total heat = 18K * 2.98 cal/(mol K) * 4.19J/cal * 0.041 mol =9 J We would expect this to equal the work done on the gas (calculated above as 21 J). The reason for the discrepancy is due to the linear approximation. If the area was calculated accurately, these two numbers would agree. 5. Batboy chills a) Heat transfer: adiabatically insulated.) Work:
Q = 0
(No heat flows into the system, as it is
W = 0 (No net work done by the system)
The energy change in the system after opening the valve is U 2 Ð U1 = ∆U = ÐW + Q (1st law of thermodynamics) = 0 So U 2 = U1, or U(T1,V 1) = U(T2,V 2).
5
For an ideal gas, U is independent of volume, so U(T1) = U(T2), meaning that T1 = T2. Unless the gas in V 1 was initially very cold (which wouldÕve required fancy refrigeration), Batboy would have nothing to fear. b) As for part (a), U(T1,V1) = U(T2,V2) 2 For our interacting gas, ∆U = cv ∆T Ð a/v ∆V, where here the ∆ now refers to the difference in T and V in each compartment before and after the valve was opened. Relative to some arbitrary initial temperature To and volume Vo, T
dV 1 o (V 1 )2
V
U1 = ∫T 1 Cv (T 1 )d T 1 + a∫V o
T
V
U 2 = ∫T 2Cv (T 1 )dT 1 + a∫V 2 o o
dV 1 , where CV (T) = heat capacity (V 1 )2
1 dV1 T2 V2 dV 1 1 = C (T )dT + a ∫ ∫ v To Vo o (V 1 ) 2 (V 1 ) 2
T
V
U1 = U 2 ⇒ ∫T 1 Cv (T 1 ) d T 1 + a∫V o
Assuming Cv doesnÕt change with temperature, CvT1 Ð a/V1 = CvT2 Ð a/V2 so (T2 Ð T1) = a/cv(1/V2 Ð 1/V1) With T2 = Ð5οC a = 0.15
T1 = 15οC Cv = 21 J/(mol K)
V 2/V1 = 2800 V2 +1, so V2 would have to be very large! 6. The Ôthermal physicsÕ (statistical mechanics) of computing a) For a system of 2 magnets, U = Ðm l, l2. The possible configurations of the magnets are: Both up:
↑ l = +1
↑ l = +1 ↑
first up, second down: l = +1
both down:
U = +m
U = +m
l = Ð1 ↓
first down, second up:
U = Ðm
l = Ð1
l = +1
↓ l = Ð1
↓ l = Ð1
U = Ðm
So we want spins to be parallel to one another, to minimize the internal energy U.
6
For N magnets, all nearest neighbours have the same interaction as for 2 magnets: Ex. ↑ 1
↓ 2
↓ 3
↑ ↓ (iÐ1) i
ááá
Interaction energy for the i
th
↑ ááá (iÐ1)
↑ ↑ ↓ (NÐ2) (NÐ1) N
magnet:
U l = Ðm (liÐ1 lI + lI lI+1) Interaction energy for the (i+1)
th
magnet:
U l+1 = Ðm(lI lI+1 + lI+1 lI+2) Note that U l + U l+1 counts the interaction between l and l+1 twice, which is incorrect. If we follow this through for the whole chain, we find: N −1
UTOT = − M ∑ li li + l i....= 1
So UTOT is maximal if all magnets are aligned in the same direction. ∴ (all spins up or all spins down. b) Energy difference = (energy before flipping a spin) – (energy after flipping a spin) If the flipped magnet was at the end of a chain, the change in energy is due to 1 bond. ∴ energy increase = +m Otherwise, a flipped spin affects 2 bonds. ∴ energy increase = +2m So, flipping a spin from the parallel state increases the internal energy. c) For 3 magnets: i)
U = Ðm(1+1) = Ð2m:
↑ ↑ ↑
ii)
U = Ðm(1Ð1) = 0: ↑ ↓ ↓
↓ ↑ ↑
iii) U = Ðm(Ð1Ð1) = 2m:
↑ ↓ ↑
↓ ↓ ↓ ↓ ↑ ↓
↑ ↑ ↓
2 configurations ↓ ↓ ↑
4 configurations 2 configurations
i) → lowest energy, low entropy ii) → moderate energy, high entropy iii) → high energy, high entropy d) F = U ÐTS i) If U > TS, a state with lower internal energy, which has a low entropy, is preferred. This means that internal energy determines the state of the system and so all the spins will tend to align in parallel. This is often called the Òordered stateÓ.
7
ii) If TS > U, a state of higher entropy is preferred; in fact, the system will want to be in the energy state with the most number of configurations available to it. This is typically the most ÒrandomÓ state of the system. iii) U = TS defines the Òorder-disorderÓ transition e) This depends on temperature. If U > TS, then the configurations ↑ ↑ ↑ ↑ ↑ 1 1 1 1 1 = 31
and
↓ ↓ ↓ ↓ ↓ 0 0 0 0 0
=0
are the easiest to code. Conversely, if TS > U, then the ÒrandomÓ states, such as ↑ ↓ ↑ ↓ ↑ 1 0 1 0 1= 21
or
↓ ↑ ↓ ↑ ↓ 0 1 0 1 0
= 10
are easiest. Other values cost energy and entropy Ð you can see how much of a free energy cost there is in constructing other states by evaluating U and S for a given T. BONUS: For
1 spin, state possible = ↑ + ↓ 2 spins, state possible = (↑ + ↓) (↑ + ↓) = ↑↑ + ↓↓ + ↑↓ + ↓↑ 3 3 spins, state possible = (↑ + ↓) N N spins, state possible = (↑ + ↓)
Let x = ↑ and y = ↓ By binomial theorem, we would know right away for x and y: N 1 N! N ( N − 1) x N − 2 y 2 + ... + y N = ∑ x N − t yt 2 t = 0 (N − t)!t! N −s and sum over s Let s = Nup Ð Ndown. Replace t by 2
(x + y) N = x N + Nx N − 1 y +
(x + y)
N
N
N!
s= −N
N + s ! N − s ! 2 2
= ∑
1 (N + s) x2
1 (N − s) y2
well, the same form also applies to writing out our combinations of ↑ + ↓ (↑ + ↓ )
N
N
N!
s= −N
N + s ! N − s ! 2 2
= ∑
1 1 (N + s) (N − s) 2 ↑ ↓2
The number of states with given energy is determined by # of up and down spins. This is given by the coefficient of the binomial term with value s. ∴ g(N, s) =
N! N+s N−s ! ! 2 2
( )( )
8
1997-1998 Physics Olympiad Preparation Program Ð University of Toronto Ð Problem Set 5 Solutions: Electricity and Magnetism 1) The Earth as Cosmic Doorknob The charge accumulated by the Earth during the time interval t is equal to 2 2 qÊÊ=ÊaSqpt, where a = 1 proton/cm .sec is the flow density; S = 4πR is the area of 6 Ð19 the EarthÕs surface; R = 6.4 × 10 m is the radius of the Earth; qp = 1.6 × 10 C is the charge of proton. The critical charge q of the Earth, which will prevent 9 protons with the kinetic energy E < 4 × 10 eV from reaching the Earth, can be found from the following equation: qqp /(4 πε 0 R) = aStqp /(4 πε 0 R) = E , where ( 4 πε 0 )−1 = 9 • 10 9 Nm 2 C −2 From this equation we have: t = ( 4 πε 0ER)/( aSqp2 ) = (Eε 0 )/( aqp2 R) = 108 /((3.14)(0.9)(1.6)(6.4)) = 3.45 • 106 sec = 40 days 9
This value is tiny compared to the age of the Earth, which is equal to 5 × 10 years. Protons from the cosmic rays continue to reach the Earth because their charge is compensated by the charge of electrons also reaching the Earth. Since their energy is extremely small, they are not considered as a component of cosmic rays. 2) C.C.? No! No! a) At a distance of 5 cm from a large sphere, we must treat only that section of the sphere near where you are standing (your closest point to the sphere). This is for the same reason that we can treat the earth as being flat locally at its surface even though it is really a sphere. E field of a spherical conductor with charge Q. NOTE: the charge on a conductor distributes itself uniformly on the conductorÕs surface. GaussÕ Law charge enclosed ε0 Q E • 4 πr 2 = ε0
∫ E • ds =
1
∴E =
Q 4 πr 2 ε o
∴ at 5 cm: 0.8 × 10 3
V Q = ⇒ Q = 2.2 × 10 −10 C 2 m 4 πε o (.05)
b) For the large sphere, 1 q1 V1 = 4 πε o r1 Small sphere, V2 =
1 q2 4 πε o r2
But since the spheres are part of the same conductor, the potentials of the spheres must be equal: 1 q1 1 q2 V1 = V2 ⇒ = 4 πε o r1 4 πε o r2 ∴
q1 q2 = r1 r2
The electric field at the surface of each sphere is given by q q E1 ∝ 21 , E2 ∝ 22 r1 r1 With the same constant of proportionality for each. ∴The ratio of the electric fields is
E1 = E2
q1 / r12 q2 / r22
q1 r = 1 q2 r2
1 r1 1 r2
but q1 q2 = r1 r2 E r ∴ 1 = 2 E2 r1
So the field is higher on the surface of the smaller sphere. A more sharply curved or ÒpointyÓ surface on a conductor has a higher electric field and therefore reaches spark discharge for a lower applied charge. I have heard that there was an ongoing disagreement between Benjamin Franklin in the U.S. and European investigators as to what the best shape for a lightning rod should be: sharp spike or small ball. The argument went like this: all this, above, is true, but the strong field around a sharp spike might cause air to ionize
2
as the rod began to charge up. If so, there would be glob of ionized air (plasma) near the rod, and plasma is a conductor Ñ so the effective conductor would have a larger radius than if a lightning rod had a ball on the top to start with! 3) Goats & Sheep Ð the Hall Effect in plasma + + ++ – + + The point here is that moving charges in a – – + +++ ++++++ – magnetic field see a force Ñ the Lorentz – – +++ +++++++ + + + + + + ++ ++ + + force: v – v v ++ + ++ ++ + ++ + + + FLorentz = q v × B + – – + ++– ++– + + + + –– The Lorentz force, on each particle, will +++++ ++ –– ––++–+–+––+++++ – – + – + be: –– ––––– –++ + + –+––+++ + ++++ ++++ Ð19 Ð1 +++ 1.6 × 10 C ¥ 0.3 m s ¥ 0.3 T +++ +++ ––– ––++–+–+–+–+ + + Ð19 – +– + – –+–– + (opposite = 1.44 × 10 N directions for electrons, ions) A block of plasma, with the electrons pulled aside from the ionsÉ So when the flowing gas becomes ionized, the electrons and ions will separate from each other in a B-field Ñ the Lorentz force pushes opposite ways for the opposite charges. Do they end up going in circles in the B-field? Probably not, because as the charges separate, they are still attracted to one another by the Coulomb force: v v FCoulomb = q E Then the net force is going to be: v v v v F = q (E + v × B) Basically, stream of electrons will move aside from the gas stream, and the stream of positive charges will also move a little in the opposite direction. The separated charges lead to an E-field, and to an attraction between the two streams. The two will balance, so that F = 0 What will be the electric field to cancel this? Ð19 1.44 × 10 N = q E, so Ð1 E = 0.9 V m How much will the charges separate from each other, if we assume they keep their original density, and just separate as a stream? We can figure out the 23 details: air at STP is 22.4 l per mole, and 1 mole = 6.022 × 10 particles. From this we find the number of particles per unit volume:
– – – – –––––– ––– –––– + –– –– – –––– ––––– – + – + – – + –––– –––– ++ + – – +++ ––– ––––––– ––– – – – – + – – –– – ++ + ––––– ––– –– – – – + + + ++ – – ––– – ––
N 6.022 × 10 23 = = 2.69 × 1019 cm −3 3 3 V 22.4 × 10 cm 17 3 With 1% ionization, this means we have 2.69 × 10 electrons per cm , or 2.69 23 3 × 10 electrons per m , and the same density of positively charged ions. If the electrons are just pushed somewhat off the ions (the ions are much more massive, and move less from the forces), what we have is something roughly like n=
3
a parallel-plate capacitor, with the two thin excess-charge slabs as the two charged plates of the capacitor. For this, we can find the field Ñ the same as between the two plates of a parallelplate capacitor. In the middle, where there are both electrons and ions, the charges of each cancel each other out (unless you give them time to move around and redistribute themselves), so the net field in the middle is just what is produced by the thin slabs of excess (unbalanced) charge. E = 4 π kc σ
where kc = 9 × 109 N m2 CÐ2 σ = charge per unit area on each plate The charge per unit area of the two slabs of excess comes from the charge density, and the distance the electrons are pushed off the ions: σ = (n e (∆x) × A) / A = n e (∆x) and the field between the plates is: [5] E = 4 π kc n e (∆x) or ∆x = E/(4 π kc N e ) Ð16 = 1.84 × 10 m A very tiny displacement (actually nonsensically tiny, given the size of an atom!). Consider, though, that if the jet is 1mm across, the potential measured across the jet from one side to the other would be 0.9 mV. 4) Shepherding fields a)For the four cases suggested in the problem: i) v=0 There is no force acting on the particle and therefore no acceleration.
B B
ii) vparallel non-zero: The cross-product is zero, therefore same result as first case.
iii) vperpendicular non-zero: The cross-product simplifies to a multiplication and the resulting force is always acting perpendicular to the motion. The particle moves in a circle. From the given equation and knowing that for perfect circular motion (such 2 as a satellite in orbit) the acting force is F = mv /r we find the radius of the circle is: r = mv/qB v out of page
B
v out of page
B
iv) We combine the results from sections iii) and ii) and get helical motion.
4
Since in all cases the velocity of the particle is always orthogonal to the acting force, the work done on the particle is 0. b) As given in the question, kinetic energy is conserved. Thus the square of the magnitude of the total velocity at any given point in time is constant: 2 2 2 v(z) =vper(z) +vz(z) =constant where v is the total velocity, vper is the component perpendicular to B (i.e. in the X-Y plane) and vz is the component parallel to B (i.e., parallel to the z axis). We want to find vz(z): 2 2 2 2 vz(z) = vper(0) +vz(0) Ð vper(z) We also know that flux is conserved: B(0) A(0) = B(z)A(z) where B is the magnitude of the magnetic field and A is area enclosed by the loop that is orthogonal to the field. Combining this with the result from part a): 2 2 vper(z) /B(z) = vper(0) /B0 This gives: 2 2 2 2 vz(z) = vz(0) Ð vper(0) (b0 z ) / B0 c) Shades of Bob, itÕs simple harmonic oscillator! KE is transferring from parallel motion to perpendicular circular motion. The particle swings back and forth along the Z axis, in a similar way to a weight on a spring, with the added circular motion around the Z axis 2
2
2
d) vz(z)=0 when z =(B0 vz(0) ) / (b0 vper(0) ) Total KE is conserved. Energy from translational KE transfers to rotational KE. 5) Science on a shoestring First we suspend the magnet from the stand (using the string). We tilt it by a small angle and let it oscillate. It undergoes oscillations (much like a torsional pendulum) such that: ⋅⋅
I θ = −µ o mBz sin(θ) ≈ −µ o mBzθ here, Bz is the horizontal component of the EarthÕs magnetic field, m is the magnetic moment, and I is the moment of inertia of the magnet. Treating it as a thin stick, I=
M d2 12
Now, the motion is simple-harmonic with a period of
5
T = 2π
I mBz µ o
Now, we align the compass to the north. Then, we place the magnet some distance r >> d away from the compass and measure the deflection angle β. North
Clearly, then, β
tan(β) =
2m B = Bz 4 π r 3 Bz
Combining, we get r >> d
Bz =
2π I µ o T r tan(β) 2 3
Ideally, one would collect a set of βs by varying r, and plot the equation (on log-log paper), and get Bz that way, but that is a lot of work (when done by hand). And thus, plugging in one set of numbers will do. To measure T, it is best to time say 10 periods, and divide the result by 10 (as long as the time does not decay too much). r is simply measures with a ruler. Measuring β might be difficult, and this might be the biggest error here. In fact, it ο might be best to set β at, say 45 , and then change r to obtain that deflection (to make measuring it easier) Ñ but one has to be careful that r >> d! To get the error, we note that δBz/Bz ≈ δX/X, where δX/X is the biggest error found (most likely for β or r; δX means error in variable X). Ð4 The accepted value is about 0.15 × 10 Tesla, BUT it varies a lot because of interference, etc. Doing the experiment outside would thus be ideal. 6) How much do you charge for a free ride? a) B circles around a straight wire in a direction dictated by the right-hand rule (which comes from the Biot-Savart law).
B
I
b) For a time-varying magnetic field in the z direction, the induced electric field circles the magnetic field direction, like the magnetic field around a wire. The direction of the induced E field is such that it resists the change in the magnetic field (Lenz' Law). (figure next page)
6
c) ∫(E•dl) around the loop = Ð∫ ((dB/dt)•ds on a surface bounded by the loop = ÐdΦ/dt, where Φ(t) is the magnetic flux through the loop at time t. In our case, making our loop of radius a, centred at the centre of the wheel, 2 ∫(E•dl) = Ð πa (dB(t)/dt) d) The total torque on the wheel about its axis: N = r × F over the whole wheel. For a piece of the wheel of length δl, r and F are perpendicular and F = qE = (λδl)E. Therefore δN = rλδlE For the total torque, we integrate around the whole wheel dB(t) N = r λ ∫ Edl = r λ − π a2 dt
B(t) increasing
direction of induced
B(t) decreasing
direction of induced
where we have used the result of part (c). The total angular momentum is L = ∫Ndt where the integral is from an initial time to just before the B field is changed to the final time t just after the B field has stopped changing. 2 L = Ðρλ πa ∫ ((dB(t)/dt)*dt)) (from to to t) 2 = Ð ρλ πa ∫ (dB) (from Bo to 0) as initially B=Bo, and finally B=0. 2 so L= ρλ πa Bo e) From part (d), we see that the value of the final angular momentum is independent of how fast the field is switched off; it depends only on the initial field value, or, more generally, it depends on the difference between the initial and final field strengths BONUS f) Marks were given for any interesting answer Ñ the question is relatively sophisticated. The torque on the Merry-Go-Round wheel comes as a result of the E-field produced as the B-field produced by the solenoid vanishes (dB/dt - 0). This induced E-field accelerates the charges into circular motion, taking the wheel to which they are bound along too. 7
XX: Some might argue that this new circulating current produces a B-field of its own; the charges (already moving) in the solenoid see this changing B-field from the Merry-Go-Round disk (dB/dt - 0) as it induces an E-field. However, this is a bit of a red-herring Ñ the solenoid has equal numbers of positive and negative charges, so it doesnÕt pick up momentum this way. Someone might argue that the B-field pushes the electrons to one side of the wire (the Hall effect, see Q. 3) so that the angular momentum they pick up doesnÕt match the positive chargesÕ, but thereÕs no hope here! ÃÃ: Basically, the wheel will accelerate when it sees the B-field changes, even if the solenoid is a long, long distance D away. Conservation of angular momentum canÕt Ôhold offÕ, or wait, until the solenoid ÔlearnsÕ that it is pushing on the wheel Ñ a time D/c at the earliest, where c is the speed of light. So the angular momentum must be in the electromagnetic field; this changing electromagnetic field must have angular momentum which is ÔleftÕ with the wheel. The changing fields the wheelÕs motion creates will reduce the angular momentum of the overall electromagnetic field, as the (kinetic) angular momentum of the wheel increases.
8
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Problem Set 1: General Due November 6, 1998 1) Pushy photons In an episode of Star Trek: Deep Space 9, Capt. Benjamin Sisco makes a hobby out of recreating an early spacecraft, from alien archaeological records. This spacecraft uses huge sails to move the ship along on the solar wind (a stream of charged particles) or by light pressure from the sun. Consider such a craft the same distance from the sun as is the earth, and only consider light pressure. a) What is the best colour for the sail? b) For a sail area the size of a soccer field on a small ship of 2,000 kg, how fast will the ship be moving after 24 hours? After 1 year? In my research laboratory we have a laser which produces a terawatt of power (1012 W) in a pulse of light about one picosecond (10Ð12 s) duration. This pulse can be focussed to a spot about 10 µm across. c) How much force from light pressure does such a pulse exert? Roughly how much momentum is transferred? What is the pressure exerted by the light? [Robin] 2) Out of sight, hope you donÕt mind In electronics, a Ôblack boxÕ is a mystery package with electrical terminals. You try to figure out what is inside it by making measurements across the terminals. a) Is the following black box possible, if only passive elements are used (i.e., resistors, capacitors and inductors, only)? Two pairs of terminals come from the box. If a battery of voltage V is connected across the terminal-pair ÔAÕ, the voltage measured across terminal-pair ÔBÕ is V/2. But Ñ reversing things Ñ when the battery is connected across terminal-pair ÔBÕ, the voltage measured across ÔAÕ is V. If it is possible, what is inside? [123 Tricky]
1
3) LeonardoÕs camera obscura Simple ray optics can be demonstrated using a pinhole camera. This consists of an opaque sheet with a small hole in it, and a viewing screen (see layout below). Rays from an illuminated object travel through the pinhole and form an image on the viewing screen. [You can make a pinhole camera yourself: take a shoebox and make a small hole in the centre of one of the small sides. Cut out the side directly opposite it and replace it with wax paper. In a darkened room put a candle in front of the pinhole. You will be able to observe the inverted image of the candle on the wax paper.] If you use something other than a small hole, the image you see will be distorted. In the following five cases, you are shown two of the three components (object, opaque sheet with hole in it, and image) and your task is to draw the missing component that is consistent with the other two. To make things simpler, I have drawn the objects from the observer's point of view (see below). Don't worry about the exact proportions or intensity, just draw the shape that is consistent with the other two shapes. (For you fancy-pants out there, ignore diffraction and anything other than simple ray optics, i.e., take it that light travels in straight lines). Example of imaging with a pinhole camera: i)
View from side: Object
Opaque sheet
Screen
Observer
ii) From observer's point of view:
Illuminated object
Opaque sheet
Image on screen
2
All the following are from the observer's view:
a) Illuminated object
Opaque sheet
Image on screen
b) Illuminated object
Opaque sheet
Image on screen
c) Illuminated object
Opaque sheet
Image on screen
d) Illuminated object
Opaque sheet
Image on screen
3
e) Illuminated object
Opaque sheet
Image on screen
[James] [Artists used pinhole cameras in their invention of perspective drawing. But their cameras were entire rooms, and the hole was in a wall. They could sketch on the opposite wall. This was the camera obscura.]
4) Heat works Consider two identical spheres made of aluminum. One of the spheres lies on a heatinsulating plate, while the other hangs on a heat-insulating thread. The same amount of heat is transferred to the two spheres. Which one will have the higher temperature? For spheres of 100g each, and a heat of 250kJ, can you estimate the difference in temperature? [123 Tricky] 5) Death of an Atom Before quantum mechanics saved the day, it was noticed that there was a pretty basic problem with the atom. If one thinks of the atom as an electron orbiting the positively charged nucleus Ñ as a planet orbits the sun, but with electric fields instead of gravity Ñ it means that the electron is accelerating because of centripetal force. But it is wellknown that an accelerating charge radiates electromagnetic waves: light, x-rays, etc. This means the electron will lose energy, and eventually be drawn into the nucleus, so atoms cannot survive by classical physics alone. How long does it survive? The power radiated by an accelerating charge is given by: dE K q 2 2 = 3 a dt c
9
2
Ð3
Ð1
where K = 6 × 10 N¥m ¥C , c = speed of light [m¥s ], q = electron Ð2 charge [C], a = instantaneous acceleration [m¥s ].
Assuming the electronÕs orbit is a circle (atom is Hydrogen) a) Find the energy radiated during 1 cycle (frequency is f) b) At this rate, how many cycles before the radius is down to practically 0? 4
c) Therefore estimate the classical-physics lifetime of a hydrogen atom. [Peter & Robin] [This radiation by accelerating charges is the principle of a broadcasting antenna for radio, TV or microwaves.]
6) Stupid bet tricks Consider the following arrangement of balls on a pool table: 4 balls are lined up (touching each other) next to a pocket. Next, the cue ball is aimed exactly between the two middle balls. My friend claims that he can sink all the balls this way. Is he right? Assume an ideal table (no rolling friction) and slippery balls. Calculate the final velocities of all balls, given the initial velocity of the cue ball is V Ð1 m ¥s , that all collisions are perfectly elastic (energy-conserving), and that all balls are identical. As a matter of fact, if the balls are hit sufficiently hard they will all sink. If you found this to be impossible, above, then explain how it actually happens. [HINT: what if the slippery assumption doesnÕt hold?] [Peter] [Try www.physics.utoronto.ca/~POPTOR, to discover if we succeeded in putting a movie of this there!]
Useful Bits of Information Al: Ð5
linear expansion coefficient = 2.31 × 10 heat capacity = 24.4 J K mole 3Ê
density = 2.7Ê×Ê10 kgÊm
K
Ð1
Ð1
Ð3
Light intensity from sun at earthÕs orbit Ð2
Solar constant = 1.353 kW m
5
1997-1998 Physics Olympiad Preparation Program Ð University of Toronto Ð Problem Set 6 Solutions: AC Circuits and Electronics 1) MishaÕs MomÕs Medallions Faraday law of electrolysis states that the mass of substance liberated in electrolysis is proportional to the charge passed: m = kq = kIt = (m0/F)It, Where k is the electrochemical equivalent of the substance; m0 is a molar mass; F = NAqe is the Faraday constant which is equal to the product of the AvogadroÕs number and the electron charge. The mass of medallion m = Vd = Ahd = m0It/F. Thus the period of time for silver-plating the medallion t = FdhA/m0I = ((9.65)(1.05)(5)(5.7)(10))/((1.1)(1.8)) = 1458 sec = 24.3 min. Therefore, Misha has to pay 4 dollars for his momÕs birthday present. 2) Tesla Yes! Edison No! The electrical power delivered to the factory P = VI = 55 kW. The transmission line conductors have a resistance of 3Ω ⇒ the voltage drop along the line U = 1,500 V. Therefore, the output voltage must be 1,500 V + 110 V = 1,610 V to deliver 110 V to the factory. The power loss is P1 = 2 UI = 1,500 kW, which is 27 times greater than the delivered power. If we use transformers, we can claculate the turns ratio n2/n1 = V2/V1 = 500 and the current in transmission line I2 = I1 . (n1/n2) = 1A. The voltage drop along the Ð3 line U = 3 V, which is only 5 • 10 % of V2 = 55 kV. The power loss in the line Ð2 now is equal to P1 = 6 W, which is only 1 • 10 % of the value of the delivered power. 3) Analog differentiation and integration a)we have q dq + R = Vin C dt dq R = Vout dt
[1] [2]
Now, for low frequencies, the capacitor is charging/discharging all the time, and most of the voltage drop occurs across it [Rcapacitor >> R, if you like]. Hence, we can drop the Rdq/dt in eq. [1]. Differentiating eq. [1] and plugging into [2] gives:
1
q ≈ Vin C d q dVin = dt C dt dq dV V = C in = out dt dt R Or,
dVin dt [condition: low frequencies; input frequency << 1/RC] Vout = RC
Plots i)
iii)
dotted line: output (cosine) normal line: input (sine)
dotted line: output normal line: input
ii)
0-line: output normal line: input
NOTE: plots are not to scale! b)Circuit: R
Vin
C
Vout
we have q dq + R = Vin C dt q = Vout C
[1] [2]
Now, for high frequencies, there is not enough time for the capacitor to be 2
charged, and most of the voltage drop occurs across the resistor [Rcapacitor << R, if you like]. Hence, we can drop the q/C in eq. [1]. Differentiating eq. [2] and plugging into [1] gives: dq R ≈ Vin dt d q dVout = dt C dt dq dV V = C out = in dt dt R Or,
1 ∫ Vindt RC [condition is: high frequencies; angular frequency of input signal >> 1/RC] Plots i) iii) Vout =
dotted line: output (-cosine) normal line: input (sine)
dotted line: output (parabolas) normal line: input
ii)
dotted lines: output normal line: input
NOTE: Plots are not to scale! 4) What goes around, comes around
V
a) If the voltage across the battery is V, then the voltage across the capacitor is ÐV in steady state.
3
b) From the rather direct hints in the question, the voltage across the capacitor looks like: dVc/dt ∝ I(t), where ∝ indicates proportional, I(t) is the total current running through the circuit as a function of time, and Vc is the voltage across the capacitor Again, from the oh-so-subtle hints, the voltage across the inductor is: Vi ∝ dI(t) / dt From KirchhoffÕs voltage law, the sum of voltages in a circuit must be zero. For this to be true for all time, the change in voltage across one device must equal the opposite change in voltage across the other device. This allows us to write: dVc/dt = ÐdVi/dt 2
2
I(t) ∝ Ðd I(t) / dt This is the same as 2 2 x(t) ∝ Ðd x(t) / dt which is the equation for a simple harmonic oscillator, as seen in a previous problem set featuring everyoneÕs favorite oscillator, Bob (no relation to the author). The solution to this is I(t) ∝ cos(kt) where is k is some constant dependent on the inductance and capacitance in the circuit. If you are not sure if this is a proper solution, substitute it into the equation and verify that the LHS=RHS. (I am being a little lax here about the boundary conditions, i.e. what is the current at I(0). By writing the solution as cos(kt), I am assuming that the initial current is non-zero, so that we are starting the clock just after the battery has been shorted and removed, not before.) c) Energy transfers from the electric field in the capacitor to the magnetic field around the inductor and back again. d) As soon as Frido tries to do something with his circuit, he will no longer have a perfect LC circuit. He must put a resistor in series with it: 2
As you know, electrical energy is turned into heat in a resistor according to RI where R is the resistance and I is the current flowing through the circuit. Without even solving the circuit equation, you can show that energy is leaving the circuit. Alas, poor Frido has been boondoggled. 5) Operation: ÔAmplifierÕ a)
i) 0V ii) Ð15V iii) 15V
b) With any op-amp circuit there are two possible results: either the inputs are different and the op-amp is in saturation, or else the inputs are driven so that 4
they are at the same voltage. Feedback allows the the second result in each of the given cases. i) The current flowing through the feed back loop must be the same current flowing out the V- line. We try to find a self-consistent situation were VÐ =0. For this to happen, the current must be 1V/100 Ohms. Thus Vout = Ð1V. This circuit inverts the input voltage ii) Same reasoning as in part (i). Vout = Ð5V. This circuit inverts and amplifies with a gain of five, the input voltage. iii) Vout = Ð1.9V. This circuits adds and inverts the input voltages. iv) Vout = 2V. This circuit doubles the input voltage (no inversion) 6) Flip-Flopping on a counter proposal a) If R and S are both set to 1, there are 2 equally stable output configurations: Q=0 and Qbar=1, or Q=1 and Qbar=0. If we switch R to 0 and back to 1, Q and Qbar will switch from the stable configuration they started in to the other stable configuration. Thus a pulse in R causes the outputs to "flip flop" between the two stable configurations. b) pulse
A
B
C
0 1 2 3 4
0 1 0 1 0
0 0 1 1 0
0 0 0 0 1
We can see that if we let the output of A be the 20 binary digit, B be the 21 digit, and C the 22, this counter gives the binary value for the number of pulses that have been fed into the input (e.g., 4 = 100 base2)
5
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Solution Set 1: General 1) Pushy Photons a) Black is good Ñ it would absorb all the light, and therefore transfer all the momentum of the light. But silver would be better Ñ when the light reflects back to where it came from, it reverses its momentum. So this would mean (pÐ (Ðp)) = 2p for the momentum transferred to the sail. b) From the useful bits of info at the bottom of the question sheets, the solar Ð2 constant is 1.353 kW m . For light, E = pc, so by taking derivatives we also have that:
dE dp = ⋅c dt dt However, power P is defined as the rate of change of energy with time, on the left-hand side, and force F is defined as the rate of change of momentum with time, on the right-hand side. So we can write: P=
dE dp = ⋅c = F⋅c dt dt
or F=
P c
or actually F = 2P/c if we let the light be reflected. A soccer field isnÕt always the same size, but has to be in a certain allowed range of sizes; roughly 100m × 50m might be reasonable. The power P incident on the Ð2 2 field is intensity times area: P = IA = 1.353 kW m ¥ 5000 m = 6.765 MW (quite a bit of power), so the force on the whole large area would be 8
Ð1
F = 2¥ 6.765 MW / (3 × 10 m s ) = 4.51 × 10
Ð2
N
Also, F = ma, so the acceleration of the 2,000 kg spacecraft would be: a = 2.25 × 10
Ð5
Ð2
ms
The velocity after constant acceleration for a time t depends on initial velocity (which is zero, here):
v = u + at
A day of seconds is (24 hr) ¥ (60 min/hr) ¥ (60 sec/min) = 86,400 s. So the Ð1 velocity after a day would be 1.94 m s , which is jogging speed. After a year, it Ð1 would be 365 times this value, about 710 m s , which is roughly 10% of the orbital speed of a low-orbit satellite. 8
Ð1
3
c) Again, if the light is reflected, F = 2P/c = 2 ¥ 1012 W /(3 × 10 m s ) = 7 × 10 N, which is the gravitational force of the smallest automobile. The force is only applied for a trillionth of a second, so the momentum transfer (the impulse) is: 3
Ð9
p ¥ ∆t = (7 × 10 N) ¥ (10Ð12 s) = 7 × 10
Ð1
kg m s
which is small, like a big dust-mote moving in a sunbeam, or a raindrop in a very misty rain. The pressure is a different story, though, because this force is applied over a tiny spot: 3
Ð6
P = F / A = (7 × 10 N) / ( (10 × 10 3
2
13
m) ) = 7 × 10
Ð2
Nm
Ð2
100 kPa = 100 × 10 N m is roughly one atmosphere of pressure. So this pressure, briefly exerted by the focussed laser pulse, is something like 700 million atmospheres of pressure. [Robin]
2) Out of sight, hope you don't mind This simple circuit works, using any two identical resistors. Worth noting: since the terminal-pairs ÔAÕ and ÔBÕ donÕt behave the same, the circuit must not be symmetric for the pairs of terminals. [Gnädig/Honyek]
A
R R
3) Leonardo's camera obscura a) Illuminated object
Opaque sheet
Image on screen
B
b) Illuminated object
Opaque sheet
Image on screen
c) Illuminated object
Opaque sheet
Image on screen
d) Illuminated object
Opaque sheet
Image on screen
e) Illuminated object
Opaque sheet
Image on screen
[James] 4) Heat works We begin with two identical spheres at some uniform temperature. One rests on an insulating table and the other is suspended by an insulating thread. To each sphere the same amount of heat energy, Q Joules say, is transferred. In the absence of external forces we would expect each sphere to have an equal temperature change. However, since gravity is present and the volumes of the sphere will increase therefore their centres of mass will move.
Since the ball on the table will have its centre of mass rise, some heat energy will be used to do work against gravity. For the hanging ball, however, the situation is reversed. Therefore the resting ball will have a lower final temperature than that of the hanging ball. Suppose each sphere is 100 g, initially at some temperature T I K (the density of aluminum is 2.7 × 103 kg mÐ3) and 250 KJ of heat is added to each. Let ∆rA and ∆TA denote the change in radius of the sitting sphere, sphere A and ∆rB and ∆TB denote that for the hanging sphere, sphere B. The initial radius of each sphere can be calculated with
1 kg [mass of aluminum] 4 3 10 πri = = 3 [density of aluminum] 2.7 × 10 3 kg m −3 1
3 × 10 −3 3 ri = ≈ 0.0207 m 27 ( 4 π)
then
Assuming a linear expansion coefficient to apply over the entire uniform heating we have (to first approx.), ∆rA = [2.31 × 10-5 ∆TA]ri = 4.777 and ∆rB = [2.31 × 10-5 ∆TB]ri = 4.777 Hence ∆TA =
1 2.31 × 10 −5
∆rA ri
∆TB =
1 2.31 × 10 −5
∆rB ri
and
The heat capacity, C, of 100 g of aluminum is simply C = 24.4
J 27 ⋅ ⋅ (100 g) K ⋅ mol 27 g/mol
= 90.3704
J K
Where we have used the molar mass of aluminum to be 27 g/mol. Clearly the centre of mass of sphere A will rise by a distance equal to ∆rA and the centre of mass of sphere B will fall a distance equal to ∆rB.
Since the heat energy transferred must balance the change in potential energy plus the energy due to the temperature change we have 250 000 = mg ∆rB + C∆TA =
9.81 ( 4.777 × 10 −7 ) ∆TA + 90.3704 ∆TB 10
Thus 250 000 = [4.777 × 10Ð7 + 90.3704] ∆TA Similarly 250 000 = [4.777 × 10Ð7 + 90.3704] ∆TB Therefore, the final difference in temperatures is ∆T = ∆TB − ∆TA 1 1 = 250 000 − −7 90.3704 + 4.777 × 10 −7 90.3704 − 4.777 × 10
[
≈ 250 000 1.170 × 10 −10 ≈ 2.925 × 10 −5
]
Therefore the difference in temperature is approx 2.87 × 10 [Gnädig/Honyek & Peter]
Ð5
K. Pretty small!
5) Death of an Atom The energy given off in a time ∆t is roughly
kq 2 a 2 ∆t c3 In 1 orbit (time T), ∆E =
kq 2 a 2T . c3 We will assume the radius of an orbit stays constant for any one cycle (the electron is actually spiralling in, so this is not really true). The radius is R ≈ 0.5 Angstroms. The electron is kept in orbit by the attraction between it and the ∆E =
2
nucleus. We have m V = kqq = ma , where m is the mass of the electron and V is R R2 the orbital speed. The charge of the nucleus is +q for Hydrogen. Plugging all this in yields the nice expression:
Ecycle =
2 πkq 2 V 3 . R c
Here, this equals about 4.4*10-24 J / cycle [this should be negative, but we will remember that this is the energy lost] The total energy of the orbit is: −
kqq 2R
compared to that for a gravitationally orbiting object: −
Gmm . R
The electron will presumably ÒdieÓ when all this energy is exhausted. If we assume that the energy loss per cycle is as in i) (not really true), we will have:
2 πkq 2 V 3 kqq = n , 2R R c where n is the number of cycles. Plugging in the stuff, n = 3.4*105. The time, then, for this to happen is simply nT (T is the period) which comes out to about 6*10-11 s. It is not really meaningful to carry the first digit because of all the approximations we made. Surprisingly, doing a more accurate analysis leads a very similar answer. Either way, this is a very short time, and we have to conclude that the classical answer cannot be right. [Peter]
6) Stupid bet tricks i) First, notice that when the balls collide all forces between them (and hence the momentum transferred) will be normal to the surfaces (that is the only point of contact between them). This is because there are supposed to be no frictional forces. Let the labeling be as follows:
If we tried to do the question directly, we couldnÕt Ð we have 4 unknowns (U, B, A, direction of B) and only 2 equations (energy, momentum).
However, since the collisions last a very small amount of time we can split the problem into the two middle balls interacting with the cue ball and THEN the two middle balls interacting with the outside balls. Doing this we get:
where the final speeds of the pair of balls are C (symmetric). Additionally, we see that the momentum of each ball must be at a 30o angle to the normal:
So,
Cx 1 = tan(30) = . Cy 3
Solving, C =
2 3 1 V (direction is 30o to y-axis), U = V [negative y direction]. 5 5
Now, the two remaining collisions (which are symmetric, so we only solve one Ð the right hand side one).
We could write down all equations and solve this by brute force, or we could simply notice that Cy will be unchanged because there are no tangential forces (the balls are slippery Ð they donÕt ÒstickÓ). Hence, B y = C y . Solving the remaining equations yields C 3 = V 2 5 3 B= V 5 A=
(the direction of B is [positive y], that of A is [positive x-axis]). So we have:
And it seems the trick shot didnÕt work Ð only the outside balls have been sunk. ii) When you hit the balls hard they deform slightly and do ÒstickÓ to one another. That means that there is a tangential force between the outside most and middle balls (remember in the solution above we assumed no tangential forces to solve this collision). Numbering the balls 1 to 4 from left to right [see picture above] yields the following: 2 has a counterclockwise spin and 3 has a clockwise spin; moreover, the 2 centre balls now also have a horizontal velocity component [why ?]. Upon hitting a bank, which is fairly soft, the spin of ball #3 will make it rebound at more or less the same angle as it came in at Ñ so 3 will go into another corner pocket. Now that you are completely confused look at the picture to make some sense of this:
[Robin is still trying to get a video picture of this pool-shot onto the website,at www.physics.utoronto.ca/~poptor ] [Peter]
1
This elevator starts with constant acceleration over two floors between the ground floor and the second floor, and then rises at a constant speed of three seconds per floor. All floors are 4 m apart. Will the bag split open Ñ should Kit have double-bagged? What should be the maximum number of cans that Kit can load up with? [Robin] 3) Sink, sank, sunk At some point in the movie ÒTitanicÓ it is said that the pressure at its sunken depth is 2 5500 pounds/inch . Given this, find the depth at which the Titanic supposedly came to 3 Ð3 rest. The ÔrealÕ answer is 12600 feet, density of water is 10 kg m (ignore air pressure). Comment on the results, i.e., truth in film-making. At another point in the movie someone estimates that the Titanic will sink in a time of roughly half an hour. Approximating the Titanic as a rectangular box of cross-sectional 2 area A = 882 × 34 feet , mass M = 40 tons (loaded), total height 80 m, and a horizontal 2 hole at the bottom of a side of the box of area AÕ = 100*0.5 m , estimate the time it takes sink Ñ that is, when the top of the box is even with the water line. Ignore air. [HINT: assume the flow stays practically at equilibrium as water goes in; you may use BernoulliÕs equation, but if you do, you have to prove it.] [Peter] 4) When bowling, always signal your lane-changeÉ Consider the problem of throwing a bowling ball with an initial spin, so that its path curves. Take the x-axis as the centre-line down the bowling alley, and y as the horizontal sideways direction. Say that when the ball is released, it has an initial spin ω r with the top of the ball moving sideways (i.e., ω points along the x-axis), and a velocity r v at an angle θ to the centre-line. Find the path of the bowling ball, by giving the x-cošrdinate and y-cošrdinate as time goes by, i.e., x(t) and y(t), where t is time (this is called Ôparametric formÕ). Assume the point of contact between the ball and the floor is very tiny. The moment of inertia of the ball will be that of a sphere. Plot some nice-looking representative paths (ω = 0, ω > 0, ω < 0) [Peter] 5) A matter of some gravity i) The moon rose in the east today at 7pm. If the lunar cycle (number of days between full moons) is 29 days, when will the moon rise tomorrow? (One other subtly ÔhintfulÕ piece of information: if the earth were moving more slowly, in orbit around the sun, the lunar cycle would be shorter.) ii) The spaceship GalileoÕs mission is to study Jupiter and its surrounding moons. To get the vessel into a Jovian orbit, NASA had the vessel fly-by both Venus and Earth to 2
increase its speed. This type of trajectory which provides acceleration without the burning of expensive fuel is called a sling-shot maneuver. Consider one of these interactions in which the EarthÕs large velocity was used to accelerate the vessel. What is the maximum speed that the vessel could have attained after this interaction, if its Ð1 speed before the encounter was 5 km s ? Galileo (mass: 1000 kg) actually did this type of maneuver twice with Earth on its way to Jupiter. How many such sling-shots would have to occur for the length of our year to decrease by one second?. [James] [Check out www.jpl.nasa.gov/galileo/index.html for some great pictures and information on this billion dollar space mission]
6) Balloon tug-o-war experiment HereÕs a neat experiment you are welcome to do with a friend (put both your names on, for POPTOR marking). ItÕs not too hard, but itÕs full of physics. You will need: 2 good-quality round balloons 1 thick drinking straw some scotch tape Have your friend blow up one balloon about 1/4 full and the other about 1/2 full. Pinch the neck of each balloon so that no air escapes. Carefully insert one end of the straw into the neck of each balloon and tape it air-tight, while still pinching. It will look like a balloon bar-bell for weightlifting. Before actually trying it, what do you think will happen when your friend stops pinching the balloons so that air can flow between the balloons? Argue for your predictions on the basis of what physics you can figure out or guess. Then, hold the straw gently and have your friend release the balloons. Were your predictions right? Try squeezing the balloons, one at a time, and see what happens. To examine this interesting phenomenon further, do the following measurement: Try to determine the air pressure inside the balloon, as a function of the size of the balloon. You will need: clear flexible tubing about 3 to 4 metres in length (see INFOBITS, below) 1 clear flexible tube about 1/2 metre in length 1 two-hole stopper with holes to fit the tubing snugly some duct tape 2 metre sticks or long rulers You might want also to use (optional) 1 500 ml (or larger) kitchen measuring cup, or beaker with volume markings two large soup cans 1 large bucket or plastic laundry tub (a nearby sink or bathtub will do fine) 3
To set up: 1. Seal one end of each piece of tubing into the stopper 2. Make the long tube into a rough ÔUÕ shape, and tape each side onto a meter stick or long ruler. Make the end farthest from the stopper to be sloping, as pictured. Be sure your setup is stable. 3. Half fill the long tube with water. This will be a manometer for you to measure pressure with. You may need more adjustments once you begin. ∆h 4 . Now fit the balloon onto the stopper. Gently fill up the balloon, blowing air into it through the short tube. If you notice air leaking anywhere you might use some Vaseline or duct tape to seal it up.
balloon
With the right amount of water in the long tube, you can measure the difference in height of the water on each side of the ÔUÕ. With the ruler or metre stick at a sloping angle you can measure along the tube, and then use trigonometry to figure out the height. What is the relationship between air pressure in the balloon and difference of water levels in the manometer? Now make a series of measurements of air pressure, for the balloon inflated to different sizes (including not inflated). How can you best measure the size of the balloon: Cast a shadow onto paper, and measure the shadow-size? Use a tape measure to find the circumference? Use the measuring cup and tub to trap air released from the balloon, and read its volume? WeÕre interested in your best ideas Ñ change anything you want. Finally make a graph of your data with height vs. balloon size, and connect the points with a smooth line. Be sure to label your axes with dimensions. From your results, can you explain the outcome of the first 2 balloon experiments? [Bonus-mark ÔsequelÕ experiments: It would be interesting to see if the pressure in the balloon, as a function of size, varied differently depending on whether you were inflating or deflating the balloon. Do you get a different answer if you measure volume of air and if you measure diameter of balloon? Would it make a difference if you filled the balloon not with air but something incompressible, like water? (How could you change the manometer in order to use water in the balloon?)] [Simal and Robin] 4
INFOBITSª Ñ Useful Bits of POPTOR Information Mass of Earth: 6. × 10 Mass of sun: 2 × 10
30
24
kg
kg 11
Distance between Earth and Sun: 1.5 × 10 m G = 6.7 × 10
Ð11
2
N m kg
Ð2
Where to get parts for experiments: there is plastic tubing at Canadian Tire, in the Plumbing section, for 17¢ per foot. DO NOT attempt to use glass tubing, unless you have teacher supervision. It is pretty easy to break it and have it go right through your hand, if your technique is wrong Ñ especially when putting it into rubber stoppers. ¥ polyethylene, 0.25Ó O.D. (outer diameter); translucent, not quite clear (stock no. 7 78380 022022 1; SKU: 039166056347) ¥ vinyl, 3/16Ó O.D.; clear, but soft, and hard to stuff into a rubber stopper (stock no. 7 78380 02032 0; SKU: 039166056224) 2-hole rubber stoppers: from chemistry class, we thought C HECK THE POPTOR WEB PAGE for other hints, and any corrections we might post: www.physics.utoronto.ca/~poptor
5
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Marker’s Comments on Set 2: Mechanics General Comments from POPTOR:: Our apologies on how late we are in getting material back to you. We hope that you find the solutions and these comments are so good that they partially make up for the delays. We have had a change in our team: Carolyn MacTavish joins us, to help us mark, make up solution sets, and run the POPTOR invitational weekend. WeÕre rushing to catch up, and you should have #3 Ð Thermodynamics returned with solutions pretty quickly this time. If you havenÕt bothered with #3 because weÕre so late, please jump right back in for #4 Ð Waves and Optics! Lots of people miss handing in one set. Peter sez: With problem set 2 finished, I am now 2 for 2 in getting the wording of my questions mixed up. The wrong revisions of questions going out, me not being able to multiply, etc. have contributed to this. Either way, you all have my apologies. In all cases, however, the mistakes have not changed the core of the questions (yet). The marking, and my comments, follow questions 3, 4 and 6 as you had them (not necessarily as I planned). Robin adds: Er, well, actually that was my fault. Sorry, Peter & all POPTOR participants.
1) Football fizix Carrie sez: Good job! This problem was done fairly well by all (avg. 6.7/10). The most common mistake was in assuming an initial velocity of 30 m/s. However, this is not the best choice since it does not yield the maximum horizontal velocity (given the constraints of the defence wall and that you are aiming for the top corner of the net-furthest from Plante). 2) Chow, baby Carrie sez: Excellent job! The majority did very well on this problem (avg. 8.2/10). The most common error was in the assumption that the bag would break if it contained exactly the weight of 16 cans (at rest).
1
stuff in Excel (or something similar) and typed the whole thing up. Most did not attempt the experimental question Ð which is perfectly understandable, since experimental questions donÕt have a long tradition at POPTOR (more like none at all). Also, they do take extra time an effort. On the other hand, I would argue that experimental questions teach you the most physics, because to do well you not only have to know your equations, but also understand what they mean. This is a purely philosophical argument, so if youÕre not into that kind of stuff you should realize that at both the national and international levels experiments are a major component of your mark (50%). To encourage people to do the experimental questions (which we will have more of) they will be weighted more than the other questions Ð each will be equivalent to two theoretical questions (plus a bonus for any interesting ideas / set-ups you come up with).
Final Comments: Peter sez: Concluding I once again want to stress how impressed I was by peopleÕs solutions to these questions considering how hard they really were. I hope you keep trying, and speaking of trying we finally got the movie of the trick shot (problem set #1) on-line (digitizing it was harder than making the shot!). Check it out, at: www.physics.utoronto.ca/~poptor/Com98/98GenXtra.html Proof positive that physics worksÉ (rules!)
3
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Solution Set 2: Mechanics 1) Football fizix Your best bet is to go for the corner of the net farthest from Plante (see figure given in question). We can reduce this to a two-dimensional problem by looking at the cross-section along the ballÕs trajectory. Since there are no air effects, the ball heads straight from your foot, over the heads of the defenders and into the upper 3m 2.5m corner of the net, as in the net you figure at right. (Yay! Cheers!) 8m 23.3m If you kick it at too high an angle, it will easily clear the heads of the defenders but will take longer than necessary to make it into the net. This gives Plante more diving time to intercept the ball. To minimize the flight time, you want the ball to have the maximum horizontal speed (covering the distance to the net the fastest), but a corresponding vertical speed that will allow it to barely miss the defenders and still go into the top corner of the net. As always in these projectile motion problems: separate the problem into horizontal and vertical components. Constraints: 1) At x = 8 m, y > 2.5 m 2) At x = 23.3 m, y < 3 m. 3) At t = 0 s, you kick the ball from x = 0 m, y = 0 m. Ð1 4) Max speed = 30 m s The equations of motion are horizontal at constant speed, and vertical motion with gravity: vx(t) = vx(t=0) = vx vy(t) = vy(t=0) Ð g * t = vy Ð g * t
(note how I am defining vx and vy)
So the ballÕs path is: x(t) = vx * t y(t) = (vy Ð (g/2) * t) * t
1
Constraint 3 is satisfied by how I set up the equations. Constraint 1: Let t = t1 when x = 8 m. 8 = vx * t1. y(t1) = (vy Ð (g/2) * (8/vx)) * (8/vx) but y(t1) > 2.5 Thus 2.5 / (8/vx) < vy Ð (g/2) * (8/vx) vy > 2.5 / (8/vx) + (g/2) * (8/vx) vy > vx * (2.5/8) + (4*g) / vx
<=Eqn 1
Constraint 2: Let t = t2 when x = 23.3. 23.3 = vx * t2. y(t2) = (vy Ð (g/2) * (23.3/vx)) * (23.3/vx) but y(t2)<3 Thus 3 / (23.3/vx) > vy Ð (g/2) * (23.3/vx) vy < 3 / (23.3/vx) + (g/2) * (23.3/vx) vy < vx * (3/23.3) + (23.3/2) * g / vx <=Eqn 2 Constraint 4: 2
2
vx +vy ≤ 30
2
<=Eqn 3
largest vx solution 3
Graph the three inequalities, to get a clear picture of how they can be satisfied, in the region above curve Ô1Õ, below curve Ô2Õ and below curve Ô3Õ: see that the maximum horizontal speed v x occurs very close to the intersection of eqn 1 and eqn 2. The intersection is at:
1 2
vx * (3/23.3) + (23.3/2) * g / vx = vx * (2.5/8) + (4*g) / vx or Ð1 vx = 20.2 m s Ð1
This corresponds to vy = 8.25 m s . To make sure that you are not hitting the crossbar, reduce vx slightly to Ð1 20.1ÊmÊs , say. This will still meet the constraints imposed by equations 1 and 2.
2
Ð1
Therefore you must kick it with an initial speed of 21.7 m s at an angle of 22 Ð2 degrees above the horizontal. (Soccer is a science!). (I am using g = 9.8 m s throughout this problem.) BONUS: Air effects can help you. By giving the ball the proper spin, you can cause the ball to deflect downward from its normal parabolic path. This will allow you to increase the initial horizontal velocity but still have the ball go in the net. If you are really fancy, you might be able to curve the ball enough so you go around the defenders instead of over (ÔcurlÕ or ÔswingÕ the ball left or right). [James]
2) Chow, baby Find acceleration a: Ð elevator has constant acceleration over two floors, Ð final velocity is one floor per 3 seconds, Ð initial velocity is zero, 2
d = 2¥4m = 8 m Ð1 v = 4m/3s = 1.33 m s u = 0
2
v = u + 2ad 2 2 (1.33) = (0) + 2a(8 m) Ð2 a = 0.111 m s Adding this to the acceleration due to gravity, the total acceleration will be: Ð2
Ð2
gÕ = 9.80 m s + 0.11 m s Ð2 = 9.91 m s
The ratio of the acceleration in the elevator to that standing on the ground is therefore: 9.91/9.80 = 1.01. The acceleration is only 1% greater. Find weight of cans with this additional acceleration: Then 15 cans of cat food accelerated in the elevator would weigh on the bag the way that 15 ¥ (1.01) = 15.15 cans would at rest. This is less than 16. Does it mean the bag would not break? No! We only know that 16 will break the bag, but we donÕt know whether a little bit over 15 would. ThatÕs because cans come in units of 1 can Ñ there was no way to test whether 15.15 cans would break the bag. Until now: Kit can be the first person to test out the bags at this weight, without having to cut up a can of cat food in the band-saw. Recommendations? Double bag, or else carry only 14 cans of cat food. The 14 would weigh in the accelerating elevator the same as 14.14 would on the ground.
3
And any weight 15.00 or under in the elevator is definitely safe, because 15 cans under normal acceleration was! [Robin]
3) Sink, sank, sunk
Consider a pill box of water of area A and height h. For equilibrium,
PA + m g = P ′A PA + A h ρ g = P ′A P′ = h ρ g + P Now extend the pill box to be a tall cylinder of height h (depth h to water level). Ignoring air pressure, P = 0, and we get: h=
P ρg
Here, h Å 12600 feet [roughly] for 5500 punds/inch2, so thereÕs good agreement. As the system starts without any water inside, the water must be accelerated (thatÕs why you cannot use BernoulliÕs equation inside the ship, which only applies to steady flows). Later, it reaches steady state. Let us use the hint and use a force approach: First note that outside of the hole the water will hardly be moving Ñ the water comes in from all directions, so that the velocity at any one point is almost 0 (but because the ocean is so big this adds up to give a large flow in the hole). Now, we will consider the system in a small time interval ∆t. Since we are in equilibrium, the water inside is stationary at first. For a small time ∆t an external force (due to external pressure) and a negative force (due to pressure of water in the ship) acts at the hole. The net force will force in a small amount of water ∆m. Additionally, a part of the force will push some of the water upwards, but as ∆tÊÐ> 0, ∆x Ð> 0 and this is negligible. ∆m is hence pushed an amount
4
1 a(∆t)2 2 1 (ρ g h − ρ g x)A′ ∆s = (∆t)2 2 ∆m ∆s = 0 +
Since the pressure due to a height z of water is ρgz (from i)) ∆s =
1 ρ g A′(h − x) 1 g(h − x) (∆t)2 = ( ∆t ) 2 2 ρ ∆s A′ 2 ∆s
From conservation of mass, and amount ∆m of water at the hole must be equal to the amount of water that causes a rise of the level in the ship, i.e., ∆m = ρ∆sA' = ρ∆xA Using this gives: 2
A 1 (∆s)2 = (∆x)2 = g(h − x)(∆t)2 A′ 2
(in the limit)
∆x A′ 1 dx g ( h − x) = = ∆t A 2 dt
As the water flows in, the system comes to equilibrium. To compensate for an increase ∆m in mass the ship must have moved (down) an amount ∆h = − ∆x
Note that this does not imply that h = x! This simply says that x and h change by the same amount. In fact, at time = 0, x=0 And h = M/(ρA) (M is the mass of the ship Ñ 40, 000 tons; this is just ArchimedesÕs principle) And so h(t) = x(t) + M/(ρA) Plugging this into the expression for dx/dt yields:
dx A′ 1 gM = dt A 2 ρA So the water level rises at a constant rate. From this we find:
5
dh A′ 1 gM = dt A 2 ρA A′ 1 gM M h(t) = t+ ρA A 2 ρA
Putting h = 80 m, we find t ≈ 40 minutes. We didnÕt catch them this timeÉ _______ This answer is pretty good Ñ only the Ò1/2Ó factor above may have to change [I did an experiment sinking a shoe box, and it looked more like it should be a Ò1Ó, which is fairly close]. Using BernoulliÕs equation gives nonsense for the time it takes the ship to sink (something on the order of 1 minute). Note that you can use BernoulliÕs equation, if you set it up the following way: take a streamline from the top of the water to the hole in the ship Ñ this flow is steady and you get something like:
Mg 1 = ρV 2 (where V is the speed of the water coming in, P the pressure A 2 at the holeÕs depth). P=
Using conservation of mass we now get:
gM dx A′ = 2 dt A ρA which is the same as the equation above, except for the factor of 2 Ñ which is just as good because we used approximations in either solution. Note also that the flow of the water is different while the hole is not completely covered with water Ñ thatÕs why the height of the hole (0.5 m) is so small, to make this effect negligible. Speaking of BernoulliÕs equation, the proof may be found in virtually any textbook (which is too bad) Ñ see e.g. Halliday & Resnick. [Peter] 4) When bowling, always signal your lane-change The math here isnÕt too hard, but one has to think about whatÕs going on or itÕs very easy to be off be a minus sign (the International Olympiad judges kill (well, almost) for errors like this). First, since we know there is friction, we see that it will oppose the ballÕs motion. Now we resolve the motion to two directions Ñ x and y Ñ and two motions Ñ translational and rotational.
6
ω y x
Y: Translational: (friction slows motion)
M a = − µ k ( Normal) = − µ k Mg a = − µk g V = Vy − µ k g t Rotational: (friction causes ball to spin faster, until Iα = τ 2 M R 2α = µ k ( Normal)R 5 5 µk g α= 2R 5 µk g ϖ=0+ t 2R The ball will continue to skid until itÕs rotating fast enough so that it can roll. Then, V = ωR. This will occur at a time given by: Vy − µ k g t* = t* =
5 µk g t* 2
2 Vy 7µ k g
The motion thus consists of two phases Ñ with skidding and without: 1 y(t) = Vy t − µ k gt 2 , 2 y ©(t) = y(t*) + V(t*)t =
t≤t* 12Vy 2 49µ k g
+
5Vy 7
t≥t*
t,
X: This is very similar:
V = Vx − µ k g t 5 µk g ϖy = ϖ − t 2R
7
When skidding stops Ð V = ωR. But we have to be careful! In this case, V changes sign, and so the above should actually read |V| = |ω|R! [How do we know it changes sign? Well, you can do what I did Ñ go through the whole calculation carrying the wrong sign. Then do the plot I asked for and get nonsense. Or you can be a lot smarter Ñ assume that V changes sign (or not), and then go back and see if your assumption holds] t2 * =
2(ϖR + Vx ) 7µ k g
1 x(t) = Vx t − µ k gt 2 , 2 x©(t) = x(t2 *) + V(t2 *)t =
t ≤ t2 * (ϖR + Vx ) 5V − 2ϖR t, t ≥ t2 * (12Vx − 2ϖR) + x 49µ k g 7
Now we can finally proceed to plot this. We have to calculate t* and t2* to know which motion occurs when. It turns out that the ball stops slipping after about 3 seconds in the xdirection, and after about 8 in the y, but by that time the ball has long left the bowling alleyÉ Plotting yields figure at right.
the [Peter]
5) A matter of some gravity i) From the information given we know that if the earth was rotating more slowly, the lunar cycle would be shorter. This means the direction of rotation of the earth is the same as the direction of the moon around the earth Ñ it has to ÔchaseÕ it instead of meeting it Ôhead-onÕ. Thus, the moon rises later every day. By how much?
8
In the time the moon has moved by angle a, the Earth has moved by angle b. The two triangles are set so that the moon rays are parallel to the surface of the earth. They are similar triangles so a = b Ð 360 ¡
ray from moon a
Also (where t is measured in hours): earth
a=t / (29*24) *360¡
moonrise
b
b= t /24 *360 ¡. Thus the time between moon rises is: 24.86 hours. Thus if the moon rose today at 7PM, tomorrow it will rise at 7:52PM. ii) This is a standard two-particle collision problem. The fact that the interaction is through gravitational forces does not change how one solves the problem. Both energy and momentum of the entire system must be conserved. We consider the system before the interaction (far from the earth) and after the interaction (far from the earth). Variables:
ve, ve+dve: velocity of the earth before and after the interaction vs,vs+dvs: velocity of the ship before and after interaction me: mass of earth ms: mass of the ship
Conservation of energy gives: 2
2
2
me * ve + ms * vs = me * (ve + dve) + ms * (vs + dvs)
2
For maximum increase in speed for the ship, the initial velocities should be parallel. We can drop the vector notation: me * ve + ms * vs = me * (ve + dve) + ms * (vs + dvs) ==> Ð me * dve = ms * dvs Substituting this into the above gives and solving for dvs gives: dvs = 2 * (ve Ð vs) / ( (ms/me) +1) We need to know the speed of the earth relative to the sun. We can write: 2
2
Fg = me * ve / r = G * msun * me / (r ) 4
Ð1
4
Ð1
This gives ve = 3 × 10 m s . Thus the final speed of the ship is: 5.5 × 10 m s . 7
There are 3.1 × 10 s in each year. For this to change by 1 s, then we must change 7 the momentum of the earth (i.e., its speed) by 1/(3.1 × 10 ) of its total. One slingshot maneuver changes the momentum by ms * dvs . The total initial 9
momentum is me * ve. be:
Thus the total number of sling-shots, N, would have to 7
N * ms * dvs = 1/(3.1× 10 ) * me * ve N ≅ 1× 10
14
Huge. The earth is not in peril from this, yet! [James]
6) Balloon tug-o-war experiment This experiment is great Ñ it shows all kinds of different physics relationships, even though it is pretty simple. It is even a little bit easier to do than to describe! What did you think would happen between the two balloons fitted to opposite ends of a straw? The first time I thought of this, I thought perhaps the balloons would end up being the same size, because the balloons are identical and used the same way, so I figured they should do the same thing Ñ and that means they would keep the same amount of air in them. Wrong! Instead, what happens is that one balloon will be bigger and the other smaller. And if you squeeze the bigger one, and make it a bit smaller than the other one, it will almost collapse, and become the smaller one while the other is the bigger one. So, they each do the same thing, but not at the same time! The balloon you squeeze to be the smaller one always expells most of its air. So there are two ÔstableÕ or fixed arrangements: the left balloon small, or the right balloon small. These will stay that way, but you can switch between them. This is technically called ÔbistabilityÕ, and it also describes the kind of room-light switch that you can snap on or snap off Ñ it can be one or the other, but not much in-between, and it wonÕt just change by itself. Bistability can be described with a potentialenergy diagram that has two local minima: there is equilibrium in one minimum or in the other, but it will cost a little energy to switch between them. ThatÕs where squeezing the balloon comes in! The rest of the experiment is about showing why there are two minima, or maybe even showing what the potential energy curve looks like for two balloons on a straw.
A ball can come to rest at either of two places on a curve like this. Therefore it is ÔbistableÕ (bi = two). It is the same for any system with a similar potential energy curve Ñ like the two balloons on a straw.
The picture below shows my own setup for this experiment. I guessed it would be a bit hard to read the fairly small pressure changes of a balloon, so I tilted the manometer arm, taped it to a meter stick, and
10
measured the surface of the water as it moved along. Since the height difference gives me the pressure in the balloon, over atmospheric pressure, tilting the tube made the water move farther along, ∆L, for a given height change ∆y (i.e., ∆L/∆y = 1/sinθ ). I blew up the balloon, through the extra smaller tube, and measured the difference in height ∆h of the water surfaces (meniscus). Actually, I measured ∆L, and converted it to a height raised, adding it to the height ∆L by which the water level in the other arm dropped. This gave me the pressure inside the balloon. To measure balloon size, I tried two methods: deflating the baloon by letting air out into a beaker turned upside down underwater, then measuring its volume; also I used a tape measure to measure the balloonÕs circumference. The second method was good Ñ it was sensitive, it was pretty easy, I could use it as I inflated the balloon or deflated the balloon, either way, and I didnÕt get as wet. Below is the curve I graphed from the data I took. Each symbol marked on the graph is a measurement I took. 5000
4000
Pressure (Pa)
ItÕs interesting that the curve for the balloon as it is being inflated is not identical to the curve for the same balloon as it is being deflated. This is because the balloon is not perfectly elastic Ñ it has a sort of ÔmemoryÕ because it takes a little while to recover itÕs shape after
3000 inflating 2000 wrinkled & deflated 1000 deflating 0 0.02
0.04
0.06
0.08
11
0.1
0.12
being stretched. So the deflated balloon ends up wrinkly, since it has not gone back to its original shape. This means that even though it returns to zero Ôgauge pressureÕ (pressure above atmospheric), the balloon is larger after being inflated. The balloon may recover its shape after a half-hour or so. This kind of general effect Ñ having a curve which is different going up than it is coming down, for the same values of radius Ñ is called hysteresis. You may see it in the future in things like magnetism (say, a nail used in an electromagnet, which stays a bit magnetized afterward), and it comes up in many places in physics. The balloon tug-of-war doesnÕt depend on hysteresis though Ñ it is due mostly to the curve of the inflating balloon, which shows a maximum pressure as the balloon is being inflated.
Pressure (Pa)
5000 When connected together by the straw, 4000 the two balloons must have the same same pressure 3000 pressure. Draw a line across the graph for 2000 this pressure, and youÕll find that t w o 1000 different sizes of balloon can have that same 0 pressure Ñ these are *0.06 0.02 * 0.04 0.08 0.1 0.12 the sizes of the two Radius (m) balloons. When you squeeze the larger balloon, the pressure of both balloons increases (they are still connected together!), so the line of the shared pressure rises, on the graph above. The large balloon is squeezed smaller, and the small balloon inflates because there is a (nearly) constant volume of air for them to share Ñ they approach each other in size. With a perfect setup, when the line touches the maximum, the balloons have the same size, and can switch which one will be smaller, then the balloon collapses under your hand to become the small balloon. Since the balloon becomes small, you arenÕt squeezing it much anymore. The pressure drops back to what it was originally Ñ only now the large balloon has become the small one, and the small one the large one!
MORE FOR THE VERY INTERESTED: You can also describe all of this in terms of the potential energy of the twoballoon system, which is to be minimimized. This potential energy has the same 12
type of curve as shown above for bistability, and the system rests in one of the local minima of the curve. You can see how the size-changeover works, in the following way: The work done when squeezing, to compress a balloon is force × distance moved (W = F ¥ ∆r), and the force is the pressure × area (F = P ¥ A). So the potential energy change, for the balloon as it changes size, is the work done on it: ∆U = Ð W = Ð F ¥ ∆r = Ð P ¥ A ¥ ∆r = Ð P ∆V, since A ¥ ∆r is the change in volume of the balloon, for a small change ∆r. This potential energy change is positive, because ∆V < 0. You do work, in squeezing the balloon and increasing the pressure, so the potential energy of the system increases. You need to increase it to get it over the hump in the potential energy, and then the system of two balloons can go over to the other minimum. Because each local minimum is ÔstableÕ, and there are two of them, the system is bistable Ñ it can be at equilibrium in two different states. [Robin]
13
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Problem Set 2: Mechanics
Erratum There are minor errors in the version of Question #3 in this problem set, as we rushed to distribute it. We will mark student solutions based on the question we distributed, but you may wish to know that the data, especially, should look more like this! The change in the hint is interesting, but does not very much alter the answer.
3) Sink, sank, sunk At some point in the movie ÒTitanicÓ it is said that the pressure at its sunken depth is 2 5500 pounds/inch . Given this, find the depth at which the Titanic supposedly came to 3 Ð3 rest. The ÔrealÕ answer is 12600 feet, density of water is 10 kg m (ignore air pressure). Comment on the results, i.e., truth in film-making. At another point in the movie someone estimates that the Titanic will sink in a time of roughly half an hour. Approximating the Titanic as a rectangular box of cross-sectional 2 area A = 882*80 feet , mass M = 40,000 tons (loaded), total height 80 m, and a horizontal 2 hole at the bottom of a side of the box of area AÕ = 100*0.5 m , estimate the time it takes sink Ñ that is, when the top of the box is even with the water line. Ignore air. [HINT: assume the system comes to equilibrium every time a bit of water goes in; you may assume that the flow is steady (i.e. has a constant velocity) Ð you may then use conservation theorems (e.g., Bernoulli's equation) Ð but be careful, as this does NOT hold initially, when the water is accelerating!] [Peter]
1
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Solution Set 3: Thermodynamics 1) Newton's Law of Cooling a)
dQ = κi A⋅ ∆T dt
With ∆T kept constant by the heater system ( (90 Ð 70) ¡F = 20 ¡F = 11 K), the heat loss is constant, too. There are different heat conductivities Ñ κ 1 for the (bottom + sides), and κ 2 for the for the top Ñ and they each contribute to heat-flow out. 2
Top, bottom area (each): 5Õ × 7Õ = 35 ft = 3.25 m 2
2
2
2
Sides area (each): 7Õ × 9" = 5.25 ft = 0.49 m Ends area (each): 5Õ × 9" = 3.75 ft = 0.35 m
2
dQ = (κ1 A1 +⋅ κ2 A2 )⋅ ∆T dt 2
2
2
= { 0.04ÊJÊKÐ1Êm Ð2 sÐ1 ¥ (3.25 m + 2¥ 0.49 m + 2¥ 0.35 m ) +É 2
Ð1 Ð2 Ð1 + 0.4ÊJÊK Êm s ¥ 3.25 m } ¥ 11K
= { ( 0.20 + 1.3 ) JÊKÐ1Ê sÐ1 } ¥ 11K = 16 J sÐ1 = 16 W The heater must be delivering an average of 16 W, then. DoesnÕt sound like too much, but this is day and night Ñ like leaving the oven light on. Or like leaving a reading lamp on for five hours a day all month. In a month, then (which is 24 ¥ 30 = 720 hrs), this amounts to 11.5 kW-hrs. At a cost of about \$0.125 /kW-hr, for me, this amounts to roughly \$1.44 per month Ñ much less than a refrigerator. (Hmm, I thought it cost me about \$10 per month for this waterbed!) b) Newton's Law of cooling tells us the rate at which heat will leave the water bed Ñ proportional to the temperature difference ∆T with the room. As the water bed loses heat, it will cool down, and ∆T will decrease, and so the rate of cooling will, too. So what we need, to start, is the relationship between temperature and heat content, to go with what we can figure out for heat flow vs. time. This relationship between heat content and temperature is what we call the specific heat: how much heat do we have to add or remove in order to make a kilogram of water go up by one degree? From INFOBITSª at the bottom of the problem set, we see that this value is: specific heat of water: C = 4.2 kJ kgÐ1 KÐ1 1
And ∆Q = C¥M¥∆T. The mass of water in the waterbed is: Ð3
M = 1 g cm
¥
Ð1 3
3
(5 ft × 7 ft × 9/12 ft) ¥ (0.305 m ft ) = 10 kg m
Ð3
3
0.74 m = 740 kg
and so the heat capacity of the whole waterbed is C ¥M = 4.2 kJ kgÐ1 KÐ1 740 kg = 3.1 MJ KÐ1 . Therefore, we can write ∆T = ∆Q/(C¥M) , for ∆Q a change in heat energy of the water in the waterbed. This lets us rewrite NewtonÕs Law of Cooling as: dT dQ =Ð /(C¥M) = Ð ∆T ¥ (κ 1 A1 +⋅ κ2 A2 )⋅/(C¥M)⋅ dt dt = Ð ∆T ¥ 1.5 JÊKÐ1Ê sÐ1 /(3.1 MJ KÐ1 ) Ð7
Ð1 = Ð ∆T ¥ 4.8 × 10 s = Ð ∆T / (574 hrs)
where ∆T is the temperature difference between bed and room air, ∆TÊ=Ê{TÊÐÊTroom}. (Heat flowing out of the bed is a drop in the energy in the bed; it gives a drop in temperature, so we have a minus sign.) Now, we have that Troom is constant so: d ∆T Ð7 = Ð ( 4.8 × 10 sÐ1⋅) ¥ ∆T = Ð∆T / (574 hrs.) dt and you may know that the exponential function A ¥exp{Ðαt} is a function with just such a derivative. In fact, this is a good solution: ∆T = (∆T)o exp{Ð t / (574 hrs.) } where (∆T)o indicates the initial value of the temperature difference. In our case, (∆T)o = (90 Ð 70) ¡F = 20 ¡F.
I turned the heat off on my own waterbed at home, and measured the temperature as time went by. The exponential curve at right shows NewtonsÕs Law of Cooling! (Actually,
I
90.0 Water temp (°F) 85.0
Water temp (°F)
So temperature differences decay, as the energy of the hotter object will flow over to the cooler one, until they are each the same temperature (∆T = 0).
y = m1+m2*(1-exp(m0/m3))
80.0
Value
Error
m1
89.057
0.37342
m2
-25.295
1.1604
m3
-102.14
10.104
Chisq
6.8694
NA
R
0.99638
NA
75.0
70.0
65.0 0
50
100
150
200
250
Time (hours)
the 2
numbers κ up from this data, so that the answer would work out to match the curve. But I messed up, and confused a calorie with a Joule, so all the numbers are 4.2× too large or too smallÉ) According to the numbers given, after two weeks (336 hrs.) the waterbed will be warmer than the room by: ∆T = (∆T)o exp{Ð t / (574 hrs.) } = 20 ¡F ¥ exp{Ð (336) / (574 hrs.) } = 20 ¡F ¥ 0.58 = 11 ¡F so, it will have dropped by 9 ¡F, or 5 K. To reheat the bed will then take: 5 K ¥ 3.1 MJ KÐ1 = 15.5 MJ which is 4.3 kW-hr. On the other hand, 16 W average for two weeks to keep the bed warm would have cost 5.4 kW-hr, or 19.4 MJ. The difference is about 25 cents worth of electricity Ñ maybe not worth the trouble of unplugging the bed. The increased energy, if the bed is left plugged in, is lost because in keeping the bed warmer, over those two weeks, the average rate of heat lost is higher.
Water temperature (°F)
90.0 In reality, the cooling is faster than these numbers I mis-invented Ñ instead of 85.0 574 hrs., the number is more like 135 hrs., in the 80.0 formulae above. As it works out, the bed would 75.0 be at room temperature within the two weeks, and it would cost a little more 70.0 Water temperature (°F) than twice as much to reheat the bed Ñ about 50 65.0 cents. The plot at right -10 0 10 20 30 40 50 shows how my waterbed Time (hours) reheated, after I plugged it back in again. I have to remember to plug it in about 2 days before my guests arrive! [Robin]
3
2) Spears into pruning hooks a) In order to ignite the ship part we'll need a temperature above a 'dull red heat', about 500 ¡C or 773 K. Using Stefan's constant σB we acquire the rate of radiation I = σ BT 4 ∼ 20 kW ⋅ m −2 Without knowledge of the thermal constants of the wood we can only estimate how much should be added to this to account for heat losses by conduction and convection. We also shouldn't assume that the shields are perfect reflectors and that the wood is a perfect absorber. A factor of 2 should give a reasonable margin of safety and )hopefully) account for all this. So an intensity greater than 40 kWámÐ2 is required in order to ignite the ship part. We are given that the shields are curved, so lets consider the effect of focussing one of the shields on a ship part. Assume the round shields are each 1m2, thus each has a diameter, D, of about 1.13 m. The focal length, f, is taken to be 100 m (distance to ships) and assume a wavelength of incident light, λ, on the shield of 500 nm (about where sun's energy peaks). Since light diffracts any time it goes through a slit, the worse the narrower the slit, light cannot be focussed to a tight spot either, without diffraction. The limit of a focal spot is given by: y = 2.44 fλ/D (derived from principles of light diffraction) yields a value y = 1.08 × 10Ð4 m. Where y is the radius of the bright central spot, created by the shield, on the ship part. So, the area of the spot is approximately 3.66 × 10Ð8 m2. The power on the tight spot is the whole power the shield intercepts, 1 kW. Thus, the intensity of light produced from focussing one shield is, I shield =
1 kW 3.66 × 10
−8
m
2
= 2.7 × 107 kW ⋅ m −2
This is much, much more than the 40 kW ¥mÐ2 loss. To determine the time consider a surface layer about 1 cm thick. Now to calculate the rate of heating lets assume that wood has a density and a specific heat equal, respectively, to 1/2 and 1/10 the corresponding values for water (since the ship is floating!) Then the thermal capacity, Ctotal, of the wood is, Ctotal = mC = ( 500 kg ⋅ m −3 )(0.01 m)(0.42 kJ kg -1K -1 ) = 2.1 kJ m −2 K −1 ∴ The rate of heating is 2.7 × 107 kW¥mÐ2 ÷ 2.1 kJ mÐ2KÐ1 or 1.3 × 107 K sÐ1
4
Assuming a temperature change of 480 K it will take about 4 × 10Ð5 sec to set fire to a ship part of area 3.66 × 10Ð8 m2 and thickness of 1 cm. If the shores are lined with soldiers holding these shields, those Romans are toast! b) First, lets look at the power required to replace that radiated from the front of the disk at its melting point. Prad = σ B AT 4 = ( 5.7 × 10 −8 W ⋅ m −2 K −4 )(7.85 × 10 −5 m 2 )(1811 K )4 = 48 W (We gave the wrong sign in the exponent, in the value for Stefan's constant at the back of the problem set. Sorry about that, and hope you found it by the crazy wrong answer it gives!) The power produced by just one mirror is 2 m2 × 1 kW/m2 = 2 kW. This should be more than enough to make up for the radiation losses. To melt the disk, first heat energy, Q 1 , is required to increase the A solar-collector farm: The mirrors on the ground temperature of the disk to the are adjustable, and redirect sunlight to the focussing mirror in the tower. melting point and then the latent heat, Q2, must be supplied in order to produce melting, where (assume ∆T = (1811 Ð 239) = 1518 K
Q1 = mC∆T
= (6.18 × 10Ð3 kg) (0.444 kJákgÐ1KÐ1)(1518 K) = 1.53 kJ and Q2 = (moles iron) E = 1.53 kJ ∴ Qtot = 5.7 kJ The power supplied by 100 mirrors is 200 kW and the power radiated at the melting point is 48 W. So we can estimate, t=
5.7 kJ Q tot = = 0.03 s Ptot (200 kW − 4.8 × 10 −2 kW)
The above assumes that Prad is constant, when it actually changes as the temperature of the disk changes. Initially radiation losses are unimportant compared with input 5
power. But as the temperature of the disk increases the radiation losses become more important. However, in the case of our disk, even at the maximum temperature, the melting point, radiation losses are negligible compared to input power. We have also assumed that the mirrors are perfect reflectors and that the iron is a perfect absorber Ñ which may be close to true for the mirrors but won't be true for the disk. Finally we have also ignored convection losses, while these are not negligible, they are difficult to calculate. [Carrie] 3) Sounds coolÉ i) We tried to write the question so as to lead you through this standard derivation. You can find it in first year physics and introductory thermodynamic books. For an ideal gas with no heat transfer, we can write: ii)
ÐPdV = CdT
(1)
From the ideal gas law: PV = nRT, and taking the differential (like the derivative): PdV + VdP = nR dT Using (1) VdP = (nR + C) dT
(2)
Divide (1) by (2) to get rid of the temperature variable: PdV −C = (VdP) (nR + C) P C V =− dV (nR + C) dP γ
To confirm that P¥V =constant is a solution, we substitute it into both sides and verify they are equal, if: γ=
(nR + C) C
This is fine since these are all constants. dP we get: ii) Using the above expression and B = − V dV B = γ P and thus γ P ρ nM ρ= where M is the molar mass of air. So, V v=
v=
γ PV nM 6
Using the ideal gas law: PV = nRT we get finally v=
Tγ R M
Given that v = γR = 300 M
300 m at S.T.P. (i.e., T ≅ 300 K), then s
so we can say that v =17 T . Therefore the colder the air, the slower the speed of sound. iii) From personal experience, I can not observe a time lag with a friend who is 20 m away, but I do notice it with sound source that is at least ~50 m away (i.e., the firing of a gun at a track meet). So I can observe a time-lag of around 0.17 s. For me to notice a time lag with a friend 20 m away, the speed would have to decrease by a factor of 2.5, corresponding to a temperature of 48 K. That is pretty cold. In fact, all the nitrogen/oxygen/carbon dioxide/etc would be frozen, so, not only would sound not travel (something the Star Trek/Wars folks havenÕt figured out yet) but you physically wouldnÕt last too long. [James] 4) Ice ice baby i) Temperature of the vapour is the same as the surface of the water. Before you turn on the pump it is 20¡C . Just before ice is formed the temperature is 0¡C. The boiling process requires heat thus cooling the remaining liquid until it freezes. Consider x kg freeze while y kg are changed into vapour. Consider that only the x kg had to be cooled from 20¡C to just above 0¡ C. Conservation of energy: 6
5
2.3 × 10 y = 3 × 10 x + 4200 ¥20 ¥ x But there is 1 l of water, which has a mass of 1 kg so: y=1Ðx and thus x = 0.86 kg Now consider that all the water (1 kg) had to be cooled from 20¡C to 0¡C: 6
5
2.3 × 10 y = 3 × 10 x + 4200 ¥20 ¥ (x+y) and thus x = 0.85 kg. Of course neither case is correct, but they are some sort of upper and lower limits for the correct answer. Since we require the answer to only one significant digit, it is 0.9 kg.
7
iii) Water evaporation is used in many instances to cool things, from wineskins to your own perspiration. For a glass of water, this process does cause the water to cool but there are other factors that do not allow it to get very cold. There is heat transfer with the rest of the room so you cannot cool the water without also cooling the room. Within the vacuum system, there is very little heat transfer from the vessel to the chamber since vacuum is a superb insulator [James] 5) Reduce, re-use, recycleÉ We have:
∆E = Q − W W = ∫ pdV
i)
W=0 3 V1( P2 − P1 ) 2 3 3 ∆E = nR∆T = V1( P2 − P1 ) 2 2 Q = ∆E =
(NOTES: all results are obtained from PV = nRT = RT here. Slope of the line in TÐP graph is R/v1) ii)
W=1J Q = ÐW = Ð1 J (edÕs. note: I think this should be +W, i.e., +1J) ∆E = 0 (cyclic process) iii)
W = P2 (V2 − V1 ) 8
Q = ∆E + W =
5 P2 (V2 − V1 ) 2
3 ∆E = P2 (V2 − V1 ) 2 iv)
W = − P1(V2 − V1 ) 5 Q = − P1(V2 − V1 ) 2 3 ∆E = − P1(V2 − V1 ) 2 v)
W=0 3 Q = − V2 ( P2 − P1 ) 2 3 ∆E = − V2 ( P2 − P1 ) 2 vi)
The net work is just the area inside the box (only on a PÐV graph!) Ð or you could just add up the work from the previous parts of the question. W = ( P2 − P1 )(V2 − V1 ) The heat is positive during the first 2 processes, so 3 5 Qincoming = V1( P2 − P1 ) + P2 (V2 − V1 ) 2 2 The efficiency is
9
e=
Wnet Qinco min g
2
= 3
V1 P2 +5 V2 − V1 P2 − P1
2
= 3
1 x +5 1− x 1− y
Here, V1 = xV2 , P1 = yP2, so 0 ² x, y ² 1. To make e big, we have to make the denominator small. To do this, we have to make the denominator of the x and y expressions big. Clearly, the maximum occurs for x = y = 0. Then, e = 2/5 = or 40 %. For a Carnot engine we would get T e = 1 − c where T c and Th are the coldest / hottest temperatures (respectively) Th occurring during the cycle. Here, PV e = 1 − 1 1 = 1 − xy P2V2 The maximum occurs for either x or y = 0, and is equal to 1. This is much higher than our cycle, and so I wouldnÕt recommend rebuilding your car engine, or anything like that. [Peter]
6) Keeping your cool a) We're given:
X moles of air inside fridge at temperature TK y moles sÐ1 escape molar heat capacity of air is cp (which we shall assume constant) the room temperature is T R
In a small finite time ∆t y∆t moles of air escapes. Therefore y∆t moles of warm air at temperature TR enters the fridge. The warm air and cold air will mix and we model the final temperature by supposing a reversible flow of heat. Since the heat gained by cold air must equal the heat lost by the warm air we have c p (X − y∆t) (T∆t − T) = c p y∆t (TR − T∆t ) Where T ∆T is the final temperature of the mixed air. Therefore
10
T∆t − T = ⇒
y∆t (TR − T∆t ) X
T∆t − T y = (TR − T∆t ) X ∆t
Then, as ∆t → 0, we have, by definition of the derivative T −T dT = lim ∆t dt ∆t → 0 ∆t = lim
∆t → 0
=
y (TR − T) X
y (TR − T) X
Thus the rate of change of temperature inside the fridge is dT y = (TR − T) dt X
y − t )e x
b) Verifying T(t) = TR + (T(0) Ð TR
is a solution of 1.
The L.H.S. is y
dT y −Xt e = (T(0) − TR ) − X dt y
− t y = (TR − T(0)) e X X
The R.H.S. is y − t y TR − TR + (T(0) − TR ) e X X
y
− t y (TR − T(0))e X X
Thus the L.H.S. equals the R.H.S. and so T(t) = TR + (T(0) Ð TR)
y − t e x
is a solution
c) Since the second term above decays to zero exponentially as t increases, T(t) approaches the asymptote TÊ=ÊTR. This is very reasonable for the unplugged fridge Ñ it goes to room temperature! d) In a small finite time interval ∆t the temperature change is T∆t − T =
y∆t (TR − T∆t ) X
11
To make this change zero the refrigerator must do work and remove the heat responsible for this change: y∆t (TR − T∆t ) X = c p y∆t (TR − T(0)) ∆Q = c p X
Where T ∆t = T(0), since the temperature inside is constant. For a Carnot refrigerator working between the cold air inside at T(0) and to warm air at room temperature TR the Ôcoefficient of performanceÕ is given by η=
∆Q T(0) = W TR − T(0)
Hence T − T(0) W= R ∆Q T(0) =
TR − T(0) c p y∆t (TR − T(0)) T(0)
Thus, by definition, the power required is P = lim
∆t → 0
(T − T(0))2 W ∆t = lim R cpy T(0) ∆t ∆t → 0 ∆t
(TR − T(0))2 = cpy T(0) Since the Carnot refrigerator is the most efficient, therefore we conclude that this value is a minimum for the required power. [Simal]
12
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Problem Set 4: Optics and Waves Due February 12, 1999 1) The land of the midnight sun (is where, exactly?) While travelling through the Yukon this past summer, we travelled across the Arctic Circle (latitude 66.7 degrees). If the earth was a simple sphere (like a billard ball) circling the sun while rotating on its own axis, a line at this latitude would experience exactly one day a year of unending daylight. North of this line would experience more than one day of continuous daylight and south of this line would have a sunrise and sunset every day of the year. a) Using this information, find the tilt of our sphere's axis relative to the plane it sweeps out as it circles the sun. b) In reality, it is possible to see the midnight sun at points south of the Arctic Circle. In each of the following cases, calculate how much farther south than the Arctic Circle you could position yourself and still be able to observe the midnight sun, by taking into consideration: i) you are standing on a hill (height: 300 m, rising from a perfectly spherical Earth). ii) the Earth has an atmosphere. For the purpose of this problem, assume that the atmosphere is 3 km high and at standard pressure and 0¡ C, and that the Earth is a perfect sphere. [James] 2) Making light of photons Since light is a wave, two identical beams can add together constructively or destructively depending on their relative phases. Interferometers, whether large such as found in a satellite array suspended high above the Earth or else small like the one on my laboratory table, make use of this fact to measure very small distances. Consider my simple interferometer: the beamsplitter reflects 50% of the incident intensity and transmits
mirror
x1 mirror x2
beamsplitter
1
1
the rest. All reflections involve a 180¡ phase change. i) What is the electric field at the output (1) as a function of the relative displacement (∆x=x1Ðx2). Consider the input field to be monochromatic and to have an electric field of E=Eocos(2π v(x/c Ð t)) where Eo is the amplitude, t is time, ν is frequency (Greek ÔnuÕ), and c is the speed of light. ii) You will notice for certain values of ∆x, there is zero electric field at (1). Where has the energy from the light gone?! iii) Put a screen at (1). For a beam from my HeNe laser (frequency ν = c/632nm), what is the smallest ∆x you could clearly distinguish? iv) If you travelled back a century, you would find that many believed that light moved in a medium, just as water waves travel in water or sound waves travel in air. This supposed medium was dubbed the ÔetherÕ. If the ether were also moving, this could change the effective speed of light, by a version of the Doppler shift. Pretend that you are a scientist at this time and you want to establish the existence of the ether. How might you use your interferometer to measure the speed of the ether relative to your lab? What limit you can place on the etherÕs speed? (For simplicity, assume you still had a monochromatic source of the same wavelength as in part iii). [James] 3) Imagining imaging When you look at yourself in a mirror, you see yourself behind the mirror by the same distance as you are in fact in front of it. Your mirror image might even appear far enough from you to be in the next room! No light really passes from the place your image seems to be, so the image is called ÔvirtualÕ. On the other hand, when you take a photograph, you need real photons to hit the film; this kind of image is called ÔrealÕ. i) Prove that any point A a distance d from a flat mirror has a virtual image AÕ which is the same distance d on the other side of the mirror. What is the magnification of this virtual image? [Hint: magnification M can be defined as the apparent size of an image divided by the size of the original object; it can also be defined using the angle between two rays of light leaving a point on the object, and the corresponding angle the rays seem to form when traced back to the virtual image.] F F' ii) Any hyperbola has two foci, which we can label F and FÕ. Show that one is the virtual image of the other, reflected in the hyperbola. Can you find a well-defined magnification? [Robin] 2
4) Correct time and temperature, at the toneÉ Consider a pendulum clock: i) If the pendulum arm is made of a material with a linear coefficient of thermal expansion α , determine an expression for the ratio of the final period over the initial period, T ′ / T , if the change in temperature is ∆Τ . ii) If the suspension system is brass and the change in temperature is 20¡ C how much time does the clock gain/lose in one day if Τ = 1.000 s? A nickel-steel guitar string, initially of length lo , is stretched to a length L such that it has a frequency f for its fundamental frequency. iii) If the temperature is increased by ∆T, determine an expression for the new fundamental frequency in terms of the old one, i.e., f ′ / f . Assume Young's modulus, Y, and the linear expansion coefficient, α , to be constant. iv) If the initial frequency is 440 Hz (ÔAÕ above middle ÔCÕ), how Ôout of tuneÕ will the guitar be, due to an increase in temperature of 20¡ C? Take lo = 0.795 m and L = 0.8 m. [Simal] 5) Toys, Toys, Toys What are the physics rules for oscillating molecules? In this question we will consider different physical models of a linear molecule (like O2 , N2 , CO2 , or some long-chain polymers) which has just been hit by a single-atom molecule (like He or Ne). What weÕll change is the model of the force between atoms, and what weÕll examine is their motion after the collision. Model a linear molecule as consisting of evenly spaced particles (atoms, balls or whatever) of mass m each. All collisions are perfectly elastic, etc. Initially, the first (left-most) particle of the chain is hit by an object of mass m, and velocity V [ to the right]. Consider first a chain consisting of just 1 particle (a monomer) Ñ i.e., we have an ordinary 2-particle collision. What is the speed and position, as a function of time, of the 1-atom chain afterwards? Part I Ð stringy bonds Consider now a 2-atom chain (a dimer) having an inter-particle spacing L. The particles are connected by an initially extended ideal (massless, non-dissipative and nonstretchable) string. The left ball [at (0,0) ] is hit by the incoming object. Find the motion of the centre of mass of the lattice. 3
Generalize to an n-particle chain (a polymer); keep it a linear (straight-line) molecule! Describe (in words and equations) also the motion of the two balls as seen from both the centre of mass frame, and an outside (laboratory) frame of reference. Sketch the corresponding position-time graphs for each ball (use one graph for each of the two reference frames). Part II Ð springy bonds This is same as Part I, except that now the two atoms are connected by a spring (with spring constant k). Describe (in words) the ensuing motion from both frames (laboratory and centre of mass) of reference; find the motion of the centre of mass and sketch the position-time graph of the left ball in both frames. [HINT: consider the centre of mass frame first]. Discuss the plausibility of the models used in Part I and Part II. BONUS: Consider that there still is a spring, as in Part II, but there is now also a dissipation mechanism. That is, there is some dissipative force of size Ðγ* (instantaneous drag force on object) = Ðγ(dx/dt). What is the motion of the centre of mass now? Find and draw a sketch of the position (as a function of time) of the left ball in both the centre of mass and outside frame. [HINT: a solution to the differential equation: md2x/dt2 = Ðkx Ð γ(dx/dt) is 2Ue
γt m
1 k γ 2 4 − t sin m 2 m 4
k γ 2 − m m
where the initial conditions are: x(0) = 0, dx/dt(0) = V(0) = U; also, 4k > γ2 ] Can you suggest a realistic dissipation mechanism for molecules? [Peter] 6) Making a shoebox spectrograph, using a compact disc Have you ever noticed the rainbow colours reflected from a compact disc? The tracks on the CD are arranged with such mathematical regularity that a CD makes a nice diffraction grating. This pretty easy experiment lets you make a spectrograph from a shoebox and a CD, one which is good enough to reveal the individual lines contained in the spectrum of light from a fluorescent light tube, streetlights, or other light sources you may have.
4
You will need: •
• • • • • •
peep-cover any compact disc (my favorites are the goldcoloured CD-R blank used for recording output slot CD-ROMs; about \$2slit 3. It saves my music collection, too!) a cardboard box the size of a box for input hiking boots; a shoebox or small corrugated cardboard CD diffraction grating box might do aluminum foil some duct tape, or other reasonably opaque tape scissors a pen or marker a utility razor-knife, or Exactoª knife (CAUTION: danger to fingers!) a ruler
To set up, read all these instructions before beginning. Refer to the illustration, too. 1. lay the box on its side, as sketched in the picture above. Measure and mark a mid-line around the outside of the box, to use for alignment (up, across, and down the middle of each face Ñ see dotted line in figure). Put a similar line around the inside of the box, also at exactly the midpoint. 2. use scissors or razor-knife to cut a lengthwise slot about 1 cm wide along the top of the box, centred on the alignment line. Cut from about the midpoint lengthwise, and go all the way to the end of the topmost panel. 3. cut a hole through the box, on the small end-panel of the box farthest from the slot you cut in (2). It should be roughly 2 cm high by 3 cm wide, and positioned toward the top of the box-end. It must be centred on your alignment line. 4. cut a piece of aluminum foil about twice the size of this hole in (3). Put it on a firm pad of paper to cut it, and along the middle of this foil make a 3cm-long cut with your razor-knife (hold the knife at a shallow angle, to pull the cutting edge, and not the blade-tip, across the foil). YouÕre making the input slit Ñ you want the narrowest slit that will be sure to pass enough light: 0.1 mm is pretty good. When done, tape the foil over the rectangular hole in the end-face, with the slit horizontal and centred. 5. tape a compact disc, label-side down, inside the box on the side box-panel. Set it up so that the hole through the CDÕs middle is centred on the alignment 5
line you drew inside the box. Cover the front half of the disc with opaque tape, so only the half farthest from the input slit will be used. The placement of this disc depends on your placement of the input slit: a ray from the input slit to the middle of the exposed section of the CD should make an angle of about 30¡ from the CD surface. You may have to try moving the disc forward and back in the box to get the input and output right, relative to the disc. The spectrograph is now complete, except for figuring out how to use it! YouÕll need to look down at the CD through the long slot in the top, and try different places along the slot. But you want the only light inside the box to have to come through the slit, so try cutting a piece of cardboard or dark cloth to cover the slot, with a 1 cm peep-hole to look through. Then you can peep at different points along the slot, and still keep stray light out by sliding the peep-hole card. To use your spectrograph, you should point it so that light will pass through the slit and onto the compact disc in the bottom. When this is exactly right, you can look at different places along the output slot to find the best spread-out rainbow, from an incandescent lamp, or many spectral lines from a fluorescent light, streetlamp, or neon light. The experimental assignment: i) Write down your own description of how you made this spectrograph, and how you made it work. ii) Describe what you see, for different light-sources such as fluorescent lights (different types?), incandescent lights, candle-flames, neon lights, streetlights of different types, a laser-pointer (caution: never stare directly into a laser pointer, or HeNe laser, or the sun. Even these simple sources can cause damage to your eyes, though you may not recognize the damage immediately). iii) Can you use the formula above, and your own observations, to figure out how many total tracks there are on a CD? [Robin]
6
INFOBITSª Ñ Useful Bits of POPTOR Information Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor
index of refraction for air at standard pressure and 0 deg C: n = 1.0003 6
radius of the earth: R = 6 × 10 m nickel steel 10
YoungÕs modulus: Y = 18.2 × 10 Pa Ð5 Ð1 coefficient of thermal expansion: α Å 1.4 × 10 K brass 10
YoungÕs modulus: Y = 9.0 × 10 Pa Ð5 Ð1 coefficient of thermal expansion: α Å 2.0 × 10 K MINI-TUTORIAL: The physics of diffraction-grating spectrographs: The input and output angles of light diffracted from a diffraction-grating depend on wavelength, and on the diffraction grating itself: n λ = 2d • (sin θi + sin θ d ) where θi is the input angle, measured from the perpendicular to the surface (the surface normal), and θd is the angle of the diffracted light, measured similarly; d is the spacing between grooves of the diffraction-grating, or tracks of the CD. The n is an integer which gives the order of diffraction, which you can investigate. The input light is scattered from very many regularly spaced grooves or tracks in the compact disc. Only at very special angles are the conditions right for constructive interference for waves coming from all of the grooves. At these special angles the light of a certain wavelength is bright; at other angles there is only random or destructive interference, and practically no intensity of light is sent on.
7
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Solution Set 4: Optics and Waves 1) The land of the midnight sun (is where, exactly?) a) At midnight on the summer solstice, the sun rays are tangent to the perfect sphere at the Arctic Circle. The Ôside viewÕ looks like: Axis for Earth rotation observer on Arctic Circle
66.7deg sun rays
equator
Once the picture is drawn it is obvious that the tilt of the sphereÕs axis to the plane it sweeps out around the sun is 66.7¡. b) i) Exaggerated for clarity: s sun rays hill
c Re
Thus c=arccos(Re/(Re+300m)) and the subtended arc opposite from angle c is 60 km. Thus you are 60 km south of the Arctic Circle. ii) Refraction causes the rays to bend at the air/vacuum interface. Modelling this as a single interface isnÕt exactly right (the interface is continuous and still causes refraction
1
due to the index gradient). SnellÕs law: sin(90¡)=n sin(a) as defined in the cartoons below:
s
Ra
s a Re+Ra Re So: Thus a=88.6¡. Solving for s gives an imaginary number (i.e., with − 1 ). What went wrong? We assumed that the refracted ray that intersected the interface exactly over the Arctic Circle would eventually intersect the ground, but it does not. If the interface were lower, the index difference larger, or the Earth bigger, it could intersect the EarthÕs surface. Thus the most southern ray to intersect the ground intersects the atmosphere at a point towards the sun:
s
Ra AC b a
s c
b As shown on the diagram, this refracted ray is tangent to the earth surface. We find s=190 km. We still have to find out how far south we are. For this triangle, the refracted angle of the ray in the atmosphere (angle: a) is 88.19¡ to the normal. The incident angle (angle: b) is 88.85¡ to the normal. Thus we find that angle c is (90Ð88.19Ð(90Ð88.85)) = 0.66¡. Using the value of Re, the corresponding arc length is 69 km. This is the distance you are south of the Arctic Circle. [James] 2
2) Making light of photons a) The output at (1) is the sum of the two beams. Remember that though the intensity after one reflection followed by transmission through the beam splitter is 0.25, the 2 electric field changes by 0.5 since I~E . The math in this problem is easier if you replace 2*cos(a) by exp(ia)+c.c. where c.c. indicates complex conjugate (i.e., in this case c.c.=exp(Ðia) ). Convince yourself that this relationship is true, using the fact (you may not know) that exp(ia) = cos(a) + i sin(a), where i = − 1 . The problem still can be done by writing cos() and sin() everywhere, but this phasor notation is easier and extremely useful! Thus the sum of the two beams at position (1) is: E 2 ⋅ x1 + x0 2 ⋅ x 2 + x 0 Etotal = 0 exp(iπ)exp i 2 πv − t + exp i 2 π − t + c.c. 4 c c where x0 is the distance from the beam splitter to position (1). Rewriting: E 2 π ⋅ v ⋅ x0 2 ⋅ x 2 + x0 Etotal = 0 exp i 2 πv −t +π+ {exp(i 2 π ⋅ v 2 ⋅ Dx) + 1} + c.c. 4 c c where Dx is (x1Ðx2). This can be rewritten in terms of cosines but for the purposes of the rest of this question, this form is easier to deal with. ii) From the above expression, you can see that there is no electric field at position (1) c when exp(i 2πv 2Dx ) + 1 = 0 ie. when Dx = n where n=..,Ð3, Ð1,1,3,.. Thus, if the ( 4v) difference in distance is 1/4 of the wavelength (recall wavelength=c/ν), no electric field is found at (1) and we have total destructive interference. Where does the energy go? Consider the electric field that is reflected back in the direction of the original beam. Due to fact that one beam undergoes three reflections and the other only one (instead of two and two), you can show using the same method as above that the MAXIMUM c reflected occurs when Dx = n ( 4v) where n=..,Ð3, Ð1,1,3,.. . Thus if the beam does not exist at position (1), it is reflected back the way it came. Energy is not lost. iii) There is some subjectivity to what it means to Ôclearly distinguishÕ but I would argue that you can clearly distinguish between a maximum and a minimum output. Maxima c at (1) occur when Dx = n ( 4v) where n=..,Ð2,0,2,.. So we can write (Dxmax − Dxmin ) =
0−c , i.e., 1/4 of the wavelength. ( 4 v) 3
The wavelength is 632 nm, so the smallest distance change you could clearly measure is 158 nm. That is pretty small. To contrast, atomic spacing in a solid is on the order of 5×10Ð 10 m, so you could measure a distance corresponding to ~300 atoms. iv) First assume the ether exists and is moving relative to your laboratory. This seems to make sense unless the ether is held to the surface of the Earth due to gravity. That would require the ether to have sizable mass. The ether must exist in space though since light travels in space, and then you would get some interesting light bending effects due to the concentration of ether around massive objects. Also you would still have ether existing in vacuum, thus you would have something with mass existing in a vacuum and now my head is starting to hurt. Even if the ether is not moving , you could put your apparatus in a vehicle, such that you would be moving relative to the ether. The trick is to rotate your interferometer and see if you can see a change in the light output at (1). If one arm is parallel to the ether motion and the other is perpendicular to it, you find that destructive interference occurs if x1 x1 2x 2 + − + 1 = 0 exp i 2πv c (c + V ) (c − V ) where V is the speed of the ether relative to your interferometer. interferometer by 90¼ and this term changes to:
Rotate the
2x1 x2 x2 − − + 1 exp i 2πv c + V (c − V ) c Define the minimum V such that by rotating your interferometer, you go from a case of destructive interference to a constructive interference. (You might argue that this is too much to require; the output intensity only needs to change by a small amount. This is really determined by the noise in your system, which is a subject we are not going to discuss here, so we take the strongest case required.). Thus: x1 x1 2x 2 1 (c + V ) + (c − V ) − c = ( 2 v)
(Destructive)
2x1 x2 x2 c − c + V − (c − V ) = 0
(Constructive)
Simplifying this gives: V2 =
c2 ( 4 v)
c x1 + x 2 + 4v
4
To make your scheme as sensitive as possible, you want to make x1+x2 as large as possible. Say 19th century technology limits you to x1+x2=1m. For a source of wavelength 632 nm, you would require V~1×10 5 m/s. Increasing x1+x2 to 100m reduces minimum V required to be observed to ~1×104m/s. [James]
3) Imagining imaging i)
From the law of reflection, θi = θr Also θi + θr is the exterior angle of the triangle AAÕP, and is therefore equal to the sum of the alternate interior angles ∠VAP and ∠VAÕP. But ∠VAP = θ i , and therefore ∠VAP = ∠VAÕP. This makes the triangles VPA and VAÕP congruent, in which case d(object) = d(image). As for the magnification, this can be determined using the angle between two rays of light leaving a point on the object, and the corresponding angle the rays seem to form when traced back to the virtual image. We have already shown that ∠VAP = ∠VAÕP, which is true of all object/image points. Thus the apparent size of an image divided by the size of the original object, or the magnification is +1. The virtual image is life-size and erect. ii) To demonstrate that one focus is the virtual image of the other we will prove that the angles the incident ray and reflected ray make with the tangent are equal. This is most easily done with the following geometry in mind. On the diagram given in the question label the following: 5
the focus F → object point with coordinates (Ðc, o) the focus F′ → image point with coordinates (c, o) the incident ray (leaving F) → line l 2 the reflected ray (`leaving' F′) → line l 1 the point of reflection on the surface of the F′ branch of the hyperbola → point A with coordinates (xo, yo) Also draw in the tangent line at point A → label it line l Then label the angle between l 2 and l → angle α the angle between l and l 1 → angle β where both α and β are counter clockwise angles. So we want to prove α = β . Suppose the curve is in `standard position' so that the equation is, x2 a2
y2
− 2 =1 b
2 x 2 yy ′ ∴ 2 − 2 = 0 → y′ = a b
b2 x a2 y
Substituting (xo, yo) yields the slope of the tangent line, x x y y b2 x m = 2 o → gives the equation of the tangent → o2 − o2 = 1 a b a yo Now we can use the formula for the tangent of the counterclockwise angle from one line l 1 to another l in terms of their respective slopes m1 and m, namely tan α =
m − m1 1 + mm1
yo − 0 where the slope of the line l 1 is m1 = x − c o
plugging in the values for the slope you get something not as bad as it seems,
6
b 2 xo
tan α =
yo xo − c
b 2 x 2 − b 2 cxo − a2 yo2 a yo = 2 o b 2 x y a y o ( xo − c ) + b 2 xo y o 1+ 2 o o a y o xo − c 2
=
−b 2 xo c + (b 2 xo2 − a2 yo2 ) ( a2 + b 2 )xo yo − a2 cyo
− b 2 xo c + b 2 a 2
b 2 ( − xo c + a 2 )
−b 2 = 2 = = c xo yo − a2 cyo cyo (cxo − a2 ) cyo where we have used the fact that a 2 + b2 = c 2 and b 2 xo2 − a2 yo2 = a2b 2 for a hyperbola. The same calculation with Ðc replacing c gives tan(−β) =
b2 cyo
so tan(β) =
−b 2 cyo
so tan(α) = tan(β), and therefore α = β. In other words reflected rays appear to come from F′ or the focus F′ is the virtual image of F. The same argument can be made for the reverse situation (F′ the object point and F the virtual image). Thus each focus is the virtual image of the other. The magnification is not well defined. This is clear when you compare the angle between two rays of light leaving a point on the object with the corresponding angle the rays seem to form when traced back to the virtual image. If for example you consider the rays drawn in the diagram (given in the question) and the corresponding angles they make with the x-axis (our second ray reflected back along the x-axis) the magnification, M M=
tan( F ′) →∞ tan( F )
Since the angle F ′ → 90o If another two rays are taken an entirely different value for M can be obtained. Thus the magnification depends on which pair of rays you are tracing out and thus cannot be well defined. A proper image isnÕt really formed. [Carrie] 4) Correct time and temperature, at the toneÉ i) Let the initial and final lengths, due to a temperature change of ∆ TK , be L and L′ respectively. The respective periods are then: T=
g L
and
T′ =
g L′ 7
Since LÕ = L(1 + α∆T), we have T′ =
g L(1 + α∆T )
= (1 + α ∆ T )
1 2
T 1
Thus the required ratio is
− T′ = (1 + α ∆ T ) 2 T
ii) Now for brass we have α = 1.9 × 10 −5 K −1 and we're told ∆T = 20°C = 20 K Since T =1.000 the new period is T ′ = (1 + 1.9 × 10 = 0.9998
−5
(10))
1 2
Since the new period is shorter the clock will measure time faster and so gain time. Since the now warmer clock measure is in just 0.9998 s. Therefore the clock gains time 1 by a factor of = 1.0002 per unit of real time. T Thus after one day (as measured by the clock initially cold) the warm clock gains 0.0002 days = 0.0002(24) (60) (60) s = 17.28 s. Therefore the clock when warmer by 20 K gains approximately 17.28 s per day. iii) In the case of a stretched string, it isnÕt exactly the change in lengths which change the frequency. The string is stretched to a length L, no matter what. Instead, as the string expands, it becomes less taut Ñ the tension in the string isnÕt as great, since the string doesnÕt need to be stretched as much. For transverse waves in a stretched string we use the equation c = f λ where c is the speed of the wave, f the frequency and λ the wavelength. The first harmonic vibrates with the only nodes at the end points of the string. This distance L is then equal to one c half the wavelength λ. Thus f = . 2L τ
µ where τ is the tension in the string and µ the mass 1 τ density per unit length. Thus f = . 2L µ
The wave speed c is given by
τ/ A = Y , where ∆l / l A is the cross sectional area ∆ l is the change in length due to the applied tension, l is the initial length and Y is ÔYoungÕs modulus,Õ a constant.
We can determine τ from the formula for Young's modulus, namely
8
So we have f =
1 Y A ∆l 2L lµ
Initially we have string-length lo,, and the string is stretched out by an amount ∆l = (L − lo ) µ =ρA
and
where ρ is the density of steel at the initial temperature. After the temperature increase ∆T we have length lÕ prior to being stretched, and stretch amount ∆lÕ l′ = lo (1 + α ∆T ) ∆l′ = L − l′ = L − lo (1 + α ∆T ) The linear mass density doesnÕt change, because we always stretch to length L. The cross-sectional area A will increase as the metal expands, but our YoungÕs modulus, assumed to be constant, is defined in terms of the initial A. (We would have been better off to assume that Y¥A was constant). Therefore f=
1 YA ∆l 1 YA {L − lo } 1 YA L = = − 1 2 L µ lo 2L µ 2 L µ lo lo
f′=
and
L 1 YA ∆l© 1 YA {L − l© } = 1 YA L − 1 = 1 YA − 1 = l© 2L µ l© 2 L µ 2 L µ l© 2 L µ lo (1 + α∆T )
It follows: L − lo (1 + α ∆T ) f′ = f (L − lo ) (1 + α ∆T ) and this is the required ratio. Now, using α = 1.1 × 10 −5 K −1 , ∆T + 20 K , f = 440 HZ, lo = 0.795 m and L = 0.8 m we have: f′ = 440
0.8 − 0.795 (1 + 0.00022) = 0.9822 (0.8 − 0.795) ⋅ (1 + 0.00022)
f ′ = 432.2 Hz, Therefore the final frequency of the now-warmer guitar is 432.2 Hz, which is out of tune by about 8 Hz. [Simal, James & Robin]
9
5) Toys, Toys, Toys The masses are equal, so after the collision the ÒlatticeÓ will move at V m/s [right]; x(t)Ê=ÊVt is the position as a function of time. First, note that momentum will be conserved, so that at any point in time we will have: mV = mL + mR V =L+R
[1]
where L and R are the velocities of the left and right balls, respectively (they clearly point in the x-direction, so we drop the vector sign). The position of the centre of mass is given by: xc.m. =
mxL + mxR 1 = ( xL + x R ) 2m 2
[2]
where xL and xR are the positions of the left and right balls (in the laboratory frame), respectively. Hence, Vc.m. =
dxc.m. 1 V = ( L + R) = dt 2 2
using [1].
In the centre of mass frame then, which moves at V/2 m/s [right], the velocities of the balls become: L′ = V −
V V = ; 2 2
R′ = 0 −
V V =− 2 2
at t = 0. (the balls are thus approaching each other) The situation is perfectly symmetric and we can right away conclude that after the two balls collide, which will take a time of L/2/(V/2) = L/V their velocities will be: L′′ = −
V ; 2
R′′ =
V 2
(the balls are going away from each other) Next, the balls will be pulled by the string, and conserving momentum and energy yields that after the collision we will have: L′′′ = L′ =
V V ; R′′′ = R′ = − 2 2
The balls therefore oscillate with a period of 2(L/V). Note that the motion is not simple harmonic, as acceleration = 0 - Ðkx. The motion may be written as 10
L′(t) =
V L 2L , t< mod 2 V V
L′(t) = −
2L V L , t> mod 2 V V
xL ′ (t) =
2L V L t, t < mod 2 V V
xL ′ (t) = −
2L V L mod t, t > 2 V V
the primes denote centre of mass frame; ÒmodÓ gives the remainder after division, so using it we can figure out what part of the period t is in. If you have never seen ÒmodÓ before, donÕt sweat it. I think it is (more or less) clear the graph looks something like this:
(the dashed line corresponds to the right ball, the solid corresponds to the left one) Since we have the motion in the centre of mass frame, and we know its velocity, we simply transform to the laboratory frame. This gives: 2L L < V V 2L L L(t) = 0, t mod > V V L(t) = V , t mod
2L L < V V 2L L xL (t) = const , t mod > V V
xL (t) = Vt , t mod
Plotting this gives:
11
The solid line is the motion of the left ball; the dashed that of the right; the thick line is the average of the two, which is also the motion of the centre of mass. The system looks somewhat like a centipede Ð first the back moves, with the front being stationary, then the front moves, and so on. Note that if we had a body of mass 2m colliding with one of mass m we wouldnÕt get the same results as here Ð our system loses energy (itÕs a soft object, like jello). In an n-particle lattice, the speed of the centre of mass would be V/n. It is instructive to try this by brute force Ð keeping the same symbols as in i) (xL, L for left object, x R, R for right) L+R=
dxL dxR + =V dt dt 2
2
dxL dxR mV = m +m + k(L − ( xR − xL ))2 dt dt 2
This is a system of differential (momentum and energy) equations that just screams out Ògo awayÓ [NOTE: if you rearrange it Ð it takes lots of work Ð you end up with something workable, but not at the high school level] LetÕs take the hint and go to the centre of mass frame. From ii), the centre of mass still moves at V/2 m/s [right]. In that frame we once again get the balls approaching at equal speeds (V/2). Since the compression on each half of the spring will be equal, we can treat the system as two springs of length L/2 (kÕ = 2k). The motion is simple harmonic with a period: T = 2π
m 2k
The motion of the left ball is xL ′ (t) =
V 2
2k m t sin 2k m
[this gives the correct initial conditions Ð xLÕ(0) = 0, L(0) = V/2] Thus, in the outside frame, the motion looks like this: x L (t ) =
V V t+ 2 2
2k m sin t 2k m
Plotting gives (not to scale!):
12
The black line is the right ball, green line is the left ball, straight line is the motion of centre of mass. The motion looks once again like a centipede, but kind of weird (nonuniform). This is very similar to ii), except that the solution is now
xL ′ (t) =
V − 2 e 2
γt m
2 1 k γ sin 4 − t 2 m m
4
k γ − m m
2
Plotting gives something like this:
13
Note that the particleÕs motion eventually decays and remains at x = 0. In the outside frame, then: Ve x L (t ) =
γt m
V t+ 2
1 k γ 2 4 − sin t m 2 m k γ 2 4 − m m
which gives (not to scale!): Here, the left ball is black, right ball red and centre of mass is green (straight line). Clearly, the last model is most realistic. After a body collides with something, we might expect its molecules to vibrate for a little bit, but not forever as in ii) or iii). ii) is most unrealistic as it experiences infinitely long vibrations and the bond between molecules is most likely not a rigid string (this implies that the force acting on the molecules occurs 14
over an infinitely short amount of time, leading to infinite accelerations, etc.). The dissipation mechanism is due to radiation Ð an accelerating charge radiates away its energy [see problem set 1]. [Peter]
6) Making a shoebox spectrograph, using a compact disc As given in the Ômini tutorialÕ under InfoBitsª, the input and output angles of light diffracted from a diffraction-grating depend on wavelength, and on the diffraction grating itself: n λ = 2d • (sin θi + sin θ d ) where θ i is the input angle, measured from the perpendicular to the surface (the surface normal), and θ d is the angle of the diffracted light, measured similarly. 2d is the spacing between grooves of the diffractiongrating, or tracks of the CD (rather than d , as given Ñ this is something of a convention). The n is an integer which gives the order of diffraction, which you can investigate.
Diffraction orders: This figure was made using a CD and a laser-pointer. You can see the laser-pointer, held at the right. The brightest spot of light is the undiffracted specular reflection (sometimes called the Ôzeroth order,Õ and then to the right are three orders or diffracted light, labelled n = Ð1, Ð2, Ð3. Try the equation yourself to see why theyÕre negative.
We used a HeNe laser from the lab, and a CD that was kicking around, and found that with normal incidence (θi =0), the 1st order diffracted angle was found to be 22 deg. Since the HeNe wavelength is 632.8 nm, the spacing between adjacent grooves on the CD works out to be approximately 1.7 µm. If you look at fluorescent lights, you can easily see the different spectral lines, of different colours, that make the fluorescent light 15
look white. The colours come from different phosphors coated on the inside of the glass tube Ñ each spectral line comes from a quantum transition in the atoms making up the phosphor. The atoms are excited by absorbing ultraviolet light, which comes from mercury atoms which are themselves excited by electrons accelerated up and down the tube. So fluorescent lights are doubly fluorescent Ñ first the mercury vapour inside, and then the phosphor coating. Your school may have a wall-chart that shows characteristic spectral lines of different materials (these are available through Edmund Scientific, among other places); a picture of part of one of ours is below.
If you look at street lights, you may find that there are different kinds. Incandescent lamps give a whole or continuous spectrum, but low-pressure and high-pressure sodium lamps are different: the low-pressure lamps give spectral lines, but in highpressure lamps the much more frequent collisions between atoms make the spectral lines spread out or broaden. Astronomical observatories care very much which sort of lights a city installs, since too much scattered light (light pollution) can wreck observations. If low-pressure sodium lamps are used, the scientists can use the frequencies between spectral lines to peer through to the deep sky. [Robin] 16
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Problem Set 5: Electricity and Magnetism Due March 5, 1999 1) Bully for you! i) Back in kindergarten, the big bully challenges you to a game of ÔNo-mercy pink leg wrestlingÕ. You decide that this endeavour is not in your best interest so you challenge him back to a Ôreal toughÕ game of point-charge placement. ÒHuh?Ó says the bully so you explain: ÒPretend you are given a set of electrically charged balls, and you must arrange them to be free, but static Ñ so that they do not fly together or apart.Ó ÒHey, what about gravity or friction or redistribution of the charges on the balls?Ó the bully demands. Unfortunately, you seem to have challenged the only bully in advancedplacement physics in the entire kindergarten class. ÒHmm, ignore them and pretend itÕs a perfect insulator, so the charges cannot moveÓ you respond, hoping that if this doesnÕt go well, the recess bell will ring before leg-wrestling becomes an option again. ÒOkay, but I go firstÓ says the bully. ÒThree balls: two with charge +4q, the other with charge Ðq. Hurry up twerp!Ó What is your answer? ii) You got him on that one but now it is your turn. You ask him ÒThree balls, all with charge q, placed on the points of an equilateral triangle. Two other balls with identical charges on them, but you get to pick the amount of charge. What is the charge and in what arrangement?Ó He screws up his face to think, exclaiming ÒThere better be a real answer for this, tweek!Ó You assure him there is one. Better think quick, what is it? [James] 2) Charged with resistingÉ A current of 1 mA flows through a conductor made of 1 mA two wires, one copper and the other iron. The crossCu Fe sections are identical, and the wires are butt-welded as shown in the figure. What electric charge naturally accumulates at the boundary between the two metals? How many elementary charges does that correspond to? [Hint: GaussÕs law] [Gnädig/Honyek] 3) ÔAscending and Descending VoltagesÕ: a circular argument ÒPsst!Ó, a man in a dark trench coat whispers to you, ÒPsst buddy, ya interested in an almost-new physics equation?Ó Being the foolhardy type, you decide to check out his 1
wares. ÒOkay buddy, IÕll even throw in one for free to show you that IÕm legit. Say ya got a wire carrying a current. That current induces a magnetic field. Take your right hand like this ya see. Point your thumb out in the direction of the current and the magnetic field curls around the wire like your fingers do.Ó You are not impressed. Heck you learned the right-hand rule in Grade 3! ÒOkay, waitasecond. I got a real good one. Real hot!Ó Whipping out a small backboard from within the long folds of his coat, he draws the following diagram while saying, Ó Say ya got a closed circular circuit, with a magnetic field going through it. If that field changes in time, there is going to be an electric field induced in the circuit, according to: v dφ v v E= where φ = B ⋅ A dt B here, E is the induced electric field around the circuit, B is the magnetic field, and A is the area enclosed by the circuit (with its unit vector pointing perpendicular to the circle, in the direction defined by the right hand rule). So waddya think of them apples?Ó
A
E
Wow! This is cool stuff, but something about it sounds fishy. ÒOkay, IÕll show you. Take a look a this.Ó He motions you into the dimly-lit alley behind him. Sure enough, your new friend has an electromagnet stashed back there. Above the end of the magnet is a circular loop of five 100 ohm light bulbs all wired in series. Even though there is no battery in the circuit, the bulbs are all glowing weakly. You whip out your magnetometer, and verify that yes, there is a uniform magnetic field running 2 perpendicular to the plane of the circuit (area=0.001 m ) and it seems to be ramping up in time: B(t) = B0 ¥ t. a) Using your friendÕs equation, what do you conjecture is the total current in the circuit? And what is the voltage across one of the bulbs? b) Darn, you think, I wish I had brought my voltmeter. So you say to your new friend: ÒLet me get this straight. There is current running through each bulb. That means that voltage must be decreasing across the bulb. Therefore as you go around the circuit the voltage keeps dropping after each bulb. You end up back at where you started but now it seems you are at a lower voltage. DoesnÕt this remind you of a print by some Dutch guy named M.C. Eaker, Esther, or something like that?Ó ÒNo, no, no, buddy. You got it wrong!Ó. What is wrong with your logic? Draw an equivalent circuit. And what is title of the picture that you are mumbling about? c) ÒOkay, now I understandÓ, you tell the huckster, Òbut there is something still wrong with your equation. If your equation was really right, those bulbs would blow up.Ó You got him there. Explain. [James] 2
4) Flux compression Ñ putting the squeeze on a B-field This is an actual experimental problem in practice Ñ a way in which very high magnetic fields can be produced, using a current-carrying circuit and high explosives. i) Consider a circuit of total area A with some current I passing through it. This causes a uniform magnetic field B perpendicular to the plane of the device. The sides of the device are suddenly imploded so as to cause the area to shrink to AÕ. What is the resulting magnetic field BÕ ?
(total area A)
(total area AÕ)
ii) If the object is a solenoid shrinking from a radius 10rÕ to a radius rÕ, with I = 1A and n = number of turns per unit length = 100 / 1 cm, what is the resulting field? [HINT: one of MaxwellÕs equations will be useful] [Peter and Bryan] 5) Scratch and dent sale on capacitors i) What is the capacitance of the earth, taking it as a perfect sphere? ii) Consider the earth not as a perfect sphere Ñ say a comet smacks into the earth and dents it, changing its volume by 3%. By what percentage does the earthÕs capacitance change? [B ONUS : extra points awarded for deriving the formula for capacitance of a sphere, given in InfoBits below] [Robin & Gnädig/Honyek] 6) Mocking mirrors The method of images is very useful for solving a variety of electrostatics problems. Consider a point charge +q a distance d away from an infinite conducting plane. Because the plane is a conductor, the charges on it will move so that the potential on the surface of the plane is constant (until this happens, there is a field in the conductor; this field makes the charges move). But you could also make an infinite constant-potential surface if you had two opposite charges (+q and Ð q) a distance 2d apart Ð the potential on their perpendicular bisector is a constant. This wouldnÕt be very important, except there is a theorem which states that the solutions to electrostatic problems are unique: if you find some electric field that satisfies the boundary conditions, it is the only solution. The twinned charges give the right
3
potential for the infinite plane, so their field around +q is the same as the field of a single +q charge above a conducting plane. i) So, what is the field at any point y just outside our infinite conducting plane?
+q d
ii) Consider two infinite conducting planes, crossing at a 90¡ angle. A charge is placed symmetrically inside one quadrant, a distance d from either plane. d What is the field just outside the conductors now? What image charges can you use to produce the same field? d +q iii) Consider now two infinite, conducting, parallel planes separated by a distance 2d, with a charge +q midway. Where do you have to place +q image charges so that the planes will be equipotential d d surfaces? [HINT: in this case you can think of the planes as mirrors, the charge +q as an object, and the image charges as images of the object formed by the two mirrors] [Peter]
INFOBITSª Ñ Useful Bits of POPTOR Information Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor Csphere =
R , k
C = capacitance, R = radius k = 8.9875 × 10 9 N m 2 C −2 = 8.9875 × 10 9 F m −1
Ecapacitor =
Q2 , E = energy, Q = charge, C = capacitance 2C
4
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Solution Set 5: Electricity and Magnetism 1) Bully for you! i) The symmetry gives it away. Put the negative ball in the middle, and place the other two to balance the Coulomb force. No matter how far apart are the positive charges, the net force on each of the three balls will be zero. ii) This is a classic lateral thinking problem. Trying to balance three balls in a triangle, with two balls sounds impossible, UNLESS you think in 3D. Place each ball above and below the plane holding the three balls. Symmetry says that each negative ball must be an equal distance from each +q ball. Now balance B the Coulomb forces. Define the equilateral triangle to have sides of length 1 unit. From the top this looks like the figure at right, from which B B 1 we can find that B = . 3 From the side, this looks like the figure at left, where angle b is opposite side B. We find that we have to satisfy two equations:
-n b
C
A
q
B
n2 3nq cos(b) = 4A2 C2 2nq sin(b) = 3 q 2 2 C
(vertical forces acting on ball Ðn) (horizontal on ball q)
Since A = C * cos(b), the first equation simplifies to: n A = 12 C q
3
The second equation simplfies to n 3 3 = C q 2 So 2 A = C 2 Recall that A 2 + B2 = C 2 . These two equations then give: A = 1±
2 3 1
and the value for n=10.4q or 0.33q. You win! [James] 2) Charged with resistingÉ Let I denote the current flowing in the wire, A the cross-section of the wire and ρ1 and ρ 2 the resistivities of the materials. OhmÕs law for a wire of length l gives U = I Ê ρ l /A which yields E = U / l = ρI/A for the electric field strength in the wire. The resistivity of copper is lower than that of iron. Therefore to give the same current in each part, the electric field strength has to be smaller in the copper than in the iron.
Cu
Fe
According to GaussÕs law, the difference in the electric field strengths implies an accumulation of charge at the boundary of the two metals (see figure). The total charge accumulated at the interface is Q = ε 0EA = ε 0 I (ρFe − ρCu ) It is interesting that this quantity depends purely on the current and material constants, but not on the cross-sectional area of the wire (i.e., of the interface). Substituting the known data, the charge is found to be Q ≅ 5.10 −21 C, which is only 1/30th of an elementary charge! Though a measurable macroscopic current flows through the wire, the accumulated charge is only a small proportion of the microscopic elementary charge. This strange result shows that classical electrodynamics (imagining charge carriers as small balls) cannot always correctly describe electrical phenomena. Only application of the more sophisticated laws of quantum theory and statistical physics can give a more accurate description. [You might be interested to note that the 1998 Nobel Prize in Physics was shared by Robert Laughlin for his theory of the fractional quantum Hall effect, and fractional charge excitations.] [Gnädig/Honyek] 3) ÔAscending and Descending VoltagesÕ: a circular argument a) Obviously, I did not miss my true calling as a screenwriter. The total induced electric field, if the given equation is correct, is simply 0.001 Bo , over a distance of 2 0.001π , so the total voltage (E ¥d) is: Bo ⋅ 0.001 ⋅ 2 0.001π , and the total current is this value divided by 500 ohm. The voltage across one bulb is 1/5 of the total. b) The picture you are mumbling about is called ÒAscending and descendingÓ (note hint in the title of the question). M.C. Escher makes use of a lovely illusion to draw a staircase with no beginning or end, and with people on it constantly descending or ascending. Actually, my colleague Julian (who is a fine violinist I might add) comments
2
that there is an additional Escher print on the same theme, with a water fall linked in a similar way, endlessly producing power.
V
R V
R
R V
Of course, we have no such ÔtricksÕ in physics, so what is going on in the circuit? The voltage ÒsourceÓ is distributed through every bit of wire, so the voltage drop across each resistor matches the voltage increase across 1/5 of the length of the wire. Thus you start out at the same voltage you began with. This is similar to the circuit at left.
c) Consider the right-hand rule in comparison to the equation your new friend is trying to sell you (another hint that I included in the equation). The induced current will create a magnetic field that will increase the flux in the loop, this will increase the induced current that will create more flux, you have a runaway effect! Any small perturbations in any small magnetic field will cause it to increase dramatically and without end. Of course this is not realistic so there must be a problem with your buddyÕs model. The real equation has a negative sign. The induced current acts against changes in the magnetic field. Mother Nature is truly a conservative force (no political insinuations intended) [James]
V
R
V
R
4) Flux compression Ñ putting the squeeze on a B-field This experiment is apparently done at Los Alamos National Laboratory, in New Mexico, USA. The two sides of the device have explosives attached to them, so that they start touching each other at the far end and finally end up being connected at the low end. The area of the loop is actually much smaller than that of the sides. The current is caused by a capacitor discharging. As hinted, the Maxwell equation we will use is that for induced emf. We have: dΦ B = −Emf dt →
Φ B = ∫ B • dA So the change in the magnetic flux causes and induced emf. In our case, the initial flux is just B¥ A, since the 3
field is uniform and perpendicular to the plane of the device. As the area of the device is shrinking, the flux is decreasing. However, we see from above that this change induces an emf in the circuit, which in turn produces a current. Which direction is this current? From the equation we see that it will oppose the decrease in the flux Ð i.e., the current will flow in a direction to oppose the decrease in flux and will thus increase it. How much does the flux grow by? It will keep growing until it equals the original flux, because then ∆B = 0, emf induced = 0, and we have reached a steady state. Thus, we see that flux will be conserved in this process and we will get Φ B = const = BA = B′A′ A B′ = B A′ Field of solenoid is B = µ 0 In using AmpereÕs Law. A direct application of i) gives 2
2
r r which gives 1.26 T in this case. Note that the field grows as the B′ = B = µ 0 In r′ r′ square of the radius, which means that we can achieve very large fields (~ 1000 T) with this technique. [Peter and Bryan] 5) Scratch and dent sale on capacitors i) Capacitance C is defined such that Q = C V, where Q is the charge on the capacitor and V its electrostatic potential. Thus we can write C = Q/V. For a spherical distribution of charge, the field and potential outside the sphere is exactly as it would be if the entire charge were concentrated at a point at the centre of the sphere. Thus at the sphereÕs surface or a vanishingly small distance beyond, V = k Q/R, where R is the sphereÕs radius. From this it follows that the capacitance of an isolated sphere is: Csphere = Q/V = Q/ (k Q/R,) = R/k (the ÔgroundedÕ capacitor plate is at infinite distance, where the electrostatic potential approaches zero) 6
9
2
Ð2
The earthÕs radius is 6.378 × 10 m, and k = 8.9875 × 10 N m C ; from this it follows Ð4 2 Ð1 Ð1 that the capacitance of the earth is 7.1 × 10 C N m , or 710 µF. Seems kind of small for such a big planet! ii) The energy of a capacitor of charge Q and capacitance C is Q2/(2C). If the change in energy of the capacitor can be found the change in its capacitance can also be calculated. The energy of the capacitor is higher when it is indented, since the surface charges move in a direction opposite to the force acting on them. Also an electrostatic field of field strength E has an energy εE2/2 per unit volume and when the capacitor is indented the electric field penetrates a volume where it was not previously present.
4
If the surface of the capacitor is only changed a little the electric field can be considered as identical to the original one near the surface. Thus, the change in energy depends purely on the change in volume and not on the actual shape of the indent. Imagine that the capacitor is hammered so that its volume decreases by 3%, but its shape remains spherical. Its radius is therefore reduced by 1% (as the volume of the sphere is proportional to the cube of its radius). The ratio of the energy of such a capacitor to the energy of the original one is the same as the ratio of the energy of the indented capacitor of the problem to that of the original one. Thus, the relative change in their capacitance is identical as well. Further, the capacitance of a spherical capacitor is proportional to its radius. The capacitance of the new capacitor (of reduced size) is therefore 1% smaller, and the capacitance of the capacitor of the original question decreases by the same amount. [Robin & Gnädig/Honyek]
6) Mocking mirrors i) We have:
So that the field at a point (0, y) is given by: r 2 cos(θ)q ) 2dq ) E(0, y) = − x=− x 2 3 2 / 4 πε o R 4 πε o y 2 + d 2
(
)
And it has no y-component. Hence, the potential, which is just the integral of the field (up to a constant) gives zero (since ∫ 0dy = 0 ), so the potential is indeed constant in the planes. ii) By comparing this to i), or noticing that the image charges correspond to images of object as produced by a mirror, a self-consistent solution is:
5
The field at any point (0, y) is r 2dq 2dq ) E(0, y) = − x+ 3 2 / 4 πε o ( y + d)2 + d 2 4 πε o ( y − d)2 + d 2
(
)
(
)
3/2
) x
which is also the field at a point (Ðx, 0), by symmetry. iii)
Consider this reasoning: to make A equipotential we have to put a charge Ðq a distance d left of it. To make A' equipotential, we put Ðq a distance d to the right of A'. But now the two Ðq charges cause A and A' not to be equipotential again. So we put a charge +q a distance 3d left / right of A / A' (respectively). And so on, ad infinitum. This is equivalent to the forming of infinitely many images when you stand between two mirrors Ñ the 1st seems a distance d away, the 2nd, a distance 3d = d + the distance of the light ray bouncing from the opposite mirror. We have:
[Peter]
6
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Problem Set 6: AC Circuits and Electronics Due April 2, 1999 1) Thief of time 100V
33 kΩ neon lamp
30 µF
A chum of mine once upon a time had built this circuit, and set it by the basement windows of his house as a burglar deterrent Ñ it makes a flashing light, and gives the impression it might be part of a burglar alarm. The flashing light comes from a small neon-filled bulb, with closely spaced electrodes. The bulb can support about 80 V across it without problems, but at a higher voltage the gas inside ionizes in the strong electric field, the resistance drops drastically, and a discharge takes place between the electrodes. The discharge electrons excite the neon atoms, which produce the light. Once electrons stop flowing, the ionized gas recovers and the resistance rises again to the original value.
i) How frequently does this particular design flash? ii) If the neon lamp goes from 220 kΩ in its normal state to about 0.5 Ω during the discharge, roughly how long does each flash of light last? iii) During a flash, roughly how much electrical energy goes into the lamp? iv) How could you easily modify this circuit to make the flash last longer? How long would the flash then be? [Robin] 2) Logic rules (!) For the questions below assume that a logical high (Ô1Õ) means a voltage of about 5 V, and a logical low (Ô0Õ) means a voltage around 0 V. Straight lines are wires and things labeled A, B, CÉ are inputs. i) Consider this attempt at a logic gate. Make a truth-table for this setup: show the output for each possible combination of inputs (use 0s and 1s). What kind of gate is it? Name at least one major problem
A B
(output)
1
this gate would have, if used in some circuit. [HINT: does the gate have, say, directionality?] To avoid the problem you may have found above, we can use a wonderful device known as a transistor. A transistor essentially functions as a switch. It has 3 electrical connection points Ð the base, the collector and the emitter. When the base is at a higher voltage than the emitter, the conduction band of the semiconductor inside shifts toward the collector and then current can flow from the collector to the emitter. Otherwise, the band is near the base and the device does not conduct between the collector and the base. Semiconductor Ñ get it? Sometimes itÕs like a conductor and sometimes it isnÕt. ii) Knowing this, can you figure out what the following circuits do? Give a basic description of each and figure out the truth tables (as described above).
V b)
a)
V
V R R
R
(output)
(output) collector
A
A
base emitter
B
[NOTE: in all circuits the resistance of the load on the output (e.g., voltmeter) is much higher than R] [Peter] 3) Small, but wireyÉ Your mission Jim, if you chose to accept it, is to connect the dots using the given components to obtain the required currents or voltages (see figure on next page). The wires and components are delicate so do not bend them. Connecting diagonal dots does work though. Remember Jim, all the components are ideal (wires have no
2
resistance, there is no voltage drop across an ammter, etc.). You have 15 seconds before this unit self-destructs. 14, 13....
e.g.) Set voltage=5V, current: I=5A.
5V wires
5V
V
A 1Ω
1Ω
A
V
10Ω
a) Set I=2mA.
1kΩ
1kΩ
A
b) Set I2=18mA. What is I1?
9V
12V
A1
9V
1kΩ
A2
9V
1kΩ
wires
wires
0.5kΩ c) The voltage across the unknown element AC changes as a function of time (see V1(t)). Set V2(t) as shown. V1(t) V2(t) t
V1
V2
AC
t Recall Jim that the current across a diode looks like: I
wires
diodes
V
d) Set I=47.5 mA. (Be careful Jim, we think this one might be harder than it looks.)
1V
13kΩ
A
13kΩ wires connectors 13kΩ unlimited supply of all three components
[James]
3
4) Is resistance futile? Given a resistance R (made of n smaller resistors of value r connected together any way you want), is it possible to add two more rÕs so that the total resistance remains R? Note that all the resistors must have both their ends connected Ð i.e., the following is illegal: The resistance from a to b obviously remains at a value R, after connecting two more resistors r, which makes the question far too boringÉ [Peter]
r
r
R
a
b
5) ÔE pluribus unumÕ Given a large number of exactly 1 Ω resistors, describe using circuit diagrams how to make a resistance of any (arbitrary) given value. [HINT: the value of the new resistance is arbitrary, but it can only be measured to a finite number of digits] [Peter] 6) Inductive reasoning
S
The circuit shown in the figure Ñ composed of three identical lamps and two coils Ñ is connected to a DC power supply. Some long time later, the switch S is opened. In order of brightness, how do the lamps rank? Explain. [Gnädig/Honyek]
C L
C L
L
Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor
4
1998-1999 Physics Olympiad Preparation Program — University of Toronto — Solution Set 6: AC Circuits and Electronics 1) Thief of time i) LetÕs use these symbols for the various bits: R = 33kΩ, the resistor after the battery RB = 220kΩ, the normal resistance of the bulb C = 30 µF, the capacitance of the capacitor in parallel with the bulb Our basic equations are: VC = VB (in parallel) VR + VB = 100 V (the whole voltage drop around a closed loop) I R = I B + IC (the current is conserved as it splits over the bulb and capacitor paths) VR = I R ⋅ R (OhmÕs law) VB = I B ⋅ RB QC = C VC (charge on a capacitor), which gives: ⇒ IC ≡ Q˙ C = CV˙C (just taking the derivative) = CV˙B Then this can go into OhmÕs law to give a voltage, VR = I R ⋅ R = ( I B + IC ) ⋅ R
(
)
= I B + CV˙B ⋅ R V = B + CV˙B ⋅ R RB and then this can be used in the voltage-sum expression VR + VB = 100 V , to reduce just to an expression for the bulb-voltage: VB + C V˙B ⋅ R + VB = 100 V RB 1 1 1 100 V V˙B + VB + − =0 C RB R RC which we can write in a simpler form for the constants, 1
V˙B + VB = b + c = 0 b=
1 1 1 + C RB R
100 V RC We can solve this several ways Ð the most straightforward is just to integrate, using dV V˙B = B , so that you get an expression d VB on one side and dt on the other, to dt integrate. c=−
A neater answer is to think about what this means Ñ what is going on. After a long time, we can guess that if the bulb hasnÕt broken down, the capacitor will be all charged up and that nothing will be changing (i.e., will have a Ôsteady stateÕ). Then there could be no changes in time, so V˙B (t ) = 0, and therefore: VB (∞) ⋅ b + c = 0 ⇒ VB (∞) = −
RB c = (100 V ) ⋅ R + R b B
Thinking about the start, when the battery is connected but no charge has yet flowed through R, we need to have VB(0) = 0. So the right solution starts at zero and asymptotically (over a long time) goes to a constant as a limit. If you already know RC circuits, or if you know 1st order differential equations, or if you integrate the equations (or if youÕre lucky!), you might guess that the change is an exponential relaxation, which if we start at zero and go to VB (∞) would have the form
(
VB (t ) = VB (∞) ⋅ 1 − e − at
)
If you put this into the equation above, youÕll see easily that it actually works:
(− a) VB (∞) e − at + VB (∞) (1 − e − at ) ⋅ b + c = 0
{(a − b)⋅ VB (∞) e−at } + {VB (∞)⋅ b + c} = 0
This has to be true for all values of t, which means each parenthesis term must be zero independently. Taking the second term to be zero gives VB (∞) , if we didnÕt already guess it above. The first term to be zero gives us a=b, so the solution is: 1 1 1 − + t RB C RB R VB (t ) = 100 V ⋅ ⋅ 1− e R + RB
or with our exact values put in:
2
(
VB (t ) = 87.0 V ⋅ 1 − e −1.16 t
)
(
VB (t ) reaches breakdown for the bulb (80V) when 80 = 87¥ 1 − e −1.16t
)
⇒ t = 2.18 s
ii) At breakdown, the capacitor discharges through the bulb, until the bulb ÔrecoversÕ. V C = VB IB+IC=0 QC = C VC ⇒ IC = C V˙C and with VB=IB RB and also (with current left-to-right taken positive) 0 = I B + IC This then gives VB + C V˙B = 0 RB 1 VB = 0 ⇒ V˙B + RC which has a solution VB (t ) = VB (0) ⋅ e
1 t RC
So a good characteristic time of the bulbÕs flash is RC = 0.5 Ω ⋅ 30 µF = 15 µs iii) The
energy
stored
in
a
capacitor
charged
to
voltage
V
is:
= 30µF ⋅ (80 V ) = 96 mJ (not enough to make the bulb explode, I hope!) It is all dumped through the neon lamp. 1 CV 2 2
1 2
2
iv) going back to (ii) we can make the flash last longer by adding a resistor R′ series with the bulb. Then τ = RC = (0.5 Ω + R′) ⋅ 30 µF which you can control by the value of R′ and we can make the bulb light up for longer. But since I B = IC = − CV˙C = − CV˙B and V˙B (t ) will be smaller with R′ added, then the current through the bulb will be smaller and it wonÕt be as bright. [Robin]
3
2) Logic rules (!) i) This is an OR gate. When all of A, B, C are low (i.e., voltage is, say, 0 V), the output is naturally low. But when either A or B or C are high, so will the output be. Problem: when either A or B or C conduct, some current flows through the circuit. But when more than one input is high, the currents actually add. If we have something fairly sensitive connected to the output, we could fry it if the current is too large. WhatÕs worse, in a computer many gates are connected together Ð but if we connect a few of these gates to the inputs of another gate, the currents will add (if we have n inputs, final current is nI. If we feed n of such gates into another similar gate, the current is now n2 I) Ð this is clearly a problemÉ ii) a) This is a NOT gate. When A is low (ca. 0 V), the base and the emitter are both at 0 Volts. The transistor is thus cut-off Ð no current flows from the collector to the emitter (it acts like an open circuit). If we now connect a device with a large resistance (>> R), the voltage drop across R is negligible and almost all voltage (Vo) goes through the device. So the output is high. [Voltage across R would be RI and that across the device rI, but since r >> R the first may be ignored. Note that the current flowing through both is the same] When A is high, the transistor conducts (with almost no resistance) Ð thus, all the voltage drop will be across R [since it is smaller than r Ð the resistance of the output device]. Hence, the output is low. So this is a NOT gate. b) The second gate (from the left) is a NOT, from above. The first gate is a NAND, by a similar argument as above. Thus, the gate is a NOT (NAND) = AND. Note that using transistors the output and current voltage are constant. There is still, however another problem, but IÕll let you figure it outÉ [Peter]
4
3) Small, but wireyÉ
A
b) Set I2=18mA. What is I1?
A1 9V
1kΩ
1kΩ
1kΩ
a) Set I=2mA.
A2 9V
9V 12V Be careful about the polarity of the 12V source!
So what is I1? The normal voltage/current laws do not help you. The only answer that I can argue would be using symmetry, and thus I1=9mA.
c)
V1
AC
V2
This is known as a full-wave rectifier and can be used to chance AC voltage into DC voltage (with an extra smoothing filter)
d) Set I=47.5 mA. total: 618 resistors I had a more complicated answer but this one, handed in by a few students, is more straightforward.
13kΩ 13kΩ
1V
13kΩ
A
13kΩ
[James]
4) Is resistance futile? LetÕs try all 6 possibilities: a)
R
r
r
5
b) r R
r
c) R r r
d) R r
r
r
R
e)
r
f) r
r R
From a) we get: R = R + 2r, which is impossible (r > 0) From b) we get: R = r/2 + R, which is again impossible From c) we get: r = r + 2R Ð impossible. 1+ 5 From d) we get: R = r 2 −1 + 5 From e) we get: R = r 2 From f) we get: 2r = 2r + R Ð impossible. So it is possible, using (d) or (e). A sophisticated aside for question 4: Note that we have a way of making resistances r that are rational. But note that both the solutions (d) and (e) in question 4 are irrational Ñ if our arbitrary r is rational then 6
it follows that R is irrational. But to make R we only used rÕs in some combinations Ð we only used rational numbers. LetÕs recall what kinds of circuits we can make using resistors: We could have resistors in parallel and series. But the equivalence formulas involve additions (series), or divisions (parallel), so there is no way to obtain a rational number. In the general case, we need to solve KirchoffÕs Laws, which are linear and create linear systems of equations. Solving such a system involves adding / subtracting rows, and multiplying and/or dividing numbers Ð once again there is no way to obtain an irrational number. And thus we must conclude that theoretically it is not possible to make the circuit from question 4, unless perhaps we could make R from an infinite number of resistors r. Experimentally, on the other hand it would be possible. For in experiments we can only measure things to a finite precision. Thus, there are no irrational values Ð since we only measure a certain amount (say 10) digits, every number is finite and hence rational. And hence we could achieve the circuits from either (d) or (e). [Peter]
5) ÔE pluribus unumÕ Within experimental error, the other resistor (call it R) can only be measured to a finite number of digits. Given any such number, we can express it as a rational number Ð i.e., in the form p/q where both p and q are integers. To create a circuit to express p/q we build the following circuit Ð we connect in series p groups of q groups containing 1 Ω resistors in parallel. q resistors
p groups
[solution from a Polish Olympiad book ÒOlimpiady Fizyczne XXI I XXIIÓ] [Peter]
7
6) Inductive reasoning When the switch is closed, currents as shown at left flow S round the circuit. (The value of the current is determined by the C C voltage of the battery and the I L L L I I resistances of the lamps.) A very short time after turning the switch off the current flowing in the coils is practically unchanged. (If this were not the case, there would be a rapid change in the magnetic flux, which would induce a very I I high voltage in the coil.) S Currents of 2I and I continue to C C flow therefore in the coils, and these determine the currents L L L flowing through the lamps 2I I I (figure at right). I
I
This means that the lamp closest to the switch suddenly flashes and the brightness of the two other lamps does not change. (This is only true for a short time, later all three lamps fade and go out.) [Gnädig/Honyek]
8
1999-2000 Physics Olympiad Preparation Program — University of Toronto — Solution Set 1: General 1) Pumpkin paradox a)
Mbig = 10000*Msmall
Therefore, for the same relative strength we need a cross sectional area 10000 times larger than the poles in the small model. So, Abig = 10000 Asmall 2
2
(1/4) π (Dbig) = 10000 (1/4) π (Dsmall) or Dbig = 100 Dsmall So Farmer Joe is going to need poles that are 100 times thicker. b) Mass is proportional to volume which in turn is proportional to area (3/2) Thus, mass is proportional to area . So, (Mbig)
(2/3)
(2/3)
/(Msmall)
= Abig/Asmall = (1000kg/0.1kg)
(3/2)
.
(2/3)
Therefore, Abig = 464 Asmall Since the big pumpkin surface area is 464 times greater than the small, Joe is going to need 4.64 L of paint. c)
Mweight/Mpumpkin = 450/1000 thus, Mweight = 9/20
The sum of the moments is zero when balanced, thus, Rweight Mweight g – Rpumpkin Mpumpkin g = 0 Therefore, Rweight = (20 Rpumpkin)/9
[Carrie]
2) Cubic quandry a) There are 6×4 = 24 equal cubes that can perfectly surround the lamp.so we can say that the fraction of power heating one cube is 1/24.
1 Qt = mc∆θ 24 24mc [seconds] ? t= Q
b) A unit area of a blackbody radiator in temperature T, emits energy by the 4 2 4 power of σT Watts. so our cube is emitting 6L σ T J/s by the assumption that our cube is in thermal equilibrium each time. Consider our case that any small change in T, will affect the emission rate and the time needed to increase the temperature accordingly. But, in this case, we are looking at changing the temperature by 1K at room temperature, i.e., from 300K to 301K. The relative 4 change is on the order of 1/300. and the change in T is about 4 × (1/300).
3014 = (300 + 1)4 = 300 4 (1+
1 4 1 ) ≅ 300 4 (1+ 4 ⋅ ) 300 300
To a good approximation, we can ignore the changes in dT/dt due to a small change in T, and we can substitute the room temperature instead. 1 Q t = mc ∆θ 24 Q ⇒ t = {mc}/ - 6L2 σ T 4 24 - 6L2 σ T 4 t +
c) In thermal equilibrium, the emission rate and heating rate are equal. so we have : - 6L2 σ T 4 =
1 Q 24
Q ⇒T = 2 144σL
1
4
[Amir]
3) Poles apart, telling First of all let us note that the North-South (N-S) designations on magnets are a little messed up. You see, the N pole is actually the North-seeking pole, meaning that it is actually the South pole (since opposites attract). We will follow these conventions in this document (it doesn’t matter how you approach this, so long as you’re clear what a North pole of a magnet means to you). After this lengthy introduction, let us get right to the point. The magnetic field of the magnet will bend the electron orbits into circles. We assume here that the field is uniform throughout the face of the monitor (clearly not true, as the field
gets smaller with distance; this solution is thus only valid for electron trajectories not far from the magnet). i) Suppose we put the N-pole (really the South pole) of the magnet on the right r r r side of the screen. The electrons will be bent according to F = − ev × B (e is the size of the electron’s charge). From this we see that the direction of deflection will r r be “DOWN” (looking directly at the monitor, v is out of the screen, B is from left to right). ii) From the voltage we can find the speed of each electron Energy = eV = 21 mv 2 , where V = 15 kV. From this, v = 7.3×10 7 m/s (Note that this is actually relativistic… we will ignore this, as the correction is small. Bonus marks will be given to people who realize this). We also have
mv 2 = e vB , where R is the radius of rotation of the electron. R
We find R = 0.41 cm. Now, each electron will actually only be deflected when it’s in the field of the magnet. In this case, this would be roughly the size of the magnet, which we cleverly enough didn’t tell you. (we assume that the magnetic field drops off really quickly at the sides of the magnet). Seeing that the turning radius is only 4.1 mm, the magnet really can’t be bigger, or there would be no picture whatsoever (all electrons would turn back). Let’s assume that the width of the magnet is indeed 4.1 mm. Then, electrons will turn by 90 degrees, and the vertical distance traveled is clearly 4.1 mm. DISCUSSION: This somewhat makes sense, but I would expect something bigger. This is mostly due to the fact that the magnetic field actually decays slowly with distance (and doesn’t become abruptly zero), whereby the force acts over a larger distance and deflects the electrons more.
y x
In the general case, we can resolve the motion into 2 directions: x (towards the screen), y (down the screen). We have:
x (t ) = − R sin(ϖt ) y (t ) = R cos(ϖt )
where v = ϖR .
We want to find a time τ when the electron reaches the screen (solid line) so –x(τ) = width of magnet, and calculate what the deflection y(τ) will be. [Peter] 4) Physics Rocks! a) To determine the critical height one must consider the equation of motion of the stone in the water. The forces acting on the stone in water are, (i) the downward force of gravity = –m*g (ii) the upward drag on the stone = 0.5 b p A v
2
(b = drag coefficient, p = density of water) and from Archimedes' principle, (iii) the upward buoyant force = p V g (V = volume of fluid displaced) Summing the forces we get, 2
p V g + 0.5 b p A v – m*g = 0 The sum of the forces is zero at equilibrium, when the velocity is the terminal velocity. Given that the diameter of the stone is 5 cm and that the density of 3 quartz is 2600 kg/m it is easy to find the following, –3
m
–5
m
A = 1.96 × 10 V = 6.54 × 10
2
3
m = 0.17 kg Plugging in the values yields a terminal velocity of 1.6 m/s (of the stone in water). Thus if the stone is dropped in air with Vo = 0 m/s then under the acceleration of gravity the distance it travels before reaching a velocity of 1.6 m/s is, 2
D = Vt /(2 g) = 0.13 m So the stone should be dropped from a height of 0.13 m above the surface of the water. b) At terminal velocity the water-drag problem is trivial! After 6 seconds the rock has fallen, 6*(1.6 m/s) = 9.6 m. [Carrie]
5) Light diversions Consider a ray of light reflected from one of the mirrors. The reflected ray is a mirror image of the continuation of the incident ray. Therefore, an easy way to trace the ray path is to make another square beside it and follow the continuation of the incident ray in that square. For more than one reflection , more squares should be drawn as shown. It is observed from the figure that as the continuation of the initial ray pass through any corners, it means that the light have gone out from the square. Therefore, the necessary condition would be tan(θ) = m /n which m and n are two arbitrary integers. Doing the first part, makes solving this part much easier. We have to build up some triangles which are image of one another. Since the cone angle is 30°, last triangle fir exactly the first one. Two regular polygons with 12 sides is produced. Analogous to condition that the rays leave through the upper hole after many reflections is that the continuation of the incident ray pass through the smaller polygon. After some simple mathematics, we figure out that D must be smaller than the width of the polygon which is d / tan 15°. [Yaser]
6) Physicists’ pipe dreams a) The water column will get narrower and narrower because the flow is constant and the speed is increasing, therefore the surface area of the water will decrease until it breaks up into droplets. b) If we consider low viscosity for water, molecules of water are freely falling down. 2
v 2 - v 0 = 2gh vπr 2 = v 0 πR 2 r=R
ν ν 2
2
r = R { v 0 /( v 0 + 2gh ) }
1
4
c) The stream will break up into droplets. It cannot get arbitrarily skinny, the way the formula gives, because: 1 . The radius can’t be less than the radius of a water molecule! 2 . Before you reach the limit (1) above, surface tension of water won't let you make it as skinny as possible. In fact, surface tension could affect the result which we obtained , but we are neglecting that! Consider the droplets have radius a . What is the height of a same-volume cylinder of radius r? The energy of surface tension for these two area ( as you could see in many books & specially at the end off this problem set :) 4 π a 3 = πr 2 ∆x 3 4 ? ∆x = a 3 /r 2 3
The surface-tension energy of these two states ( water tube & droplet ) are :
σ 2 π r ∆x
and σ 4 π a 2
substituting delta-x, we have and comparing them: a3 σ 8π 3r
a 3 3r and σ 8π 〈 3r 2a
it means that, if 3r/2a is less than 1 , then the surface energy of droplets is less than that of a water cylinder, so it’s energetically favourable for the water to break into droplets instead of staying in a cylindrical shape. What does determine the actual radius of a droplet? Several factors may affect that — for example the circular motion of water flow. But the most important one is density of water. It will break into droplets when the weight of the droplet has the same order of magnitude as the surface tension. It means: ρ = 4πa 3 / 3 g = 2πrσ
using previous result, we will get:
r=
4σ = 2 ↔10 −3 m = 2 mm 4ρg
and then the height at which we reach this redious is: h = v 02 / 2gr 4
(R 4 − r 4 )
d) For the experimental part, I just used a tap with diameter of 6 mm and tried to see at what height the water breaks up. After that, I fixed the height and measured the volume of the poured water and the time interval. Here are my results:
Height of break up Volume (cm) (ml)
Time needed (s)
v 0 = V /( πR 20 T )
5-6 cm
100 ml
78
4.5 cm/s
7-10 cm
100 ml
62
5.7 cm/s
16-20 cm
200 ml
50
14.7 cm/s
26-30 cm
300 ml
61
17.4 cm/s
Due to large error I have, the experimental behaviour is not exactly the same as our expectation, but `h’ is still proportional to v 20 . [Amir]
1999-2000 Physics Olympiad Preparation Program — University of Toronto — Problem Set 2: Mechanics Due December 17, 1999 (revised date) 1) Going for a spin Cally is playing on the whirly-ride (like a merry-go-round, but non-motorised) in her neighbourhood park. To make it go faster she pushes against the ground using one of her feet. a) If she accelerates the whirly-ride from rest to a speed of 15 r.p.m. in 3 seconds, what is the corresponding angular acceleration? b) By what fraction does the angular velocity change when her 20 kg kid brother Matt leaps radially onto the edge of the whirly-ride? Assume the initial moment of inertia 2 (just Cally and the whirly-ride) is 2000 kg m and that the whirly-ride has diameter 5 m. c) Since one of her all-time-favorite past times is tormenting her little brother, Cally decides to speed up the merry-go-round in an effort to fling her little bother from the ride. Assume that Matt is sitting at a radius of 4.5 m, that the coefficient of static friction between him and the floor of the whirly-ride is 0.7 and that he is naively not holding on! At what speed will Cally’s little brother start to slip? . [Carrie] 2) Bubble bonanza a) What is the presure inside a soap bubble of radius ‘r’ if the surface tension of the soap solution is σ? b) What will happen if two bubbles are brought to touch each other? (assume that one of them is bigger than the other, and there is still a bubble-film between them c) What will be the end result if the wall between the the two touching bubbles breaks, and the two bubbles merge? d) What if those two bubbles were attached to opposite ends of a tube (e.g., a drinking straw) like a weightlifter’s dumbell? [Amir] 3) Siphoning cellar My outside water faucets in my new house don’t work, and I’m reduced to running a hose from the laundry tubs out my basement window to fill my above-ground swimming pool. If I forget and leave the hose-end in the pool, then when I disconnect 1
my hose from the tap, water from the pool will siphon back into my basement. The setup looks like this:
1.75 m
1.0 m
1.5 m
10 m
Water in a 2 cm-diameter hose flows with a drag force Fdrag on the hose which is proportional to the length of the hose and to the speed of water flow. This drag results in a pressure drop along the hose. This pressure drop, per unit length, is proportional to v by the formula: ∇p =
∆p 32 Uη = ∆x d2
which is described more fully in INFOBITS™ below. a) If I forget the hose when I leave for work, and it siphons for 12 hours while I’m gone, how much will the water level in the pool drop? Estimate, or calculate exactly. b) When the last of the water drains out of the pool, then the water in the hose becomes of shorter and shorter length. This changes the drag on the water. See if you can approximately calculate the flow rate of water out of the hose as it empties. [Robin] 4) Gimme a brake... The Bumble Bee ride at Super Fun Fun World consists of carts (painted yellow and black of course), which seat 4 people, that traverse a windy, hilly track. The figure below illustrates one section of the ride. On a particularly unusual Tuesday there is a massive power Aieeee! failure at Super Fun Fun World (which consequently isn’t so fun fun). The power fails just as 6m one of the carts 4m nears point A with a velocity of 0.5 m/s. Point A
5m
Point B
2
a) If no emergency brakes are applied with what speed will the cart hit the bumper at point B? Take it that you have drum brakes, which press on a hub which is half the wheel diameter. Each drum presses onto a 30° arc of the hub. The thickness of the hub is the same as the wheel. For steel wheels on steel rails, the 30° coefficient of static friction is 0.78, 30° and the coefficient of kinetic friction is 0.42. You can assume that only about 3 mm of the rail is in contact with the wheel. b) At the bottom of the first hill the emergency brakes are applied radially to either side of the wheels. If the brake pads have a static coefficient of friction 0.85 and kinetic coefficient 0.65, what is the optimum force to apply to the brake pads such that the braking force is a maximum? c) With the brakes applied (assume with optimal constant force) how far up the second hill will the cart go? d) With the brakes held on, will the cart go back down the hill?
[Carrie]
5) It’s a ball? What a gas! This question is the beginning of a kinetic model for an ideal gas. It can lead you to figure out the ideal gas law PV = nRT. a) A ball with mass m and velocity v collides with a stationary wall. If friction is negligible, and bouncing is perfectly elastic, the ball bounces back with the same speed. But what happens if the wall, itself, moves toward the ball with velocity u? Consider all velocities be in the same direction, perpendicular to the wall. b) Going through (a) you can find out that the momentum has been changed. This means that the wall did some work on the ball. Can you show explicity that the difference in kinetic energy of the ball is exactly the same as the work done by the wall? BONUS: Can you use this model to show how much the temperature of a gas of atoms behaving like such elastic balls goes up as it is compressed — as long as no heat is allowed in or out? [Yaser]
3
6) Dimensional thinking A cylindrical vessel of water is leaking via a small hole of radius ‘r’. We want to keep the height of the water constant. The best way is to add water at some rate R (kg per second). a) Can you guess what should be R in terms of ρ (the density of water), H (height of the water level in the vessel), r (radius of the hole), g (acceleration due to gravity) and A (the cross-sectional area of the vessel). [HINT: Use your intuition, or common sense, and –2 watch that you keep the correct units in each stage (i.e., acceleration has units L T , where L is distance and T is time)] b) Can you prove your formula with physics theorems?. [Amir]
4
INFOBITS™ — Useful Bits of POPTOR Information Coefficient of resistance for pipe flow: There is a drag on the inside of a pipe, for a liquid flow through it. This drag means that there’s a pressure drop as you go along the pipe — it is why small plumping-pipes lead to poor ‘water pressure’ for a shower or similar. The pressure gradient (change in pressure per unit length) is: ∆p λ 1 = ρU 2 ∆x d 2 where p is pressure, ρ is fluid density, d is pipe diameter, U the average speed of liquid flow, and λ is a dimensionless constant which is equal to: ∇p =
λ=
64 Re
for laminar (smooth) flow
Re is called the Reynold’s number. (usually a value of a few 1,000 for laminar flow) The Reynolds number for a pipe can be found from: Re =
ρUd η
putting all this together, we see that: ∇p =
∆p 32 Uη = ∆x d2
C HECK THE POPTOR WEB PAGE for other hints, and any corrections we might post: www.physics.utoronto.ca/~poptor
5
1999-2000 Physics Olympiad Preparation Program — University of Toronto — Solution Set 2: Mechanics 1) Going for a spin α=
∆ω 15 ⋅ 2 ⋅ π / 60 = = 0.5rad / s 2 ∆t 3
Angular momentum is conserved (but energy is not!!) I oω o = ( I o + mr 2 ) ω ω 2000 = = 0.9% ω o 2000 + 20(2.5)2 Ignoring the fact that Matt is sitting off the whirly-ride, what happens is that the force of friction is providing the centripetal acceleration necessary to keep him on the ride. The force is ≤ µmg = 137.2 N . Thus, at maximum, v v2 = 1 rad s ±1 = 0.7 ⋅ 9.8 , v = 6 m s ±1 and ω = 4.5 4.5
[Carrie]
2) Bubble bonanza a) Consider 2 hemispheres of a bubble of radius r. They are repelled by a force of F = ∆PA = ∆Pπr 2 , where A is the area of the projection of half a sphere and ∆P = Po − P (difference between outer (atmospheric) and inner pressure). The hemispheres are pulled together by surface tension (which is given as force per unit length); moreover, there are 2 surfaces to a bubble, so the total force here is F = 2(2πrσ ) . Solving yields P = Po +
4σ r
b) Note that pressure is inversely proportional to the radius. Thus, the smaller bubble will actually have a higher pressure inside! This means that the film between the 2 bubbles will be pushed into the larger bubble.
1
If the film breaks, the bubbles will either break (but that’s so boring), or merge. Using the fact that the numbers of moles inside the bubble stays constant, assuming that temperature stays constant and using the ideal gas law, one gets PV = PV 1 1 + P2 V2 4σ + P r 3 = 4σ + P r 3 + 4σ + P r 3 o o 1 o 2 r r1 r2 (P, V, r refer to the final bubble, the remaining variables refer to the 2 initial bubbles) The equation could in theory be solved; note that for Po = 0, the final bubble will be bigger than either of the 2 initial bubbles… The system is obviously not in equilibrium. Since the smaller bubble is at a higher pressure, it will in fact contract and force air into the bigger bubble. Thus, we will end up with a membrane at one end and a big bubble at the other. [Peter, Amir]
3) Siphoning cellar Solutions for this problem will be provided with Set #3
[Robin]
4) Gimme a break... First I have to comment that this question was so hard that nobody got it right, including some of us here at POPTOR… a) Ok, that was easy. Conserving energy yields 1 2
mv 2 = 12 m(0.5)2 + mg(6 − 4)
From this, v = 6 m/s. b) This is where things get difficult. Clearly we want to stop the cart using static friction, as these coefficients are higher than those for kinetic friction, and thus the force will also be larger. Something has to slip, however (you can’t just stop the cart dead) – since the coefficient of kinetic friction between the wheels and the rail is lower than that between the brakes and the wheel, we want to avoid letting the wheels slip (it will yield the lowest force). Now, if we have an object of mass m lying on a level plane, where the coefficient of static friction is µ it certainly is true that the maximum force we can apply so that the object doesn’t start moving (and kinetic friction doesn’t kick in) is FMAX = Ffriction = µmg . But a rolling object is different!! For one, it is not stationary, it is already moving. So what force can we apply? Well, we have to ensure that the cart keeps on rolling, i.e. V(t) = ω(t)R, and not slipping…
2
I am also going to assume that there are 2 forces acting on the wheel – each of magnitude F/2 - pointing in different directions. We have to analyze the linear and F/2 rolling motions (about the axle) while F/2 the brakes are applied, knowing we Friction start with the cart rolling (m is the mass of the cart, M the mass of the wheel, I the moment of inertia of, F the applied force, R the radius of the wheel and µ the coefficient of static friction between the wheel and the track). ma = −µmg Iα = µmgR − FR V
From this we find v(t) = vo + at = vo − µgt ω(t) = ω o + αt = ω o −
F / 2 ⋅ Rt µmgRt + I I
We have that vo = ω o R at t=0, but to ensure that this is true at later times, we clearly need that µg =
F /2 ⋅ R µmgR − R I I
The moment of inertia is that of the wheel, i.e., of 2 cylinders, one of radius R and the R 2 5 2 1 1 other of radius R/2. Thus, I = 2 MR + 2 M = MR2 (I’ve assumed for simplicity 2 8 that both sections of the wheel weigh the same amount; it would probably make more sense to assume that their densities are the same, but oh well…) F/2 =
5 8 8 µMg + µmg ≈ (0.78)mg 8 5 5
(assuming the wheels are not very heavy) This is the force felt by the wheel; the actual applied force is F = Fa (0.65)
(the brakes have to be slipping)
Solving for the applied force gives Fa = 2 ⋅ 18.8m = 38m NOTE: most people put that F = 2 µmg , which is less than the above. Plugging this into the equations above we can see that V (t) ≠ ω(t)R , and so it seems that the cart slips… but you can do some experiments at home to convince yourself otherwise. What’s going 3
on here? Well, the force of friction is actually Ffriction ≤ µmg , and what will in fact happen is that it will adjust itself to a lower value so that the cart doesn’t slip… The above force is the maximum that can be applied so that the cart doesn’t slip; a bigger force and it’s all over… c) The above force does work ( W = Fa d ) to slow the cart down. It starts with an energy 2 1 2 m( 0.5) + mg6 . After a distance d it will stop; here, d turns out to be 1.6 m. Thus, the cart never even makes it to the hill… d) If the cart was on the hill, we need it to be in static equilibrium. From the above diagram we see that this amounts to mg sin θ ≤ 0.78mg cosθ , or θ ≤ 38o . Eyeballing the diagram, this seems to be false, and thus the cart will probably fall… [Peter] 5) It’s a ball? What a gas! a) Doing the question in the reference frame of the wall really simplifies things. In the rest frame of the wall, the ball velocity is the same before and after the collision but opposite in the direction. In this system, the ball’s initial velocity is v + u. So the ball bounces back with v + u in reference to the wall. This velocity is v + 2u in reference to the laboratory. b) The change in kinetic energy =
1 2
m(v + 2u)2 − 21 mv 2 = 2mu(u + v)
The work done by the wall = Fx But F =
∆p m(v + 2u − ( − v)) 2m(u + v) = = ; x = ut ∆t t t
Therefore work = Fx = 2 mu(u + v), same as the change in kinetic energy above… BONUS: (solution based on Feynman’s); “< >” denotes an averaging Let us work in 1D (results in 3D are very similar); let ρ be the density (#) of atoms per unit volume. Assume the compression is slow (so the gas stays in equilibrium). Then, the wall is practically stationary. The force exerted by the wall = change in momentum / ∆t is F = 2ρmv 2 A We want to average this over all the speeds, v. Notice that only _ of all atoms (balls) are moving toward the wall – we do not want to average over them, as they do not contribute to the force on the wall. This yields P = F / A = ρ < mv 2 > Multiplying by the volume, V gives, PV = N < mv 2 > 4
In 1D we have that 21 kT = 21 < mv 2 > and thus PV = NkT If you want to know how temperature changes with volume, you can note that this is an adiabatic process and look up the formula in any book (or derive it using the above). [Peter, Amir] 6) Dimensional thinking a) First of all, figuring out how much water to add is equivalent to figuring how much water is actually leaving. There a couple of ways to do this, but dimensional analysis is perhaps the most instructive. First, since the hole is small, the amount of water flowing out in a short time will change the volume in the vessel by very little; thus, the area A is irrelevant. Now, we need to combine the remaining variables into a flow rate R that has units [kg/s]. We now assume that the formula is of the form R = Cρx H y r z g w The units must combine to [kg/s]. It’s pretty easy to see that x=1 and w=1/2. Now, it would seem logical that the flow is proportional to the area of the hole, i.e., r2 . Since the flow also grows as H to some power (not as inverse of H), the only logical choice is to set z=2 and H=1/2. Thus, R = Cρr 2 gH , where C is some constant. b) Using Bernoulli’s Law to find the flow rate of the water leaving the vessel leads one to 1 2
ρv 2 = ρgH + 21 ρU 2
Here, v is the speed of flow in the pipe and U is the speed of water flow in the tank. For a large area A (compared to the area of the hole), the U term is negligible. However, we don’t have to make this assumption. Since the flow is steady, the continuity equation applies and we have UA = vπr 2 . The rate R is R = ρvπr 2 . Solving gives R = ρπr 2
2 gH πr 1− A 2
2
.
(for A >> πr 2 , this is indeed equivalent to a), if C = π 2 )
[Peter, Yaser]
5
1999-2000 Physics Olympiad Preparation Program — University of Toronto — Problem Set 3: Thermodynamics Due January 14, 2000 1) Clocks a’ Rockin’ A friend of mine knows someone who overclocked his computer, that is, he made it run faster than it was originally designed to. The entire system remains the same, except now twice as much data is transferred per unit time, as before. Thus, in effect, the current has doubled. My friend claims that the new system is so hot, that the cooling system’s temperature has to be around freezing. Consider a resistor R, placed in surroundings with room temperature 20 °C, which has been overclocked as described above. If the original system causes a temperature rise of 13 °C, what temperature will the new system heat up to? [Peter] 2) Literally linear Consider a rod with the linear coefficient of expansion λ and initial length of L. It starts out at temperature T 1 . a) Heat the rod to a temperature T2 — what is its new length? Then cool it back to its initial temperature T1 .— what is the length of the rod after this cycle, assuming the usual formula for linear expansion and contraction? Go through the calculation in a little detail — don’t just guess, or use common sense. b) Do this heating and cooling, again and again for n times. What is the length of the rod now? Does your answer seem strange? (say ‘yes!’) Can you explain (or guess!) what has made the answer weird? You may think that this happens because we assumed that the linear heat expansion is the same in all temperatures. This is not the issue we’re looking for. Try to find another reason. [Yaser] 3) We will break dividing walls A thermally insulated cylinder, like a vacuum flask, contains a single species of ideal gas in two separate compartments, divided by an insulating wall. The pressure, volume and temperature of gas in each part are : P1, V1, T1 and P2, V2, T2. What is the final P, V and T of the system, if we remove or break the separating wall? [Amir]
P1, V1, T1
P2, V2, T2
1
4) The nonaligned axis There are special crystals which have different coefficients of linear heat expansion along two different directions, x and y. Say that the coefficients have values Le 1 and Le 2 in the directions x and y. Such crystals do exist, and are termed anisotropic. Consider a square-sided rod cut from a block of y such a crystal, with the rod’s longitudinal axis making an angle of 30° with the x axis. What is the linear heat expansion of this rod along its length? (Ignore the third dimension) Does this rod expand properly, or does it somehow bend? [Yaser]
3 0° x
5) Patent, or pshaw? Here’s an idea: venetian blinds which are painted black on one side and silver on the other. Assume that the blinds are installed between two panes of glass, with vacuum in between (so air and its convection has no effect in this question), and assume that the reflectivity of the silver side is 90%, whereas that of the black side is 10%. a) How well does it work to turn the silver side of the blinds outward when the room is already too hot, so as to reflect the light, while turning the black side outward to –2 absorb heat when the room is too cold? Take it that sunlight is 1kW m , and find the steady-state temperatures due just to thermal radiation (Stefan-Boltzmann law). b) Is there any change you can see that would improve the blinds’ function to control radiation of heat into the room? [Robin] 6) Tea? For who? Berto used an electrical heater to heat 3 kg of an unknown liquid in a container (don’t ever try heating any unknown liquids at home, though!). The heater delivers 1000 watts of power. The initial temperature of the liquid is 20 °C. As the liquid warmed up, Berto wrote down its temperature in different times. In Tables 1 & 2 below, some of the data collected by him are shown. a) Neglecting the heat capacity of the container, find the specific heat of the liquid. b) Berto turned off the electrical heater when its temperature reached 60 °C. Then, he started writing down the liquid temperature as it was cooling . You can see in the table, below, the liquid temperature for some different times. Using these data, try to find the heat capacity of the liquid to a better approximation.
2
Hint: You should consider the loss which is the heat transfer between liquid and room in your calculation in part (a). c) Sir Isaac Newton, in the 17th century, showed that objects cool at a rate proportional to the difference between their temperature and the ambient temperature, such as room temperature. Try to find the proportionality constant for (b). Table 1: Liquid temperature during heating time(min)
0
3
4
5
6
temperature (°C)
20
34.1
38.5
42.9
47.2
Table 2: Liquid temperature during cooling time(min)
0
8
13
19
26
temperature (°C)
60
50.2
45.1
39.8
35.2
[Yaser & Carolyn]
INFOBITS™ — Useful Bits of POPTOR Information Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor Specific heat of water: 4.2 kJ kg –1 K–1 –8 –2 –4 Stefan-Boltzmann constant σ = 5.7 × 10 W m K –2 –2 Solar constant = 1.353 kW m ; at surface of the earth, about 1 kW m
3
1999-2000 Physics Olympiad Preparation Program — University of Toronto — Solution Set 3: Thermodynamics 1) Clocks a’ Rockin’ First, I have to say that this was not an easy question — probably way too hard for #1. The key is to realize that the resistor is radiating energy with a temperature T, with a background temperature 20°C. Most radiating objects behave more or less like black bodies, and I will assume this to be the case here as well … Running at input power P, the computer’s resistor reaches a steady state temperature T at which the input power is balanced by thermal radiation power (from the StefanBoltzmann law). Since all objects radiate thermally (unless they’re at zero temperature), the resistor also absorbs a little bit of thermal radiation from the room-temperature surroundings at To = 20° C or 293 K. So the power balance is: P = σ{(T )4 − (T o )4 } When the resistor is "overloaded", the power (P’) goes up by a factor of I2 = 4 (i.e., P = I2R, where R is constant). The new temperature T’ will radiate more power to balance this. P ′ = 4 ⋅ P = σ{(T ′)4 − (T o )4 } thus, dividing these equations: (T ′)4 − (T o )4 =4 (T )4 − (T o )4 The question tells us that the temperature rises by 13° C, so we know: T’ – T = 13 K therefore, (T + 13)4 − (293)4 =4 (T )4 − (293)4 One way to solve this is to do it graphically, plotting the left and right side of:
{
(T + 13)4 − (293)4 = 4 (T )4 − (293)4
}
and seeing where they cross. This isn’t hard to do, with a computer. Or you can use a spreadsheet program to try a whole lot of guesses.
1
2 10
10
( T + 1 3 )4 -T o 4
function values
1 10
4 ( T4 - To 4 )
10
5 10
9
0 10
0
-5 109
-1 101 0 280.0
290.0
300.0
310.0
320.0
Temperature The solution is T = 297.8 K, or normally about 5° C above room temperature. With the overclocking, the temperature is therefore about 18° C above room temperature, or 38° C. If we wish to cool the system down to ∼20°C (room temperature), and our cooling system is the same size as the resistor, the cooling system all by itself has to be at Tc ~ 0°C, for then 0 + 38 ≈ 20o C 2 so the cooling system is indeed at 0° C when the resistor (actually the processor) is off. In case you are curious, this is a true story — you may even verify it yourself (I would suggest using someone else's computer, though …).
2. Literally linear a) Let T2 − T1 = ∆T > 0 (1st) heating: 2
L1 = L(1 + λ∆T) (2nd ) cooling: L 2 = [L(1 + λ∆T)][1 − λ∆T] = L(1 − λ2 ∆T 2 ) Every cycle (heating and then cooling the length of the rod is multiplied by (1 – λ 2∆T2) ∴ after n cycles L2n = L(1 – λ 2∆ T2)n (this can be proved by induction, if you like) λ∆T better be less than 1 (otherwise the length of the rod becomes negative), and hence:
lim (L 2n ) = 0
n→∞
This clearly isn't right - whatever happened to conservation of mass??? The explanation is that this is actually a differential statement ( dL = L ⋅ λ ⋅ dT ) valid only for tiny ∆L and ∆T. (otherwise "L" in the equation is not actually constant). Because of this, only 1st order terms (i.e., ∆L and ∆T) are important — 2nd order (∆L2, ∆T2) should be ignored. Hence, 1 – λ2∆T2 ≈ 1 and there is no problem. If λ were constant, one could write: dL = Lλ dT and solve this: dL = λ dT L dL ∫ L = ∫ λdT ln(L) = λ (∆T) + C λ∆T
L = Lo e
(you can check for yourselves that the length of the rod is indeed conserved after a cycle when using this formula) Expanding this yields: L ≈ Lo (1 + λ∆T + (λ∆T)2 + …) and ignoring 2nd order terms and higher gives: L ≈ Lo (1 + λ∆T) Unfortunately, λ varies with T (among other things), and this is not really true …
3
3) We will break dividing walls Clearly, V = V1 + V2
(1) ("conservation of volume")
and
(2) ("conservation of mass")
n = n1 + n2
The vessel is insulated, and hence energy has to be conserved. E = αnkT = E1 + E2 = αn1kT1 + αn2kT2 ⇒ nT = n 1T1 + n2T2
(3)
since PV = nRT, we get from (3): PV = P1V1 + P 2V2 and P =
P1V1 + P2 V2 V1 + V2
using 2 yields: PV P1V1 P2 V2 = + T T1 T2 (P1V1 + P2 V2 ) =T P1V1 P2 V2 + T1 T2
4) The nonaligned axis The x and y directions expand independently.
4
x ′ = x(1 + Le1∆T) y ′ = y(1 + Le 2 ∆T)
y
x = L cos(30°) y = L sin(30°) Assuming straight:
x
the
rod
expands DL = Sqrt ( Dx^2 + Dy^2) Dy= Y*H2*(T2-T1)
L’ – L = Lα∆T
Dx = X*H1*(T2-T1)
and (L’ – L)2 = (x’ – x)2 + (y’ – y)2 ∴ (L’ – L)2 = L 2 (Le12 cos2(30) + Le22 sin 2(30)) (∆T) And hence α2 = Le12 cos2(30) + Le22 sin 2(30)
L
Y= LSin 30 30
X = LCos 30
In fact, from the diagram we can see that the expanded part of the rod is at a different angle to the horizontal, and thus the rod must bend to compensate.
Dy Tan Theta = Dy /Dx = Tan 30 * (H2 / H1) Dx
5) Patent, or pshaw? Here's the situation: In steady state, nothing is either heating up or cooling down. So the net power being absorbed as radiated heat must equal the power being emitted as radiated heat (we’re ignoring convection and conduction here). The blinds absorb the sunlight, certainly. More subtly, though, there is radiated energy from the roomtemperature room inside, and also radiated energy from outdoors — because neither has zero temperature! So all of these go into the absorbed power, keeping track of what the absorption coefficient is on each face.
5
Let the ‘outside reflectivity’ be written as R, regardless of whether it’s the silver or black side. Likewise, let the ‘inside reflectivity’ be r, regardless. The absorption is then (1–R) or (1–r). outside reflectivity = R outside temperature = Tout inside reflectivity = r inside temperature = Tin –2 incident sun power = 1000 W cm area of each side of blinds = A Absorbed power = absorbed sunlight + absorbed outdoor radiation + absorbed indoor radiation –2
4
4
= 1000 (1–R) W cm • A + (1–R) σ Tout • A + (1–r) σ Tin • A Radiated power 4
4
4
4
= eout σ Tblinds • A + ein σ Tblinds • A = (1–R) σ Tblinds • A + (1–r) σ Tblinds • A where e is the emissivity of the object, and equals the absorption a ( = 1– R or 1– r ). Both sides of the blind radiate, of course. These two powers must balance: 1000 (1–R) W cm
–2
4
4
• A + (1–R) σ Tout • A + (1–r) σ Tin • A 4
4
= (1–R) σ Tblinds • A + (1–r) σ Tblinds • A and we solve for the temperature of the blinds (when they’re hotter, they’re sending more heat to the room, of course). Tblinds
4 1000 (1 − R)/ σ + (1 − R) Tout + (1 − r ) Tin4 = (2 - R - r)
1/ 4
Then when we have the black side outward R = 0.1, and r = 0.9; when the silver side is outward, R = 0.9, and r = 0.1. So we get two different temperatures for the blinds. If we take the indoor temperature to be 20° C, and outdoors to be 0° C, we find: silver side out: Tblinds = 307.5 K or 34.4° C black side out: Tblinds = 383.0 K or 109.9° C The ratio of absolute temperatures is 1.25 — 25% hotter (Kelvin) with black out. Interestingly, if we ignore the room and outside temperature, we get the answers we’d have if both were zero — the answers we’d have basically in deep space! silver side out: Tblinds = 204.7 K or –68.5° C black side out: Tblinds = 354.5 K or 81.3° C
6
The ratio of absolute temperatures is 1.7 — 70% hotter (Kelvin) with black out. However, the temperature is actually much lower in this case, just because the blinds are sitting in ‘deep space at zero degrees’. This isn’t the whole story, though! Even if the blinds are hotter with the black side outwards, how efficient are the blinds at radiating into the room from the silver side of them? 2
silver side out (black side in): power/m into room: = (1–0.1) σTblinds = 0.9• 5.7 × 10
–8
Wm
–2
K
–4
4
–2
4
–2
• (307.5 K) = 459 W m
black side out (silver side in): Tblinds = 383.0 K or 109.9° C = (1–0.9) σTblinds = 0.1• 5.7 × 10
–8
Wm
–2
K
–4
• (383.0 K) = 122 W m
Though the temperature is higher, the temperature factor of 1.25 leads to an increase of 4 2.44 in T . The emissivity factor, however, is 1/9th — more important than the temperature. So, really, the silver side out leads to more power radiated into the room! This offers plenty of control over the temperature – so the patent is indeed working, just not the way I thought it might! b) To make them work even better: you could see what happens when you make the black even blacker and the silver more reflective; you could use another set of blinds set up parallel, too — with both silver sides facing outwards, the effect would be even greater (can you calculate this? What then do you do when you put the outside black blinds outwards? Do you do the same with the inside blinds or do you leave them open, neither side outwards?) You could also use a fan, so you could convectively cool the blinds when they are black side out, and put the heat into the room more efficiently. Lots of answers are possible… [Robin]
6) Tea? For who? The best way to analyze data is to plot it, whenever possible. a) Assume no power loss. Then, dQ dT = 1000 = mc dt dt Solving, 1000t = mc(T (t ) − 20) . Plotting T(t)-20 versus t (the time in seconds) should yield a straight line with slope 1000/(mc). Note that the line has to pass through (0, 0), as when t=0 we know T(t) is exactly 20°C! Power =
Many commercial packages (e.g., Excel) can fit data, but they will not give you an errorestimate on the slope. Instead, it’s best to plot the data yourself, and then fit the 7
steepest and shallowest lines that still fit the data. The best fit is then the average of the two slopes, and the error is the slopes of the steepest line minus the average slope. I’ve done this in Excel; see below
30
y=(0.077 +/- 0.001) t
25 20 15
Series1
10
Series2
5
Series3
0 0
100
200
300
400
1000 = (0.077 ± 0.001) and hence c = ( 4330 ± 60) J /(kg oC ) . We have used mc ∆c 0.001 the fact that = where ∆c is the error in c. Note that all errors are rounded off to c 0.077 1 significant figure, and all results are rounded off to the same decimal place as their errors.
We thus have
b) The system clearly loses power. This means that our value for c is too high, as we assumed that the liquid absorbed all 1000 W of power, when it in fact did not. Let’s assume that the system loses power at a steady rate. Our equations become (1000 − L)t = mc(T (t) − 20) and − Lt = mc(T ′(t) − 60) , where T’(t) is the temperature during cooling. Plotting this data shows that it is not in fact a straight line (so L is not a constant), but we won’t worry about this here. Plotting and fitting the data as in a) yields 0 -5
0
500
1000
1500
Series1
-10 -15
Linear (Series1)
-20 -25 -30
2000
y = -0 .0171x 2 R = 0.9757
8
(I have only plotted the average slope above). This yields a value c of (3600 ± 100) J /(kg oC) ∆T = − k(T ′(t) − 20) . This may be solved, but calculus is required. The ∆t easiest way to proceed is probably to plug in all values for T’(t), find a bunch of k’s (4,
c) We have that
in fact) and then average them. Doing this gives k ≈ 5 ⋅ 10 −4 1 / s . This value could be used again to obtain a revised value for c; it will not be much different from the current value, however. For completeness, let us solve the above equation. dT dx dx = and our equation reads = − kx . We also know that Let x = T − 20 ; hence, dt dt dt x = 40° C at t = 0. Rewriting the above and integrating both sides gives
dx = − k ∫ dt or ln( x ) = − kt + C x
To make C fit our initial condition we finally arrive at T − 20 = 40 exp( − kt ) This can be fitted e.g. by taking ln() of both sides – plotting ln(T-20) versus t yields a straight line with slope 0.0006 1/s… Since the Carnot refrigerator is the most efficient, therefore we conclude that this value is a minimum for the required power. [Peter & Yaser]
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1999-2000 Physics Olympiad Preparation Program — University of Toronto — Problem Set 4: Optics and Waves Due February 25, 2000 (revised date) 1) Bob-bob-bobbin’ along A terrifically important concept in waves and oscillations is that of the harmonic oscillator. We want to use some basic principles of the harmonic oscillator in later POPTOR questions, so have a look here at the basics. The ‘generic’ harmonic oscillator that you’ll see in university, and forever after that on into graduate school and beyond, is a mass resting on a frictionless table and attached to a spring. A property of springs, for small stretches or compressions, is that the force exerted by the spring is proportional to the amount of stretch. This is called a linear restoring force and the force is represented F = –k x, because the force tends always to restore the position of the mass to its original neutral position (unstretched spring). a) Write the force equation for such a mass-on-a-spring (inertial force and restoring force). Since acceleration is the time-derivative of velocity, and velocity in turn the time-derivative of position, this actually is a ‘second-order linear differential equation’. b) You can do pretty simple derivatives. Suppose that sin(ωt) is a solution of this particular differential equation. For that to work, what must be necessary? c) So, what is the rate of oscillation of a 100g mass on a spring that has a spring –1 constant of 80 N m ? [Robin] 2) Keeping up an image The distance between a screen and a light source lined up on a table is 120 cm. Moving a lens between them, sharp images can be obtained at two different positions, producing two different images. The ratio of sizes of these two images is 1:9. a) What is the focal length of the lens? b) Which image is the brighter one? Determine the ratio of the brightness of these two images. [Gnädig/Honyek] 3) Shifty radar Officer Smith and Officer Wesson are on patrol monitoring speeding along Hwy 401. They are using a hand-held radar gun which is set to detect the speed of approaching 1
cars, sending out waves at a frequency of 1000 MHz. After several hours of speed monitoring Wesson turns to Smith and asks, "How does this thing work anyway?" To which Smith replies, "Well now, are you familiar with the Doppler effect?" a) Finish Smith's explanation. Suddenly the two officers are distracted from the conversation, coffee and donuts by the sounds of tires screeching and horns honking. They look up to see a milk truck racing down the 401. Smith gives Wesson the eye and Wesson fires the radar gun. b) If the observed frequency difference is 330 Hz, how fast is the milk truck going? [Carrie] 4) To air is human A cylinder of radius r contains n moles of a monatomic ideal gas. A tight-fitting piston of total mass M is fitted into the cylinder, sealing it. The room temperature is T. Consider that the cylinder is really well thermally insulated, so all changes are ‘adiabatic’. a) What is the new resting height h (height of the piston from the base of cylinder)? Now, we attach or add a small mass m to the piston at rest. As you may guess, it'll start to oscillate. b) If the piston’s oscillations damp out (e.g.., due to friction) what would the final h be? Suppose the motion is not damped, and that height from (b) is the equilibrium height of piston and the piston will oscillate around that equilibrium point. c) Each oscillator can be modelled by a system of mass and spring. What is the corresponding mass and spring coefficient for this example? d) Find the minimum amount of h in this oscillation.
[Amir]
5) Bending the truth Here’s an actual test used to find the index of refraction of a material: Consider a piece of paper with a drawing of a group of parallel lines. Consider also a polished cylindrical rod made of some unknown glass (for which we want to find the index of refraction). Set the rod down on the paper, at some angle θ such that 0 < θ <π/2.
?
?
?
?
?
?
?
?
?
?
a) How will the lines appear as viewed through the glass rod? Make a sketch. b) Find the index of refraction of the glass, using details from what you see in part (a). [Amir] 2
6) Cool abrrrrations! Consider a beam of light incident on a convex lens. We expect all the parallel rays to converge at a focal point. But, in reality, the visible beam of light consists of different wavelengths from violet to red. Since the index of refraction of the lens depends on the wavelength we are faced with a defect called chromatic aberration — different wavelengths will focus at different places (see figure at right).
Chromatic Abberation
NOT TO SCALE white Light
Red light
blue light
a) Using the lens-maker’s formula, find a formula that gives the change in the focal distance of a lens due to a small change in the index of refraction. b) One way to minimize the chromatic aberration is to put two lenses of different materials together, to make one new lens. Suppose one part is made of Light Crown Glass and the other from Heavy Crown Glass. Below is a table showing the index of refraction vs. wavelength for these two different kinds of glass. We are looking for a lens with f = 10 cm. What should be the focal lengths of the two Light and Heavy Crown glass lenses so as to minimize the chromatic aberration? Index of refraction of two kinds of glass in terms of wavelength Wavelength (nanometer)
488
520
568
632
694
890
Light Crown Glass
1.4877
1.4859
1.4836
1.4813
1.4796
1.4758
Heavy Crown Glass
1.5793
1.5766
1.5735
1.5704
1.5681
1.5634
Hint : Lens-maker formula 1/f = ( n–1) * ( 1/R1 + 1/R2) f : focal length of the lens n : index of refraction of the lens material R1 & R2 : The radii of curvature of the two sides of the lens
[Yaser]
Remember to check the POPTOR web-page for hints and any possible corrections! www.physics.utoronto.ca/~poptor
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1999-2000 Physics Olympiad Preparation Program — University of Toronto — Solution Set 4: Optics and Waves 1) Bob-bob-bobbin’ along This question was a sneaky way to get you solve a second order differential equation quite painlessly. a) The forces involved included the restoring force, –kx and the inertial force, ma, d2x where a = 2 . These two balance each other (the net force is zero). Thus the force dt equation for our system is, d2x m 2 = − kx dt Which is a second-order ordinary differential equation. b) We are given x = sin(ωt) as a solution to our force equation. Taking the first time derivative we obtain, dx = ω cos (ωt) dt Differentiating again with respect to time yields, d2x = − ω 2 sin (ωt) 2 dt Plugging this expression for the acceleration into our force equation, along with x=sin(ωt) yields
(
)
m −ω 2 sin (ωt) = − kx = − k sin (ωt) 2 Therefore, solving for ω we get ω =
k . m
So this oscillation x = sin(ωt) isn’t always a solution — only for a certain ω will there be a solution. So we find that the frequency of oscillation depends this way on both the spring constant of the spring (its strength) and the mass of the bob attached. c) Putting the values k = 80 N/m and m = 100 g into our equation for ω, gives us a value ω=
80 = 28 /sec, or 4.5 Hz 0.1
[Carrie]
1
2) Keeping up an image Since the object distance u and the image distance v can be exchanged in the lens law 2 and their ratio shows the magnification to be (v/u) = 9 (or 1/9), then ν / u = 3 (or 1 / 3) . Thus, the object distance is 30 cm (or 90 cm) and the new image distance is 90 cm (or 30 cm). The focal length can be calculated from the lens law: ƒ = 22.5 cm. If the same amount of light passed through the lens in both cases then the 9 times smaller image would be 81 times brighter, as the smaller image occupies a surface 81 times smaller on the screen than the larger one. However, the lens placed at a greater distance receives only one ninth of the light reaching the nearby lens, therefore the small image is only nine times brighter that the large one. It can be shown in general that in such cases the small image is as many times brighter than the large one as the large one is larger. [Gnädig/Honyek]
3) Shifty radar a) The radar gun uses the Doppler shift to detect the speed of cars. For light the correct doppler effect involves relativistic considerations. For cars, which have relatively small source velocities, we skip those. Thus we have for the officers viewing at rest, fL ≈ (1 + v/c)f S Where fL is the frequency perceived by the officers (or listeners at rest), v is the velocity of the source, c the speed of light, and fS the source frequency (1000 MHz in this case). If waves are reflected from a moving car the reflected frequency, fR, is that of the a source moving with twice the car velocity. So, fR ≈ (1 + 2v/c)f S. The waves which are reflected from the moving cars are beat against the transmitted waves, yielding a frequency difference given by, F = fR - fS ≈ 2vfS/c Solving for the velocity of the car we get, v ≈ Fc/2fS. b) Plugging in the numbers to our velocity equation we get, v = [(330 Hz)(3 × 108 m/s) / (2 ( 109 Hz))](1 km/1000m)(3600 s /1 hr) = 178 km /hr !!!! Book him, Dano!
[Carrie]
2
4) To air is human a) We know that in an adiabatic process PVγ is constant. P0 =
nRT0 nRT0 = V0 Ah 0
P1 = P0 +
Mg A
γ
γ
P1V1 = P0 V0
γ
γ
⇒ P1 h1 = P0 h 0
P ⇒ h1 = h 0 0 P1
1/ γ
P0 A = h0 P0 A + M g
nRT0 P0 A h1 = P0 A P0 A + Mg
1/ γ
1/ γ
b) Exactly the same as part (a) you can see that: nRT0 P0 A h2 = P0 A P0 A + (M + m)g P2 = P0 +
1/ γ
(M + m)g A
c) The force acting in a mass-spring system is –k ∆x. Consider that we change the equilibrium state by moving the piston x away from equilibrium. The force acting on the piston to push it back will be: F = P0 A – PA + (M + m) g Since P(h2 + x)γ = P2h2γ = P1h1γ = P0h0γ P = P2
γ
h2
(h 2 + x) γ
and since x is quite small compared to h2, x P = P2 1 + h2
−γ
γ x ≈ P2 1 − h2
γ x ⇒ F = P0 A + (M + m) g − P2 A 1 − h2 ⇒ F = γ P2
A x= kx h2
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the spring coefficient of the system will be (M + m).
γ P2 A and the corresponding mass will be h2
d) Since the harmonic oscillator goes up and down with the same distance, and the upper level of this oscillator is (h 2 – h1) above the equilibrium, the lower level will be h2 – h1 under the equilibrium. h min = h1 − (h 2 − h1 ) = 2h1 − h 2
5) Bending the truth a) A rod is a cylindrical lens which basically magnifies in one direction only, and leaves the other direction unchanged, so the line will be magnified in the direction perpendicular to the rod, and won't change in the direction of parallel to the rod. b) Since we are looking far from the rod, the outgoing beams (going to your eyes) are parallel. The point A is emitting light in all directions, but we are concerned about the beam that will go out vertically. The beams are sketched at right. The point A will be seen as A’ and we can AH′ find the ratio of by trigonometry. First AH we assume that AH is approximately equal to the BH' line: HA = BH′ = R sin(θ − (ϕ − θ)) = R sin(2θ − ϕ) HA ′ = R sin ϕ Magnification =
HA ′ sin ϕ = HA sin(2θ − ϕ)
We neglect rays which pass either far from the centre, or at large angles from the axis sin θ ≅ θ . (the paraxial approximation). In this case, θ and ϕ are small and sin ϕ ≅ ϕ sin ϕ= n sin θ
ϕ = nθ
⇒ Magnification =
nθ n = 2θ − nθ 2−n 4
In practice: The rod is magnifying in the direction of the line perpendicular to the axis of rod. The point A will be seen as A′ . HA ′ = m (Magnification) HA HA ′ = OH tan(α + β) HA = OH tan(α)
HA ′ tan(α + β) n = = tan(α) 2−n HA
⇒ n=
2 tan(α + β) tan(α) + tan(α + β)
Where α is the angle between rod and parallel lines and β is the angle between bent lines and parallel lines. One can find n by measuring α and β. [Amir] 6) Cool abrrrrations! a)
1 1 1 = (n − 1) • + f R1 R2 ∆
1 1 1 = ∆(n) • + f R1 R2 =
1 ∆(n) • f (n − 1)
b) When two different lenses are placed together, their inverse focal lengths add up: 1 1 1 = + f f1 f2 We want the aberration to be minimum. So ∆ =>
1 =0 f
1 ∆(n1 ) 1 ∆(n2 ) + • =0 • f1 n1 − 1 f2 n2 − 1
Consider two wavelength 488 nm and 890 nm (nanometers). ∆(n1 ) = 1.4877 − 1.4758 = 0.0119 ∆(n2 ) = 1.5793 − 1.5634 = 0.0159 Now we have two equations, two unknowns 5
1 1 1 1 + = = f1 f2 f 10 1 0.0119 1 0.0159 • + • =0 f1 (1.48 - 1) f2 (1.57 - 1) Therefore f1 = 2.5 cm f2 = – 3.3 cm
[Yaser]
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1999-2000 Physics Olympiad Preparation Program — University of Toronto — Problem Set 5: Electricity and Magnetism Due March 17, 2000 (revised date) 1) Vile away the hours To complete his giant, super-evil-destructor ray gun the great, evil scientist Dr. Vile requires a copper sphere with a charge of exactly +3 µC. He has a copper sphere of mass 2 g. a) What fraction of the electrons must be removed from his copper sphere to give it a charge of 3 µC ? Having removed the correct number of electrons Dr. Vile places the 3 µC copper sphere near two other spheres of charges +7 µC and +5 µC. The three spheres are arranged in an equilateral triangle formation, with sides of 9 cm. b) What is the resultant force (and its direction) on the 3 µC copper sphere? [Carrie] 2) Capacity for thought Two parallel-plate capacitors with equal capacitance C are charged to voltage V. The distance between the plates is d and their area is A. There is no dielectric between the plates. Connect the two capacitors by two wires so as to make a circuit. Obviously, there is no current in the circuit. Now, imagine that the plates of one capacitor move apart with velocity v and the plates of the other capacitor move toward each other with the same velocity v. At this time, there must be some current in the circuit. Find its value. [Yaser] 3) Dielectric City Consider two parallel-plate capacitors C1 and C2 which carry corresponding charges Q1 and Q2. a) What is the total energy of the two capacitors, taken as a system? Now, we connect the two positive plates to each other and two negative plates to each other. b) What will be the new energy of the system? 1
c) What happened to the energy of the system when the plates were connected? d) What is the energy difference between an empty capacitor and a capacitor full of water, if each carry the same charge? e) What will happen if I put a charged-up capacitor on the surface of water, as shown in the figure? Why? How high will the column of water be when it rises into the space between capacitor plates?..... [Amir]
4) Super loopy Two identical superconducting circular loops are positioned coaxially (a line passes through both their centres, perpendicular to each) and far apart. The loops have self-inductance L. Equal currents I are passing through each of these loops in the same direction. Now, bring the loops together. a) What is the current in each loop? b) Find the difference between initial and final energy of the system? [Yaser] 5) Keeping it together... A narrow beam of electrons, of radius r and all moving at the same velocity v much less than speed of light (c), produces a charge current I. We assume that the beam has cylindrical symmetry. a) What is the electric field at the edge of the beam? b) What is the magnetic field at the edge of the beam? c) What is the radial (diverging) velocity of electrons at the border of the beam, after the beam has traveled a longitudinal distance 100 times r? d) What is the diverging angle for I = 1 mA and r = 1 cm and v = 1000 m/s? [BONUS: Since the B field depends on electron speed, can you find the speed for which the force due to B-field entirely cancels the force due to E-field? Hint: think ‘frame of reference’] [Amir]
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6) A feel for fields a) Prove that it’s impossible to have a magnetic field which increases along the z-axis but which has only a z-component. This field must have a radial component! By choosing a short cylinder, show that for a cylindrically symmetric B-field: Br =
r dB 2 dz
A loop with electrical resistance R is falling in a cylindrically symmetric magnetic field. The centre of the loop as it falls is exactly aligned with the axis of the cylindrically symmetric field, and perpendicular to it. Along this axis, the field z-component is changing as dBz/dz. The radius of the loop is r and its mass is m. b) Write the equation of motion governing the fall of the loop, and sketch the graph of its velocity vs. time. c) What is the terminal velocity of the loop? [Yaser] [Hint: you may want to use the fact that magnetic field lines are always closed loops — they cannot start or end at a point, as E-fields do...] INFOBITS™ — Useful Bits of POPTOR Information Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor
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1999-2000 Physics Olympiad Preparation Program — University of Toronto — Solution Set 5: Electricity and Magnetism 1) Vile away the hours 22
i) The 2.0 g sphere contains 1.99 × 10 atoms. The charge on the nucleus of each atom 22 23 is 29 e. Thus the sphere contains (1.99 × 10 )•29 = 5.77 × 10 electrons. One electron –19 –6 –19 has a charge of 1.6 × 10 C. The electrons removed = (3 × 10 C)/(1.6 × 10 C per 13 electron) = 1.875 × 10 electrons. So the fraction of electrons removed is (1.875 × 10
13
23
–11
) / ( 5.77 × 10 ) = 3.25 × 10
b) The figure at right illustrates the configuration of the charged spheres with q1 = 7 µC, q2 = 5 µC and q3 = 3 µC. Let the positive x direction be as shown, and the y direction perpendicular to it, towards the upper left. The net force on q3 is the vector sum of the repulsive forces F 1 and F2 . The magnitude of F1 is given by: F1 =
k q1 q3 r2
9 × 10 9 N m2 C −2 ) • 7 × 10 −6 C • 3 × 10 −6 C ( =
(0.09 m)2
= 23.3 N Similarly F2 is: F2 =
k q2 q3 r2
9 × 10 9 N m2 C −2 ) • 5 × 10 −6 C • 3 × 10 −6 C ( =
(0.09 m)2
= 16.7 N r r r The vector sum F = F1 + F2 has components along x and along y. In the x direction: Fx = F1x + F2 x = 23.3 N + 16.7 N • cos(60°) = 31.7 N and in the y direction, 1
Fy = F1y + F2 y = 0 N + 16.7 N • sin(60°) = 14.5 N Thus the magnitude of the resultant force is:
(31.7 N )2 + (14.5 N )2
F=
= 34.9 N The direction is at an angle arctan(14.5/31.7) = 25° from the x axis.
[Carrie]
2) Capacity for thought The charge of each capacitor is Q=CV. When the capacitors are connected to each other and the separation between the plates change, the total charge is still conserved so 2Q=q1+q2 . Since the capacitors are parallel to each other, the voltage across them is q q equal to V = 1 = 2 . We know that the capacitance is inversely proportional to the C1 C2 separation of the plates. Therefore C1d1 = C2 d2 . Since the plates are moving together or apart by speed v for the two capacitors, we have d1 = d + vt Therefore
d2 = d − vt
C1 d2 d − vt = = C2 d1 d + vt
On the other hand, q1 =
C1 d − vt q2 = q2 . C2 d + vt
Substitute the value of q2 = 2Q − q1 in the above formula we have q1 = Q
d − vt d
and
q1 = Q
d + vt d
The current in the circuit is the rate of change of charge, which is equal to I=
dq2 dq Qv =− 1 = . dt dt d
[Yaser]
3) Dielectric City a) The energy stored in a capacitor with capacitance C and charge Q is TOTAL energy of a system consist of two capacitors is: E1 =
Q2 . So the 2C
Q12 Q2 + 2 2C1 2C2
2
b) At first, I’ll find the equivalent capacitor and the total charge on it. Those two capacitors are connected as parallel capacitors and the total charge on the right-hand-side plates is conserved (there is no external current) we have q 1 + q2 = Q1 + Q 2 (or similarly for the left plates: (–q1 ) + (–q2 ) = (–Q 1 ) + (–Q 2 )) Cequivalent = C1 + C2 Qtotal = Q1 + Q2 Now I can find the new energy of the system of one equivalent capacitor. (Q1 + Q2 ) 2 Qtotal 2 E2 = = 2Cequivalent 2(C1 + C2 ) c) One may find the difference between E2 and E1 as: Q12 Q22 (Q1 + Q2 ) 2 E1 − E2 = + − 2C1 2C2 2(C1 + C2 ) =
C2 (C1 + C2 )Q12 + C1(C1 + C2 )Q22 − C1C2 (Q12 + Q22 + 2Q1Q2 ) 2C1C2 (C1 + C2 )
C22Q12 + C12Q22 − 2(C1Q2 )(C2Q1 ) = 2C1C2 (C1 + C2 ) =
(C2Q1 − C1Q2 ) 2 >0 2C1C2 (C1 + C2 )
which is always equal or larger than zero. The energy loss is because of the emission of accelerated particles due to connection of two wires with different voltages. When you connect the first two wires (say negative plates) nothing will happen and you will just have one ground to measure voltage on each part of the two circuits. You can then calculate the voltage difference of two positive plates. Connecting the other two wires will create a large electric field between two wires (when they are separated by a small distance) and then the electrons will be accelerated. If we could put a resistor between two capacitors, this energy could have been changed
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into heat. Often, the energy loss can be found in the form of electromagnetic waves in visual range (you’ll see it as a spark!). d) Capacitance of a capacitor full of water is ε times capacitance of an empty capacitor. A (Recall the formula for capacitance C = ε ε o , where ε is susceptibility of water.) c Energy of full and empty capacitor can be found by using that formula: Eempty = E full =
Q2 2Cempty
Q2 Q2 = 2C full 2 ε Cempty
and the energy difference is: ∆E = Eempty − E full Cf = ε εo
Q2 1 = ( 1− ) Ce ε
A1 A ; Ce = ε ε o 2 d d
Ecapacitor =
(total charge)2 2(equivalent capacitance)
in which I've assumed that an equivalent capacitor have the total charge (initial charge). Q2
⇒E= 2
ε0 ( ε A1 + A2 ) d
Since A1 = wh and A2 = w(H-h) we can write: Q 2d Ec = 2 ε 0 w( ε h + H − h) On the other hand, gravitational energy of water is: Eg = mg ×
h h ρw d g 2 = ρgwhd × = h 2 2 2
And total energy of the system is: Etotal = Ec + Eg =
ρw d g 2 Q 2d + h 2 ε o w(H + ( ε − 1)h) 2
Since the energy tends to be minimized in physics world (and defines the equilibrium state):
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−Q 2d(ε − 1) dEtotal =0⇒ + ρw d g h = 0 dh 2ε o w(H + (ε − 1)h)2 ⇒ Q 2 (ε − 1) = 2ε oρw 2 gh(H + (ε − 1)h)2 3
2
Q2 2 h 1 h h ⇒ + + − =0 H ε − 1 H (ε − 1)2 H 2ε o (ε − 1) ρ w 2 g H 3 *Imaginary-valued solutions have no physical meaning, here. *Answers with
h > 1 mean that the capacitor will be completely filled. H
h < 0 mean that the water won't rise at all. (in our equation this will H h could be negative). not happen, but if one could find a medium with ε <1, H
The condition that capacitor is completely filled with water is: 1+
2 1 Q2 + − ≥0 ε − 1 (ε − 1)2 2ε 0 (ε − 1)ρw 2 gH 3
[Amir]
4) Super loopy The main point of this question is that the loops are superconductors. Superconductors have no resistance. Therefore, when the loops are brought together, the magnetic flux, which passes through each of them, must not change. Otherwise there would be a voltage across each loop, which would cause an infinite current! To avoid infinite current, the currents in the loops change so as to keep the flux fixed. When the loops are far apart the flux passing through each loop is equal to Φ i = LIi which is because of the self-inductance of each loop. When the two loops are brought close together, the mutual inductance is equal to the self-inductance. The flux passing through each loop is combined in equal parts. One is from the self-inductance and the other is from the mutual inductance. Therefore Φ f = LI f + LI f = 2LI f To avoid infinite current, Φi = Φ f ⇒ I f =
Ii 2
The energy of an inductance is equal to
5
U=
1 2 LI 2
Therefore the difference in the energies of the system is equal to ∆U = U f − U i =
1 2 1 2 1 2 1 2 3 LI f + LI f − LIi − LIi = − LIi2 2 2 2 2 4
It can be shown that this is equal to the work needed to bring the loops together. Here the work is negative because the loops attract each other.. [Yaser] 5) Keeping it together... i) I'll find the density of electrons inside the beam. I = ρv (πr 2 ) ⇒ ρ =
−I π r 2v
The electric field at the edge of beam can be found by using Gauss’s law. E=
ρ ⋅ π r2 ρr λ −I ^ = = = r 2πrε o 2πε o r 2ε o 2π ε o r v
Using Ampere's law, I can find B: B=
µo I 2πr
The radial force on electrons on the edge is F = − e(E + v × B) µ I v I ⇒ F = e − o ^ r 2πr 2πε o r v ⇒F=
Ie (1 − µ o ε o v 2 ) 2πε o r v
as you may know, the quantity µ o ε o = ⇒F=
1 where c is the speed of light. c2
Ie Ie v2 (1 − 2 ) ≈ 2πr ε o v 2πr ε o v c
The change in radial momentum can be found by using
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∆Pr = Fr ∆t =
Ie 100 r • 2πε o r v v
∆Pr =
100 I e 2πε o v 2
∆vr =
100 I e ∆Pr me 2πε o me v 2
in which I've assumed that Fr , r and v won't change too much during this process. One may find the ratio: 100 • 10 −3 • 1.6 × 10 −19 ∆vr 100 I e = = v 2πε o me v 3 2π • 8.85 × 10 −12 • 9.1 × 10 −31 • (107 ) 3 ∆v tanϑ = r ≈ 0.3 v ⇒ ϑ = 0.3 radians ≈ 17° (The speed in the problem set was wrongly written as 1000 m/s. It is 10,000 km/s. Sorry!). BONUS : In part (c) we found that the force acting on electrons at the border is F=
Ie 2πε o v r
v2 1 − c2
putting this force to be zero will result: v2 =0 ⇒ v=c c2
1−
[Amir]
6) A feel for fields a) Try to draw field lines, which have to be denser as the z changes. The field lines must bend, which shows the presence of a radial component. Use Gauss’s Law for this part. The magnetic field lines that enter a cylinder must go out so as to make a closed loop. B z (z+∆z) r
∆z
B z (z)
Br
The flux from the top and bottom surfaces is Φ1 = πr 2 (B( z + ∆z) − B( z) = πr 2
dBz ∆z dz
The flux from the sides is 7
Φ 2 = 2πr∆zBr For the total flux to be zero we should have Φ1 = Φ 2 ⇒ Br =
r dBz 2 dz
b) There are two forces acting on the loop, gravitational force and the magnetic force. The rate of change of the magnetic flux respect to time is dΦ πr 2 dBz 2 dBz = πr = dt dt v dz where v is the velocity of the loop falling down. For the current in the loop we have V dΦ/ dt πr 2 v dBz i= = = R R R dt The force exerted by the magnetic field, which is due to radial component of the magnetic field and upward, is equal to F = i l Br = i (2πr) Br Substitute the value of current and magnetic field we already derived we have F=
2
π 2 r 4 dBz v R dz
Therefore the equation of the motion of the loop is 2
d2v π 2 r 4 dBz m 2 = mg − v R dz dt The magnetic force is negative because of Lenz’s law. The magnetic force is proportional to v and negative, causes damping. Initially, the velocity increases but reaches a terminal velocity which is when the acceleration is zero.
v vf
t
mgR d2v = 0 ⇒ vf = 2 4 2 π r (dBz / dz)2 dt The velocity of the loop versus time is plotted. [Yaser]
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1999-2000 Physics Olympiad Preparation Program — University of Toronto — Problem Set 6: AC Circuits and Electronics Due April 3, 2000 1) Ohm my! You are given the circuit shown, which starts with a battery of voltage V and a resistor R across the battery. If I add another resistor r in series with R, what should be the value of r so that the power dissipated across it should be a minimum? Similarly, what should be the value if r is added in parallel with R? [Robin]
R
r
2) A current affair There are two circuits shown below, with the bottom part of the diagram held at ground. If the input is a current turned on, then off, as graphed, what is the output voltage vs. time?
R
I
I
Vo
C t I C
R
Vo
[Yaser] 3) Pandora’s box o’ electronics Consider a black box with 3 terminals (A, B, C). Only batteries & resistors are present inside. Given these measurements below (taken with a multimeter, between pairs of 1
Voltage: A A
Resistance: B
C
0V
1.6 V
A
1.6 V
B
B
A
C
B
C
99 Ohms
nonsense (< 0) nonsense (< 0)
C
Current: A A
B
C
0A
0.015 A
B C
0.031 A
terminals), what is inside? Draw a circuit diagram, and specify the values of components. [HINT: the above values, since they are experimental, have an error-bar in the last digit] [HINT 2: measuring resistance across a battery will result in an error] [Peter]
4) Only logical Logic gates are used to electronically perform Boolean or Logical operations on input signals. The simplest is the NOT gate which is symbolised by the diagram at left. The response of the NOT A C gate can be summarized in the following 1 0 truth table, at right. So if the input signal (A) is true or 1 the NOT gate response (C) is 0 1 false or 0 (and vice versa). Another type of logic gate is the AND gate with symbol shown at left and truth table at right. Thus the output C is true only when both A and B are true, otherwise C is false.
A
B
C
1
1
1
1
0
0
0
1
0
0
0
0
The OR gate has symbol shown at left. a) What do you suppose is the truth table for this gate? b) Build an OR gate using NOT and AND gates. c) Build an AND gate from NOT and OR gates. BONUS d) Build an EXCLUSIVE OR gate (where C is true if one of A or B is true but not both), using AND, OR and NOT gates. … [Carrie]
2
5) Net worth i) Consider an infinite square array of resistors r, sketched conceptually at left. Consider any 2 neighbouring points (separated by a resistance r) — the equivalent resistance is r /2. Show why. ii) Consider now an array of equilateral triangles (arranged so that at any junction there are 6 resistors r coming off). What is the equivalent resistance across any arm? iii) Consider an array of hexagons (arranged so that at any junction there are 3 resistors r coming off). Now what is the equivalent resistance across any arm? iv) Consider a network of regular polygons in which, at any point, N resistors r are coming off. What is the equivalent resistance? [BONUS: can you draw such a figure in 2 dimensions for N = 5 ?] [Peter] 6) My current analysis... Roughly sketch the impedance of this circuit, at right, as a function of the frequency ω of the sinusoidal driving source (the circle with the squiggly bit). A qualitative description will do. NOTE:
1 1 = α2 ≠ = β2 L1C1 L2C2
[BONUS: find the impedance as a function of frequency exactly] [Peter]
Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor
3
1999-2000 Physics Olympiad Preparation Program — University of Toronto — Solution Set 6: AC Circuits and Electronics 1) Ohm my! You might be able to do this one by inspection: if r Æ ∞ then no current flows in the circuit, so no power is dissipated through r. On the other hand, as r Æ ∞ power dissipation increases for a constant current. So a few equations are in order. In the series circuit V = I (R + r)
(voltage V across resistors in series, current I through resistors) 2
P = I • Vr = I r
(power P dissipated in r, current I through r same as R)
2
V = r R + r For an extremum: 0=
2
dP V 2r = 1− dr R + r R + r
which happens at r = R. However, this is a maximum, not a minimum. This is easy to see by graphing it, by finding the second derivative, or by simply noting: 2 lim V r = 0 r → 0 R + r 2 lim V lim 2 r lim 1 r = V = V2 =0 2 r → ∞ R + r r → ∞ (R + r ) r → ∞ 2(R + r )
…so if it is zero at the origin and at infinity, has one extremum, and cannot be negativevalued, it must be a maximum. The minima you’re asked for correspond to r = 0 and r = ∞. 2
How about in parallel? It’s even easier to see: Pr = Vr I r = (Vr ) /r. The voltage Vr = Vo the battery voltage. Clearly this power decreases monotonically as r increases — the miminum is at r = ∞. [Robin]
1
2) A current affair Usually questions like this one are given with a specified voltage applied — which would be easier! Here it is a current that is specified (without telling you how they made sure such a current was produced, at whatever voltage needed). This makes some parts tricky, unfortunately. For the first case, all the current must go through the resistor, and onward to charge the capacitor. Therefore, the capacitor charge goes up linearly respect to time, and the q I t voltage across it would be V = = 0 . After time τ no current flows, so the charge C C remains on the the capacitor (except perhaps for leakage due to some large resistance through the capacitor which ideally should be infinite. Leakage can happen through moisture in the air, too).
I(t) I0 τ
t
V I0 τ/C τ
t
For the case where the capacitor and resistor are connected in parallel, the current Io is divided through both of them. Initially most of the current will pass into the capacitor and increase the capacitor voltage as it charges up (since we’ve assumed a constant current flow, this the source voltage will have to increase to match the capacitor voltage, to keep current constant). As this voltage increases, the current through the resistor in parallel also will increase, according to V = IR, and this will make the current through the resistor increase — a bigger share of the current flows through the resistor. Even if the current were constant for all time, the capacitor cannot charge up forever: at some voltage the current drain through the resistor would be equal to the whole current supply, and the capacitor will not take any of the current. Then the circuit will reach ‘steady state’. This helps us figure out how to solve the following equation which describes the circuit (a famous physicist once said: “Never write down an equation until you’ve figured out what the answer should be.”) 2
I o = I capacitor + I resistor dQ V + dt R dV 1 =C + V dt R =
Where V is the voltage across both the capacitor and the resistor (in parallel), and Q is the charge on the capacitor. The current Io is constant, so we can solve this equation either systematically or by guessing from what we figured out about voltages, above. We try a solution which has exponentials in it (because the derivative of the exponential is again an exponential, which maybe would cancel with the original). In steady-state, the dV/dt term will be zero (i.e., nothing is changing), so finally V(t Æ∞ ) = Vo = Io R. If the voltage approaches this value asymptotically, then maybe it should have the form:
(
V(t) = Vo 1 − e αt
)
Try it in the differential equation: it works, as long as α = –1/RC. Sounds OK, because if R Æ 0 (short circuit) or C Æ 0 (no capacitor), then it V takes no time at all to reach the final state. In fact what happens is that the voltage follows this curve V(t) until the current is turned off. Then the charge on the capacitor leaks away through the resistor until it goes back to zero, exponentially with the same time-constant α = 1/RC. [Yaser & Robin]
I0 R t
τ
3) Pandora’s box o’ electronics There is no unique solution to this problem (for example, a 100 Ω resistor could also be drawn as two 50 Ω resistors, etc.). Trial and error always works, and here’s a possible solution. The battery has a value of 1.5V, resistance at A is 75 Ω, resistance at B is 25 Ω and the resistance at C is also 25 Ω. Using these values we first of all see that the resistance from B to C is 100 Ω (within error of the 99 Ω given); measuring resistance from A to C or B to C yields an error, as we’re going across the battery [an aside: the way a multimeter measures resistance is to send a small current across the circuit, measure the potential difference created and find the resistance from Ohm’s Law. However, when a
3
battery is in the way it will change the current, and perhaps even reverse it, which will yield in very strange values shown by the multimeter]. The current from A to B is 0, and the voltages from A to C and B to C will be 1.5 V (within error of 1.6V), as given [the internal resistance of the multimeter is extremely high when measuring voltage]. The current from A to C (remember that the resistance of the multimeter is very small when measuring current) is 1.5 / 100 = 0.015 A and the current from B to C will be 1.5 / 50 = 0.030 A, again within error of the given values. If you like a more systematic approach, try this: • First, try to keep things as simple as possible (i.e., start with the simplest components, like resistors and batteries, adding more complicated ones later). • Next, the most complex circuit possible is when all terminals are interconnected. Now, it may be shown that such an ‘outer’ triangle is equivalent to an ‘inner’ triangle, just as I’ve drawn before. So, we re-draw the diagram to the one I had first (this is not necessary; I’m only doing it so that this solution is the same as the one above). Now, it is pretty clear that there has to be at least 1 battery between A-C and B-C. The easiest way to satisfy this is to put one at C. • As to the remaining resistors, you can either guess the values or solve 3 equations and 3 unknowns, as below: R A + R B = 100 R A + R C = 1.5 / 0.015 = 100 RB + R C = 1.5 / 0.030 = 50
(RA, RB and RC are the resistances in the arms A, B and C, respectively, in the 1st diagram). And this indeed does it… [Peter] 4) Only logical The truth table for the OR gate is shown at right: If either A or B is true (1) then the output is true (1). b) Using NOT and AND gates the following OR gate can be constructed:
A
B
C
1
1
1
1
0
1
0
1
1
0
0
0
4
c) Using NOT and OR gates the following AND gate can be constructed:
BONUS d) One possible EXCLUSIVE OR gate built out of AND, OR and NOT gates:
[Carrie] 5) Net worth iv) Let me do the general problem first. Consider an infinite, homogenous grid where at any junction N resistors are connected together. Note that a voltage source is really equivalent to a current source, i.e., something which injects & draws current. Consider injecting a current I into the junction. Because the whole system is Nsymmetric, the current will split evenly and I/N will flow into every resistor. See the example, above left. Consider now drawing a current I from a junction. Again, a current I/N will be drawn from every resistor. Now superimpose the 2 current flows, offset by 1 arm (see right). We 5
will have a current I coming in, and a current I coming out one arm away — equivalent to a battery. The current in the given arm will be I/N + I/N = 2I/N. The resistance is r. Thus, the voltage drop is 2Ir/N which must also equal RI, where R is the equivalent resistance. From this we see that R = 2r/N. This also solves the remainder of the questions… This question is inspired by a Polish Olympiad question, from way back when… Answers: i) r/2 ii) r/3 iii) 2r/3 iv) 2r/N BONUS: it is not possible to draw such a figure in 2D; in 2 dimensions, the only possible figures are those for which N = 3, 4 or 6. To convince yourself of this you can note that at each junction the sum of all the internal angles has to be 360 degrees (of course, you don’t have to be able to draw this figure for part (iv) to hold…). [Peter] 6) My current analysis... Here’s what you have to know: i) when the driving frequency ω is small, the current is almost steady (DC). The inductor behaves like a piece of wire, with zero resistance. The capacitor, on the other hand, doesn’t let any current through and behaves like a break in the circuit. ii) when ω is very large, the opposite is true: the capacitor lets current through but the inductor doesn’t (the e.m.f. that builds up inside is very large) iii) an LC or RLC circuit will resonate at a frequency of ω 2 = power dissipated is maxi-mum, and the impedance is minimum.
1 . At this point the LC
Now, the circuit in question contains 2 circuits (the R, L2 , C 2 circuit and the C1 , L1 circuit), each with their own resonant frequencies. We can thus expect that the overall impedance will have minima near the resonant frequencies. In fact, the impedance of the R, L 2 , C2 circuit will be higher than that of the other circuit, due to the resistor R. Also, the impedance will never quite reach zero, as the two resonant frequencies are not equal ( α ≠ β ). From all this, we ‘guesstimate’ the curve on the following page (note that I have arbitrarily assumed a < b). For an analytical treatment I will employ the complex number technique. What happens here is that a sinusoidal driving source (e.g., cos(ωt) ) can be conveniently represented 6
as a complex number (cos(ω t) = Re(exp(iωt)), where Re denotes the real part). This is useful as derivatives of the exponential are really easy to calculate. This, in the end, is useful to solve the differential equations present when dealing with capacitors and/or inductors. The gist of all this is the fact that the resistor, capacitor and inductor, in a circuit driven by a sinusoidal source, behave as resistors whose generalized ‘resistance’ (called impedance) is a complex number (the number has to be complex because capacitors and inductors change the phase of the current; the phase is represented by the angle the impedance makes with the real axis in the complex plane). The values turn out to be: Resistor R: Z = R 1 iωC Inductor L: Z = iωL
Capacitor C: Z =
2
(Z is the impedance; i = –1) Finally, the standard Kirchoff rules apply to impedances (i.e., they’re added when in series and the reciprocals of the sum of their reciprocals is taken in parallel). After this crash course, let’s apply this to our circuit. The total impedance will be 1 Z = (iωL1 ) + + iωC1
α2 = iωL1 1 − 2 + ω 1+ R
1 1 1 + 1 R iωL2 + iωC2 1 1 β2 iωL2 1 − 2 ω
Real nasty… now, what we really want to plot is the magnitude of this thing, |Z|. This yields some pretty ugly expressions, and not much can be done so simplify them. 7
You can take my word for the fact that this yields very similar results to the graph above; the second minimum isn’t really a minimum, more of an inflection point, but it’s close enough for this rough analysis. [Peter]
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2000-2001 Physics Olympiad Preparation Program — University of Toronto — Problem Set 1: General Due October 30, 2000 1) The path of least resistance (...and highest voltage) You are given 32 identical 3-volt batteries, each of which has an internal resistance 2 ohms (i.e., even if you short-circuit them with a wire you still have 2 ohms appearing in such a circuit). How can you connect all these batteries together so as to get highest possible current through a 4-ohm resistor? [HINT: Think of the batteries set up in n rows and m columns...]
[Yaser]
2) The last straw... a) When you suck on a straw, you create a partial vacuum in your mouth. If a person could make a partial vacuum in their mouth equal to 50% of atmospheric pressure, what’s the farthest possible vertical distance above their drink of water? b) Is it reasonable that your lungs might produce a vacuum 50% that of the atmosphere, by inhaling? What else is at work? c) Does the size of the straw (its diameter) have any effect on the maximum height? To answer parts b) and c) it is suggested that you test the effects on height using different-sized straws. Once you have collected the data you can determine the approximate pressure — this is the empirical approach. Materials:
– a bucket or pitcher of water – a few ~1.5 m lengths of plastic tubing of different diameters (Canadian Tire stores sell this, for example) – a measuring tape
The plan — If you suck up water from the bucket up the tube, like a long straw, at some point you can no longer raise the water level within the tubing. Measure the height of the water level within the tubing above the water level in the bucket. Try it also by inhaling, to draw up the water, instead of sucking. Record your result and repeat the experiment for tubes of different diameter. Once you have collected the data you can determine the approximate vacuum pressure your lungs can achieve by themselves, and compare that to the best overall partial vacuum you are able to produce otherwise. [Sal] 1
A 2 g spherical bullet 9 mm in diameter is fired from a gun at 250 m s into one end of a thermally insulated cylinder. The cylinder, 15 m long and with a diameter of 1 m, is pressurized to 20 atm with Argon gas at room temperature. While passing through the cylinder, the bullet experiences a drag force which slows the bullet: 1 Fdrag = − ερAv 2 2 where
ε = coefficient of drag (a constant) ρ = density of gas A = cross-sectional area (area of an object’s shadow) v = speed 2
Ignore gravity, and assume ideal gas behaviour. Eventually, the bullet hits the opposite end of the cylinder and stops. a) How long does it take for the bullet to reach the opposite end of the cylinder? b) Calculate the bullet’s velocity just before hitting the cylinder wall. c) Calculate the change in temperature of the Ar gas in the cylinder due to the bullet’s motion before impact. [HINT: If you can integrate, you can do this; if you get stuck, check the POPTOR webpage for ways to solve the quite simple differential equation you will get]. [Brian] 5) Bubbling ideas Suppose we have a perfectly round soap bubble of initial radius Ro, with internal air pressure p, and external air pressure po. a) What is the change of pressure inside the bubble if the bubble size is increased by a small amount? Now put a charge Q on the bubble, spread evenly over the surface. b) What is the new radius of the bubble after charging it up? [HINT: perhaps you know or can find the field outside a spherical charge distribution, and figure out the field inside too. But what about the field right on the bubble wall, neither inside nor outside? The right answer is a kind of averaging... [Peter] 6) Magnus matters What does it take to pitch a curveball in baseball? Besides some talent, it takes just a bit of physics — the Magnus effect. The drag force on a ball depends on how quickly it is moving through the air. When a ball is moving through the air and also spinning, the airspeed on one side of the ball is greater than on the other. So it isn’t strange that the force on one side of a spinning ball can be greater than on the other, and the ball can be pushed sideways — curve in its path. Here’s a simple experiment to find out about this. You need: a cardboard tube of the type that is found at the centre of a roll of paper towels a skinny elastic band, at least 5 cm long a sticky note, like Post-It™ brand We want to drop the tube through the air while it is spinning. To do this, cut the skinny elastic band so that it’s now just an elastic string. Attach one end of the elastic to the 3
sticky-note, using tape (duct tape works well for me). Then stick the sticky note onto the cardboard tube right in the middle (as a kind of anchor) and wind the elastic around the tube. Now hold the tube horizontally while holding the free end of the elastic. Let go of the tube while keeping hold of the elastic — before it really falls, the elastic will pull on the tube and spin it up, and the sticky-note will peel off at the last.. It works best if you can release it over a balcony or from a ladder. a) What happens to the tube as it falls? Does it make a difference which way the tube is spinning as you drop it? How does this compare to the way the tube falls if it isn’t spinning at all? How do you explain the effect? How do you imagine the spinning motion changes the forces on the tube? b) Measure how far the tube falls, and how much it moves sideways. You may need to make several trials, measuring each time, and average your answers together to get a reliable result. If you can figure out the error in your measurement, that’s worth extra points. Can you tell what path or trajectory the tube takes as it falls — curved or straight? You might want to use a video camera and then look at the tape frame-byframe to analyse things. c) Use the results from (b) to figure out what the angle of descent is for the tube, relative to the vertical. You should probably find that the falling tube pretty quickly reaches terminal velocity, when the air drag-forces exactly match the force of gravity (for a parachutist in free-fall this is roughly 200 km per hour!). At what speed does the cylinder fall? Can you use this to figure out roughly how quickly the tube moves sideways? d) Theory — this just takes some algebra, though it looks a little hairy. Bonus points! Use the Fdrag formula from question 4; for anything shaped like a cylinder, the coefficient of drag is ε = 1.0. Can you use this to predict the terminal velocity of the falling spinning cylinder (you may also have to weigh your tube)? How does this compare to what you measured? You can likewise look at the sideways-motion terminal velocity. What must be the sideways force, to give the sideways terminal velocity you found above? e) BIG BONUS: For a given cylinder and spin-rate, can you roughly predict the angle, relative to the vertical, at which the spinning tube should fall? [Robin]
4
POPBits™ — Useful bits of information Drag force: Fdrag = ε ρ A v
2
ε = coefficient of drag (a constant) ρ = density of fluid A = cross-sectional area (area of an object’s shadow) v = speed
where
εsphere = 0.5 εcylinder = 1.0 C v (Ar) = 12.5 J mole
–1
K
–1
–1
Mm (Ar) = 39.948 g mole
(heat capacity at constant volume) (density of argon, per mole; molar density)
ρair = 1.2928 g L–1
5
2000-2001 Physics Olympiad Preparation Program — University of Toronto — Solution Set 1: General 1) The path of least resistance (...and highest voltage) If two identical batteries are connected to each other in series, the equivalent resistance is the sum of their individual resistances. The voltage across them would also be the sum of the individual electromotive forces. When they are connected in parallel, the total resistance is half the individual resistance and the voltage would be the same as individual voltages. If we connect all the batteries in series, we could get a high voltage but at the same time the total resistance is big and lowers the current. We could reduce the resistance by connecting all in parallel but then the total voltage is small. The optimum case is to set the batteries in n rows and m columns. If the voltage of each battery is V and its resistance is r then the voltage in each row is mV and the resistance is mr. So we have n equivalent batteries each having voltage mV and internal resistance mr. Therefore the total votage is mV and the total resistance is mr/n. Then the current that passes through a resistor R is equal to I=
mV V = mr R r + R+ m n n
We have to maximize the above equation with the condition N=nm. Therefore we should minimize the denominator, which is the sum of two terms having the product of Rr/mn or, equivalently, Rr/N. We know that the sum of two numbers which have a constant product is minimized when they are equal. Therefore R/m = r/n. It means mn= 32 and m/n=2. So m=8 and n=4.
[Yaser]
2) The last straw... a) By creating a partial vacuum in your mouth, atmospheric pressure forces the liquid up the straw. The difference in pressures will work only to a certain height h until the weight of the water in the straw will balance the differential-pressure force: P/Patm = 0.5, P = 0.5Patm = 5.05 x 104Pa Patm = P + ρgh h = (P atm –P) / ρg = (1.01 x 105 Pa – 5.01 x 104 Pa) / {(1000 kg/m3 )(9.8 m/s2 )} 1
h = 5.15 m Experimental Results Note: Inhaling is different than sucking By sucking, resting, then continuing to suck, your tongue acting as a piston can produce a vacuum effect capable of raising the liquid in the tubing higher than a height your lungs are capable of sustaining. I did the experiment both ways. Experiment by inhalation At the hardware store I picked up 3 1.5 meter tubes with diameters 1.58 cm, 1.90 cm, 2.54 cm. The tubing was taped along side my washing machine so as to get the tubing as straight as possible. One end of the tubing was submerged in a bucket of water . The other end was brought to my mouth. I exhaled, and then I inhaled as much as possible. As the water level rose in the tubing, it became more and more difficult to breathe in . This was not the result of me running out of lung capacity, because I was able to continue breathing in if I took my mouth away from the tubing. Once the water level stopped rising I quickly plugged the hole of the tubing with my tongue. The level of the water in the tubing was then recorded. My lungs, I felt, where getting tired after a few attempts so I rested every 3 to 4 attempts. 5 attempts for each tube were performed. Below are the results. Diameter of tubing (cm):
1.58
1.9
2.54
Attempt 1
112
99
106
Attempt 2
105
114
105
Attempt 3
112
115
111
Attempt 4
106
108
117
Attempt 5
100
109
103
107 ± 2
109 ± 3
108 ± 2
Average
Table 1. The heights of water within a tube achieved by inhaling on differentsized tubing. b) This result would suggest that the you could not raise the water level within the tubing to 5m simply by inhalation.
2
The results above show that the different average heights for each tube-diameter are within each other’s error. The diameter of the tube had no bearing on the height attainable, as one expects. The weight of the water within a larger tube will exert greater force against the atmosphere, but this force is distributed over a greater surface area due to the increased tube diameter. The resulting pressure exerted by the weight of the water, Force/Area, is therefore not affected. These results suggest that your lungs are only capable of producing a partial vacuum of 0.895 atmospheres — 0.105 ATM below atmospheric: 1.08 = (P – Patm) / ρg = (1.01 x 105 Pa (1 – C) / {(1000kg/m3)(9.8m/s2 )} solving for C, C = 0.895 Why does this differ from our earlier calculations? Because experiment shows that a typical person’s lungs cannot deliver 0.5 atmospheres of partial vacuum! Experiment by sucking on tube It would seem that the vacuum your mouth is capable of producing relies on another mechanism besides lungs capabilities. As mentioned earlier your tongue acting as a piston may be able to produce a greater partial vacuum in your mouth. To test this idea we repeated the experiment by sucking. (Unfortunately we initially misguided you by suggesting a short length of the tubing to use — sorry about that!) What I did was I found a tube longer than 5 m and repeated the experiment, in a stairwell this time. With a bucket of water on the ground floor I placed the tubing in the bucket and walked up a few flights of stairs. I then began to suck on the tubing, pausing occasionally to catch my breath. By placing my thumb over the opening of the tubing I maintained the water level in the tubing while resting. I continued this pattern of sucking and resting until I could no longer raise the water level. Measurements are shown below. 1.3 cm diameter tubing
Height in meters
Attempt 1
4.5
Attempt 2
4.2
The results are close to those calculated earlier! With each suck I was able to draw the water higher and higher. Not being restricted to one breath I was able to take advantage of the mechanisms my tongue is responsible for. There finally reached a point where the partial vacuum, created in my mouth, could no longer raise the water level beyond the 4.5 meter. This is the height that determines the vacuum capabilities of
3
my mouth. Lungs & diaphragm muscles, in fact, are not nearly as sturdy as mouth muscles, though a little baby crying might leave you wondering... [Sal] 3) Fermi, fer you Folks don’t have to get the following answers – it only matters that you reason well, and work with whatever scraps of information you already know. It isn’t OK to look up significant answers for things. You’re supposed to work like on a desert island, with what’s in their heads. Clever resourcefulness, inventiveness but mostly reasoning from common ideas and facts is what gets marks here. a) How many piano tuners do you figure there are in Toronto? • about 3.5 million people in Toronto people: 3.5 M • about 20% of households have a piano pianos: 700 k • about 10% of these get their pianos tuned regularly, say 2x per year, others maybe once every two years, (2 x 70 k + 0.5 x 630 k) = tuning vists/year: 455 k • say ~1.5 hours to tune a piano + 0.5 hours driving time ⇒ in an 8-hour day maybe 4 visits. If a tuner works 280 days/year tuning visits/year/worker 1,120 Total tuners in whole Toronto area:
Concert pianos probably are the specialty of a small handful of tuners; likewise some people have relatives who make a hobby of it, or have their pianos tuned by people ‘off the books’. So I think this number is probably good within a factor of two. How else can one get an idea? Try the Yellow Pages, which lists the fairly large tuning services, some of which employ more than one person; then figure that a certain number don’t pay for ads, or list in regional Yellow Pages, and it probably agrees with this number. b) I saw lots of meteors streak across the sky this summer — they moved through an arc of about 45 degrees of the sky in a little less than a second. How fast do meteors go? Well, angular speed is only good if we can guess the height of the meteors. That can’t be any higher than the height of the atmosphere, which isn’t of course a sharp edge. I think the International Space Station orbits at something like 240 miles, if I recollect right. Also that the Russian space station Mir is going to be in trouble when it drops to about 120 miles. Now, a person can still breathe on top of Everest at 10,000 m, about the same height as airliners fly through the air, so there’s a reasonable amount of air at 6 miles up (maybe roughly 50-70% of an atmosphere?) A meteor would burn in a lot less air than that. I think the space shuttle is in a fireball, with protection from its heat-tiles, at 40 miles up, so I think meteor showers must show up at heights of roughly 20-40 miles up.
4
And for one step? Well, how many steps in a year? Hmmm, well I walk about 600 m to the streetcar from home, and about 500 m at the other end – 2.2 km just getting to work. My steps are about 1 m per pace (I’m tall), so that’s 2,200 steps then. I walk for about 1 hour net, between classes and offices, during the day while I’m at work, and another 20 minutes at lunch. I figure I take about 1 step per second (very roughly), so that’s 1 1/3 hours x 3600 seconds/hour x 1 step/second = 4,800 steps. I take my sneakers off at home. So I walk about 7,000 sneaker-steps each day, roughly, 300 days per year because I don’t always wear sneakers. So a little over 3 million steps in 18 months to 22 remove about 20 moles (20 x 6 x 10 ) of atoms. 17
So with each step I must be leaving behind roughly 4 x 10 atoms! Yikes, one day people ought to be able to figure out where I walked! They just need to make a combination upright vacuum cleaner and mass spectrometer! REMINDER: all these values are very rough – within a factor of 2 or 3 either way is reasonable. The important thing is the reasoning, as Sherlock Holmes might once have said. [Robin] 4) Number four with a bullet The Physics – The key to this problem is realizing that the acceleration of the bullet is not constant while traveling though the cylinder of gas: From the drag equation Fdrag = − 21 ερAv 2 we know that bullet experiences a force that is a function of its velocity. Since F = ma and the mass of the bullet is constant, the acceleration has to be a function of velocity. Therefore, the following constant acceleration equations cannot be used: x(t) = x(0) + v(0)t + 21 at 2 (1) v(t) = v(0) + at (2) including the variations on equations (1) and (2): v(t)2 = v(0)2 + 2 a( x(t) − x(0)) x(t) = x(0) + 21 (v(t) + v(0)) x(t) = x(0) + v(t)t − 21 at 2
In this question, only the following fundamental definitions apply: dx v= dt dv a= dt F = ma
(3) (4) (5)
(6) (7) (8)
7
The Solution – Given quantities converted to SI units where appropriate: Variable Name
Definition
Value
v(0) m r L R T P ε Mm C
Initial velocity of bullet Mass of Bullet Radius of bullet Length of cylinder Radius of cylinder Initial temperature of gas Pressure of argon gas in cylinder Drag coefficient for a sphere Molar mass of argon gas Specific heat of argon gas
250 m/s 2.00 x 10-3 kg 4.500 x 10-3 m 15.0 m 0.500 m 300 K 2.027 x 106 Pa 0.500 39.948 g/mol 312.5 J/kg/K
Quantities to be calculated: Variable Name v(t) t V A ∆T M • • •
Definition Final velocity of bullet after traversing the cylinder Time it takes for the bullet to traverse the cylinder Volume of cylinder Maximum cross-sectional area of bullet Change in gas temperature Mass of argon gas in cylinder
Calculate maximum cross-sectional area of bullet: A = π r 2 = π (4.50 × 10 -3 )2 = 6.362 × 10 -5 m/s Calculate cylinder volume: V = π LR 2 = π (15.0)(0.500)2 = 11.78 m 3 Calculate density of argon gas in cylinder: n P = V RT n PMm (2.027 × 106 )(39.948) ρ = Mm = = = 3.246 × 10 4 g/m 3 = 32.46 kg/m 3 V RT (8.31451)(300) Calculate the mass of argon gas in the cylinder: M = ρ V = ( 32.46)(11.78) = 382.4 kg PV = nRT (Ideal Gas Law) ⇒
Part A • Calculate the time it takes for the bullet to transverse the cylinder: 1 ma = Fdrag = − ερAv 2 2 dv since a = , then dt
8
m
dv 1 = − ερAv 2 dt 2
dv ερA 2 =− v = −β v2 dt 2m
where β =
this is a differential equation – what function v(t) can be differentiated once to equal itself-squared, multiplied by a constant – β ?
ε ρ A (0.5)(32.46)(6.362 × 10 −5 ) = = 0.2581 m -1 − 3 2m 2(2.00 × 10 )
(see A Guide to Solving Simple Ordinary Differential Equations (ODE’s) on the POPTOR webpage for hints etc. for this problem set to understand the following steps) dv v2 v( t )
= − β dt t
dv
∫ 2 = −β ∫ dt v( 0 ) v 0 1 1 − = βt v(t) v(0) v(0) β t + 1 1 1 = βt + = v(t) v(0) v(0) v(t) =
v(0) v(0) β t + 1
equation for the velocity of the bullet as a function of time
We can’t solve for time t since we don’t know what v(t) is. However, we do know what the distance is so use the definition for velocity: since v =
dx , then dt
dx v(0) = dt v(0) β t + 1
this is another differential equation
(again see A Guide to Solving Simple Ordinary Differential Equations (ODE’s) ) dx = v(t)dt = L
t
0
0
v(0) dt v(0) β t + 1 1
∫ dx = v(0)∫ v(0)β t + 1 dt
L=
v(0) 1 ln(v(0)β t + 1) = ln(v(0)β t + 1) v(0)β β
ln( v(0)β t + 1) = Lβ
9
v(0)β t + 1 = e Lβ t=
e Lβ − 1 v(0)β
equation for the time it takes the bullet to transverse the cylinder
substitute in the numbers: t=
e(0.2581)(15.0) − 1 = 0.726 s (250)(0.2581)
Part B Using the equation for velocity and the value for time found in Part A: v(t) =
v(0) (250) = = 5.23 m/s v(0)β t + 1 (250)(0.2581)(0.726) + 1
Part C the change in the bullet’s kinetic energy is converted to heat in the gas: ∆EK = ∆Q
(
)
1 1 1 m v(t)2 − m v(0)2 = m v(t)2 − v(0)2 = MC∆T 2 2 2 ∆T =
(
m v(t)2 − v(0)2 2 MC
)
equation for change in temperature of argon gas
substitute in the numbers: (2.00 × 10 −3 )((250)2 − ( 5.23)2 ) ∆T = = 5.23 × 10 −4 K 2(382.4)(312.5) FINAL ANSWERS Question
Part A Part B Part C
t = 0.726 s v(t) = 5.23 m/s –4
∆T= 5.23 × 10 K
[Brian] 5) Bubbling ideas a) The temperature of the air inside the bubble is always the ambient temperature. Therefore, pV = constant, where V is the volume of the bubble. So if the volume changes by a small amount ∆V then (using the product rule) we have 10
∆( pV ) = ∆p ⋅ V + p ⋅ ∆V = 0 ⇒ ∆p = − p
∆V V
b) If we put a charge on the bubble, the bubble size increases as each part of the bubble is repelled from every other part. This will reduce the internal pressure. The electric field outside the bubble at the surface is Q ( 4πεR02 ) where R0 is the radius of the bubble. The field inside is zero. We could say that the field exactly on the bubble is the average of these two fields Q (8πεR02 ) . A full-theory calculation gives exactly this answer. Therefore, the electrostatic pressure is PE =
Eσ∆A Q Q Q = = 2 2 ∆A 8πεR0 4πR0 32επ 2 R04
where σ is the surface charge density, E is the electric field on the bubble and ∆A is the area of a small piece on the bubble. This pressure should be equal to the change in the internal gas pressure, following its change in size PE = − ∆p → ∆V =
V Q Q → ∆R = 2 4 P 32επ R0 96επ 2 R03 P
[Peter & Yaser]
6) Magnus matters a) If the tube is not spinning, it just falls straight down. If it is spinning, then it doesn’t fall straight, it scoots out to one side or the other, depending on which way it was spinning. If the tube is spinning so one side moves backwards with the passing air, and the other side moves forwards against the passing air, the tube swings away from the side that moves forward against the passing air (greatest relative speed). When the tube falls relatively slowly, as it does for us, the flow around the tube is pretty well-behaved — no very serious turbulence (which would make the problem tougher). We call the flow laminar, which means it flows in smooth layers. Now, the air is very slightly viscous — as syrup is more viscous than water, and water is more viscous than alcohol. That means that as the tube moves, it carries a layer of air along with it, close to its surface. It also means that when the tube rotates, it carries the air with it then. Imagine the tube holding still, and the air rushing past (as in a wind tunnel). The side of the tube rotating forward against the oncoming air will actually slow the air slightly; the side of the tube rotation with the oncoming air will carry the airflow a little faster. This means the air is slower on the side of the tube that rotates against the flow, and faster on the other side. Bernouilli’s principle (which really is just a conservation of energy formula for fluids!) means that the air pressure will be higher on the side moving against the 11
airflow, and less on the other side. The net pressure difference is a force, which pushes the tube sideways — the Magnus Force. Bernouilli’s principle is a big part of how an airplane wing generates lift out of the forward thrust of the engines. It’s also the reason why a strip of paper, held below your lower lip, will rise as you blow air across the top surface. [EXTRAS: When the object moves very quickly (like a fast baseball, or many real airplane wings), it isn’t any more exactly the Bernouilli effect, but a bit more complicated. What happens there has to do with the transition that airflow makes when it isn’t smooth, but becomes turbulent and begins to detach from the flow around the tube. When the relative airspeed is higher, the flow detaches earlier in the stream around an object; where the surface moves back with the airflow, it delays this boundary-layer separation. The result is that the air flowing past a spinning body is tilted, in the stream past the object. So the air in the wake of the spinning tube actually kicks over to one side. The momentum change of that air results in the force pulshing sideways. (Look for a streaming-video description of this on the archive of the Science of Baseball website, from a show on the Discovery Canada: www.exn.ca)] b) I recorded this on my computer, using a cheap CCD web-camera and shareware NIH Image 1.62 [http://rsb.info.nih.gov/nih-image], which can put a timestamp on each frame to help analyse motion. Here are a few frames in the sequence:
You don’t have to do this, though — you can get very nearly as much by finding where it hits the ground, measured from where you let it go, and using a stopwatch to time it. b) I measured from the video frames on my computer. If you use a video camera, you can measure from the TV screen. You might want to use a dry-erase felt pen (but remember to erase it afterwards!); but be careful about using any markers at all on a 12
computer monitor, since they are sometimes anti-reflection coated with a material that dissolves (they used to often dissolve with ammonia-based cleaners like Windex too! I’m not sure this still is true, so check the manual for your display for warnings!) In the graph on this page, each marker on the trajectory is a measurement at a different time (the blue ones are a little more than 0.1 second apart). The path is curved: it starts off going to the right, because of the elastic band I used to spin it, I think. Then it drops, but as it gains speed it begins to push to the left. c) The angle of descent was about 45°, so the tube moves at the same speeds in the downwards direction and the sideways direction. It almost reaches a constant speed (terminal velocity) after 1 second, when it hits the floor, but doesn’t quite get there. If you used a balcony in your school, you probably did better than I did. I was able to –1 extrapolate that the terminal velocity would be around 3.8 m s . d) At terminal velocity, there is no acceleration — all forces balance. Then the 45° trajectory indicates that the net force is at 45° down and to the left, so in this case the Magnus force is about the same as gravity, at terminal velocity. People have patented ‘sail’-steamships with tall rotating cylinders for 0.5 sails; they could use wind-power to make the ship tack into the wind at 45° this way. 0
From POPBits™, at the end of question-set #1: 2
-0.5
The mass of my cardboard tube was 25 g, and 2 its cross-sectional area was 125 cm (see comments below). Balancing forces, and therefore setting ma = F drag at terminal velocity, we can solve simply for vterminal: m a = ε ρ A vterminal
-1.5
2
–2
-2
0.025 kg • 9.8 m s
= 1.0 • 1.2928 g L 2 2 • 0.0125 m • vterminal Thus vterminal = 3.9 m s
-1
y (m)
Fdrag = ε ρ A v
–1
3
–3
• (10 L m ) -2.5
–1 –1
-3
as compared to 3.8 m s measured above (this -1.5 -1.0 -0.5 0.0 0.5 worked unusually well for such a formula). x (m) The sideways Magnus force is comparable to the force of gravity, so the x-component of velocity is about the same as this value, and the final terminal velocity is the resultant, about √2 times larger.
13
e) B IG B ONUS : You can come fairly close to the right answer with this slightly handwaving reasoning (you can also have a look in The Physics of Baseball (2nd Ed.), Robert K. Adair, Harper Collins, NY, 1994, ISBN 0-06—95047-1). The drag force is: Fdrag = ε ρ A v
2
which can be described as a pressure-differential ∆P (front – rear) multiplied by the area presented to the wind or rushing air. Since this pressure-differential depends on the relative speed, if an object spins there will be a difference in the force from side-to-side across the ball. Then 2
2
FMagnus = ∆Pleft-right A = ε ρ A (vright – vleft ) now, vright = v + 2πr • f vleft
= v – 2πr • f
where f is the rotation frequency (revolutions per second) of the object. Substituting: 2
2
2
2
(vright – vleft ) = (v + 2πr • f) – (v – 2πr • f) = 8π r v f so FMagnus = ε ρ A (8π r v f )
For our case at 45° falling angle, Fdrag = FMagnus so we can figure out the spinning frequency f: (8π r v f ) = v
2 –1
which we solve to find f = 6.3 rev s
I do think the elastic band setup I used could provide this spin-rate. But, in fact, this hand-waving argument fudged a little on the appropriate use of areas and direction of airflow. Even so, any error is by a constant factor: the formula shows the appropriate dependence on speed v and spin-rate f. Comments on my own setup Cardboard tube: circumference 14 cm --> diameter d = 4.5 cm length 28 cm 2 cross-sectional area presented to wind (rectangle) A = 125 cm mass 25 g
14
(I didn’t have a weigh-scale at home, so I took a 25 cm rod, a handle from a cat-toy, and hung a plastic grocery bag tied to a string on one end, and just a piece of string on the other. The rod had a piece of string tied around the middle to suspend the whole thing to tilt freely. I slid the string along the middle of the rod until the grocery bag and loose string balanced each other evenly. Then I taped the cardboard tube to the loose string, which tipped the balance over. I then dripped water drop-by-drop into the plastic bag until it balanced the cardboard tube again — it took 25 ml, which weighs 25 g.) [Robin]
15
2000–2001 Physics Olympiad Preparation Program — University of Toronto — Problem Set 2: Mechanics Due November 27, 2000 1) Take your best shot Evangeline, an Olympic biathalon competitor-in-training (who hopes to use her keen grasp of kinematics to win gold), keeps a water supply on her training course: on top of a 10 meter-high water tower platform sits a 10 meter-tall cylindrical drum of water, 3 meters in diameter. Unfortunately, the pipes to the drinking fountain below have frozen solid. Evangeline skis up to a point 5 meters away from the base of the tower, and with her last round she uses her target rifle to shoot a small hole in the wall of the water drum. Where, precisely, on the drum should the hole be so that she can have a drink without moving from the place where she stands? [Sal] 2) The fountain of (misspent) youth Sukumar has accidentally backed his pickup truck into a fire hydrant, and broken it off –1 at ground level (oops!). Water gushes straight upward at a rate of 150 kg s , and with a –1 speed of 20 m s . Suku tries to fix things by holding a nearby garbage can over the flow, but finds himself and the can lofted into the air on the water flow. If Suku weighs 60 kg, and the garbage can weighs 2.5 kg, what is the height that he’s waving to his friends from, when the police get there? [Yaser] 3) Huh? Hmmm… Consider three containers, as illustrated at right, each with the same area A at its base, and each with height H. The first one is a cylinder, and the other two are tapered. The area of the tops of the other two vessels are 2A and A/2. a) If all three containers are filled with water, what's the water pressure at the bottom of each? What total force does this then exert over the whole bottom plate of the container? b) If the containers of water are 1
each put on a weigh-scale, individually, what weight will the scale show for each? Treat the containers as being very thin-walled, and essentially massless. Only the bottom plate touches the weigh-scale, so why is the measured weight different from the force found as area × pressure you calculated in part (a) ? Remember that balance shows the force which is exerted on it. [Yaser] 4) All aboard! –1
a) A freight train traveling with an initial speed of 65 m s , cuts its engine and coasts 8 kilometers along a flat track before finally coming to a stop. Calculate the train's coefficient of rolling friction. Is this a value for the whole train, or for each railroad car? b) A locomotive is pulling an empty 15.0-m-long hopper-loader car at a constant speed –1 of 1.00 m s . The mass of the locomotive and empty car is about 75,000 kg. As the hopper-car moves along the track it passes under a grain dispenser that pours grain into –1 the car, at a rate of 1000 kg s . The grain dispenser only operates when the car is beneath it. How much work does the locomotive do while the hopper-car is being filled? Assume that the coefficient of rolling friction is the same as calculated in part (a). [Brian] 5) Java jump-up (eeyow!) A few years ago McDonald’s™ was sued by a woman who was badly scalded when the cup of hot coffee she was holding between her thighs — while driving — spilled. The woman won the suit, and since then their coffee cups include a printed warning on them that the presumably hot coffee inside is really hot. So when the makers of the Java Jumpstart Commuter Mug* recently came out with a new model, we at POPTOR suggested they have it evaluated by Deepee-Deetee Consultants Inc., a group of OAC Physics students, who evaluated the mug’s stability on the dashboard of a moving car. The physics-consultants found that the mug's stability depends significantly on how much coffee is inside. Find the height or coffee level in the mug for which the mug of coffee is least likely to tip when the car accelerates. The mug is 20 cm tall, has a radius of 6 cm, and weighs 200 grams (1/4 of which is the weight contribution of the bottom of the mug) [Sal] [*a fictitious company. If you find the first true storey unlikely, compare with other true stories of human fallibility at the irreverant Darwin Awards homepage, http://www.darwinawards.com]
2
6) Weight! Bob! In science, either experiment or theory can lead the way in discovery. But it pretty much isn’t finished science until both experiment and theory come together. Here’s one example where a simple experiment can tell you the answer you might not be able to guess theoretically — string and steer you in the right way to think, in order to understand.
pulley
Two identical weights are connected by a 1 m string. To begin, one weight is set on a frictionless table, and the second is held as in the picture, with the midpoint of the string between them draped over a massless frictionless pulley.
If the second weight is dropped, which happens first: • does the first weight slide along until it hits the pulley, or • does the second one swing around until it hits the side of the table? Try this in an experiment, to find the true answer, and then see how well you can explain it by theory. Try whatever you like to make things practically frictionless as given — but tell us what you did in your experiment, to approximate these ideal conditions, and then carefully describe what you measured. HINTS: To make the first weight practically frictionless on the table, you might try two child’s wooden cars with wheels. Or you might be able to use an air-rail at school for the table and tie together two identical air-rail cars for the weights. Or perhaps a dryice puck on the table would be practically frictionless, and you could tie it to an equal weight to drop at the side. The frictionless pulley can be nearly any small lightweight wheel (no inertia!), or perhaps teflon tubing rotating on a nail — or you might use teflon-coated electronics wire to connect the weights, and then a smooth round edge on the table to slide over with little friction. [Yaser, Robin]
C HECK THE POPTOR WEB PAGE for other hints, and any corrections we might post: www.physics.utoronto.ca/~poptor
3
2000–2001 Physics Olympiad Preparation Program — University of Toronto — Solution Set 2: Mechanics 1) Take your best shot Due to water pressure in the tank, water squirts out horizontally from the bullet hole at some speed, depending on the ‘head’ or height of water above where the hole is made. The water then falls a certain vertical distance. The horizontal distance the water reaches depends on the squirt-speed and the time it takes the water to fall — a time that only depends on height above the ground. Let the hole the rifle makes in the tank be the origin of a vertical axis which we will set to zero. Let h be the height above the hole to the top of the tank. As water flows out through the bullethole, the height of the water in the tank must drop. In a time ∆t, the volume of water flowing through the bullet-hole is: Aflow × distance-flowed = Aflow (vflow × ∆t) where Aflow is the area of the hole, and vflow is the flow-speed of water at the hole. Likewise the water in the tank will drop by a volume: Atank × distance-dropped = Atank (vtank × ∆t) Equating these, we get: vtank = vflow × (Aflow /Atank ) As the water drains from the tank due to the bullet hole, the kinetic energy put into the squirting water simply equals the potential energy of lowering all the water in the tank from the hole up. In a small time ∆t this is: 1⁄2 Mflow vflow
2
= Μtank g h
1
2
1⁄2 (ρ Aflow (vflow ∆t)) vflow = (ρ Atank h) g (vtank ∆t) Where ρ is the density of water. We can cancel terms, and use vtank we found above, to get: 2
vflow
=2gh
This is the initial horizontal speed. The initial vertical speed is zero. The time of flight for the water leaving the tank before it hits the ground 5 meters away from the base of the tank is given by: 2
t = 2 (20 – h) ⁄ g This time needs to be exactly the value that lets the water squirt 5 m sideways, at constant speed vflow , before hitting the ground. vflow t = 5 m therefore, 4 h (20 – h) = 25 Solving for h we get the result that in order for the water to reach the athlete 5 m away from the base of the tank, she must shoot a hole approximately 32 cm down from the top of the tank. [Sal] 2) The fountain of (misspent) youth Assume that Suku is at height H. The speed of the water at this height, before the collision with the garbage can, is 2
2
v = vo – 2 g H –1
where vo = 20 m s is the initial speed. After the collision, the water stops its motion, since the collision between a liquid and any surface is always inelastic. The change in momentum of the water during the collision acts provides the a force and compensates for gravity in order to hold the garbage can and Suku up. This force is equal to: F= ∆m * (v – 0)/∆t = 150 (kg/s) * v where ∆m is the mass of the water which has a collision at a certain time. ∆t is the collision time. Both ∆m and ∆t are supposed to be infinitessimals (vanishingly small numbers) but their ratio is actually equal to the rate of the water flow, which is 150 kg/s. This force is equal to the weight of the garbage can plus Suku. F = (60 + 2.5) • 9.8 = 150 v = 150 • From which we can find H = 19.5 m. Yikes!
400 - 2 • 9.8 • H [Yaser]
2
3) Huh? Hmmm… a) The water pressure at the bottom of all the containers is the same and equal to (ρ*g* H) where ρ is the density of the water. The pressure is dependent on the height of the water and not on the amount of the water. The total force which is exerted by the water on the bottom plate, by this pressure, is equal to the product of the pressure and area of the plate and is the same for all. b) The weigh-balance shows three different values for the containers full of water. For the non-cylindrical containers, this value is not equal to the product of the water pressure and the bottom-surface area. For the cylindrical container, the water pressure on the side walls is everywhere sideways, and contributes nothing to the weight. This isn’t true for the other containers. For the tapered containers, not only water pressure exerts a force over the bottom plate: the rigid walls of the container also exert force. For the one in which the area of the top is bigger, this force is downward and therefore the total force on the bottom plate is more than the product of pressure and area. For the other container, in which the area of the top is smaller, this force is upward, and therefore the total force is less than than the product of the pressure and the area. The easiest way to find the right answer is to imagine the sloped wall in a series of stairsteps, and then take the limit of the steps becoming vanishingly tiny. Each step has a vertical part and a horizontal part. Pressure on the vertical part, which is a tiny cylinder, gives an outward force everywhere; these radial forces add up to zero net force. Pressure on the horizontal part, like a shelf around the bottom of the little cylinder, gives a net downward force, and these side-wall forces add up to exactly the weight of the water which lies above the sloped walls (i.e., outside the cylinder lying above the circular base). The net force can be found by adding all the contributions. Each horizontal step is actually an annulus — a flat ring lying between two circles of radius r and r + dr. For a wall sloped at an angle θ from the vertical, the radius r of the vessel as a function of vertical position z from the base is: r = ro + z ⋅ tan θ It’s easy to find dr then (as the differential of r): dr = dz ⋅ tan θ The area of the annulus is then: A = 2πr ⋅ dr = 2πr ⋅ dz ⋅ tan θ = 2π(ro + z tan θ) ⋅ dz ⋅ tan θ The downward force on this small annulus is then: 3
dFy = p( z) ⋅ A where p(z) is the pressure, which is a function of z because z gives the depth. The pressure is simply the weight per unit area of all the water lying above, and so p( z) = ρg( H − z) Thus dFy = p( z) ⋅ A = ρg( H − z) ⋅ 2π(ro + z tan θ) ⋅ dz ⋅ tan θ The total force down on the side walls is then the sum of all these tiny contributions, for all values of z from z=0 to z=H. This is just the integral (siimpler than it seems — the integrand is only a polynomial, actually): z= H
∫ dF
Fy =
y
z=0 z= H
=
∫ ρg(H − z) ⋅ 2π(r + z tan θ) ⋅ dz ⋅ tan θ o
z=0
z= H
= 2πρg tan θ
∫ (H − z) ⋅ (r + z tan θ) ⋅ dz o
z=0 z= H
= 2πρg tan θ
∫ {Hr + (H tan θ − r ) ⋅ z − tan θ ⋅ z } ⋅ dz o
2
o
z=0
H = 2πρgg tan θHro z 0 +
{
= 2πρg tan θ
1 2
1 2
H
( H tan θ − ro ) ⋅ z 2 0
}
− 31 tan θ ⋅ z 3
H 0
H 2 ro + 61 H 3 tan θ ⋅
Naturally, as θ goes to zero (as for a cylinder), this vertical wall-force contribution goes to zero. It is obvious that this down-force on the sidewalls is the same as the weight of the water in the ‘extra’ region outside the cylinder — going back to the term for pressure, all we have done is simply to add the weight of the water lying above every little horizontal annular step. Since we considered every little horizontal step, we’ve considered the whole volume of water in the extra region. [Yaser & Robin]
4
4) All aboard! a) Assume the train experiences constant acceleration (deceleration). Known quantities converted to SI units where appropriate: Variable Symbol
Description
Numerical Value
X(0)
Initial displacement of train
0.00 m
X(t)
Final displacement of train
8000 m
V(0)
Initial velocity of train
65.0 m/s
V(t)
Final velocity of train
0.00 m/s
Unknown quantities: Variable Symbol
Description
µ
Coefficient of rolling friction
the train experiences a rolling frictional force (proportional to its normal force) that is opposite to its direction of motion: F = −µmg F = ma ma = −µmg µ=−
a g
(1)
constant acceleration displacement equation: X(t) = X(0) + V(0)t +
1 2 at 2
since X(0) = 0.00 m, then X(t) = V(0)t +
1 2 at 2
(2)
constant acceleration velocity equation: V(t) = V(0) + at since v(t) = 0.00 m/s, then V(0) + at = 0 t=
−V(0) a
(3)
5
substitute equation (3) into equation (2) to find the train’s constant acceleration: X(t) = −
V(0)2 V(0)2 + a 2a
2 aX(t) = V(0)2 − 2V(0)2 a=−
V(0)2 2X(t)
(4)
substitute equation (4) into equation (1) to find the coefficient of rolling friction: V ( 0) 2 (65.0)2 µ= = 2 gX(t) 2(9.81)(8000) µ = 0.0269 SOLUTION FOR PART (B): Note that although the rolling frictional acceleration µg is constant, since F = µgm, the rolling frictional force will not be constant if the train’s mass is changing – that is: F(t) = µgm(t). Known quantities: Variable Symbol
Description
Numerical Value
V
Constant velocity of train
1.00 m/s
m
Initial mass of train and hopper car
75000 kg
l
Length of hopper car
15.0 m
r
Mass transfer rate of grain
1000 kg/s
µ
Coefficient of rolling friction
0.0269
Unknown quantities: Variable Symbol
Description
m(t)
Mass of train and hopper car as a function of time
m(x)
Mass of train and hopper car as a function of displacement
W
Work done by the train 6
mass of train and hopper car as a function of time: m(t) = m + r t convert m(t) to m(x) using the fact that t = m(x) = m + r
x : V
x V
in order to maintain a constant velocity the train must exert a force that balances the rolling frictional force:
∑ F = Ffriction + Ftrain = 0 Ftrain = − Ffriction = −(−µ mg) = µ mg since m is a function of displacement: x F(x) = Ftrain = µ g m(x) = µ g m + r V the definition of work is: l
W = ∫ F(x)dx 0
l l x W = µg ∫ m(x)dx = µg ∫ m + r dx V 0 0 l
x2 W = µg mx + r 2V 0 (15.0)2 l2 ( . )( . ) ( )( . ) ( ) 0 0269 9 81 75000 15 0 1000 W = µg ml + r = + 2V 2(1.00) W = 3.27 × 10 5 J
[Brian]
5) Java jump-up (eeyow!) Given that the weight of the mug is M and that the weight of the bottom of the mug is M –3 the height of the centre of mass (yc.m.) of the mug with coffee of density ρ (~1 g cm ) 4 filled to height h is given by,
yc.m. =
2 3 MH 2h 4 2 + ρπR 2
( M + ρπR 2 h)
Define a constant a = ρπR 2 such that the mass of the coffee contained is m = ρπR 2 h ≡ a h 7
∴ yc.m.
3 ah 2 MH + 8 2 = ( M + a h)
The maximum stability of the mug of coffee is achieved for the lowest centre of mass. To find the minimum of yc.m. we need d( yc.m. ) = dh
3 ah 2 a h( M + a h) − ( a) MH + 2 8 ( M + ah)2
=0
3 ah 2 MH + 8 2 3 ah 2 + Mh − MH = 0 8
∴ h( M + ah) =
Solving for h we get 1 1 2 3 2 h= − M ± M + MHa 4 a
the negative solution can be ignored. 1 1 2 3 2 h= − M + M + MHa 4 a
Given: M = 200 g,
H = 20 cm
a = ρπ R 2 = 36π g / cm
Solving for h we get, 1
1 3 2 h= −200 + 40000 + (200)(20)(36π) 36π 4 h = 3.6 cm The mug is most stable with coffee filled to a level h = 3.6 cm.
[Sal]
8
6) Weight! Bob! In an experiment, the table-based weight hits the pulley before the free one smacks into the side of the table.
string
How to explain this? • It’s easiest to analyse the forces in x (horizontal) and y (vertical) components.
pulley
• The force on either block comes from tension in the string • There’s only one tension, so the forces on the block are equal in magnitude, though they may change in time • The difference is that the force on the left-hand block is always directed along the string, towards the pulley. The force on the right-hand block is also always directed along the string, but this vector changes as the angle of the string changes. • The left-hand block moves horizontally towards the pulley with an acceleration: aleft = T/m where m is the mass of the block, and T is the tension in the string • The right-hand block moves horizontally towards the wall with an acceleration: aright = T cos(θ) /m where m: mass of the block, T: tension in the string, θ: the angle the string makes with the horizontal • Before the right-hand block falls, there is no tension in the string. Once it does begin to move, then it makes an angle θ >0, so aright < aleft • The left-hand block is always accelerated to its collision with the pulley faster than the right hand block to its collision with the wall. The left-hand block hits first. [Yaser, Robin]
9
2000-2001 Physics Olympiad Preparation Program — University of Toronto — Problem Set 3: Thermodynamics Due January 15, 2001 1) Bye, metallic bar! Mechanical thermostats for home heating, and flasher units for car turn-signals (and Christmas tree lights!) often use this principle: Two strips, copper and iron, are bonded together, side by side. They share the same elastic properties (e.g., Young’s modulus), and the same length l, and diameter d. They don't have the same linear thermalexpansion coefficient, though — if we heat them both to a higher temperature, one expands more than the other, and they bend. If the change in temperature is ∆T and the linear thermal-expansion coefficients are λFe and λ Cu, find the angle of deflection. [Yaser] 2) Maxing Maxwell Have you ever wondered how fast molecules are flying about, maybe air molecules in your lungs? The atmosphere we breathe is composed of particles of different gases, the kinetic energy of these particles being related to their thermal motion. These gas particles collide with one another constantly, affecting their velocity. To speak of the velocity of a particle is therefore not very practical. What we can speak of though is a probability distribution of velocities — how likely it is to have one velocity or another. A result known as Maxwell's Distribution describes how particles are distributed over different speed-ranges: M P(v) = 4π 2π RT
3/2
v2 e−M v
2 /2RT
where M is the molar mass of the gas, T is the absolute temperature of the gas and R is a constant. The distribution gives the probability (maximum 1.0 = 100%) that a particle in the gas will have a velocity that falls between v and v+dv, where dv is an incremental. a) Calculate the most probable velocity for molecules of oxygen at T = 300 Kelvin. –1 –1 –1 Oxygen has a molar mass M = 0.0320 kg mole . R = 8.31 J mL K . b) Now graph the Maxwell Distribution.
[Sal]
1
3) A pressing concern There is cylinder filled by a gas, with area A. In the cylinder, there is piston with mass m on top which makes the height of the piston from the bottom of the cylinder a distance h. How much weight should we add onto the piston to reduce the height to half of its initial value? Assume the air pressure to be Po and the temperature of the gas is always room temperature. [Yaser] 4) Ringing the sphere An aluminum sphere is heated to 100 °C. The diameter of the sphere, at this temperature, is 25.4508 mm. The sphere rests on top of a 20 g copper ring which has been cooled to 0 °C. At this temperature the inside diameter of the ring is 25.4 mm, so the sphere cannot pass through the hole. A long time later, when the system reaches equilibrium (when the temperature of the sphere and ring are equal), the aluminum sphere is barely able to slip through the copper ring. Assuming there’s no loss of heat to the surroundings, only conduction between the sphere and ring, calculate the mass of the aluminum sphere. Linear coefficients of expansion –1
Cu = 431.8 nm K –1 Al = 584.2 nm K Specific Heats –1
–1
Cu = 386 J kg K –1 –1 Al = 900 J kg K
[Sal]
5) Reality Check A 1-liter thin-walled stainless steel container is filled with helium gas at standard temperature and pressure. The container sits on a hotplate so that temperature of the container and gas can be controlled. a) Not only the gas, but the container too, will expand when heated! Calculate the maximum pressure of the gas in the container, and the temperature at which it occurs, assuming ideal gas behaviour and linear thermal expansion of steel. What is the volume of the container at this temperature? The equation for thermal expansion in one dimension is:
∆x = α x ∆T In this equation ∆x is the amount of expansion, α is the expansion coefficient, x is the initial length and ∆T is the change in temperature. –5
-1
α (steel) = 1.1x10 K
2
b) Do the values you calculated in part (a) make sense? If not, explain possible reasons why not. [Brian] 6) Chillin’ Sam likes to enjoy a tall glass of cold lemonade on a hot summer day. He would like to know in advance how much ice he’ll need to put into a pitcher of lemonade in order to lower its temperature by the desired amount. a) Calculate the volume of water Sam needs to freeze in order to make ice enough to lower the temperature of a litre of room-temperature lemonade by 10 °C. The temperature of the freezer Sam uses to make the ice is –10 °C. Assume the lemonade is initially at room temperature (20 °C), and is basically water, and that all the ice has melted before the temperature of the lemonade is measured. b) Now try it! As an experiment, measure the change in temperature of a litre of water caused by a small ice cube. You may want to use a vacuum flask (e.g., Thermos™ bottle) to do this experiment, so that heat energy is conserved. Compare your measurements to theory, using the equations from part (a). Comment on your results. density
ρ (water) = 998 kg m ρ (ice) = 917 kg m
–3
–3
specific heat C(water) = 4190 J kg C(ice) = 2220 J kg latent heat
–1
–1
–1
K
–1
K
Hf(ice) = 333000 J kg
–1
[Brian]
Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor
3
2000-2001 Physics Olympiad Preparation Program — University of Toronto — Solution Set 3: Thermodynamics 1) Bye, metallic bar! 1) If the change in temperature is ∆T then the final lengths of the two strips are lFe = l(1 + λ Fe ∆T )
lCu = l(1 + λ Cu ∆T ) Since they are bonded together, they have to bend in order to have different length. If they are assumed as an arc in which has a (subtended) angle of α, the angle of deflection from a straight line, measured at either end, would be α/2. If the radius of the curvature, measured to the midpoint between the two strips, is R then we have (α in radians): d lFe = R + α 2 d lCu = R − α 2 Combining all the formulas above, and eliminating R from them, we obtain the angle of deflection in radians to be α /2 =
l∆T (λ Fe − λ Cu ) 2d
[Yaser]
2) Maxing Maxwell The most probable velocity is the one for which there is a peak in the velocity distribution curve. To find the maximum of this distribution we find the local maximum: take its derivative with respect to velocity and set that equal to 0. Solving for the velocity we get, 1/2
v = (2RT/M)
Oxygen gas at a temperature of T = 300 Kelvin –1 with a molar mass of 0.0320 kg mol has a most probable velocity of, v ~ 395 m s
–1
1
A plot of the velocity distribution of oxygen at a temperature T = 300 Kelvin is shown above. [Sal] 3) A pressing concern If the initial pressure inside the cylinder is P i, and the piston is at its equilibrium position, we must have a balance of forces: Po A + mg = Pi A When we put an additional mass M on top of the piston then we have Po A + (m + M ) g = Pf A where Pf is the final pressure. Since the temperature inside the cylinder is constant and equal to room temperature, the product PV of pressure and volume must finally turn out the same before and after putting the weight. Knowing that the height is reduced to half its initial value, we get h PV = Pi ⋅ h A = Pf ⋅ A 2
Pf = 2 Pi
Substituting Pf in above formulas we get the weight needed to be Mg = Po A + mg
[Yaser]
4) Ringing the sphere The relative change (per degree Celsius) in the diameter of the copper ring and aluminum sphere is associated with their respective linear coefficients of expansion. After some time the copper ring, initially at a temperature of 0 °C, will reach equilibrium with the aluminum sphere at some mutual temperature Teq. The changes in temperature for each will result in changes ∆D in diameter D: ∆D = (Tf –Ti) D α where Tf and Ti are the final and initial temperatures of the copper ring or aluminum sphere. α is the linear coefficient of expansion of each metal. We are told that when the system reaches equilibrium the diameter of the Al sphere equals the inner diameter of the Cu ring. DCu = DAl Therefore, DCu(initial) + ∆DCu = DAl (initial) + ∆DAl Solving for the shared temperature Tf, we find that the system reaches equilibrium at a temperature Tf = 50.38 °C.
2
We are also given that the system reaches equilibrium with no loss of energy to the surroundings. This means that heat is only transferred between the metals — the heat lost by the aluminum sphere as it cools is absorbed by the copper ring as it warms. Therefore, CCu MCu (∆TCu) = CAl MAl (∆TAl) Where CCu , CAl and Mcu , MAl are the specific heats and masses of the two metals. We are given the mass of the copper ring, and the changes in temperature of both metals were calculated above. We can therefore solve the equation above for the mass -3 of the aluminum sphere: MAl = 8.71 x 10 kg. [Sal] 5) Reality Check It is very important to fully understand where the equations we use come from. Sometimes students are tempted to use the “equation hunting” method when solving problems. This is a very dangerous practice since many of the equations we use (such as those found in textbooks) can be: (1) (2) (3) (4)
A limited case of a more general equation An approximation to a more complicated expression Derived under certain assumptions All of the above
The “equation hunting” method always generates interesting solutions, but not always correct ones – in fact, some pretty crazy answers can result as you will soon find out! Always ask yourself if the answers you calculate make sense. Known quantities converted to SI units where appropriate: Variable Name
Variable Definition
Variable Value
To
Initial temperature of gas
300 K
Po
Initial pressure of gas
101325 Pa
Vo
Initial volume of gas
1.00 × 10 −3 m 3
α
Linear expansion coefficient for steel
1.10 × 10 −5 K-1
Unknown quantities: Variable Name
Variable Definition
T
Temperature of gas at maximum pressure
P
Maximum pressure of gas
n
Number of moles of gas in the container 3
In this question we will assume ideal gas behavior (the “ideal” in ideal gas should raise some red flags here) and linear thermal expansion of an object in one dimension. •
Generate an expression for the volume of the container as a function of temperature: change of length in one dimension as a function of the change in temperature ∆x = α xo ∆T total length in one dimension as a function of the change in temperature
[
]
x = xo + ∆x = xo (1 + α∆T ) = xo 1 + α (T − T0 )
(1)
definition of volume V = x3
(2)
substitute (1) into (2):
[
] 3 = Vo[1 + α(T − To )] 3
V = x 3 = xo3 1 + α(T − To )
(3)
where Vo = xo3 •
Generate an expression for the pressure inside the container as a function of temperature: ideal gas law P=
nRT V
(4)
substitute (3) into (4) P= •
nRT
[
]
Vo 1 + α (T − To )
(5)
3
Solve for the temperature of the gas at maximum pressure: at maximum pressure
dP =0 dT
[ [
]
nR 1 − α (2T + To ) dP nR 3αnRT = − = 3 4 4 dT V 1 + α (T − T ) Vo 1 + α (T − To ) Vo 1 + α (T − To ) o o
[
]
[
]
]
solve for T:
[ ]=0 4 Vo [1 + α (T − To )]
nR 1 − α (2T + To )
4
T= •
1 1 1 1 − To = − 300 = 4.53 × 10 4 K − 5 2 (1.10 × 10 ) 2 α
(6)
Calculate the number of moles of gas in the container (this is a constant): PV (101325)(1.00 × 10 −3 ) n= o o = = 0.0406 mol RTo (8.31451)(300)
Calculate the maximum pressure – substitute (6) into (5): P=
nRT
[
= 3
(0.0406)(8.31451)(4.53 × 10 4 )
)]
3
] 3 = (1.00 × 10−3 )[1 + (1.10 × 10−5 )(4.53 × 104 − 300)]
3
]
Vo 1 + α(T − To )
[
(
(1.00 × 10 −3 ) 1 + (1.10 × 10 −5 ) 4.53 × 10 4 − 300
P = 4.58 × 106 Pa = 45.2 atm •
Calculate the volume at maximum pressure – from (3)
[
V = Vo 1 + α (T − To )
V = 0.00334 m 3 = 3.34 L So the temperature of the gas and container would be roughly 40,000K, pressure is around 40 atmospheres and the volume of the container has tripled. Note that the melting point of steel is about 2,000K. Clearly the container would either melt or rupture before reaching the temperature, pressure and volume calculated above. Note, too, that for large changes in temperature, it is usually incorrect to assume a linear change in length. [Brian] 6) Chillin’ The physics: the ice absorbs heat (thermal energy) from the lemonade until the temperatures of the ice and lemonade become equal. There are two types of energy involved in solving this problem: (1) thermal-motion energy and (2) crystallization energy. The first is the kinetic energy associated with the movement of molecules or atoms in liquids, solids and gases. Crystallization energy is the bond energy required to take apart a crystalline solid molecule by molecule or atom by atom – in this case, it is the energy required to convert ice (a solid) to water at constant temperature. Crystalline energy is also called latent heat of melting.
5
Known quantities converted to SI units where appropriate: Variable Name C(water )
Variable Definition
Variable Value
Specific heat of water
4190 J/kg/K
Specific heat of ice
2220 J/kg/K
H f ( H 2 0)
Heat of formation for H20
333000 J/kg
V (water )
ρ(water )
Density of water
T (water )
293 K
T (ice)
Initial temperature of ice
263 K
283 K
C(ice)
Tf
1.00 × 10 −3 m 3 998 kg/m3
of
Unknown quantities: Variable Name m(water ) m(ice) V
Variable Definition Mass of lemonade Mass of ice Volume of water needed to freeze in order to make ice
Solution: •
Calculate mass of lemonade: m(water ) = ρ(water )V (water ) = (998)(1.00 × 10 −3 ) = 0.998 kg
Calculate mass of ice: As with all thermodynamic problems we need to begin our solution with a conservation of energy equation (energy cannot be created or destroyed). If we consider the ice and lemonade as a closed system (that is, no heat can get in or out of the container holding the lemonade and ice), then the change in internal energy of the ice plus the change in internal energy of the lemonade must equal zero: ∆E(water ) + ∆E(ice ) = 0
(1)
The heat absorbed by the ice from the lemonade takes place in three steps: 6
(1)
Heat is absorbed to take the ice from its initial temperature up to 0 °C (273 K)
(2)
Heat is absorbed to convert the ice to water at constant temperature 0 °C (this is the crystalline energy)
(3)
Heat is absorbed to raise the resulting icewater at 0 °C to its final temperature
from equation (1) m(ice)C(ice)(273 − T (ice)) + =0 m(water )C(water ) Tf − T (water ) + m(ice)H f ( H 2 0) + m(ice)C(water ) Tf − 273
(
)
(
)
solving for m(ice):
m(ice) =
(
) C(ice)(T (ice) − 273) − H f ( H 2 0) − C(water )(273 − Tf )
m(ice) =
(0.998)( 4190)(283 − 293) = 0.105 kg (2220)(263 − 273) − ( 333000) + ( 4190)(273 − 283)
m(water )C(water ) Tf − T (water )
Calculate the volume of water Sam needs to freeze in order to make enough ice to lower the temperature of the lemonade by 10 degrees: V=
V=
m(ice) ρ(water )
(0.105) = 1.05 × 10 −4 m 3 = 0.105 L . (998)
[Brian]
7
2000-2001 Physics Olympiad Preparation Program — University of Toronto — Problem Set 4: Optics and Waves Due February 16, 2001 (revised date) 1) Sub hubbub In our deep-sea research submarine POPSub there is a frosted (not clear) light bulb of radius R. With you in it, the sub is at a height H from the bottom of the sea, examining life on the ocean floor. The light bulb is at a height h from the square (L × L) window which is d thick and made of a glass with index of refraction n.
h d H
L
a) Calculate the area at the bottom of the sea which is illuminated. b) As you look out, how big does the spot look?
[Yaser]
2) “Out, out damned spot!” Each time there’s a solar eclipse, a certain number of people suffer eye damage, or even are blinded, because they look directly at the sun. It isn’t even necessarily safe at maximum, when the sun is blocked. To be safe, one needs to use the sort of goggles that prevent the same kind of eye-damage in welders, whose torches can be very very bright; there are other special filters and glasses, but some you might think are good aren’t actually enough, since, e.g., invisible infrared or ultraviolet may pass through and cause damage . When you think about using a magnifying glass, in the sun, to burn wood or set fire to paper, it really isn’t very surprising your eyes are in danger. Direct sunlight is about 1 –2 kW m , at the earth — how about calculating the intensity of light focussed from the sun using a lens? Your eyeball is roughly the size of the circle you make with thumb and forefinger placed tip-to-tip, as when you make an ‘OK’ sign. Make a guess, from that, of roughly what the focal length is of the lens-system of your eye, and let’s figure the numbers: a) Estimate the size of your eye’s pupil when it’s at its smallest, and figure out the power in watts that would shine through if you looked straight at the sun (don’t do this 1
b) There’s a different ‘picture’ of how this works: that the single bang of a firework ‘bomb’ is a sound with many many different frequencies in it, rather than a single note — something like white light, as opposed to coloured light. What frequency would you expect to hear, standing in the same place as (a), if all the different wavelengths of sound simply diffracted from the ‘grating’ made of building-ribs? [Robin] 4) A high-tech diet, high in optical fiber... The internet and most telephone communication depends most heavily not on copper wire, nowadays, but on optical fiber. Rather than electrical pulses in wire, for signals, optical fiber communications use laser pulses propagating in tiny flexible glass strands. Think of the following model of a laser pulse propagating in an optical fiber. Take a glass rod with a circular cross-section of radius r. Say that light is launched into the fiber through the end, and that all rays lie within a cone of angle θ measured from the axis of the rod. The index of refraction of the fiber is n. a) If the input pulse is very short-duration, what is the duration of the pulse of light coming out at the end of the fiber? Assume the length of the fiber to be L. b) If the input light is a train (series) of pulses with duration τ, and time-separation ∆t, what should be the maximum length of the fiber in order that the output pulses are still distinguishable. This is a real issue for communications, since we need to know the fastest digital signal that can be sent. c) If the peak intensity of a single input pulse is I, what is its output peak intensity. [Yaser] [check the POPTOR web-site for an example of sound in a tube doing the same thing as this...] 5) Wave goodbye Hong and her brother Liang have a clothesline in their back-yard. They mentioned to me their experiments with their giant one-string guitar... If they hit the cable sharply with a stick, what they see is two waves that run away from the impact-place, one in either direction. They want to know why — can you explain for them? What they see is called a travelling wave. The shape of the wave stays the same, as it simply moves along the wire as if it were sliding. A travelling wave has the form: f ( x , t) = f ( x + ct) or f ( x , t) = f ( x − ct) If you look closely, you’ll see that the top one is a wave with shape f(x) travelling to the right (increasing x with time), and the lower one is the same shape, but travelling to the left (decreasing x with time). In fact, almost any shape will work. 3
There are some very interesting properties of such travelling waves: 1) if you have a shape that works as a wave, then the same shape would work as a wave if it were bigger or smaller by any factor a. 2) if you have two different shapes that work as travelling waves, then if you add the two together, the sum will also work as a wave. These are properties of what we call a linear system. They’re true, for instance, for sound waves in air (except for explosions and such). If they weren’t true, we couldn’t listen to music (think about that...). a) Say that Hong and Liang suddenly put a ‘dent’ f(x) in the shape of the clothesline by quickly striking with the stick. How do you figure out the two waves that run away in either direction? Look for a trick or idea that uses properties (1) and (2) above to help prove what happens; you don’t have to use much math. b) Say that the clothesline is tied firmly at both ends. Then the position of the cable at the end cannot move. What happens when a wave travels right to the end of the cable? Try the same kind of ideas that work for (a). c) Say that at one end the clothesline is tied to a ring that slides up and down an upright pole; say that this slides perfectly without friction., and the ring has negligible mass. This is a different boundary condition. What happens when a wave travels right to the end of this cable? [Sal/Robin] 6) Ten yards for interference… Many people have home theatres these days and although technology makes having a home theatre relatively simple, in order to get the maximum effect one still has to take into account the geometry of the setup. Consider a basic home theatre setup, with distance 2d between the speakers, and a distance L from the speakers to the opposite wall of the room. Assume the speakers are point sources, that each puts out exactly the same single-frequency pure sine wave (this must be a pretty boring movie, or a test of the Emergency Measures Systems…) and that the walls do not reflect any sound. a) Find a general expression giving the position(s) P of maximum intensity along the opposite wall as a function of speaker spacing d and room length L. Give the position(s) P in terms of distance y from the centreline between the speakers. Do not assume d<< L. b) If L = 5 m, d = 3 m and both speakers are producing an in-phase 320 Hz sound wave, calculate the position in one direction of the first three points of maximum intensity along the opposite wall, as measured from the centreline between the speakers. The speed of sound in air is 331 m/s.
4
HINT: This question involves solutions to an equation that can’t easily be written as a tidy equation y = f(x), but more like 0 = f(y) + g(y). You’re welcome to solve the equations graphically — plotting, and then looking for roots. You’re also allowed to solve by hand iteratively, or get a computer to solve the equation numerically. [Brian]
INFOBITS™ — Useful Bits of POPTOR Information You don’t need this to solve question 5, but it’s interesting. Waves satisfy a differential equation (an equation of derivatives) called the wave equation: ∂ 2 f ( x , t) ∂x 2
1 ∂ 2 f ( x , t) − 2 =0 ∂t 2 c
It is actually a very simple equation — you just need to take the derivative twice. The unusual notation is because these are partial derivatives. The partial derviative with respect to x means that you take the derivative with respect to the x-variable (totally ignoring the t-variable). So, for instance, ∂ sin( x + 3t) = cos( x + 3t) ∂x but ∂ sin( x + 3t) = 3 cos( x + 3t) ∂t
(using the chain rule)
You just pretend the other variable doesn’t even exist. A travelling wave can always be described by f(x,t) = f(x ± ct), if f is any function that can be differentiated properly. It gives the shape f(x) in space, travelling in the direction of increasing x for f(x – ct) and decreasing x for f(x + ct).
Remember to check the POPTOR web-page for hints and any possible corrections! www.physics.utoronto.ca/~poptor
5
2000-2001 Physics Olympiad Preparation Program — University of Toronto — Solution Set 4: Optics and Waves 1) Sub hubbub This question is supposed to be the easiest one in the set. If you want to consider the size of the window and the light bulb, you’ll find the geometry a bit fussy without teaching very much. The reasonable approximation is that the light bulb is tiny compared to other sizes, like the distance from window and therefore its size should be neglected. A thick window also can be solved as we do below (with an extra refraction surface), but let’s take the window to be thin compared to other distances, like h and the height H from the bottom of the sea. The area at the bottom of the sea which is illuminated, is square in shape, each side is equal to an unknown, X. The angles that the rays traveling in the sub and sea makes with the normal are α and γ. These ray which hit the edge of the window determine the size of the illuminated area at the bottom of the sea. X is equal to X = L + 2( H + d)tan γ From Snell’s law sin γ =
sin α nw
⇒ tan γ =
sin α 2 nw − sin 2 α
where nw is the index of refraction of the water. From the geometry of the system we have sin α =
L L2 + 4 h 2
⇒ tan γ =
L 2 2 2 (nw − 1)L2 + 4nw h
1
Thus, 2( H + d) X = L 1 + 2 2 2 (nw − 1)L2 + 4nw h If you were in the sub you would see the illuminated area formed by the continuation of the ray, which is in the sub. Therefore, the length of any side of the square is X ′ = 2( H + d + h)tan α =
L( H + d + h) h
[Yaser]
2) “Out, out damned spot!” Most of the focussing comes from the curved shape of your eyeball at the cornea, and the lens of the eye mostly ‘touches up’ the focussing. Makes little difference for this question. a) The focal plane of the eye is located approximately 22 mm behind the cornea or front of the eye. The iris controls the size of the pupil and on a bright day a typical pupil size is 2 mm in diameter. Approximating the eye size of focal length of sun lens as a “thin lens” eye (3 cm) we can use Newton’s size of distance to sun image form of the thin lens equation (a similartriangles law, actually): Hi / Ho = f / x Where:
Hi Ho f x
height of the image; 8 height of the object (radius of Sun ≅ 7 x 10 m) focal length of the lens 11 distance of the object (distance Sun-Earth ≅ 1.5 x 10 m)
The size of the image on the focal plane of the eye is; –4
Hi = (Ho • f) / x ≅ 10 m The area of the image on the focal plane is therefore; A ≅ π •10
–8
m
2
The pupil of the eye, approximately 2 mm in diameter under bright conditions will allow into the eye approximately; P
3
2
≅ ( 3 • 10 W/m ) • (π •10
–8
2
–3
m ) ≅ 3 • 10 W
2
This is about the power of a high-school HeNe laser. The resulting intensity in the focus on the retina of the eye is: I
≅ P/A ≅
5
2
3 • 10 W/m
About 300 times concentrated from direct sunlight. If your pupil has not closed down –2 fast enough, this can be much higher — more like 1000 kW m . b) In the problem above, the image size was determined by geometry — actually by the angle the sun subtends at your eye, which is the angular size, responsible for parallax. When you consider a laser with practically parallel light rays, the focus should be essentially a point, except for the fact that there is diffraction of the light, which makes the final focal spot slightly blurry. The formula for this is: d=2λf/a where d is the focal spot diameter, λ is the wavelength, f is the focal length, and a is the beam diameter at the lens. Often the quantity (f/a) is called the f-number. For our case, λ ~ 700 nm (red light), f = 3 cm, a = 1 mm; then d = 42 µm. The laser power of 5 mW all 6 –2 goes into this spot, for an average intensity of 3.6 × 10 W m . [EXTRAS : How do you understand this formula for diffraction? Only an infinite plane wave propagates without any change at all — but it extends infinitely! For any real wave, there is a general relationship: ∆x ∆k ~ 1 where ∆x is the uncertainty in the position, and ∆k is the uncertainty in the wavevector, related to the momentum; it’s direction is the direction of propagation. For a plane wave, ∆x is infinite, so ∆k can be zero — you know exactly where the wave is going (but not at all any special place where it is). When a plane wave passes through a slit, or the pupil of your eye, the hole size sets a limit on ∆x, and so ∆k cannot be zero. The wave doesn’t lose any momentum, going through the hole, so what happens is the direction of k spreads out — the wave propagates not just in a straight line, but it diffracts from the aperture. The same thing applies for the focal spot, if you trace backwards: light from a tiny focal spot spreads out faster than from a big focal spot. So tiny focal spots go together with smaller f-numbers. tanθ = (a/2)/f where θ is the half-angle of the spreading out of the light. Then the formula for d can be written as d • tanθ ~ 1 It makes sense, when you realize that d = ∆x (the width) and tanθ = ∆k (the spreading).] [Sal & Robin] 3
3) Musical buildings
θ2
a) The question is pretty easy if you let the distances be large. Then the rays are parallel, and the solution is much like the way you solve for interference in a thin layer, like oil on water — you need to examine the pathD differences.
θ1
The figure at left shows the basic setup of angles for incoming and outgoing sound waves; I draw them as rays here. The figure at right illustrates the pathlength differences: the straight lines across the path mark where paths are equal, and the extra distance one ray must travel is marked by the fat line (in red, if you have a colour version of this). This extra pathlength is from two parts: extra distance coming in, and extra distance coming out from the building-grating.
∆l1
D
D
∆l2
θ2
The figure at left shows the details for the incoming, and the figure at right shows the detail for the outgoing distance. The total extra path-length is ∆l1 + ∆l2 : ∆l1 ∆l 2 = sin θ1 and = sin θ 2 D D ∆l = ∆l1 + ∆l 2 = D ⋅ (sin θ1 + sin θ 2 )
θ1
At the speed of sound Cs, one reflection will be later than the other by a time T: T=
∆l D = ⋅ (sin θ1 + sin θ 2 ) Cs Cs
In fact, all reflections are separated in time by this — the ‘bang’ of the fireworks burst comes to the listener as a series of pulses, and T is the period of this repetitive signal. The frequency is then: ν=
1 Cs = T D ⋅ (sin θ1 + sin θ 2 )
4
For a building with ribs about 2 m apart (roughly the size of a window), with Cs = 300 –1 m s , and θ1 = 22° and θ2 = 45°, this gives ν = 140 Hz. That’s a fairly low note, but not as low as the famous 40 Hz organ note in Bach’s Tocatta and Fugue in D-minor which isn’t actually present on many recordings… It isn’t necessary to assume that the distances L1 and L2 are very large. If they aren’t, and interesting thing happens: not all the periods T between notes are exactly the same, but they increase or decrease as the ‘bang’ ripples off all the ribs of the building. (This is not difficult to show — just re-draw the diagrams above, with rays coming from a point.) As a result, the note that you hear actually shifts a little while you’re listening. The technical term for this is ‘frequency chirp’, like the whistle of a bird may rise or fall. b) The fireworks ‘bang’ has many frequencies in it, so another way to imagine this same problem is to compare it to white light hitting a diffraction grating and spreading out — like a room light diffracting from a compact disc. In that case, it’s easy to find what frequency you’ll hear, in a given place. It’s just the same wavelength formula as for a diffraction grating: nλ = D ⋅ (sin θ1 + sin θ 2 ) where n is the order of diffraction (check this out on a CD, and you’ll see several rainbows in the colours from a point-source light). In terms of sound, the note you hear is usually determined by the lowest frequency, or longest wavelenth — the fundamental. The other wavelengths are shorter; these higher frequencies are then multiples of the fundamental, called the overtones. These overtones determine the timbre or basic sound-quality of an instrument, like flute vs. oboe. For the fireworks, the fundamental frequency is: C Cs ν= s = λ D ⋅ (sin θ1 + sin θ 2 ) which is the same formula we found before.
[Robin]
4) A high-tech diet, high in optical fiber... a) There are many paths light can take down the fiber. That ray of light which moves in a straight line parallel to the axis of the fiber reaches the other end of the fiber earlier that some ray which has an angle θ with respect to the axis, and which travels more in a zig-zag pattern to get to the end of the fiber. The time it takes for the straight ray to travel a distance L along the fiber is t1 =
nL c
where c is the speed of light. The other ray has to travel a longer distance: 5
L cosθ and it gets to the other end at a time t2 =
nL c cosθ
The duration of the pulse at the output of the fiber is usually measured as the width of the pulse measured halfway down from the peak of the pulse — the full-width at halfmax (FWHM). To find this exactly, one needs to know how the input light goes into the different L rays, but it will be roughly Td half of the time delay between the first rays to arrive and the last rays to arrive: t −t nL 1 −1 τd = 2 1 = 2 2c cos θ where τ d is the pulse duration. This effect is usually called modal dispersion. We assumed in our calculation that the initial duration of the input pulse is much smaller that the output pulse, i.e., much smaller than this spreading–out by different rays. Otherwise the time-stretching adds to the original pulse duration the way that two sides of a right-angle triangle give the hypotenuse: t
τ = τp2 + τd2 b) Here we again assume that the initial pulse duration is much smaller than the dispersion broadening. In order that we can tell the pulses apart, after they broaden, they shouldn’t overlap with each other much. Therefore, the duration of the output pulses should be smaller than roughly half of the time separation between them. τ d ≤ ∆t 2 ⇒ L ≤
c∆t n(1 cos θ − 1)
c) The energy of a pulse is equal to the integral of the intensity in terms of the time. If the pulse looks similar to a Gaussian pulse, we can approximate the energy of the pulse by the product of the peak intensity and the pulse duration. If the fiber is lossless, we expect the energy of the output pulse to be equal to the input pulse. Therefore,
6
Iτ in out out I peak τ = I peak τ d ⇒ I peak = ∆t in out where I peak = I and I peak is the peak intensity of the input and output pulses. We assume that the length of the fiber is the maximum length such that the output pulses are distinguishable. [Yaser]
5) Wave goodbye a) Let us assume that the shape of the “dent” after the stick strikes the clothesline is given by f(x, t=0). An example of such a “dent,” frozen in time, is shown in the figure above. From the properties of waves given in the question, such a f(x, t=0) can be the instantaneous result of the addition of two identically shaped pulses, each half the size of the original “dent” — one pulse travelling to the left and the other to the right with the same speed. f(x, t=0) = 1/2 f1 (x)+ 1/2 f2 (x) where
f1 (x, t =0) = f(x, t= 0) and f1(x, t ) = f1(x +vt) f2 (x, t =0) = f(x, t= 0) and f2(x, t ) = f2(x –vt)
Once a solution, always a solution: the dent created instantaneously immediately decomposes into these two travelling waves after the line is struck. This explains the two waves you will see emerge, after impact, travelling in opposite directions. b) The fixed ends of the clothesline impose some conditions on the evolution of the waves. For all times t, the actual wave functions f1 , f2 must be constant at the fixed ends. f1 (x = L1 , t ) = 0
[1]
f2 (x = –L2 , t ) = 0
[2]
where L 1 and –L2 are the positions of the fixed ends relative to the position of impact. The following trick can satisfy conditions (1) and (2) while revealing something familiar from experience. For any wave f 2 , imagine a matching wave which is a negative version, and travelling in 7
the opposite direction toward the fixed end in some imaginary extension of the wire (see the figure below). There is no fixed point in this picture. As the wave and its negative counterpart approach the spot which ought to be fixed, the waves will always add up to zero — satisfying the condition for the fixed point! What you observe mathematically is the two waves passing each other and leaving the tiedpoint fixed. Since the point in the middle never moved, we could have nailed it down — it’s our fixed end. So when a real wave goes into a fixed end, the resulting wave is the negative wave coming back from the fixed point, just the solution you’d have from the doppleganger. c) Using a similar trick as in part (b), we consider superimposing a waves f2 with a positive clone coming the other way. This will ‘explain’ or model for us reflections at a sliding ring on a pole. No longer do we have restrictions (1) and (2): at the sliding ring, they’re replaced by the condition that there can be no vertical forces — only horizontal ones (or the massless ring would immediately move in response and the vertical force would vanish). With only a horizontal force possible, the tension in the string, for the right solution, must give a horizontal force: so the string itself must be horizontal. That’s fine — our doppleganger wave now is always a mirror image of the real one, and therefore symmetric. When these two add, the tangent to the string at the place of the pole is always horizontal; this is true for any smooth wave adding with its mirror image. The waves, as they add, rise to a height of twice the amplitude of each. So the ring at the pole rises to twice the height of the wave that travels to it, and then the wave coming back will be identical to the one going in. [Sal]
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6) Ten yards for interference… The Physics: In this question, we do not assume the spacing 2d between the speakers is very small when compared to the sound wave paths distance L from the speakers to the back wall. If the difference in path length speaker 1 from speaker 1 to point P and speaker 2 y to point P is an integral multiple of the d sound wavelength, there will be big center of constructive interference. d
speaker 2
screen TV L
room back wall
Speaker 1 to point P:
d1 =
Speaker 2 to point P:
d2 =
a) Path lengths from speaker 1 to point P and speaker 2 to point P can be determined using the Pythagorean theorem:
(y − d)2 + L2 (y + d)2 + L2
Therefore, the difference in path length is: y ≥ 0:
δ = d2 − d1 =
y ≤ 0:
δ = d1 − d2 =
(y + d)2 + L2 − (y − d)2 + L2 (y − d)2 + L2 − (y + d)2 + L2
For constructive interference, the difference in path length must be an integral multiple of the wavelength (δ = mλ ): y ≥ 0:
(y + d)2 + L2 − (y − d)2 + L2
= mλ = m
v f
(1)
y ≤ 0:
(y − d)2 + L2 − (y + d)2 + L2
= mλ = m
v f
(2)
where m = (0,1,2,K , ∞ ) In equations (1) and (2), m is the order of the interference (i.e., n=2 means they interfere by being two waves out of phase, f is the frequency of the sound wave, v is the speed of sound the medium and λ is the wavelength of sound in the medium. b) We need to solve for y using equations (1) and (2) when L = 5 m, d = 3 m, f = 320 Hz, v = 331 m/s: y ≥ 0: y ≤ 0:
331 320 331 (y − 3)2 + 52 − (y + 3)2 + 52 = m 320
(y + 3)2 + 52 − (y − 3)2 + 52
=m
9
The first maximum occurs when m = 0: y ≥ 0:
331 (y + 3)2 + 52 − (y − 3)2 + 52 = (0) 320 = 0
y ≤ 0:
331 (y − 3)2 + 52 − (y + 3)2 + 52 = (0) 320 = 0
by inspection, both equation yield y = 0 at the first maximum. The second and third maximum occur when m = 1: y ≥ 0:
331 331 (y + 3)2 + 52 − (y − 3)2 + 52 = (1) 320 = 320
(3)
y ≤ 0:
331 331 (y − 3)2 + 52 − (y + 3)2 + 52 = (1) 320 = 320
(4)
Not everyone will find it easy to determine a regular closed form solution for y in equation (3) and equation (4) (some did!). There are, however, a number of easy ways to solve these equations: •
plug in values for y in the equation and iteratively converge to zero in on a solution
graph the left hand side and right hand side the equation to see where the curves intersect
use a math program to solve the equation numerically.
I have solved equation (3) and equation (4) numerically using a math program called Maple. The answers I got are: y ≥ 0:
y = 1.02 m
y ≤ 0:
y = –1.02 m
Therefore, the three points of maximum intensity closest to the centerline are: y = {–1.02 m, 0 m, 1.02 m}
[Brian]
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2000-2001 Physics Olympiad Preparation Program — University of Toronto — Problem Set 5: Electricity and Magnetism Due March 9, 2001 (revised date) 1) Bridging the gaps Consider an insulated wire of length L wound on a metal cylinder. The insulation breaks at some hidden place, and the wire inside touches the cylinder. Using a battery, an ammeter and two variable resistors, figure out where along the wire is the break. [Peter] 2) Roadkill this question In the circuit drawn at right, all circuit elements lie on the edges of a cube. All resistors have identical resistance R , all capacitors have identical capacitance C. The voltage between points A and B is V. Figure out what charge is produced on the capacitor next to B. [Yaser]
B A
3) Loopy circuits
A B
C
D
A wire loop is laid out on a plane to make the closed circuit shown at right. A uniform magnetic field B is oriented perpendicular to this plane — sticking directly out of the page. This magnetic field changes at the rate dB/dt. What is the induced electromotive force in the wire? Find the answer in terms of the area of each section, A, B, C, and D. [Yaser]
4) Use the Force, Luke! A mass spectrometer is a device that measures the mass of charged particles (called ions) by steering particles with the same mass to the same point in space. Mass spectrometers are analogous to prisms, for light: a prism spreads a beam of white light 1
into its component colors (or B into page wavelengths); in a particle detector mass spectrometer a beam of ions with different masses is spread out according to mass. There are many uses for mass y spectrometers such as in radiocarbon accelerator dating, where mass E spectrometers d measure the ratio of numbers of carbon 13 14 isotopes C and C. The figure at right shows the geometry of a simple mass spectrometer: A given ion is accelerated by an electric field to a specific energy — and hence velocity. As the ion leaves the accelerator, it enters a region containing a uniform magnetic field that is perpendicular to its direction of motion. As the ion moves through the magnetic field, it experiences a centripetal force that causes its path to curve along a circle. A detector set along the y-axis, in this figure, can determine the intercept of the ion-path and that axis. a) Derive an expression relating the energy, charge and mass of the ion to its location (along the y-axis) on the detector. Assume non-relativistic energies. b) What minimum path-radius can this apparatus accommodate. c) If B = 2 Tesla, what is the minimum measurable mass for a ion with charge q = +1 e (where e is the magnitude of charge on an electron), accelerated to an energy 25 eV? What is the maximum mass? Do you think there may be any difficulty in trying to detect extremely heavy ions? If so, why? Note: an electron volt (eV) is another unit for energy. [Brian] v v v F = q v × B is the force experienced by a charged particle moving through a HINTS: magnetic field. In this equation, q is the charge of the particle, v is the velocity of the particle and B is the magnetic field. The right-hand rule for cross products may be useful in determining which way the particle curves.
2
5) A ringing charge A thin ring of radius R positioned in the x-y plane has a positive linear charge density + ρ [C/m] on the top half and an equal-magnitude negative linear charge density –ρ [C/m] on the bottom half. Derive an expression for the magnitude and direction of the electric field at point P, a distance d away from the center of the ring.
y
+
+
+
+
+
+ +
+
x
R
P
– –
– –
z
d
– –
– –
HINT: what is contribution to the electric field at point P from any small segment of the ring? [Brian] 6) Scoping out electrical charge Two identical spheres of mass m are identically charged, then each suspended from the same point by two non-conducting threads. Take each thread to have zero mass and fixed length L. The electrostatic forces acting on each object will act to push them apart. THE THEORY a) Derive an expression for the charge q on the spheres, found from their mass, the length of strings, and the angle θ between the threads. THE EXPERIMENT You will need the following: 1) 2 pieces of light string or sewing thread, and sewing needle 2) Styrofoam™ (or similar) packing material — packing ‘peanuts’ or blocks 3) utility knife or paring knife or razor blade 4) A protractor or ruler 5) A bit of tape, or push-pin Procedure: From the Styrofoam block cut out two cubes and try to cut the corners off to make something nearly spherical. Try fairly small cubes — about a centimeter or less. Using the sewing needle and thread, thread equal lengths of thread through the centers of the spheres. Using the tape or push-pin, attach the free ends of the thread together and
3
suspend the whole thing freely — I tied the two ends together and then taped the end to the edge of my bookshelf. Charge the spheres up — I rubbed mine against my hair, but you could rub a balloon against a sweater and then touch it for a few seconds to the styrofoam. The suspended charged balls will now hang with some space between them. b) Now use the results of your measurements, and part (a) above, to determine the charge on the Styrofoam spheres. [Brian]
INFOBITS™ — Useful Bits of POPTOR Information Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor
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2000-2001 Physics Olympiad Preparation Program — University of Toronto — Problem Set 6: AC Circuits and Electronics Due April 6, 2001 (revised date) 1) Inducting Henry A circuit contains an inductor L 1 = 70 mH in series with a resistor R1 = 50 Ω, both of which are in parallel with a R2 = 350 Ω resistor and a 90 V battery. The switch is closed for 110 µs and then opened.
L1
R1
R2
What is the value of the current in both resistors at the moment the switch is opened at 110 µs? [Sal]
90 V +
2) Inducing Resistance a) Graph the current that would flow from a power supply in the circuit at right (across terminals marked with circles) if the following voltage were applied: 0 V(t) = Vo
C
t<0 t≥0
b) What condition on L, R, and C will make the impedance seen by the power supply one which is purely resistive? (You may want to see the POPTOR Primer™ at the back of this set). [Robin & Cambridge Tripos]
R
L
R
3) Rectifying a situation In its simplest description, a diode is an electrical component that permits current to flow in only one direction. It can be D used as a rectifier to change alternating current (AC) to direct C current (DC). RL upply a) In the circuit shown at right, a step-down transformer converts 110V AC to 6.3V AC. D is a diode
Transformer
1
and the load resistor R L is 100 Ω. If the frequency of the AC power supply is 60 Hz, plot the voltage across R L versus time. b) In order to get a smoother output voltage, capacitors can be used as shown in the circuit at right. If Cs is 330 µF, plot the voltage across RL for three values of R L, so that
D1 AC Supply
Cs
RL
Transformer
RLCs infinitely large RLCs = 1/6 sec RLCs = 1/60 s [Yaser]
4) Zeners and the art of electronics maintenance A Zener diode maintains a more or less constant voltage across it in the reverse direction independent of current passing through it for a wide range of currents. This property may be used to stabilize a power supply against variations in input supply voltage and in load current drawn. Assume that there is a Zener diode, which the I-V curve is as below. This diode is placed in the circuit, which is shown. Transformer gives an output voltage reduced from the AC supply by a factor of 11. Rectifier is used to change the AC voltage to DC. a) If Rs is 100 Ω, for RL = 100 Ω, what is the voltage across it if the voltage of AC supply is varied from 85 volts to 130 volts?
C upply
I
5V V
0.7V
Vs IL
Rectifier
Rs
RL
Iz
b) Assume that the voltage of AC supply is 110 volts. What is the voltage across RL if we vary its resistance from 50 Ω to 150 Ω? [Yaser]
2
VL
5) Active courage a) By replacing an op-amp with its ideal model (see the POPTOR Primer™ below), derive an expression for Vout as a function of Vin and use this expression to calculate the V ratio out . Vin V b) What is the limit of the equation out you derived in part (a) as the gain A Vin approaches infinity? What effect does the op-amp have on the output signal with respect to the input signal? c) Is the equation in part (b) a good approximation to the equation in part (a) if the gain A is equal to 10,000? (less than 1% is good) d) Why doesn’t the op-amp amplify the input voltage by the op-amp gain A? e) BONUS (Worth an insane number of marks): When the op-amp gain A is very large, what does the voltage on the negative terminal V 1 become? What happens to the voltage V1 when V2 is non-zero (not grounded)? [The result of this bonus question illustrates the real usefulness of op-amps: op-amps in negative feedback mode allow one to arbitrarily set the voltage of any point in a circuit without sinking current (i.e., with infinite effective resistance). This is the basis for circuits such as the Miller Integrator.] [Brian]
6) Freaky Filters for “Phreaky” Physicists Various types of frequency filter appear in many devices. Here are some examples: Electrical – radio antenna Mechanical – automotive shock absorbers Optical – sunglasses Acoustical – earplugs Data – Net-Nanny This problem will concentrate on understanding how simple first-order electrical filters work. But before getting into solving the actual problem, you’ll need to understand the useful concept of impedance: See the POPTOR Primer™ below, for a briefing, if you don’t know about impedances very well. For the two filters on the next page, answer the following questions.
3
jwC
R Vout Vin
+ –
jwC
Vout Vin
+ –
R
Filter A
Filter B
a) Derive expressions for
Vout (these should complex functions of ω). Vin
b) From part (a) produce expressions for the magnitude and phase of function of frequency.
Vout as a Vin
Vout when ω = 0 and as ω approaches infinity Vin (HINT: you may want to use l’Hôpital’s Rule)? c) What is the magnitude and phase of
d) Two-way speakers usually have a tweeter for high-frequency sounds and a woofer for low to mid-frequency sounds. The reason why two-way speakers are used is because a single speaker cannot efficiently produce both high and low frequency sounds (FYI: speakers act like filters too!). The tweeter and woofer each have a separate filter that filters the incoming signal from the amplifier. If you were building your own two-way speaker, which filter (A or B) would you use on the tweeter and which filter would you use on the woofer? Since the magnitude of the filter responses do not cut off sharply, the tweeter and woofer will have some overlap in the sound frequencies they produce. Use the equations for magnitude developed in part (b) to calculate the crossover frequency for the two filters if R = 10,000 Ω and C = 0.02 µF. BONUS (Worth a reasonable number of marks): What effect will the phase have on the music played through our two-way speaker? [Brian]
Remember to check the POPTOR web-page for hints and any necessary corrections! www.physics.utoronto.ca/~poptor
4
POPTOR Primer™ in AC Electronics Active Circuits For many of you this may be your first encounter with an active circuit element. Unlike resistors, capacitors, inductors and diodes which are passive circuit elements, active components like op-amps and transistors have the ability to put energy into a circuit in a controlled manner based on electrical input from the circuit itself! Op-amp stands for “operational amplifier” and is one of the simplest of all the active elements. The figure at V 1 – right shows the circuit symbol for an op-amp. An opVout amp is a three-terminal device that amplifies the difference in voltages between the “+” and “–” input V 2 + terminals by a factor A and outputs the result (“A” is called the gain or amplification factor). The gain for a typical op-amp is usually between 10,000 and 1,000,000. Electrical characteristics of a “real-life” op-amp are very complex. However, for most applications it is more than reasonable to represent the op-amp by a simple model. The circuit at left shows a model V1 – Vout of an ideal op-amp – that is, if we could build a perfect op-amp, this + A(V2 – V1 ) model would completely describe – its electrical response. Circuits V2 containing op-amps can be + analyzed using standard circuit techniques (such as Kirchoff’s Voltage and Current Laws and Ohm’s Law) by replacing all the op-amp symbols with its model. R2 The circuit on the right shows an op-amp in a negative feedback R1 configuration. All negative – feedback means is the output is fed V1 back into the “–” input terminal; V + Vout in – in this case, the negative feedback V2 occurs through resistor R2 . + If you go on to take electronics courses at university or college, you will learn more about negative feedback. In the meantime, for a motivating explanation, negative feedback is used to 5
eliminate non-linearity in circuit response and increase stability – but you do not need to know this in order to solve this question! AC Impedance Ohm’s Law V = IR relates the voltage V across a resistor R to the current I through the resistor. But circuits can include capacitors and coils in addition to resistors. How then can we analyze such circuits? Is there an Ohm’s Law for capacitors & coils? The answer is “yes” – it is the impedance version of Ohm’s Law, V = IZ, where Z is called the impedance (boldface denotes a complex number, here). The impedance Z can be thought of as a complex-valued “resistance” having the form a + jb where a is the “real” part and b is the “imaginary” part [1]. Don’t get scared of complex-valued impedances. Complex numbers are only used to simplify the bookkeeping when determining things like magnitude and phase as you will soon find out [2]. The great thing about impedances is that the familiar techniques used to analyze DC circuits carry over! When we want to extract a physically meaningful number from the impedance version of Ohm’s Law, we just have to take the magnitude and phase of the complex expression. There are a few things about impedances you need to know about: The phase of a positive imaginary impedance is 90° and the phase of negative imaginary impedance is –90°. Similarly, the phase of a positive real impedance is 0° and the phase of a negative real impedance is 180°. The phase of an impedance with real and complex parts can be anywhere between 0° and 360°. Experiments show that when a sinusoidal (sine-wave) voltage is applied across a capacitor, the current through the capacitor is 90° out of phase with respect to the applied voltage and increases proportionally with frequency. Similar experiments show that when a sinusoidal voltage is applied across an inductor (coil), the current through the coil is –90° out of phase with respect the applied voltage and decreases inversely with frequency. (N.B.: the frequency of the applied voltage and current are the same for both the capacitor and coil). Without proof (or you can take this to be an experimental result −j [3]), the impedance of a capacitor is jϖ C and the impedance of a coil is where C is ϖL the capacitance, L is the inductance and ω is the frequency. 1. In circuit analysis the engineer’s “ j” is used to denote √(–1), instead of using the mathematician’s “i” which can be confused with current. 2. In principle, we can solve these problems without complex numbers. However, doing so would induce a really bad case of trigonometric diarrhea. 3. The impedances of a capacitor and coil can be derived from Maxwell’s equations. However, Maxwell’s equations themselves cannot be derived, but are based upon empirical results.
6
2001-2002 Physics Olympiad Preparation Program – University of Toronto –
Problem Set 1: General Due 9 November 2001 Since this is the first problem set, a few comments before starting: • If you want to know what kind of physics you might need to solve these problems, look at the Physics Olympiad syllabus at http://www.jyu.fi/ipho/syllabus.html. You don’t need to know the syllabus by heart to do the problems, but you do need to be able to recognize what you need to know so you can look it up. • Don’t forget to look at the information given in the POPBits™ section at the end of the problem set, it sometimes has information helpful or necessary for particular problems. • Pay attention to words like “estimate” or “about”. They indicate that the expected answer is not exact because either the input data is not precisely known or because approximations or simplifying assumptions are necessary. Much of real physics is learning how to turn insoluble exact problems into soluble approximations. • “Nothing ventured, nothing gained”: Whether you finish a problem or not, please make sure your reasoning and analysis are clear. If you write down nothing, it is easy for us to mark – we just give you zero – but pretty boring. Your basic ideas may be right even if you make a mistake or get stuck. • Now, on with the show!
1) Let the sun shine in! Hold a small, flat, square mirror toward the sun and send the reflected light onto a wall. Describe how the shape of the bright spot on wall changes as you move the mirror closer to and farther from the wall. What information about the sun can you obtain with this experiment? Write us your measured value and its error. (Hints: I made my mirror by masking a small round mirror with masking tape and paper so that only a 0.5 cm square was exposed; this way you can also easily experiment with different size or shape mirrors by just changing the tape. The mirror does not have to be an exact square, but it does have to be flat, so don’t use a mirror which magnifies or reduces, as do many round makeup or hand mirrors. If it is more convenient, you can leave the mirror fixed and look at the reflection on something like a piece of flat cardboard which you move closer to and farther from the mirror.) [Yaser]
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2) Ping-Pong! A ball will bounce back from a wall with the same speed it hits, if the collision is elastic. What happens if the ball (which is moving with a speed v) hits a “wall” (e.g. a table tennis paddle) that is moving (with a speed u) toward the ball? (a) What is the velocity of the ball after it bounces back from the wall? (Hint: Think of a frame in which the speed is the same before and after the bounce.) (b) What is the change in the kinetic energy of the ball if its mass is m? (c) Try to show that your answer to part (b) is equal to the work done by the wall on the ball. (Hint: Assuming the duration of the collision is ∆T, find the force exerted on the ball and its displacement.) [Yaser] 3) Don’t break the window! You have a ball of mass m attached to a massless string. The maximum tension the string can handle before breaking is T. (What we are actually talking about is a yo-yo that has been left outside until its string starts to rot, but we have translated this into “textbook physics”.) (a) If you hold one end of the string at a height h above the ground, what is the maximum length the string can have so that you can still swing the ball in circles around you without the ball touching the ground or the string breaking? (b) If you do swing it around in circles slightly too hard and the string breaks, for what length string will the ball fly the longest horizontal distance before hitting the ground? [Alex] 4) Cowabunga! First a well known joke: Farmer Smith was not satisfied with the yield of his milk cows, so he decided to called in an animal psychologist, an engineer and a physicist to try to improve matters. All three inspected the farm and the cows and made their recommendations. The animal psychologist went first, “If you paint the milking shed green the cows will be happier and happy cows will give more milk.” Then came the turn of the engineer. “If you narrow the milking stalls by 10 centimeters you will be able to add an extra stall and thus be able to milk an extra cow in the same time.” Farmer Smith was very happy so far, now it came to the turn of the physicist. She got out a black board, drew a circle and said: “First, we assume a spherical cow.” (a) Estimate the capacitance of a cow. (Hint: To help any city-slickers, I’ll point out that a typical run-of-the-mill Holstein1 dairy cow weighs about 700kg and is more or less neutrally buoyant in water. ) (b) Estimate the resistance of a cow. (Hints: People and cows have similar resistivity. Sometimes different approximations are necessary for different parts of a problem.) (c) When a cow walks across a carpet in winter, it can easily charge itself up to 10KV, and it can get a nasty shock if its nose touches a doorknob. Estimate the peak power in such a shock. (Hint: The total resistance of a mammal in an electrical circuit is normally dominated by 1
http://www.holstein.ca/
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1
http://mpfwww.jpl.nasa.gov/MPF/index1.html
Page 3 of 4
POPBits™ – Possibly useful bits of information Constants and units1,2 astronomical unit (mean earth-sun distance) 12C
atomic mass unit: (mass atom)/12 elementary (i.e. electron) charge electron volt Newtonian gravitational constant solar luminosity speed of light in vacuum standard acceleration of gravity at the earth’s surface Stephan-Boltzmann radiation constant tropical year (2001)
au u
149 597 870 660±20 m (1.66053873±0.00000013)×10-27 kg
e
(1.602176462±0.000000063)×10-19 C
eV GN LÍ
(1.602176462±0.000000063)×10-19 J
c g
σ yr
(6.673±0.010)×10-11 m3/kg/s2 (3.846 ± 0.008)×1026W 299 792 458 m/s 9.80665 m/s2 (5.670400±0.000040)×10-8 W/m2/K4 31556925.2 s
Radioactive half-lives of some isotopes3 14C
(Carbon-14) (Potassium-40) 238Pu (Plutonium-238) 40K
5730±40 years 1.277±0.001 × 109 years 87.7±0.1 years
Resistivity of human tissue4 blood bones fat muscle nerve
1.7 Ω·m 160 Ω·m 27 Ω·m 7.0 Ω·m 2.5 Ω·m
Great excuse for a party Birthday of Marie Curie5 (1867) and Lise Meitner6 (1878)
November 7
1
http://physics.nist.gov/cuu/Constants/index.html http://pdg.lbl.gov/2000/contents_sports.html 3 http://ie.lbl.gov/education/isotopes.htm 4 http://www.acusd.edu/~mmorse/BMES2000.shtml 5 http://www.mariecurie.org/mariecurie 6 http://www.users.bigpond.com/Sinclair/fission/LiseMeitner.html 2
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2001-2002 Physics Olympiad Preparation Program – University of Toronto –
(Right) At small distance between the mirror and the wall in which the circles form the square . (Left) At large distance between the mirror and the wall in which the circles overlap and the whole image looks like a circle.
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2) Ping-Pong! (a) A table tennis paddle is much heavier than the ping-pong ball. Therefore we can assume that it’s mass is infinity compared to the ball. Therefore, in the wall’s frame (or that of a table tennis paddle!), the velocity of the ball is the same before and after it hits the wall, but in the opposite direction. The velocity of the ball in this frame is v+u and so it bounces back with v+u relative to the wall. Therefore its velocity relative to the ground is v+2u. (b) The change in the kinetic energy is 1 1 ∆K = K f − Ki = m(v + 2 u) 2 − mv 2 = 2 mu( u + v ) 2 2 where m is mass of the ball and K is the kinetic energy. (c) The change in the linear momentum of the ball is ∆P = Pf - Pi = m(v + 2u) - (-mv) = 2m(v + u) Therefore if the collision time is ∆T, then the force is ∆P 2 m( u + v ) F= = ∆T ∆T The distance that the ball moves during the collision is equal to the distance that the wall moves which is u∆T . Therefore the work done by the wall on the ball is W = Fu∆T = 2 mu( u + v ) which is equal to the change in the kinetic energy of the ball as we expect. 3) Don’t break the window!
r
L
h
string
d
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FT Fc Fg
The faster you spin the ball around you, the higher it will go, but the more stress we put on the string. Looking at the above diagrams, we see that h − d Fg mg g = = ⇒ d=h− 2 2 r Fc mω r ω This is independent of the length of the string, and as we intuitively feel, d→h as ω→∞. The tension in the string is, however, proportional to the length of the string: L F F mgL FT = = T = T ⇒ = mω 2 L h − d Fg mg h−d (a) Rearranging the above equation, the length of the string F (h − d) L= T mg So L is a maximum when FT is a maximum and d is a minimum, i.e. when FT=T and d=0. Th ∴ Lmax = mg (b) The distance the ball flies is equal to its horizontal velocity, v=rω, multiplied by the time it takes to fall to the ground, t. If the starts to fall at a height, d, then the time it takes to hit the ground is 1 2d d = gt 2 ⇒ t= 2 g What we need to do is express the distance of travel, s=vt, in terms of L, and find the value of L which maximizes s. 2d 2d s = vt = rω = r 2ω 2 g g The maximum distance will occur for the maximum tension, FT=T, and by geometry
r 2 = L2 − ( h − d ) 2 and using previous results F T ω2 = T = mL mL
and
d=h−
g
ω
2
=h−
g mgL =h− FT T mL
we have
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2 mgL T 2( h − mgL T ) 2 s = L − h − h − T mL g
2 mg hT = 2 L1 − − L T mg Looking at this we easily see that the maximum distance will be when we maximize hT L − L mg This is just a parabola in L with an obvious maximum at hT Lmax = 2 mg For our interest, we can now calculate the maximum distance
smax =
T2
h 2
m 2 g2
−1
Note Some people assumed the motion of the ball was vertical instead of horizontal. You can’t actually “swing the ball in circles around you” in a vertical plane, but semantics aside, let’s see what happens if you swing the ball in a vertical plane beside you.
v ball
L
θ
string
h path of ball for part(b)
d s • The ball must orbit fast enough so the centrifugal force can overcome gravity at the top of its orbit, so the velocity of the ball at the top of the orbit must be such that v2/L>g. • If the string breaks, it will break at the bottom of the orbit since the tension must be a maximum since gravity and the centrifugal force add together at that point. To figure this out in detail, let’s assume you have got the ball going and it is now orbiting with no further energy input from you. The constant total energy of the ball is
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E tot = E potential + E kinetic 1 = mgL cosθ + mv 2 2 where we have set the gravitational potential energy zero at the height h. The velocity of the ball thus depends on where in the orbit it is E − mgL cosθ v 2 = 2 tot m Not surprisingly, the maximum velocity is at the bottom of the orbit (θ=180°). The tension in the string must balance the sum of the centrifugal force plus the component of the gravitational force along the string: v2 − mg cosθ L 2( E tot − mgL cosθ ) − mg cosθ = L 2 E tot = − 3mg cosθ L As expected, this shows the tension is a maximum at the bottom of the orbit and a minimum at the top; to minimize the tension we want Etot to also be a minimum. The tension at the top (θ=0°) cannot be less than zero or the string will not be taut and the ball will fall. So we want 3 E tot = mgL 2 ∴ FT = 3mg − 3mg cosθ = 3mg(1 − cosθ ) The tension at the bottom must be less than the breaking tension, T, so we must have T > 6 mg This is independent of the length of the string, so as long as the string is strong enough to support 6mg, the maximum length of the string is just set by the height you hold the end of the string above the ground, i.e. Lmax = h If we take the same string and swing it horizontally, we have Lmax=(T/mg)h=(6mg/mg)h=6h, so you can swing a much longer string horizontally instead of vertically. Even the weakest string that can support the weight of the ball, i.e. T>mg, has Lmax>h if swung horizontally. FT = m
Finally, let’s check how far the ball will fly if the string breaks when swinging vertically. Since the tension is maximum at the bottom, the ball will fly horizontally from the bottom of the orbit and the time to hit the ground is t=
2( h − L) 2d = g g
Again we want to maximize s=vt,
Page 5 of 12
s = vt = 2
E tot − mgL cos(180°) 2( h − L) m g
3 mgL + mgL h − L = 42 m g = 10 L( h − L)
So the maximum value is L=h/2, and the maximum distance the ball will fly in the vertical orbit case is 5 smax = h 2 Comparing this with the horizontal orbit, we see that the minimum strength string strong enough to be swung1 vertically (i.e. T>6mg) will fly exactly the same distance if swung horizontally or vertically. In the vertical case, however, the maximum distance is independent of the string strength, but in the horizontal case the maximum distance increases with the strength of the string without limit (except for mundane factors such as air resistance and the size of the earth). 4) Cowabunga! (a) For this question, it is convenient to assume that the cow is a sphere, since that is a shape we know how to deal with and we only want an estimate. The capacitance of an conducting object is the ratio of its electric charge to its electric potential or “voltage”, i.e. Q C= φ By Gauss’s Law and symmetry, the potential at the surface of a conducting sphere with charge Q and radius r is the same as the potential a distance r from of a point charge Q: q φ= 4 πε0 r (Where, by convention, V=0 at infinity defines the zero potential.) The capacitance of a sphere of radius r is therefore: Q C = = 4 πε0 r φ The radius of a sphere with the same volume, V, as a Holstein2 cow (mass M=700kg, density ρ=1000kg/m3) is 1 / 3 1 / 3 1 / 3 3 V M /ρ 700 kg /1000 kg / m = = = 0.55 m r = 4 4 π 4 π π 3 3 3 The cowpacitance is thus 1 2
Try saying “the minimum strength string strong enough to be swung” 5 times fast. http://www.holstein.ca/
Page 6 of 12
(
)
C = 4 πε0 r = 4 π 8.854187817 × 10-12 F/m 0.55 m = 60 pF (While writing this problem, I discovered that the standard capacitance of an isolated cow – I am not making this up – is 0.1nF, in good agreement with our estimate. If the cow is standing in a field, the earth acts as a conducting plane and this increases the capacitance to about 0.2nF, but we don’t care about a factor of 2 in this problem.) Note that you had to provide the value of ε0; we do not always give the values of basic constants available in any appropriate physics textbook. (b) A sphere is not a useful approximation for resistance, since it is hard to calculate the resistance of a sphere. We’ll assume a cube, since the resistance of a cube of side length L and resistivity ρ is very simple: L L ρ R=ρ =ρ 2 = L A L For cow-size cube M 1 / 3 cow L = = 0.88 m ρ cow The capacitance of a cow just depends on the cow being a conductor of a certain shape, but resistivity depends on the internal structure of a cow. My rough estimate of the average resistivity is based on the values given for people’s insides in the POPBits™ is ρcow~5Ω·m. The resistance is dominated by the low resistance pathways (e.g. the blood vessels), so the high resistivity of bones and fat are largely irrelevant. So my estimate of the resistance of the cow is just 5Ω / m R= = 6Ω ~ 10Ω 0.88 m This is much cruder than our capacitance estimate. This is because parallel capacitances add, so the size of the object pretty much determines the total capacitance; parallel resistances add inversely, so any small scale structure (e.g. blood vessels, noses) can have a big effect. The resistance is also sensitive to how the cow is connected in a circuit. In this case the resistance of the cow is probably dominated by the cow’s nose and is probably an order of magnitude higher than our simple estimate since the cow’s nose is an order of magnitude smaller than the whole cow. (Note: This seems consistent with the 150Ω I get when I hold one probe in each hand and measure my resistance with a multimeter, I have not yet tried to measure the resistance of my nose.) (c) The peak power is just the initial power, since the cow’s voltage drops as it discharges: 2 V peak V2 (10kV ) 2 ~ 10 MW Ppeak = I peakV peak = = initial = 10Ω R R No wonder static hurts! Note: By mistake, a draft pdf version of this problem set was originally posted with 600kg cows at 20KV, instead of 700kg cows at 10KV as in the print version. This is an approximate problem so either set of numbers are equally good. 5) Symmetry Cubed! (a) By symmetry, the resistance between any two vertices must be the same, so let’s choose vertices to measure the resistance between A and B.
Page 7 of 12
A C B
D
By symmetry, C and D will have the same potential so no current will flow between them, so that resistor can be ignored and the tetrahedron is equivalent to
A C B
D
So the total resistant between A and B can be calculated by adding the 3 paths in parallel, i.e. 1 1 1 1 2 = + + = RAB R 2 R R R R ∴ RAB = 2 (b) First we calculate the resistance between A and D.
A
B
E
F H
G
C
D
There is lots of symmetry, so it easy to see that G, E, & B will have the same potential and can treated as the same point, and similarly C, H, & F are at the same potential and can be treated as the same point, so the circuit is equivalent to
Page 8 of 12
A
B
E G
F H
C
D
So we see that
R R R 5R + + = 3 6 3 6 (c) Now we want the resistance between A and C. By symmetry, E, F, G, & H are all at the same potential. As with part (a), the circuit is not affected by ignoring the resistors connecting points at the same potential, so our equivalent circuit is: RAD =
A
B
E
F H
G
C
D
So
1 RAC
1 RAEC RAGC −1 R + 1 + 1 + R CD AB RBFD RBHD 1 1 1 = + + 2R 2R −1 R + 1 + 1 + R 2R 2R 1 1 = + R ( R + R + R) 3 ∴ RAC = R 4 =
1
+
1
+
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(d) Now we want the resistance between A and B. This is the toughest one yet, but there is still some symmetry, and we see that vertices E & G are at the same potential and can be connected without changing the circuit, and F & H are at the same potential and be connected. Our new equivalent circuit is
A
B
E, G
F, H
C
D
A
B R R/2
R/2
A
B
R/2
R
E,G
F,H
R/2
R/2 R/2
R/2
R/2
E,G
F, H
R C
D
2R
A
B R R/2
R/2 2R/5
G
F, H
Page 10 of 12
R A
B
5R/12
A
7R/5 ∴ RAB =
B
7 R 12
Note: You can, of course, solve all these systems using Kirchoff’s rules and Ohm’s law to write down systems of linear equations which can be solved by substitution or matrix algebra. This is fine, but we thought using symmetry was more fun. 6) Is it hot enough for you? (a) The activity A is N τ1 / 2 /ln 2 where N is the number of atoms and τ1/2 is the half-life of the radioactive isotope. The number of atoms for each isotope is simply its mass divide by its atomic weight: 15 atomic mass units (amu) for Carbon-14 and 40 amu for Potassium-40. For a 70kg person the masses of each isotope are: A=
mC14 ~ 70 kg × 19% × 10−12 ~ 1.33 × 10−11 kg =13ng mK 40 ~ 70 kg × 0.3% × 0.0117% ~ 2.46 × 10−5 kg Therefore the activities are
AC14 ~ AK 40 ~
1.33 × 10−11 kg 1 = 2193 / s (5730 yr /ln 2) (14 amu) 1.66 × 10−27 kg / amu
(
2.46 × 10−5 kg
)
(
1
)
(1.277 × 10 yr /ln 2) ( 40 amu) 1.66 × 10−27 kg / amu 9
= 6363 / s
So the total activity is Atotal=8600 decays per second. (b) About half the energy is deposited in the person’s body, therefore the power deposited by each isotope is: 1 PC14 = AC14 0.16 × 10 6 eV 1.60 × 10−19 J / eV = 2.8 × 10−11W 2 1 PK 40 = AK 40 1.3 × 10 6 eV 1.60 × 10−19 J / eV = 6.63 × 10−10 W 2 So the total activity is about Ptotal=0.7nW.
( (
)( )(
)
)
(c) In thermal equilibrium you will radiate as much heat, P, as you produce, so your temperature, T, will be P 1 / 4 T = εAσ Page 11 of 12
where A is your surface area (about 2m2), and σ is the Stefan-Boltzmann constant. Your emissivity, ε, is less than the black body value of 1, but it doesn’t matter too much since the temperature only varies as the 4th root, so we’ll assume it is 1. So
1 / 4 0.7 nW T ~ = 0.3K 2 -8 2 4 2 m 5.670400 × 10 W/m /K The cosmic microwave background radiation (http://background.uchicago.edu/~whu/beginners/introduction.html) is 2.7K so it will keep you warmer than than your internal radiation.
(
)
Page 12 of 12
University of Toronto 2001-2002 Physics Olympiad Preparation Program Problem Set 2: Mechanics Due Monday 17 December 2001 (Note small extension to due date!) Welcome back. • This and the next 4 problem sets each concentrate on different areas of physics, but this does not mean that you do not need to use a broad range of physics. For example, every question on this problem set requires knowledge of mechanics1, but knowledge of mechanics alone may not be enough to solve every problem. • Each problem set has an experimental question. They typically require a few common items, but you should look at them early in case you have to find something you don’t immediately have. Your Physics teacher may be able to help. • If you can’t do all of a question, do the parts you can do. • Unless otherwise specified, assume everything takes place on the earth’s surface and gravity points down in all our diagrams. • We often provide links to websites, but the problems do not require any information from these websites. Often the links are just for fun. 1) Salty log! Salt water is denser than fresh water, and in the ocean you will sometimes find a sharp vertical discontinuity in salinity (known as a “halocline”) between fresher water on top and saltier water underneath. This often happens near coasts where fresh water runs into the sea or where glaciers or sea ice are melting. Variations in seawater salinity and temperature drive the circulation of deep ocean waters2 and have a major impact on climate. Consider a log washed down a river and out to sea. Eventually the log becomes saturated with water and it starts to sink, but if it reaches a halocline it may float on the boundary. If the log has a uniform density ρL, and the (assumed) uniform densities of the surface and deep water are ρS and ρD, what fraction of the volume, V, of the log will be above the halocline in the fresher water. [Isamu] 1 2
At least “mechanics” as defined by us. http://www.oceansonline.com/watermasses.htm
Page 1 of 4
2) Don’t leave me! (a) Show that the Sun’s gravitational force on the Moon is more than twice the force of the Earth on the Moon. (The distance from the moon to the earth is about 400000 km.) (b) So why doesn’t the Sun pull the Moon away from the Earth? [David] 3) Singing in the rain! If you are suddenly caught out in the rain, is it better to run or walk to the nearest shelter? (a) First consider the case where there is no wind and you have no umbrella. Assume it is a distance L to the shelter, the raindrops are falling with a constant velocity u, and the rain is falling at a constant rate of n mm/hour. Calculate how wet will you get (i.e. how mach rain will hit you) if you walk with a constant speed v to the shelter. What speed, v, should you walk or run to minimize how wet you get? (Your maximum speed is vmax.) (b) What is your answer if you have an umbrella? (c) What is your answer if you have an umbrella and the rain is failing at an angle θ in to the direction you are walking? i.e. θ=0° is the rain falling vertically, θ=90° is the rain blowing horizontally into your face, θ=–90° is the rain blowing horizontally into your back. (Assume the vertical component of the velocity of the rain is still u.) Hints: Simplify a person as a rectangular cube. An umbrella prevents water from hitting the top of the cube, but not the front, back, or sides. By “rate”, I mean that after one hour an open container left out in the rain will have a layer of water n mm deep. [Yaser] 4) Sproing! A block of mass m sliding on a ramp is connected to a spring that has a negligible mass and a force constant of k, as shown in the figure. The spring is unstretched when the system is as shown in the figure, the block has an initial velocity v down the ramp, and there is friction between the incline and the block. The static and kinetic coefficients of friction are µs and µk; you can assume the friction is small but not zero.
k
v
(a) Estimate how much smaller the kinetic energy of the block will be the next time it passes α through the initial point. (b) About how often is the block’s velocity zero? (i.e. What is the frequency that v=0?) (c) Plot qualitatively the position of the block as a function of time until it permanently stops. (d) Where is it more likely for the block to permanently stop: at a point higher or at a point lower than the initial point (where the spring is unstretched)? Explain your answer. [Isamu]
Page 2 of 4
5) Fun before television! Professor Bailey lived on a steep hill when he was a kid, and he and his friends used to throw balls down the hill to try to get them into the woods at the bottom with only one bounce. (Fortunately they had some helpful dogs who thought it was great fun to do the hard part of finding the balls and – usually – bringing them back.)
S1
S2 If you are standing on top of a smooth linear hill with aθslope of tanθ and throw a ball out horizontally with an initial velocity v, (a) how far down the hill, S1, will the ball hit on its first bounce? (b) What is the ratio, S2/S1, of the distance (S2) travelled by the ball on its second bounce over the distance (S1) travelled on the first bounce? Hints: Assume the bounces are perfectly elastic. Ignore air resistance. [Alex] 6) Soup’s on! Roll over Galileo! Dr. Zed loves science, but even he sometimes gets misquoted in the press. As reported in the Summer 1996 issue of the popular Canadian children’s magazine “Chickadee”3, Dr. Zed likes to race empty and full soup cans against each other down a ramp. The magazine said that full cans always won because they were heavier. (a) Is it true that full cans roll faster than empty cans? Was the magazine’s explanation correct? Note: The empty and full cans in each race are identical except – Duh! – the empty cans are empty and the full cans are full, and both the top and bottom of the empty cans are removed by a traditional can opener that leaves the rims on the can. The ramp is not too steep, so the cans roll on the rims without slipping. Dr. Zed used condensed cream soup so the contents of the full cans are jelly-like and act more or less like a solid as long as they are not subject to too high a stress. Dr. Zed likes soup a lot, so he has a big supply of empty and full soup cans. He also has cans full of all sorts of other stuff – beans, tomato paste, pumpkin pie filling – if he wants some variety for his can races or for his dinner.
3
http://www.owlkids.com/
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(b) What is the relative acceleration, afull/aempty, of the two cans? i) Measure it experimentally. (Explain your methods and present your data.) ii) Calculate it theoretically. iii) Discuss any differences and give possible qualitative explanations for any discrepancies (larger than your uncertainties) between your experimental data and your theoretical calculation. (c) What happens if you race a can of condensed Cream of Mushroom (or any other soup which is a non-fluid jelly in the can) against the same size can of chicken or vegetable broth (or consommé or any other soup with a water-like consistency in the can)? i) Which can is first? What is the experimentally observed relative acceleration, acream/abroth, of the two cans? ii) Give a qualitative theoretical explanation for your results. Hints: Measure the relative accelerations either by timing each can separately or by racing them and seeing where the second can is when the first can reaches the bottom – whatever works best for you. Many cheap digital wristwatches have stopwatch functions. The second method works best if you have two people. For a ramp I just used a shelf propped on some books. Don’t forget to tell us any possibly relevant experimental parameters, e.g. the size of the cans you used and the slope and length of your ramp. [David]
POPBits™ – Possibly useful bits of information Constants and units4,5 astronomical unit (mean earth-sun distance) Earth mass Newtonian gravitational constant Solar mass standard acceleration of gravity at the earth’s surface tropical year (2001)
au 149 597 870 660±20 m M ⊕ (5.974±0.009)×1024 kg GN (6.673±0.010)×10-11 m3/kg/s2 MÍ
(1.98892 ± 0.00025)×1030kg
g
9.80665 m/s2
yr
31556925.2 s
David’s Maxim of the Month “A measurement without an uncertainty6 is just a number; a measurement with an uncertainty is data.” Great excuse for a party Nobel Prize awards7 (Eric Cornell, Wolfgang Ketterle, Carl Wieman)
December 10
4
http://physics.nist.gov/cuu/Constants/index.html http://pdg.lbl.gov/2000/contents_sports.html 6 Also known as an “error bar”. 7 http://www.nobel.se/physics/laureates/2001/index.html 5
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2001-2002 Physics Olympiad Preparation Program – University of Toronto –
Solution Set 2: Mechanics 1) Salty log! If the log has a density greater than the fresher surface water but less than the deeper saltier water, it will float at the halocline such that the net sinking force on the fraction, f, of the volume of the log above the halocline will be equal and opposite to the upward buoyancy force on the fraction, 1-f, below the halocline, i.e. fV ( ρ L − ρ S ) = (1 − f )V ( ρ D − ρ L )
∴ fρ L − fρ S = ρ D − ρ L − fρ D + fρ L ∴− fρ S = ρ D − ρ L − fρ D ρ − ρL ∴f = D ρD − ρS As a quick check of our calculation, we can consider the case before the log begins to sink and it is less dense than the fresh water. In this case it will float on the surface with a fraction in the water ρ − ρL ρ f = 1− S = L ρ S − ρ Air ρ S which is exactly what we expect. 2) Don’t leave me! (a) The ratio of the forces of the earth and the sun on the moon are GN mMoon mSun 2 FSun rSun = FEarth GN mMoon mEarth 2 rEarth mSun =
mEarth
2.0 × 10 30 kg 2 rSun 2 rEarth
=
(1.5 × 1011 m)
6.0 × 10 24 kg
2
= 2.4
(400000km) 2
(b) This question is not trivial, since the three-body problem is not exactly soluble for a inverse-square law force. We cannot write down the exact general equations of motion for an arbitrary system of three (or more) bodies interacting by gravity, so we cannot prove that the earth and the moon will never part company, even if we assume they are ideal point masses. But if they do part company, it certainly won’t be for a very, very, very, very long time. This is because the acceleration of the moon due to the sun is the same, on average, as the acceleration of the earth due to the sun, so once they are moving
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together around the sun (as they do) they will continue to move together. The maximum difference in acceleration between the earth and the moon towards the sun is much smaller than the acceleration of the moon due to the earth. GN mSun GN mSun 2 − rSun aEarth − Sun − aMoon − Sun (rSun − rEarth ) 2 = GN mEarth aMoon − Earth 2 rEarth mSun 1 2 − 1 2 r r r − ( ) Sun Sun Earth = mEarth 2 rEarth ≈ =
3 2 mSun rEarth 3 mEarth rSun
2 × 2.0 × 10 30 kg( 400000 km)
(
6.0 × 10 24 kg 1.5 × 1011 m
)
3
3
= 0.013
For the moon to separate from the earth, it would have to gain or loose angular momentum from the sun or the earth and there is no simple mechanism to do so. 3) Singing in the rain! (a) We make the approximation that we are vertical rectangular cube with a top surface area T and a front (or back) surface area F. The time it takes us to reach the shelter is t=L/v, so our “top wetness” (i.e. the amount of rain landing on our top) is WT = nTt = nTL/v and the rain we sweep out of the air onto our front is WF = ρFL = (n/u)FL where, ρ, is the volume fraction of the air taken up by rain drops. Our total wetness is W = WT+WF = nTL/v + (n/u)FL = nL(T/v + F/u) So our minimum wetness is when we run as fast as we can, i.e. v= vmax (b) If we have an umbrella, then WT=0 and our total wetness will just be W = WF = (n/u)FL So wetness is independent of our speed, v. (c) If we have an umbrella, and the rain is falling at an angle (along our Front/Back plane) then the amount of rain reaching our front or back is not just the amount we sweep out as we walk, but also includes a contribution due to the horizontal movement of the rain, W = WF = (n/u)FL + (n tanθ)Ft = (n/u)FL + (n tanθ)FL/v = nFL (1/u + tanθ/v )
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where n tanθ is the rate at which the rain is falling horizontally. If θ = 0 then we just recover our answer to part (b) as expected. If θ > 0, then the rain is blowing in our face, (1/u + tanθ/v )>0, so the best we can do is run as fast as we can, i.e. v= vmax, as for part (a), and our total wetness will be W = nFL (1/u + tanθ/vmax ) If θ < 0, then the rain is blowing from behind, (1/u + tanθ/v )>0, so we can keep dry if we just walk at the horizontal speed of the rain so that it is falling vertically in our reference frame, i.e. our optimum speed is v= –u tanθ = |u tanθ| and our total wetness is W=0. (Note: We want to get to the shelter so we assume v is positive. If our goal is just not to get wet, an alternate forθ > 0 is just to walk away from the shelter with speed v= –u tanθ = –|u tanθ|) 4) Sproing! The total force on the block along the ramp is the sum of the spring, gravitational, and frictional forces: 2
d x F = m 2 = kx + mg sinα - µ k mg cosα dt where x is the displacement of the block along the ramp from the position where the spring dx dt is the unit direction of velocity of the block along the ramp. i.e. force is zero, and vˆ = dx dt The frictional force always opposes the direction of motion, and we have made the usual (but not always accurate) textbook assumption that sliding friction is independent of speed. Without the frictional damping term, it is just an harmonic oscillator
d2x
= kx + mg sin α dt 2 which you are expected to recognize after some rearrangement mg sin α d 2 x + mg sin α k k = x + 2 m k dt and know the solution to be mg sin α x + = x 0 sin(ωt), ω = k m k mg sin α ∴ x = x 0 sin(ωt) − k where x0 is a (as yet unknown) constant. The block’s velocity (in the zero friction limit) is thus dx = ωx 0 cos(ωt) dt m
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v . The ω minimum (i.e. lowest) and maximum positions of the block (in the zero friction limit) are mg sin α mg sin α xmin = − x 0 − xmax = x 0 − , k k which will occur at times π 3π ωtmin = + 2 nπ , ωtmin = + 2 nπ , n = 0,1, 2,… 2 2 The total kinetic (T) and potential (U) energy of the block are: 1 dx T = 2 m dt 2
We know the initial velocity at t=0 is v (I probably should have said v0), so x 0 =
1 U = 2 kx2 + mg sinα x where we have chosen the (arbitrary) zero for the gravitation potential energy to be the initial point x=0 so that the total potential energy is zero at the initial point.. The initial kinetic energy is 1 T0 = 2 mv2 and we can find the maximum extension points where the block’s motion reverses itself since the velocity and hence kinetic energy at these points is zero so all the energy must in the form of potential energy, i.e. 1 2 1 2 mv = kxmin + mg sin α xmin 2 2 1 1 2 −mg sin α ± ( mg sin α ) − 4 k − mv 2 2 2 ∴ xmin = 1 2 k 2 =
−mg sin α ±
(mg sin α ) 2 + kmv 2
k By definition, xmin must be less than zero, so only one of the two solutions is possible, i.e. xmin =
−mg sin α −
(mg sin α ) 2 + kmv 2
k (a) The frictional force is small, so the motion of the block is damped only very slowly. The kinetic energy lost is just the integrated frictional energy loss. On the way down to its minimum position the frictional energy loss is
W =
∫ 0xmin µk mg cosαdx = µk mg cosαxmin
−mg sin α − = µk mg cosα
(mg sin α ) 2 + kmv 2 k
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On the way back up to the initial point the block will lose the same amount of energy since it travels the same distance, so the kinetic energy will be smaller by an amount 2 2 −mg sin α − ( mg sin α ) + kmv ∆T = 2W = 2µk mg cosα k (b) The block’s velocity is zero twice every oscillation cycle, i.e. at a frequency 1 k 2ω/(2π)=ω/π= . π m (c) 0.8 0.6
Relative x amplitude
0.4 0.2 0 0
5
10
15
20
25
30
-0.2 -0.4 -0.6 -0.8 -1 -1.2 Time (in oscillation cycles)
(d) Because of gravity, the minimum potential energy position of the block is below the initial position (which is the position where the spring force and potential energy are zero). The springs will oscillate about a position below the initial position (i.e. from our mg sin α solutions about the middle point of the oscillation is x middle = − ), so the block as k it slows down will be more likely to stop below its initial position. 5) Fun before television! (a) Lets choose an x-y coordinate system in which the positive x direction points vertically down and the positive y direction points horizontally away from the hill. Before the first bounce has a constant horizontal velocity and constant vertical acceleration and the path of the ball as a function of time is
Page 5 of 15
y=vt ,
1 x = 2 g t2
The parameterization for the slope of the hill is x/y = tanθ and the ball will bounce when its coordinates intercept those of the hill, i.e. when 2 2 1 y 1 x x1 = g 1 = g 1 2 v 2 v tan θ 1 ∴ gx12 = x1v 2 tan 2 θ 2
v 2 tan 2 θ g The distance down the hill for the first bounce is ∴ x1 = 2
S1 = x1 + y1 = x1 + 2
2
2
x12 tan 2 θ
= 1+
v 2 tan 2 θ v 2 tan 2 θ = 2 g g sin θ tan 2 θ 1
2
v 2 tan θ g cosθ (b) This is pretty much the same as part (a), except a bit messier. First we need the ball’s vector velocity ( x˙, y˙ ) before the bounce. y˙1 = v , x˙1 = gt To get the x component we need the time of the bounce: =2
t1 =
2 x1 2 v 2 tan 2 θ 2v tan θ = 2 = g g g g
∴ x˙1 = gt1 = 2v tan θ y˙1 By energy conservation in an elastic collision the speed ( v1 ) of the ball must be the same x˙1 before and after the bounce, so looking at the θ φin angles in the bounce, we can see that the x˙ b α velocity components before ( x˙, y˙ ) and after φout ˙y b ( x˙ b , y˙ b ) the bounce are related by θ y˙1 = v1 cos φin , x˙1 = v1 sin φin , φin = θ + α y˙ b = v1 cos φout , x˙ b = v1 sin φout , φout = −θ + α These can be solved in various ways, some of them involving messy trigonometry, but I think the easiest is to think about how each component of ( x˙, y˙ ) “bounces” separately and then adding them back together, i.e. y˙ b = x˙1 sin(2θ ) + y˙1 cos(2θ ), x˙ b = − x˙1 cos(2θ ) + y˙1 sin(2θ )
So the path of the ball after the collision is given by 1 y = y˙ b t + y1, x = gt 2 + x˙ b t + x1 2
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α
We can make things easier for ourselves by choosing to redefine our coordinate system such that x1 = y1 = 0 , since the distance bounced can only depend on the initial vector velocity , the acceleration due to gravity, and the angle of the slope. So y y = y˙ b t ⇒ t= y˙ b 2 y 1 1 y x = gt 2 + x˙ b t = g + x˙ b y˙ b 2 2 y˙ b
Once again the bounce will occur when the ball’s coordinates intersect that of the hill (using our redefined coordinate system) 2 y 1 y ∴ y tan θ = g + x˙ b y˙ b 2 y˙ b
∴y =
x˙ 2 y˙ b 2 tan θ − b g y˙ b
2( x˙1 sin(2θ ) + y˙1 cos(2θ )) = g =
− x˙ cos(2θ ) + y˙1 sin(2θ ) tan θ − 1 x˙1 sin(2θ ) + y˙1 cos(2θ )
2
2 2(2v tan θ sin(2θ ) + v cos(2θ )) −2v tan θ cos(2θ ) + v sin(2θ ) tan θ − 2v tan θ sin(2θ ) + v cos(2θ ) g
(
)
2v 2 tan θ 1 + 2 sin 2 θ g So distance travelled along the slope on the second bounce is y S 2 = x 2 + y 2 = tan 2 θ + 1y = cosθ and the ratio compared to the first bounce is =
(
S2 y g g 2v 2 = = 2 y= 2 tan θ 1 + 2 sin 2 θ S1 S1cosθ 2v tan θ 2v tan θ g
(
= 1 + 2 sin 2 θ
)
)
This does not depend on v or g; it is 1 when the slope is near horizontal and goes to infinity when the slope is near vertical, both of which make sense since in the former case the ball picks up very little gravitational energy so every bounce should be the same length, and in the latter case the ball picks up a lot of gravitational energy before bouncing. 6) Soup’s on! Roll over Galileo! (a) I practically choked on my breakfast cereal when I read the magazine’s explanation in 1996. One of the most famous experiment’s in all of science is Galileo dropping balls of different weights from the Leaning Tower of Pisa to show that gravitational acceleration is independent of mass. (For a virtual version of this experiment, see
Page 7 of 15
http://www.pbs.org/wgbh/nova/pisa/galileo.html. For a more modern version, check out the video of David Scott dropping a hammer and a feather on the moon at http://cass.jsc.nasa.gov/expmoon/Apollo15/apo15g.avi.) Whether Galileo actually dropped balls from the Tower is somewhat controversial, but we know for sure that Galileo studied acceleration of gravity using balls rolling down inclined planes. From Newton’s Laws (of Gravity and Motion) I knew heavier cans should not roll faster, except for effects such a friction and air resistance. Jumping up from the table, I immediately took a bookshelf (825±10 mm long) and put one end on some books. I took some full cans and empty cans (from our recycling bin) and quickly established that full cans do roll faster than empty cans, but it has nothing to do with how heavy they are. A full 284ml can rolls just as fast as a full 796ml can; similarly a small empty can rolls at about the same speed as a large empty can. The difference must have to do with how the mass is distributed, not how much their is. I must admit I find it hard to believe that Aristotle actually thought acceleration was proportional to mass. Yes, a flat piece of paper and a rock do fall at very different speeds, but all you need to do is drop a small pebble and a large rock to see that acceleration isn’t even close to being proportional to mass. (Even a crumpled up piece of paper hits the ground at almost the same time as a rock if dropped from chest height.) Aristotle was not an idiot (e.g. see http://www.batesville.k12.in.us/Physics/PhyNet/AboutScience/WasAristotle.html), but in this case he seems to have created a classic example of how “common sense” can produce spectacularly wrong but long lasting dogma. Note: The answers I give below is far longer and more detailed than we expect from you, and even so I have left out much of the details of what I did. (b) For this problem set, I redid my 5 year old experiment more carefully. i) In order to measure the acceleration I used 796mL cans of Campbell’s Cream of x ω Mushroom and Consommé soup; the cans were D=2R=101±1mm in diameter and 118±1mm in length. (All length measurements with a Canadian v Tire Mastercraft 7m long tape measure.) I removed one end from some cans, made soup, then removed the other end. The empty L h cans weighed mempty=70±5g (on a kitchen scale) and the full cans weighed θ mfull=0.9±0.1kg. (The kitchen scale only went to 250g, so I had to empty the can and measure the contents in several portions which is why the error is large. I roughly checked the calibration of the scale using some water, but the quoted uncertainties are based on how reproducible my measurements were.) After playing around with various cans and ramps (“playing around” is an essential part of good experimental technique – it is how you get a feel for how to do things), I decided to use a ramp made from a L=825±10mm long Ikea wooden
Page 8 of 15
bookshelf resting on VHS video cassettes. I used the cassettes instead of books since they were all the same size (each about 30mm thick). I used this ramp because it was straight, wide, not as slippery like laminate surfaces (so the cans always rolled instead of starting to slide), and it was short enough so I could release the can with one hand and catch with the other. (This last point saved me a lot of time since I didn’t have to get up to fetch the can after each run.) When I raced two cans, the full Cream of Mushroom can beat the empty can by 10±2cm, 8±3cm, 13±3cm for ramp heights of h = 120±5mm, 60±5mm, and 182±5mm respectively. I found I could not release the cans simultaneously and see their separation accurately at the bottom (I was alone sine my kids lost interest after about 30minutes), so I decided it was better for me to time the cans one at a time with the stop watch function on my \$30 Casio AQ-140 wrist-watch. After more playing around the best method seemed for me to hold the watch in my right hand which also blocked the can at the top of the ramp. I would start the watch with my thumb while raising my hand which released the can. I would stop the watch when the can hit my hand at the bottom of the ramp. I hoped that if I was consistent my reaction times would mostly cancel out. In order to have a consistent set of data (e.g. what would happen if my reaction times changed) did runs alternating an empty can, a full Cream of Mushroom can, and a full Consommé can so I would have the data for part (c) as well. I rolled each can about 10 times for a given ramp height. I did this for 3 ramp heights (h=120±5, 60±5, and 182±5mm), before realizing that my timings depended on where I was looking when I released and caught the cans, and also that by lifting my hand up to release the can, I might sometimes give the can a little push. I decided to continuously look at my catching hand from release to catch, and I pulled my hand away from the can and up to avoid pushing the can (although there is still the possibility of air movements “pulling” the empty can after my hand). I started again and did a consistent set of measurements (“Can Runs”) which are given in a table at the end of this solution set. The times (in seconds) are given for 5 runs for each type of can (Empty, Cream of Mushroom, Full Consommé). The data are presented in the order I took them, but notice that I did not just steadily increase (or decrease) the height of the ramp, to avoid the possibility of a correlated time bias (e.g. maybe I got tired and this affected how I measured the time). I was actually quite pleasantly surprised by how reproducible my measurements were; I didn’t realize my reflexes are consistent to 0.1s. If constant forces (e.g. gravity) on the cans are dominant, then the mean acceleration of the cans is just given by a=
2L t2
So since L is a fixed distance, a full aempty
t 2 empty = t full
Page 9 of 15
The date for the full and empty cream of mushroom soup is summarized in the table and chart below: a full h a (in h a empty σ( full ) Correct a empty videos (in mm) σ(h) tempty σ(tempty) tfull σ(tfull) ed 0 10 2 4.72 0.07 8.35 1.33 0.32 0.05 0.30 1 29 2 3.02 0.05 3.22 0.13 0.88 0.04 0.87 2 60 2 2.17 0.03 2.05 0.05 1.11 0.03 1.13 3 90 3 1.83 0.06 1.66 0.03 1.21 0.05 1.26 4 120 3 1.57 0.03 1.43 0.05 1.20 0.05 1.25 5 152 4 1.43 0.04 1.30 0.03 1.22 0.04 1.28 6 184 4 1.31 0.04 1.17 0.04 1.25 0.06 1.33 8 243 5 1.20 0.03 1.04 0.05 1.32 0.08 1.44 11 337 5 1.02 0.04 0.94 0.03 1.17 0.06 1.25 2.00
a(full)/a(empty)
1.67 1.33 1.00 0.67 0.33 0.00 0
100
200
300
Height of ramp (cm)
For shallow ramp heights the full can is slower than the empty can, but for steeper ramps the relative acceleration seems to flatten out approaching about 4/3. If I average the last 5 points, I get a full aempty ii)
= 1.23 ± 0.05
The empty can is a hollow cylindrical (radius R) with a moment of inertia: Page 10 of 15
I empty = mempty R 2
If the cream soup contents rotate uniformly with the can, then the contents of the can are a solid cylinder with a moment of inertia
1 I contents = mcontentsR 2 2 The total moment of inertial of the full cream of soup can is the sum of the moments of the metal shell and the contents, i.e.
(
)
(
)
1 1 1 I full = mcontentsR 2 + mempty R 2 = mcontents + 2 mempty R 2 = m full + mempty R 2 2 2 2 where m full = mcontens + mempty . There are various ways of calculating the acceleration, but I like to follow the energy. When the can rolls down the ramp (height h, length L, angle with the floor θ), the can gains both translational and rotational kinetic energy, i.e.
1 1 T = mv 2 + Iω 2 2 2 Since the can rolls without slipping, the angular velocity is ω=v/R, so
I 1 1 v2 v2 T = mv 2 + I 2 = m + 2 2 2 R 2 R The can’s kinetic energy after it has rolled a distance x down the ramp must equal the gravitational potential energy it has lost, i.e. v2 I m + 2 = mgx sin θ 2 R v2 I ∴x = m + 2 2 mg sin θ R ∴
but v ≡
dx v d2x I m = + dt mg sin θ dt 2 R2
dx , so dt a=
d2x dt
2
=
dv g sin θ = I dt 1 + mR 2
Page 11 of 15
a full aempty
1 + I empty 2 m empty R = = I full 1 + m R 2 1 + full
2 m 1 + empty R 2 m 4 empty R = 1 m full + mempty R 2 3 + mempty 2 m full 2 m full R
(
)
So in the limit where we ignore the weight of the empty can compared to the full can, we have
a full aempty
m → empty
m full
→0
4 = 1.333 3
or using my measurements for my cans a full aempty
=
4 = 1.300 ± 0.004 70 ± 5 g 3 + 0.9 ± 0.1kg
(Note: If you solve the torques with respect to the point of contact and you use the wrong moment of inertia - the one with respect to the center of the can- you get a fraction for the two accelerations that is really close to my experimental result: 1.23. This is a coincidence.) iii)
Our theoretical calculation is very close to the experimentally observed value for larger slopes, but several factors could be responsible for the difference and the odd results for small slopes: friction, a constant offset in my timings, air resistance. Air resistance would have a larger affect at higher speeds, so we would expect it to affect the higher ramp data more, which is not what we see. I investigated the vary large variations or the shallowest slope (h=10mm, which had no videos and only a thin book propping up the ramp), by trying another can. This can rolled slowly in jerks and almost always stopped and did not reach the bottom of the slope. A little playing around showed that this can preferred to rest in one orientation, apparently because its centre of gravity did not lie on its axis. Since mushroom soup has chunks, this is easy to understand if the mushroom chunks were resting on one side inside the can. The original can probably had a smaller imbalance which was enough to affect its results.
(c) The data at the end of the solution set also includes data for a consommé can.
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The consommé was clearly the fastest. The plot below shows the data for the relative accelerations of the consommé compared to both the Cream of Mushroom and the empty can.
a(consommé)/a(X)
i)
6.00 5.67 5.33 5.00 4.67 4.33 4.00 3.67 3.33 3.00 2.67 2.33 2.00 1.67 1.33 1.00 0.67 0.33 0.00
X=empty X=cream
0
100
200
300
Height of ramp (cm) If we again only consider the last 5 data points, then the average measured ratios are
aconsommé = 1.66 ± 0.09 aempty aconsommé
= 1.35 ± 0.08 acream of mushroom ii) The consommé flows freely inside the can. If friction and viscosity were zero, the can would roll down the slope but the consommé would not rotate within the can. This means that almost all the gravitational potential energy gained by the can would go into translational, not rotational, motion so it would roll with maximum possible acceleration. aconsommé =
g sin θ
I 1 + consommé m 2 consommé R
=
g sin θ
2 1 + mempty R 2 m consommé R
=
g sin θ mempty 1 + mconsommé
Page 13 of 15
aconsommé 2 = → 2 mempty mempty aempty =0 1 + m consommé mconsommé
1 m full + mempty 1 + 2 m full = mempty 1 + mconsommé
(
aconsommé acream
)
3 m + empty 2 m 3 full = → m empty mempty =0 2 1 + m consommé mconsommé
So data indicate the consommé is not quite as fast as our model, which is reasonable since the consommé undoubtedly does rotate somewhat..
Page 14 of 15
Data from Can Runs for Problem 6 (19 December 2001) height (in videos) 5 height in mm 152 ± 4
Empty (s) 1.45 1.4 1.43 1.48 1.39
Cream (s) Consommé (s) 1.31 1.11 1.33 1.09 1.28 1.06 1.26 1.07 1.3 1.08
3 height in mm 90 ± 3
1.9 1.82 1.76 1.78 1.88
1.67 1.68 1.63 1.62 1.7
1.39 1.45 1.4 1.36 1.37
1 height in mm 29 ± 2
3.04 3 3.02 2.95 3.1
3.39 3.15 3.34 3.09 3.15
2.4 2.5 2.45 2.4 2.29
6 height in mm 184 ± 4
1.36 1.32 1.27 1.31 1.27
1.13 1.21 1.19 1.12 1.19
0.95 0.97 0.97 1.05 1.07
2 height in mm 60 ± 2
2.15 2.13 2.18 2.21 2.16
2.05 2.03 2.09 1.98 2.11
1.68 1.67 1.64 1.69 1.7
4 height in mm 120 ± 3
1.57 1.53 1.62 1.59 1.56
1.43 1.41 1.37 1.49 1.47
1.19 1.23 1.22 1.19 1.24
8 height in mm 243 ± 5
1.25 1.17 1.18 1.19 1.2
1.01 1.05 0.99 1.03 1.13
0.93 0.94 0.92 0.94 0.93
11 height in mm 337 ± 5
0.99 1.03 1.06 1.01 0.96 1.05
0.91 0.95 0.91 0.91 0.99 0.96
0.79 0.82 0.78 0.84 0.9 0.88
0 height in mm 10 ± 2
4.8 4.78 4.66 4.71 4.66
8.4 7.99 6.85 10.49 8.01
3.77 3.84 3.93 3.95 3.73
Page 15 of 15
University of Toronto 2001-2002 Physics Olympiad Preparation Program Problem Set 3: Thermodynamics Due Monday 14 January 2002
1) How much heat could a heat pump pump if a heat pump could pump heat? The heat in my house (and many other old houses in Toronto) is provided by an ancient coal boiler converted to burn oil. It is starting to leak water so we are considering what to replace it with, and one possibility is an air heat pump. (Geothermal heat pumps1 work better, but our back yard is too small to fit the pipes.) As its name implies, an air heat pump pumps heat by absorbing some heat (Qout) from the outside air (temperature Tout) and supplying some heat (Qin) to the air inside the house (temperature Tin). The heatpump works cyclically and for each cycle the heat pump motor does an amount of work (W). Assume the heat pump has the maxiumum possible efficiency. (a) What is the relationship between Qin, Qout, and W? (b) How much heat will be pumped into the house for every joule of work done by the motor if the inside temperature is 20°C and the outside temperature is –20°C? (c) If the outside temperature was absolute zero, the heat pump would still pump heat into the house. How is this possible and where does the heat come from? [David] 2) Relatively heavy When I have a fever, I sometimes feel lethargic and heavy. Estimate the fractional amount I am heavier if my temperature increases from 37°C to 39°C? Hints: In addition to writing down the most famous equation in all of science, did you know that Einstein had over 40 patents on refrigerators?2 All experimental evidence is consistent with General Relativity’s principle that inertial and gravitational mass are equivalent. Assume I neither gain nor lose any molecules as I heat up, and you may make the crude approximation that I am a bag of ideal gas water molecules. (My kids sometimes think I am full of gas, but not ideal.) [David]
1 2
http://www.earthenergy.ca/tech.html http://gtalumni.org/news/magazine/sum98/einsrefr.html
Page 1 of 5
3) Jet cooled! The New Zealand physicist Ernest Rutherford1, who discovered the atomic nucleus in 1911, is often described as the greatest experimental physicist of the 20th century. His great experimental tradition still lives on in New Zealand and manifests itself in such recent accomplishments as the jet powered beverage cooler2. This cooler consists of a tank of liquid petroleum gas (LPG) in a container of water. When gas is released from the tank, the water is cooled very quickly. To avoid accidental and explosive accidental ignition of the very flammable gas, the released LPG is burned in a jet engine with a very satisfying 125dB roar. The specifications of the cooler are a bit sparse, but it appears to be able to cool about 10 litres of water from 14°C to 2°C in about 5 minutes. (a) What is the average rate (in Watts) that the cooler removes heat from the water? I don’t have any LPG in my lab (and I have enough sense not to risk explosion, fire, and deafness just to cool a drink), but I do have a 50 litre bottle of helium gas (atomic weight 4.003) at a pressure of 170 Atmospheres and a temperature of 21°C. (b) If I release gas from my bottle and allow it to expand adiabatically to atmospheric pressure, estimate how cold (in °C) will it get? (c) If I then use this cold gas to cool water, how much gas (in litres measured at 170 Atmospheres and a temperature of 21°C) do I need to release in order to cool 10 litres of water from 21°C to 9°C? Hint: Assume the water and gas are brought into thermal contact and left to adiabatically reach equilibrium. [David] 4) Perpetual motion? Of the many fascinating perpetual machines proposed through history, that proposed Johannes Taisnierus in 1570 and discussed by Bishop John Wilkins of Chester3 in his book “Mathematical Magick, or the wonders that may be performed by mechanical geometry” is one of the simplest.
The magnet fixed on the top of the column was to draw the iron ball up the ramp until the ball reached the hole in the ramp. The ball would then fall and be returned to the ramp’s base, and the procedure would begin again.4 1
http://www.nobel.se/chemistry/laureates/1908/ http://www.asciimation.co.nz/beer/ 3 http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Wilkins.html 4 A modern “perpetual motion machine” involving rolling balls and magnets can be found at http://www.theverylastpageoftheinternet.com/magneticDev/finsrud/finsrud.htm. 2
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Let’s consider an modern electrostatic version of this device, as shown in the diagram below. The electric dipole (point charges ±Q, separated by a distance d) at the upper left is aligned with the path of the ball (charge –q, mass m) rolling up the ramp. The length L (>>d) is the distance from the centre of the charged ball to the centre of the dipole. When the ball is over the hole, its centre is a distance d from the +Q point charge. d -Q
+Q
d
L ? ? -q
θ
m
trap door (lets ball move up through hole, but not down) If you really made such a device, describe and discuss what would happen when the ball is released (from the initial position shown in the figure) until it either stops or it finishes one complete cycle of an infinitely repeated loop. Does your answer depend on the ramp angle, θ, or any other parameters (possibly including parameters not listed here)? Assume that the device is as well made as is humanly possible, and feel free to make any realistic minor design changes that might improve the performance of the device. [Isamu] ?
?
5) A slow day on the Bruce Trail!1 If all else fails, a pleasant walk is better than moping in bed, and random walks are particularly important in physics. A one-dimensional random walk is when you walk along a line, and for each step the direction – forwards or backwards – is determined randomly. If the probabilities for going forwards or backwards are equal, then after N steps you will find yourself on (root-meansquare) average a distance of N steps away from where you started. This is only recommended for hikers who aren’t in a hurry to get anywhere. A classic example of a random walk process is diffusion, and the classic elementary school demonstration of diffusion is dye2 dropped into water. I think these demonstrations are great, but it is not obvious to me that diffusion is the dominant process distributing the dye in most such demonstrations. One website3 I looked at said it took 7 minutes for dye to diffuse throughout a glass of cold water, but only 4 minutes for a same-sized glass of hot water. (a) Show that such a result cannot be due to random walk diffusion. Suggest possible explanations for the reported result (or convince me that I am wrong in saying that the result cannot be due to random walk diffusion). 1
http://www.brucetrail.org/ The “dye” is usually food colouring, but you can use ink, juice, pop, …. 3 http://hastings.ci.lexington.ma.us/staff/SLee/science/molecules.html 2
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Hints: Calculate the ratio of temperatures, Thot/Tcold, of the water in the two glasses if it the results are due to random walk diffusion. The mean speed of the molecules depends on the molecular mass and on the temperature, and the mean step size is roughly given by the mean spacing between water molecules. Diffusion in 3-dimensions can be treated as 3 independent one-dimensional random walks. (b) Do the experiment yourself by dropping some dye into hot and cold glasses of water. Do your results agree with the results I report from the elementary school website? (c) Design, do, and describe an improved experiment to study the diffusion of dye in room temperature water. Is the movement of the colour in your container in your new experiment consistent with random walk diffusion? Hints: Here are some of the “improvements” I tried: I found a spot where my water containers were be safe from being moved or shaken or subject to sudden temperature changes. (I discovered the top of my fridge or a south facing window are NOT good places.) I waited for the water to be motionless and in thermal equilibrium before adding the dye. I tried adding the dye to the bottom, not the top. Chunks of leftover Hallowe’en lollipops gently dropped into the containers initially seemed to work well (and were used to bribe young children not to mess with my experiments). The candy dissolved in the water and released its colouring, but a dense bottom layer of sugar water was also created which seemed to affect the diffusion.1 Next I tried injecting food colouring into the bottom of a container with an eyedropper2, but often the dye leaked out before I reached the bottom. I also tried putting a few drops of food colouring into a tiny paper pocket which I weighted with a nickel and taped the edges shut before dropping into the container. [David]
1
Note: A good experimenter always reports any interesting effects observed, even if they don’t match initial expectations. 2 Actually a plastic straw with the top taped shut since I didn’t have an eyedropper.
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POPBits™ – Possibly useful bits of information Constants and units1,2 astronomical unit (mean earth-sun distance) atomic mass unit: (mass 12C atom)/12 Boltzmann constant
au u
149 597 870 660±20 m (1.66053873±0.00000013)×10-27 kg
k
(1.3806503±0.0000024)×10-23 J/K
elementary (i.e. electron) charge
e
(1.602176462±0.000000063)×10-19 C
electron volt
eV GN ε0
(1.602176462±0.000000063)×10-19 J
Newtonian gravitational constant permittivity of free space Planck constant solar luminosity speed of light in vacuum standard acceleration of gravity at the earth’s surface standard atmospheric pressure at the earth’s surface Stephan-Boltzmann radiation constant tropical year (2001)
h LÍ c g
(6.673±0.010)×10-11 m3/kg/s2 8.854187817×10-12 F/m (6.6260755±0.000004)×10-34 J·s (3.846 ± 0.008)×1026W 299 792 458 m/s 9.80665 m/s2
atm 101325 Pa
σ yr
(5.670400±0.000040)×10-8 W/m2/K4 31556925.2 s
Interesting facts about water Specific heat = 4186J/kg/K Freezing temperature at standard atmospheric pressure = 273.15K Elephants can smell water more than 5 km away. Great excuses for a party Birthday of the quantum3 (1900) Isaac Newton’s birthday (1642)
December 14 December 25
1
http://physics.nist.gov/cuu/Constants/index.html http://pdg.lbl.gov/2000/contents_sports.html 3 http://www.nobel.se/physics/educational/tools/quantum/energy-1.html 2
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2001-2002 Physics Olympiad Preparation Program – University of Toronto –
Solution Set 3: Thermodynamics 1) How much heat could a heat pump pump if a heat pump could pump heat? (a) By energy conservation, the work must provide the difference between the heat pumped from the outside and the heat delivered to the inside, i.e. |W|=|Qin|-|Qout|. (b) The maximum efficiency possible is for a Carnot engine, ∴
Qhot T = hot Qcold Tcold
Qin T = in Qout Tout
To convert from °C to absolute temperature in °K, we just add 273°. The heat pumped per joule of work is
Q −1 T −1 273° − 20° −1 Qin Qin = = 1 − out = 1 − out = 1 − = 7.3 273° + 20° W Qin − Qout Qin Tin (c) The motor’s work is converted to heat, so the heat pump always provides at least W heat. Rereading the question, one could argue that pump does not “pump heat into the house” (especially if the motor is inside the house), so maybe “provide heat to the house” would be better wording. 2) Relatively heavy Another easy one, as long as you remember E=mc2. Some students (equally correctly) used the heat capacity of an ideal gas or water, but this is how I did it. The total energy of a molecule is the sum of its rest mass and its thermal energy. (We can define its potential energy to be zero.) An ideal gas molecule has no internal or rotational degrees of freedom, so its only thermal energy is its kinetic energy and the total energy is 3 E = mc 2 = m0c 2 + kT 2 3kT ∴ m = m0 + 2 2c Where m0 is the rest mass of the molecule at T=0. In this approximation, my mass is just proportional to my molecules masses, so the fractional increase in my mass going from T1=37°C to T2=39°C is 3kT2 3kT1 3k m0 + − m0 + 2 (T − T ) 2 T2 − T1 m(T2 ) − m(T1 ) 2c 2 2 1 c c 2 2 = = = 3kT 3kT m(T1 ) 2 m0c 2 m0 + 21 m0 + 21 + T1 2c 2c 3k
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As is well known (i.e. you are supposed to know it) water is H2O and hence has an atomic mass of 18, so m0=18u, where u is the atomic mass unit. Thus m( 39°C ) − m( 37°C ) 39°C − 37°C = 2 − 27 m( 37°C ) 2 18 × 1.661 × 10 kg (299792458 m / s)
(
(
)
3 1.3807 × 10−23 J / K
)
+ ( 37°C + 273°K )
= 1.5 × 10−14
So the fractional mass increase is about 10-14. (You get the same order of magnitude no matter which way you do it.)
3) Jet cooled! (a) The average cooling rate is just dQ (10 l × 1kg / l)(14° − 2°)K J = (4186J / kg / K ) = 1674 = 1.7kW dt s (5 min × 60s /min) where we have used the well known density of water (1kg/litre). (b) The reason I chose helium1 for this problem is that it is a noble gas, and noble gases2 act almost like an ideal gas at normal temperatures. For an ideal gas undergoing adiabatic expansion, P(1−γ)Tγ is a constant, where P is the pressure, T is the absolute temperature, and γ=5/3 is a constant. (1−γ ) 3 −1 γ P 170 atm 5 i ∴ Tf = Ti = (273° + 21°)K = 38K = −235°C 1atm Pf (c) To cool 10 litres of water by 12°, we need ∆Q = (10 l × 1kg / l)(21° − 9°)K ( 4186 J / kg / K ) = 502320 J of heat removed. The thermal energy of an ideal gas molecule is just 3kT/2, so using the ideal gas law, our cold expanded helium gas has a thermal energy 3 2 Q = N kT = PV . 2 3 To provide the required ∆Q, we must have 2∆Q . N= 3k∆T where ∆T =9°C-(-235°C)=244K is the temperature increase of the gas as it reaches equilibrium with the water. Using the ideal gas law the volume of gas (at 170 Atmospheres and 21°C) required is 2(21K + 273K )(502320 J ) NkT 2T∆Q V= = = = 23 × 10−3 m 3 = 23l . P 3P∆T 3(170 atm × 101325Pa/atm)244 K
1 2
http://www.webelements.com/webelements/elements/text/He/key.html http://www.xrefer.com/entry.jsp?xrefid=490629&secid=.-&hh=1
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That corresponds to about 4 cubic metres at atmospheric temperature - enough helium to inflate about 300 11" balloons. Once again, many students used (equally correctly) the heat capacity of an ideal gas. My answer corresponds to using the heat capacity at constant volume, but you get 14l if you use the heat capacity at constant pressure. Rereading how I worded the question, assuming constant pressure probably does make more sense (since the gas is being released and expands in air at atmospheric pressure), but we accepted both answers. 4) Perpetual motion? When the ball is released, it will not move unless the attractive force of the dipole can overcome gravity and static friction. We will assume that we can make it such that the static friction is very small, so we must have Qq Qq − > mg sin θ 2 2 4 πε0 ( L − d 2) 4 πε0 ( L + d 2) 2 Ld Qq > mg sin θ ∴ 2 4 πε0 2 2 L − d 4 Since L>>d, this requirement becomes Qq d > mg sin θ 2πε0 L3 This is possible for appropriate charges and masses, although it may not be easy. (For example, everything would have to be constructed from insulators and the experiment done in a vacuum to prevent charged particles in the air being attracted to our charges.) Once the ball starts to move, it will roll up the ramp until it hits the barrier where it will be deflected down. (If there is just an wall perpendicular to the direction of the ball, the ball will bounce backwards with a strength proportional to the elasticity of the collision, so I added a deflector to make sure the ball goes down in the right direction.) The ball will roll (if the deflector is just right, otherwise it will bounce) around the circular return path. Because of the 2nd Law of Thermodynamics (e.g. friction, inelasticity, …) the ball will not quite get back to its initial position. (For example, even if friction is essentially zero, the ball will not have quite enough energy to raise the flap since in the initial position the flap was down, and the flap has some mass which must be raised against gravity.) Since the ball can’t quite get back to its starting point, it will roll back down the circular return path and end up oscillating back and forth with small and smaller amplitude until it finally stops.
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Note that the oscillations may include bouncing back down the ramp, but because of the energy losses, the ball never can get back to its start. The exact path depends on the amount of friction and the inelasticity of any bounces. The equilibrium position depends on the strength of the dipole. If it is strong enough to overcome gravity, the ball will stick at the top of the ramp. (It will stick the first time it reaches the top of the ramp if the collision with the wall Fdipole θ is inelastic enough, but we have assumed we are doing a good job in designing this device.) If we want the ball end up with the ball finally Fgravity stopping on the circular path, we must build the dipole such that Fdipole sin θ < Fgravity 1 Qq 1 sin θ < mg i.e. − 4 πε0 d 2 (2 d ) 2 3Qq sin θ < mg ∴ 16πε0 d 2 This requirement combined with our earlier requirement to get the ball started gives Qq d 3Qq sin 2 θ > mg sin θ > 3 2 2πε0 L 16πε0 d
∴d >
(
3 sin 2 θ 8
)
1
3
L
Since we want L>>d, this means the angle θ must be pretty small. When the ball stops on the circular path, it will stop slightly to the left of the bottom of the ramp. 5) A slow day on the Bruce Trail! A couple of comments about experimental questions on POPTOR. First, the important thing is make measurements and then to interpret them to the best of your ability. Second, in many ways, one never “completes” a real experiment, since one can always improve the experiment in some way.3 Third, it is traditional in schools and universities – and this is also the case for POPTOR – that experiments do not usually receive a weight in marking proportional to the effort required to do them well.4 (a) The density of liquid water doesn’t change very much between freezing and boiling temperatures, so the number of steps to diffuse over the whole glass is pretty much 3
An interesting book discussing these kinds of issues is “How Experiments End”, see http://www.fas.harvard.edu/~hsdept/faculty/galison/how_experiments_end.html. 4 For POPTOR, we “correct” for this by keeping a separate record of how people do on experiments.
Page 4 of 8
constant independent of temperature. i.e. The mean free path that a diffusing dye molecule travels before colliding with a water molecule is about the same for both temperatures. The change in diffusion times is dominated by the change in mean molecular velocities at different temperatures. The diffusion time, τ, is inversely proportional to the mean velocity, i.e. 1 τ∝ v Using the relationship between kinetic energy and velocity, and temperature and kinetic energy for an ideal gas we have 1 τ∝ ⇐ E kinetic = 12 mv 2 E kinetic 1 ⇐ E kinetic = 23 kT T So the diffusion time is inversely proportional to the square-root of the absolute temperature. The maximum temperature range of liquid water is from freezing (273K) to boiling (373K), so the maximum ratio of diffusion times is T τ 373 max cold = max hot = = 1.17 273 τ hot Tcold So the maximum ratio of diffusion times in water is about 1.2, which is much less than the reported ratio of 7min/4min=1.75. One obvious possible reason for the reported results is that hot water was moving more which mixes the dye faster. One might expect convection in the hot water driven by temperature differences at the water surfaces. Another possibility is that the measurement errors were very large, i.e. people aren’t very good at judging when the dye has completely diffused, especially if they have prior expectations about what should happen. The whole point of the demo was to show that diffusion is faster in hot water, and a 17% (or less) difference is not subjectively “large enough”. (b) I reproduced the experiment using identical cylindrical glass glasses 10±0.2cm tall and 6.3±0.2cm in diameter (measured with a plastic ruler). I used water from my taps and the hot water was 50±1°C and the cold water was 15±1°C, where the uncertainties are just my reading errors since I didn’t calibrate my rather cheap outdoor thermometer. 7:35pm Food colouring added to each glass. The hot water clearly had more swirling. In the cold glass the dye when mostly to the bottom and then swirled slowly. Clearly the residual motion from adding the water to the glasses had not stopped. 7:38pm Movement is clearly visible in the hot water, consistent with convection. The is only slow movement in the cold water, and the dye is in thin strands.. 7:40 The dye is almost uniformly distributed in the hot glass, but it is clearly slightly darker at the top than the bottom. More clear is a dark vertical swirl along the axis of the glass, which is wider and darker at the bottom. The cold glass is still has clearly visible strands of colour. ∝
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7:51
Can still see dark central region in hot glass, and strands in cold, but mixing is steadily progressing. 8:02 Dark central region in hot glass is fading, but still just visible. The strands in the cold water are still visible, but fading. 8:12 Both hot and cold are still fading and mixing, but otherwise the same 8:31 Both hot and cold look uniform to me. It took about an hour before both hot and cold glasses looked uniform to me, and there was clearly non-diffusive mixing at the beginning. There was no obvious difference in the diffusion. (c) I used the same glasses as in part (a), and as mentioned in the question, I let the water reach equilibrium before adding the dye. The candy dropped to the bottom was interesting in that after dissolving the dye slowly diffused up a couple of centimetres, darker at the bottom than the top, but then all movement stopped until I threw it out after a week. There was a constant 2 cm layer of coloured water at the bottom, and no sign of movement. (Within the 2cm layer it was darker near the bottom.) When I swirled the glass before throwing out the water I could clearly see the sugar (because of the change in index of refraction). My guess (and it is only a guess) is that so the candy formed a network of molecules which stopped the diffusion. The thing that worked best was a better eyedropper than the one I mentioned in the question. This one had a narrow tip and very little food colouring (my preferred dye) leaked out while I gently inserted it, and I could deposit a nice layer of dye at the bottom. The first time I did this, I put it in a window and I noticed a beautiful (but very slow) convection cycle starting with water on the warm (room) side moving up about 1.5cm after an hour, but no movement on the cold (window) side. Once put in a uniform temperature position, I observed the following: 11:30pm Food colouring (red and blue) added to two glasses. It formed a thin layer on the bottom of each glass, but initially not all the bottom was covered. 11:39pm Dye has almost covered the bottom of each glass. 11:49pm Dye covers bottom of both glasses uniformly with a layer 1-2mm thick. (The range is because it is hard to determine the top of the layer.) 12:20am Dye layers now 2-3mm thick 12:43am 4-5mm thick. 1:15am 5-6mm thick 2:03am 6-8mm thick 5:03am There is a light green-blue colour throughout the blue glass, but also a dark 5-10mm thick layer at the bottom. The red glass has a uniform pink colour througout, with a similar darker red layer on bottom. 7:15am The bulk colour is a bit darker maybe, but the dark 5-10mm thick layer at the bottom is still there. 8:00pm The liquid is uniform blue in the “blue” glass, but the red still has a darker layer on the bottom. 8:00am One glass is a unifrom blue, the other a uniform red.
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It is very hard to measure the progress, to see if it follows a square-root time spread. This is because diffusion does not have a sharp edge. Nevertheless the time scale for diffusion of the food colouring throughout my 10cm tall glasses is about a day, with the red taking a bit longer than the blue. We can’t confirm the square root time dependence, but we can estimate the diffusion time to see if it is consistent. The molecular weight of water is 18, so the density of water molecules is ρ H 2O 1kg / m 3 n= = = 3.35 × 10 25 m−3 -27 mmolecule(H 2O ) 18(1.66053873 × 10 kg) and the mean free path between collisions should be about
λ=
(
mmolecule(H 2O ) 1 = = 3.35 × 10 25 m−3 3 3 ρ H 2O n
)
−1
3
= 3.1 × 10-9 m = 3.1nm
The mean velocity of the water molecules is calculated from the kinetic thermal energy, i.e. 1m v2 2 molecule(dye)
∴ v =
= 23 kT
3kT mmolecule(dye)
For water molecules diffusing at room temperature
∴ vH 2 0 =
3kT mmolecule(H 2 0 )
=
(
)
3 1.3807 × 10−23 J / K (273 + 20K ) 18(1.66053873 × 10-27 kg)
= 637 m / s
and the mean time between collisions is
τ=
λ v
The mean number of steps to diffuse a distance L is about L 2 steps = λ and the mean time to diffuse a distance L is about
ρ H 2O L 2 L2 t = τ = = L2 3 λ mmolecule(H 2O ) λv
mmolecule(dye) 3kT
For water this is L 2 L2 L2 s t = τ = = = 5.1 × 10 5 2 L2 λ λv 3.1nm(637 m / s) m
For our 10cm tall glass we expect t = 5 × 10 5
s 2
(0.1m) 2 = 5 × 10 3 s
m So the water molecules take well over an hour to diffuse through the glass, and the heavier dye molecules should take even longer. A quick search on the web and we can find
Page 7 of 8
that most common food colourings have molecular weights5 of 400-1000, with most around 500, so scaling by the square root of the molecular mass (from the above formulae) we expect their diffusion times to be about 500 tdye ≈ 5 × 10 3 s ≈ 7 hr 18 which is quite consistent with what I saw. (Note that the diffusion time tells us the average time for a molecule to travel the 10cm, but some molecules will be slower and some faster, so it will take longer than the diffusion time to get a perfectly uniform colour.)
5
http://www.kiriya-chem.co.jp/eng/syokuyou/chem-2.html
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University of Toronto 2001-2002 Physics Olympiad Preparation Program Problem Set 4: Optics and Waves Due Monday 11 February 2002
1) Yes, Officer! A car driven by a physicist is stopped by a police officer for running a red light. The physicist explains that the light appeared yellow because of the Doppler shift. (The wavelengths of red and yellow light are about 690 nm and 600 nm respectively.) Does the physicist deserve the ticket? [David] 2) You push, I’ll pull. Radiation pressure is a very important factor for interstellar dust clouds1, comet tails2, and stellar and planetary system formation3. Once a cloud of gas and dust collapses and a star ignites, its radiation blows away smaller particles. How big must a spherical dust particle be to be currently attracted by the sun instead of being repelled. i.e. What is its minimum radius, R? Hints: Assume the dust has a typical mineral density of about density 3 g/cm3 and that the dust totally absorbs the sunlight. [David] 3) 500 channels - but nothing to watch! For years I have been waiting for my favourite TV show, Fraggle Rock4, to be available again in Toronto. (That part isn’t made up, but from here on I may wander just a teeny bit from reality.) I recently learned that the show is playing on a small station broadcasting on Channel 2 (57 Mhz).which is not on Toronto cable. The station is 50 km away, but fortunately my house is on a hill and with a good antenna I can pick up the station by direct line of sight. One day in the middle of the show I notice that the signal is fading in and out, going from maximum to minimum and back again 8 times per minute. I then learn from the radio that one of my atmospheric physics colleagues has lost control5 of a humongous balloon6 and it is currently just halfway between me and the TV station. The balloon is 20 km up and slowly rising at a 1
http://www.seds.org/messier/more/m045_i349.html http://antwrp.gsfc.nasa.gov/apod/ap001227.html 3 http://www.seds.org/messier/more/m042_hst2.html 4 http://www.henson.com/television/series/fraggle_home.html, http://home.no.net/fraggel/fragglerock.htm 5 http://www.atmosp.physics.utoronto.ca/MANTRA/press.html 6 http://www.msc-smc.ec.gc.ca/events/balloon/launch/launch17_e.cfm 2
Page 1 of 4
constant speed, and the signal I am receiving is the sum of the direct line of sight signal plus the reflected signal from the balloon. How fast is the balloon rising? [David] 4) An intense experience! Nonlinear optical materials have an index of refraction that is dependent on the intensity of the incident beam. For a linear non-linear optical material, n=n1+n2I, where I is the intensity of the beam, n is the index of refraction, and n1 & n2 are positive, real constants. Most laser beams have a Gaussian distribution of intensity in the xy plane normal to the direction of the beam, z. If for simplicity we only consider one axis we have I ( x) = I 0 e
x2 w2
1
http://www.microscopy-uk.org.uk/mag/art98/watermic.html
Page 2 of 4
Hints: The drop should be more or less circular as seen from the top. I have used shiny transparent tape, transparent food wrap, transparent CD or audio cassette cases…. (Plastic seems to work better than glass.) You want the drop to be as rounded as possible, not smeared out, e.g. like the drop on the left, not like the one on the right.
I use a drinking straw to gently deposit my drops on the plastic. Your droplet lens is fairly sensitive to vibration, so it is better to support your piece of plastic so you don’t have to hold it and you can put things underneath your lens without touching it, but this isn’t necessary. Make sure you have good lighting; I used either bright daylight or a flashlight. With a little practice, I could put samples (e.g. salt crystals) on the bottom of an audio cassette case and consistently put the right sized drop on the top which gave the right focal length. The magnification is conventionally defined as the apparent size compared to the size seen at the naked eye at a distance of 25cm from the eye. I found pieces of linear graph paper very useful for studying magnification and distortion; I darkened some of the lines to make them more visible if looked at out of focus while looking through a droplet at another piece of graph paper in focus. Overhead lights provided a convenient light source for the focal length measurement. [David]
Page 3 of 4
POPBits™ – Possibly useful bits of information Constants and units1,2 astronomical unit (mean earth-sun distance)
au u
149 597 870 660±20 m (1.66053873±0.00000013)×10-27 kg
k
(1.3806503±0.0000024)×10-23 J/K
elementary (i.e. electron) charge
e
(1.602176462±0.000000063)×10-19 C
electron volt index of refraction of water Newtonian gravitational constant solar luminosity (i.e. light output) solar mass
eV nH20 GN LÍ MÍ
(1.602176462±0.000000063)×10-19 J 1.33 (6.673±0.010)×10-11 m3/kg/s2 (3.846 ± 0.008)×1026W
speed of light in vacuum standard acceleration of gravity at the earth’s surface Stephan-Boltzmann radiation constant tropical year (2001)
c g
atomic mass unit: (mass Boltzmann constant
12C
atom)/12
σ yr
(1.98892 ± 0.00025)×1030kg 299 792 458 m/s 9.80665 m/s2 (5.670400±0.000040)×10-8 W/m2/K4 31556925.2 s
Great excuses for a party High energy Nobel birthdays: Hideki Yukawa3 (1907) Samuel Chao Chung Ting4 (1931) Abdus Salam5 (1926)
January 23 January 27 January 29
1
http://physics.nist.gov/cuu/Constants/index.html http://pdg.lbl.gov/2000/contents_sports.html 3 http://www.nobel.se/physics/laureates/1949/press.html 4 http://www.nobel.se/physics/laureates/1976/press.html 5 http://www.nobel.se/physics/laureates/1979/salam-bio.html 2
Page 4 of 4
2001-2002 Physics Olympiad Preparation Program – University of Toronto –
Solution Set 4: Optics 1) Yes, Officer! If the driver is relativistic, the police officer clearly should issue a ticket, so we can just use the non-relativistic Doppler formula for frequency (f ) or wavelength (λ): f ′ v λ v = 1 + or = 1 + f c λ′ c 690 nm v ∴ −1 = 600 nm c m km v = 0.15c = 4.5 × 10 7 = 1.6 × 10 8 ∴ s hour So the driver is clearly speeding and deserves a ticket. The exact relativistic Doppler shift differs by only a factor of 1 − (v c ) = 0.99 . 2
2) You push, I’ll pull. The momentum and energy of the photons are h hc pγ = and Eγ = λ λ The number of photons produced per second by the sun are L L λ Nγ = = Eγ hc A dust particle absorbs the momentum of any photon hitting it, so the pressure on a dust particle at a distance r from the sun is the total momentum of the photons per unit area per unit time, i.e. N γ pγ L hλ L P = = = 2 2 2 4 πr hc 4 πr λ 4 πcr The total light force on a spherical dust grain of radius R is
FL = P × Area = PπR 2 =
LπR 2
=
L R 2
4 πcr 2 4 cr 2 The mass of a spherical dust grain of radius R and density ρ is
m = 43 πR 3ρ The dust grains will be attracted to the sun when the light pressure force is weaker than the gravitational attraction of the sun, i.e. when
Page 1 of 5
∴ FL < FG ∴
L R 2
L R 2
4 cr 4 cr
2
< GN
mM r2
4 πR 3ρM < GN 3 2 2
r 3L ∴R > = 1.9 × 10−7 m 16πcGN ρM So the dust grains must be larger than 0.2µm to fall into the sun. This is why the clouds of gas and small dust are blown away when a star ignites. Another nice example of the effect of radiation pressure on interstellar dust is the Trifid Nebula (http://antwrp.gsfc.nasa.gov/apod/ap990608.html). 3) 500 channels - but nothing to watch!
r
r 2
2
x
d
d 2 2 The distance between me and the TV transmitter are fixed, so the interference indicates that the signal path length is increasing by 8 wavelengths per minute. The path length of the reflected signal is r = 2 x 2 + ( d / 2) = 4 x 2 + d 2 = 4 (vt + x 0 ) + d 2 2
2
where t=0 is defined to be when the balloon has a height x 0 = 20 km . The balloon rises with a constant velocity, v=dx/dt, so 4 (vt + x 0 )v dr d 2 = 4 (vt + x 0 ) + d 2 = 2 dt dt 4 (vt + x ) + d 2 0
We are interested in the time t=0, when the rate of change of pathlength is dr 2v = dt t= 0 2 1+ d 2 x0 The wavelength of the signal is
c 3 × 10 8 m / s 100 = = m f 57 × 10 6 / s 19 So the balloon is rising with a velocity of
λ=
2 2 1 8λ 1 dr 1 100 25 = = 0.56 m / s m 1 + 1+ d 2x = 1 + 50 km (2 × 20km) 15s 19 0 2 dt t = 0 2 60 s 16 Note: Some people used differences instead of derivatives to get almost the same answer. This works fine in this case since the change in distance is so small compared to the initial distance. v=
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4) An intense experience! (a) 1.2 1
I/I0
0.8 0.6 0.4 0.2 0 -4
-2
0
2
4
x/w The width is determined by the parameter w. (b) The beam focuses down because the index of refraction is proportional to the intensity and so is higher nearer to the axis. When passing through varying indices of refraction, light always bends in the direction of higher index of refraction. This follows from Snell’s Law or the law of optical distance given as a hint in part (c). In this case there is positive feedback (i.e. the light bends towards the region of high index of refraction which increases the intensity there which increases the index of refraction, …), so the light will form a thin beam. After it focuses, it diverges and again focuses and the focusing and diverging will be repeated in the medium, but the net result is a beam. (See, for example http://focus.aps.org/v8/st26.html.) (c) The variation in n drops over a distance w, so we can make the approximation of triangular focusing L2 = w 2 + f 2 At the edge the index of refraction is n1, while at the centre it is n1+n2I0, so using the hint that all the light meeting at a focus will have travelled the same optical distance, we have n Ln1 ~ f n1 + 2 I 0 2 where we use the average index of refraction between the centre and the edge.
w L
f
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n ∴ w 2 + f 2 n1 ~ f n1 + 2 I 0 2 2 n n2 ∴ w 2 + f 2 n12 ~ f 2 n1 + 2 I 0 = f 2 n12 + n1n 2 I 0 + 2 I 02 2 4
(
)
∴f ~
wn1
→
n1 w n2 I 0
n1>> n 2 I 0 n n1 + 2 I 0 n 2 I 0 4 (d) If n2<0, then the beam will blow up instead of focusing, since the index of refraction will now be lower in the high intensity region.
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Diameter (mm) 3.1±0.3 3.9±0.3 5±1 6±0.4 7.7±0.5 12.5±0.5 18±1 5±1 3±0.3
Focal Length (mm) 4.4±0.6 4.8±0.6 6.0±0.8 8.2±0.8 13.5±1 29±3 54±6 6.0±0.8 3.5±0.5
Magnification 40±5 32±4 30±3 22±3 17±2 9±1 4±1 30±3 40±3
(a) As you see from the above table, I found a 5 mm drop to have a focal length of 6±1mm. (b) A 5mm drop has a magnification of 30±3. (c) In general, the larger diameters gave larger focal lengths, larger field of view, and smaller magnification. There are several different kinds of distortion. When looking at graph paper, the lines were more curved near the edges for small drops, but for large drops it was less likely all the field of view would be in focus and the drop was less likely to be circular which lead to more distortion. One thing I noticed was that difference support plastics made different height drops. CD cases made very high (i.e. thick) drops, while the drops were lower on other plastics (e.g. overhead transparencies, name tag holders, …). I assume this depended on how hydrophobic the plastic is. I could also make the drops lower by waiting for them to evaporate or poking them with a paper clip. Shorter drops had longer focal lengths and less magnification. You can see a nice example of a water droplet microscope in the journal Physics Education 36 (March 2001) 97-101 (http://www.iop.org/EJ/S/1/NTO803278/abstract/0031-9120/36/2/301). If you cannot access the IOP website, it is also available (although the beautiful colour photos are only in black and white) from the Applied Spectroscopy Laboratory at the University of Indonesia at http://www.geocities.com/spectrochemical/int_papers/ip34.pdf.
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University of Toronto 2001-2002 Physics Olympiad Preparation Program Problem Set 5: Electricity and Magnetism Due Monday 11 March 2002 1) Purple Haze! My old friend Steve studies top quarks using the CDF1 experiment which has a very large superconducting solenoidal magnet with a uniform horizontal field of 1.5 Tesla. In addition to physics, Steve also loves guitars.2 Is it safe for Steve to play his guitar inside the CDF magnet? In particular, is he likely to shock himself or burn his fingers on the E4 string if it is vibrating with a peak-to-peak amplitude of 1mm? Hints: The E4 string (330 Hz) on Steve’s guitar is 66 cm long, 0.23mm in diameter, and has an electrical resistivity of 12 µΩ cm, a thermal conductivity of 80 W/m/K, a linear mass density of 0.3g/m, and a specific heat of 500 J/kg/K. (You may not need all this data, but we didn’t want to leave out anything potentially useful.) The string vibrates sinusoidally in its lowest mode, and both ends of the string are connected to pins at constant temperature (21°C). Assume that any heat produced in the string can only escape through the ends of the string or through Steve’s fingers, i.e. thermal radiation and convection are negligible. The time scale for heat flow out of the wire is much longer than the oscillation period of the wire. Thermal conductivity and electrical conductivity are completely analogous and the equations have identical form, i.e. electrical conductivity relates charge flow and voltage differences, thermal conductivity relates heat flow and temperature differences. [David] 2) Relatively electric! Near the surface of the earth there is typically a vertical electric field of about -120 V/m. (The field can reach 20 kV/m underneath a thundercloud - this is why your hair may stand on end just before you get hit by lightning - but we’ll ignore this interesting but irrelevant safety tip for the rest of the problem.) (a) Why aren’t people electrocuted by the typical 200 V potential difference between the air at their feet and the air at their head? (b) What is the typical surface electric charge density on the earth’s surface? Give your answer in units of electron charges per metre squared. (c) What is the typical surface magnetic field (magnitude and direction) at the equator produced by this surface electric charge density, as measured by a magnetic field meter just above the surface of the earth 1 2
http://www-cdf.fnal.gov/cdf.html http://webug.physics.uiuc.edu/courses/phys398/
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i) if it is attached to the earth and rotates with it? ii) just above the surface of the earth, letting the earth rotate under it? (d) Are either of your answers to the two parts of (c) consistent with the observed magnitude (typically 30µT) or observed direction of the surface magnetic field at the earth’s equator? Hints: My coordinates are such that an electric field pointing up is positive, pointing down is negative. The surface of the earth, cows, and people, all have similar (non-infinite) resistivities. Don’t worry about the fact that the meter sitting above the surface of the earth would be moving supersonically, e.g. assume it is in a jet flying in an eternal sunrise, or sunset, or noon. The north pole of a magnet is the one which is attracted to the north pole of the earth. [David] 3) Spheres within spheres An insulating sphere of radius R with uniform volume charge density of r has an electric field inside r ρ rr where r is the distance from the center of the sphere (r
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5) Refrigerator Art Experimentally and quantitatively determine the arrangement of north and south poles on a flat, flexible refrigerator magnet. Describe what you did and what your conclusions are. Hints: We are talking about the flexible magnetic cards which usually have advertising on one side and a dark smooth surface on the back. We don’t care which poles are north and which poles are south, but what is the arrangement of the poles. I figured it out just by playing around with two similar fridge magnets and making a measurement or two with a ruler, but there may be other methods. [David]
POPBits™ – Possibly useful bits of information Constants and units3,4 Boltzmann constant Earth equatorial radius elementary (i.e. electron) charge electron (or positron) inertial mass electron volt Newtonian gravitational constant permittivity of free space permeability of free space speed of light in vacuum standard acceleration of gravity at the earth’s surface Stephan-Boltzmann radiation constant
k R⊕
(1.3806503±0.0000024)×10-23 J/K
e me
(1.602176462±0.000000063)×10-19 C
eV GN ε0
µ0 c g
σ
6.378140×106 m (510998.902±0.021) eV/c2 (1.602176462±0.000000063)×10-19 J (6.673±0.010)×10-11 m3/kg/s2 8.854187817×10-12 F/m 4π×10-7 N/A2 299 792 458 m/s 9.80665 m/s2 (5.670400±0.000040)×10-8 W/m2/K4
Interesting (?) Integrals ( a is a constant)
∫ sin(ax )dx = − 1a cos(ax )
∫ cos(ax )dx = 1a sin(ax )
∫ sin2 (ax )dx = 2x − 41a sin(2ax )
∫ cos2 (ax )dx = 2x + 41a sin(2ax )
Great excuses for a party Galileo’s birthday (1564) Discovery of radioactivity (1896) 3 4
February 14 March 1
http://physics.nist.gov/cuu/Constants/index.html http://pdg.lbl.gov/2000/contents_sports.html
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2001-2002 Physics Olympiad Preparation Program – University of Toronto –
Solution Set 5: Electricity and Magnetism dx
1) Purple Haze! y This can be done using the Lorentz force x law, but we’ll use Faraday’s Law. Assuming the string is oscillating sinusoidally, the induced electromotive force produced by the string sweeping out an area A in the magnetic field B is πx dA d L a pp ϕ=B =B ∫ sin(ωt) sin dx L dt dt 0 2
app L
L L πx ω cos(ωt) cos = −2 BLa ppν cos(ωt) =B π L 0 2 where ω=2πν is the radial frequency and ν=330Hz is the frequency of oscillation. The magnitude of the peak e.m.f. is then ϕ p = −2 BLa ppν = 2(1.5T )(0.66 m)(0.001m) 330 s−1 = 0.7V This is not enough voltage to shock Steve, but is the string temperature safe?
a pp
By Joule’s and Ohm’s Laws, the power produced in a segment, dx, of the string with voltage drop dV across its resistance dR is 2 a pp πx ω cos(ωt) sin dx 2 B AB 2 a 2ppω 2 πx L dV ) ( 2 cos2 (ωt) sin 2 dx dP = = = ρ L 4ρ dR dx A where A is now the cross-sectional area of the string and ρ is it’s resistivity. The average power produced over one complete oscillation of the string is dP =
AB 2 a 2ppω 2 4ρ
2 πx
sin dx cos (ωt) = L 2
AB 2 a 2pp (2πν ) 4ρ
2
sin
2π cos2 (ωt) dt ∫ 0 dx L ∫ 02 π dt
2 πx
1 1 2 π t − sin 2 ω ( ) AB 2 a 2ppπ 2ν 2 2 πx 2 4ω AB 2 a 2ppπ 2ν 2 2 πx 0 = sin dx = sin dx L L 2π 2ρ ρ Since heat is being produced, the string will heat up. By symmetry, the middle of the string must be the hottest point, and the net heat flow in the right half of the string (in the figure) is to the right, and the flow in the left half is too the left. The string will heat up until it reaches a steady state where the heat flowing through each segment just matches the joule heating of the
Page 1 of 8
part of the string whose heat must flow out through the segment (i.e. integrated from the midpoint to the segment): x x AB 2 a 2 π 2ν 2 πx ′ dQ dT pp = −kA = ∫ dP = ∫ sin 2 dx ′ L dt dx L / 2 2ρ L /2
AB 2 a 2ppπ 2ν 2 x L L AB 2 a 2ppπ 2ν 2 x ′ L 2πx ′ x 2πx = = sin sin − − − 2ρ 2ρ 2 4 π L L / 2 2 4 4 π L (k is the thermal conductivity of the string.) 2 2 2 2 dT B a ppπ ν L 2πx ∴ = L − 2 x + sin dx 8 kρ π L The temperature profile of the string is thus x x B 2 a 2 π 2ν 2 dT L 2πx ′ pp T ( x ) − T (0) = ∫ dx ′ = ∫ L − 2 x ′ + sin dx ′ ′ dx 8 ρ π k L 0 0 2 B 2 a 2ppπ 2ν 2 Lx − x 2 − L cos 2πx − 1 = 8 kρ 2π 2 L The peak temperature of the string is for x=L/2
B 2 a 2ppν 2 L2 π 2 T ( x) = + 1 + T (0) 8 kρ 4
(1.5T )2 (0.001m)2 (330 Hz )2 (0.66 m)2 π 2 + 1 + (21K + 273K ) = 8(12 × 10 −6 Ω ⋅ 10 −2 m )(80W / m / K ) 4 = 5100 K
This means that Steve will burn his fingers, and if he tries to keep on playing with an insulated pick the string will melt (the element with the highest melting point, 3800K, is carbon, but a carbon fibre or diamond string would burst into flames first; the metal with the highest melting point, 3695K, is Tungsten, see http://www.webelements.com). I think Steve should keep his guitar out of the CDF magnet. Alternate shorter approach from Alex: The area swept out by the string is
A=
660mm
x 2 dx = 420 mm ∫ (1mm) sin 660πmm 0
and this area is swept out twice per cycle, so the average e.m.f. is
ϕ ≅B
dA = 1.5T ⋅ 2 ⋅ 330 Hz ⋅ 420 mm 2 = 0.42V dt
The resistance is
Page 2 of 8
(
)
−6 ρL 12 × 10 Ω ⋅ cm 66cm ϕ ≅ BR = = = 1.9Ω A π (0.023cm )
So the total power is
V2 ϕ ≅P= = 93mW R Half the power is lost through the end of the string (with most power being produced in the middle). Thus
P σπd 2 ∆T = 2 4 L /2 LP ∴ ∆T = = 4600 K σπd 2 ∴Tmiddle = Tend + ∆T ≈ 4900 K 2) Relatively electric! (a) People are much better conductors than air, so they short out the field so their whole body is at a constant potential. The energy density of the atmospheric “battery” is so low that only an infinitesimal surface current flows through the body. (b) The earth is a conductor, so it will have zero field inside it and a charge density on E=200V/m its surface which can be calculated from the electric field at the surface using σ Gauss’s Law. Near the surface of the earth it looks like a plane, so we can just use a E=0 Gaussian box with top (and bottom) area A, i.e. r r r r r qinsidebox r = E top ⋅ Atop + E bottom ⋅ Abottom + E sides ⋅ Asides ε0
σA = EA + 0 ⋅ A + E sides Asides cos π2 ε0 σ ∴ =E ε0 So the surface charge density is ∴
8.854 × 10−12 F / m)(−120V / m) ( σ =ε E = = −6.6 × 10 6 electron ch arg es / m 2 0
1.6 × 10−19 C / e Note that the surface charge density is negative. (c) i) In the rest frame of the earth’s surface, the magnetic field generated by the surface charge density is zero, since it isn’t moving in this reference frame. ii) In this frame, the charge is moving with a velocity
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2πR⊕ 2π 6.378 × 10 6 m m = = 464 1day 24 hours × 3600 s / hour s underneath the observer. This corresponds to a current sheet with a current density σv. By symmetry and the Biot-Savart Law, the magnetic field must run at right angles to the velocity and parallel t to B North the surface. The v Surface charges ⊗I direction of the magnetic up field can be determined L to be from south to north using the right hand rule. so we can use Ampere’s Law to calculate the magnetic field: r r ∫ B ⋅ ds = 12 µ0 Ienclosed v=
loop
∴ BL = 12 µ0 Lσv ∴ B = 12 µ0σv = 12 µ0ε0 Ev =
Ev 2c 2
= 12 β
E = c
1 2
(120V / m)(464 m / s) = 0.3 pT
(3 × 108 m / s)
2
(d) So the magnetic field is the right direction, but much too weak. 3) Spheres within spheres (a) The electric field in the region of overlap is ρr −ρr2 ρ E = E1 + E 2 = 1 + = (r1 − r2 ) 3ε0 3ε0 3ε0 Choosing sphere 1 as the origin, and defining the x axis to be the direction from the centre of sphere 1 to the centre of sphere 2, then ρ E= ( x1 − x 2 , 0, 0) 3ε0 ρ ∴E = d x√ 3ε0 (b) i)
The charges cancel where the spheres overlap, so the surface charge density is just σ(θ )= ρ t(θ ), where t(θ ) is the thickness of the uncancelled charge layer at the surface. The maximum charge thicness is obviously at t(0 )=d, σ0=ρd.
y +ρ
-ρ x r2
r1
R1
R2
L2
L1
R
θ
+ d
Using the cosine rule and the fact that the centre of each sphere is ofset by ±d/2 from their combined centre, we can calculate the thickness in the limit where d<
Page 4 of 8
t(θ ) = L2 (θ ) − L1 (θ ) =
(L2 (θ ))2 − (L1 (θ ))2
d 2 2 Rd d 2 2 Rd cosθ − R 2 + − cos(π − θ ) = R2 + − 2 2 2 2 d d ≅ R1 + cosθ − R1 − cosθ R R = d cosθ where we have used the approximation that 1 + ε ≅ 1 + 12 ε when ε<<1. So the surface
charge density is
σ = ρ d cosθ = σ 0cosθ
and the electric field strength inside the (overlapped) sphere is σ ∴ E = 0 x√ 3ε0 ii)
Outside the spheres, this charge configuration is just an electric dipole with dipole moment p = Qsphered = 43 πR 3ρd = 43 πR 3σ 0 The electric potential outside the spheres is just the dipole potential p⋅ r p V ( r) = = cosθ 3 4 πε0 r 4 πε0 r 2 so at the surface of the sphere 4 πR 3σ Rσ 0 Rρd Rσ 0 V ( r) = 3 cos = cos = cos = θ θ θ 3ε0 3ε0 3ε0 4 πε0 R 2
4) Does it (anti)matter? Conduction electrons are free to move, so if they are at rest they must feel no net force. In a piece of copper on the surface of the earth, the conduction electrons feel the force of gravity, but they don’t all fall out the bottom of the copper, so there must be a force balancing gravity. This force is an equal and opposite electrostatic force generated by the electrons themselves sagging under the force of gravity. i.e. They start to fall but this very quickly causes the bottom of the copper to become slightly negatively charged relative to the top, and the electrons will stop moving when the charge imbalance is such that it exactly balances gravity everywhere inside the conductor. The electrons natural arrange themselves for this to happen, and this is just the same explanation why the electric field inside a conductor is always zero (except they never mention that the effect of gravity is to create a tiny non-zero electrostatic field). For a positron, the electrostatic force is the equal and opposite to that on a electron, but the gravitational force is equal and in the same direction (according to General Relativity), so if the net force on a conduction electron is zero, the net force on a positron must be
Page 5 of 8
apositron = 2g =19.6m/s/s straight down. (g is the standard acceleration of gravity at the earth’s surface.) 5) Refrigerator Art Playing around with two fridge magnets, I noticed that if you stick them together like this (where the white area is the printed face and the dark area is the magnetic part), and then gently pull them like this, they make a “clack, clack, clack” sound. Studying this effect more closely it is clear that what is happening is that they start to repel each other as I pull them
and then they are attracted,
and then repelled, and so on,
,,
,
and each time they are attracted together they make a nice “clack!”. This only happens when I pull them along their long axis, i.e. like this
not when I pull them like this
Page 6 of 8
This must mean that fridge magnet must consist of many magnetic domains (essentially little dipole magnets) whose North and South poles must be arranged like
N S N S N S N S N
(I can’t tell which is north and which is south, but they must be alternating.). The positions where the fridge magnets were attracted together were at 5, 9, 13, and 17 mm, so the width of the domains is about 4 1/2 mm. (Your magnets may differ, although all the ones I tried seemed similar.) Looking from the side, the domains are probably arranged like this NSNSNS NSNSNS NSNS SNS NSNSNS NSNSNS N
(A) or less probably like this (B)
N
SS
NN
SS
NN
SS
NN
SS
N
(C) N
SN
SN
SN
SN
SN
SN
SN
S
.
It cannot be like this
since that us equivalent to this
N
S
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(assuming all domains are equal strength magnets) which would not give the correct “clack, clack behaviour. To check that arrangement (A) is correct, I held the ends of the fridge magnets close to each other. They either attracted or repelled, e.g. NSNSNS NSNSNS NSNS SNS NSNSNS NSNSNS N
NSNSNS NSNSNS NSN SNS NSNSNS NSNSNS
When I flipped them over, they did the opposite e.g. NSNSNS NSNSNS NSNS SNS NSNSNS NSNSNS N
(This took some care to see since the forces are weak and the main observable affect is not the repulsion but the fact that the magnets tend to push to one side to try to align North to South poles.) These observations are not consistent with arrangement (B), in which case the force should not change direction.
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University of Toronto 2001-2002 Physics Olympiad Preparation Program Problem Set 6: Circuits and (a bit of) Modern Physics Due Monday 8 April 2002 • You need a multimeter or ohmmeter for the last problem. If you don’t have one, ask your Physics Teacher or a friend. • Please be prompt with this problem set; we will be selecting students for our exciting POPTOR training weekend right after the due date.
1) Everyone has the capacity to learn the quadratic formula! Calculate the equivalent capacitance between points A and B of the following infinite circuit, if all the capacitors are equal and of value C.
A
B Hint: If you got rid of the two left-most capacitors (one horizontal and one vertical), what would the capacitance be of the remaining infinite chain? [Isamu] 2) I always resist getting a shock You may have noticed warnings on television sets or computer monitors not to mess about inside even if the power is unplugged. This is because there are some large capacitors inside which are charged up to high voltages – a potentially lethal combination. Large inductors can be equally lethal but in a different way. If you have a circuit with a current flowing through a large inductor, if you try to turn it off by opening a switch there can be spectacular sparks generated by huge voltages across the switch. I work on the ATLAS1 experiment at the CERN2 laboratory and we have a very large inductor. It is a superconducting solenoidal magnet 5.3m long and 1.2m in radius. The solenoid is wrapped uniformly with about 1200 turns of superconducting wire which carry a current of 8000A. (a) Estimate the magnetic field at the centre of the solenoid. 1 2
http://pdg.lbl.gov/atlas/atlas.html http://public.web.cern.ch/Public/, “Where the Web was born!”
Page 1 of 4
(b) When we talk about “how big” a resistor is, we need to specify both its resistance (in Ohms) and its power rating (in Watts). (The power rating tells us how much power the resistor can dissipate continuously without melting, blowing up, bursting into flames, … whatever.) How big a resistor would you need to put in series with the solenoid so that if you had to switch the magnet off suddenly, the voltage across the switch would not exceed 80V? (c) With the resistor in parallel, how long will it take for the voltage across the open switch to drop to a safe value of 8V? [David] 3) Cracked! After buying a new TV, I connect it to an satellite dish with a 10m long rubber coated copper wire. I then discover that I have trouble picking up any channel below Channel 19 (500MHz). (I switched to satellite after dealing with that pesky balloon in Problem Set 4, but I seem to be doomed to never manage to get Fraggle Rock.) When I measure the resistance of the wire with a DC ohmmeter I discover it has infinite resistance so I decide it must have a hairline crack of width d (see figure, not to scale). The reason the low frequency channels can’t get through is that d
10m the wire has an RC time constant of about 1/(500MHz)=2×10-9s. What is the width of the crack, d? Hints: Assume the sides of the crack are parallel and straight, and that the crack is filled with rubber. The resistivity of copper is 1.7×10−8Ω·m; the dielectric constant of rubber is κ=7. Assume all resistances and capacitances other than the wire’s are negligible. [David] 4) Gravity isn’t Bohring Some cosmologies allow the possibility that other universes exist in which the forces and constituents differ from our universe. Imagine a universe in which there is no long range electromagnetic force, so the only long range force is gravity. Assume the proton, neutron and the electron exist with exactly the same masses as in our universe, but there are no electromagnetic forces between them. A hydrogen atom would still consist of an electron bound to a proton, but it would only be bound by gravity. (a) What would be the energy difference (in electron volts) between the ground state and the first excited state of a hydrogen atom in the imaginary universe without electromagnetism? Assume the value of Planck’s and Newton’s constants (and all the other laws of physics except for electromagnetism) are the same in the imaginary universe as in our universe. Page 2 of 4
(b) Assume the current size of the imaginary universe is about the same as ours (radius ~ 1010 light years). Can gravitationally bound hydrogen atoms exist in the imaginary universe? [David] 5) Resisting the light When a light bulb heats up, does its electrical resistance go up or down? Find an unbroken 120V incandescent light bulb; any typical size (15W, 25W, 40W, 60W, 75W, 100W) will do. (If you – temporarily – take a light bulb from a lamp, make sure that the lamp is turned off, unplugged, the bulb has cooled down before removing it, and you have permission from the owner of the lamp.) (a) Measure the resistance of the light bulb when it is at room temperature. (Just use an ohmmeter or multimeter; ask your physics teacher for help if you don’t have one at home.) (b) Calculate the resistance of the light bulb when it is turned on and hot under normal operating conditions. (Do not try to measure the resistance while the light bulb is plugged into a circuit! Just calculate the resistance from the bulb’s voltage – 120V - and power rating, e.g. 60W.) (c) Assuming your measurement and calculation are accurate, answer our original question: When a light bulb heats up, does its electrical resistance go up or down? [David]
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POPBits™ – Possibly useful bits of information Constants and units3,4 astronomical unit (mean earth-sun distance)
au u
149597870660±20 m (1.66053873± 0.00000013)×10-27 kg
k
(1.3806503 ± 0.0000024)×10-23 J/K
e me
(1.602176462 ± 0.000000063)×10-19 C
(1.602176462 ± 0.000000063)×10-19 J
permittivity of free space
eV GN ε0
permeability of free space
µ0
4π×10-7 N/A2
Planck constant
h
(6.6260755 ± 0.000004)×10-34 J·s
atomic mass unit: (mass 12C atom)/12 Boltzmann constant elementary (i.e. electron) charge electron (or positron) mass electron volt (eV) Newtonian gravitational constant
(510998.902 ± 0.021) eV/c2 (9.10938188 ± 0.00000072)×10-31 kg (6.673±0.010)×10-11 m3/kg/s2 8.854187817×10-12 F/m
(4.13566727 ± 0.00000016)×10-15 eV·s proton mass solar luminosity speed of light in vacuum standard acceleration of gravity at the earth’s surface Stephan-Boltzmann radiation constant tropical year (2001)
me
(938271998 ± 38) eV/c2
(1.67262158 ± 0.00000013)×10-27 kg (3.846 ± 0.008)×1026W 299 792 458 m/s
c g
σ yr
9.80665 m/s2 (5.670400 ± 0.000040)×10-8 W/m2/K4 31556925.2 s
Great excuses for a party Most famous physics class demonstration ever (Oersted discovers that electric currents generate magnetic fields)
3 4
April 1820
http://physics.nist.gov/cuu/Constants/index.html http://pdg.lbl.gov/2000/contents_sports.html
Page 4 of 4
2001-2002 Physics Olympiad Preparation Program – University of Toronto –
Solution Set 6: Electronics and Modern Physics 1) Everyone has the capacity to learn the quadratic formula! Since this is an infinite chain, removing the two leftmost capacitors still leaves the same chain with the same total capacitance CAB.
A C
C
CAB
B From this we see that we must have 1 1 1 = + C AB C C + C AB
∴ C (C + C AB ) = C AB (C + C AB + C )
2 ∴ C AB + C AB C − C 2 = 0
−C ± C 2 + 4 C 2 −1 ± 5 =C 2 2 We can’t have a negative capacitance, so 5 −1 C AB = C 2 ∴ C AB =
2) I always resist getting a shock (a) We can estimate the magnetic field using the formula for an infinitely long solenoid: B = µ0 nI N N 1200 ∴ B ≈ µ0 I = 4 π × 10-7 2 8000 A = 2.3T L A 5.3m So the magnetic field will be about 2 Tesla. (b) Since the resistance is in parallel, no current flows in it as long as current can flow throught the superconducting (and hence zero resistance) solenoid, but when the solenoid is switched off, all the current flows through the resistor and the voltage across the resistor is the voltage across the switch. The initial current is just the current originally flowing through the solenoid, and the maximum voltage is just the initial voltage: Vmax = I 0 R and the required resistance is V 80V R = max = = 0.01Ω I0 8000 A Page 1 of 4
The peak power dissipated is P = I 0Vmax = 8000 A × 80V = 0.64 MW To be safe we should round up the size, so what we need is a 0.01Ω, 1MW resistor. This is not something you find at your local Radio Shack1. (c) The resistor and the solenoid form an LR circuit with an exponential time constant, τ=L/R. The inductance per unit length for an infinite solenoid is L = µ0 n 2 × Area l
so the inductance of the solenoid is approximately L = µ0
N2 l2
( )
πr 2 l = πµ0
N 2r2 l
and the time constant is τ=
L N 2r2 = πµ0 R Rl
= 4 π 2 × 10−7
kg ⋅ m 1200 2 (1.2 m) C 2 (0.01Ω)5.3m
2
= 154 s
The current decay is exponential, so to drop to one tenth of the original voltage will take τ 1 = τ ln10 = (154 s)2.3 = 356 s 10
i.e. about 6 minutes. 3) Cracked! The broken wire is equivalent to two resistances and the crack is a dielectric filled parallel plate capacitor, all in series. Since they are part of the same wire, their resistivity, ρ, and cross-sectional area, A, are the same. The resistances of the two pieces of wire are
L R1 = ρ 1 A
L R2 = ρ 2 A
And the total resistance is
L + L2 L R = R1 + R2 = ρ 1 =ρ A A where L=10m is the total length of the wire. The capacitance of the crack is
C = κε 0
A d
where d is the width of the crack and κ is the dielectric constant. The RC time constant is
1
Page 2 of 4
L A L τ = RC = ρ κε 0 = κε 0 ρ A d d So the crack width is
d = κε 0 ρ
L τ 10 m
= 7 ⋅ 8.854 × 10-8 F / m ⋅ 1.7 × 10-8 Ω ⋅ m
2 × 10-9 s = 5 × 10-9 m = 5nm 4) Gravity isn’t Bohring (a) The energy levels of an atom are determined by the quantization condition that orbital angular momentum is quantized in units of Planck’s constant over 2π, i.e. h Ln = mev n rn = n = nh n = 1, 2, 3,… 2π Using a simple Bohr atomic model for our gravitational hydrogen atom, the attractive gravitational force must balance the centripetal force, i.e. mem p mev n2 GN = rn rn2 Using our quantization condition and solving this gives rn =
n 2 h2 GN me2 m p
The total (potential+kinetic) energy of the electron is thus mem p E n = 12 mev n2 − GN rn mem p mem p = 12 GN − GN rn rn = − 12 GN2 =
m 2p me3 n 2 h2
−0.26 × 10−77 eV
n2 The energy difference between the ground (n=1) and first excited state (n=2) is 1 1 ∆E = −2.6 × 10−79 eV 2 − 2 = 2 × 10−78 eV 2 1 This is 79 orders of magnitude smaller than the corresponding energy in our universe (10.2eV).
Page 3 of 4
(b) The smallest radius of a gravitationally bound hydrogen atom is that of its n=1 ground state r1 =
h2 GN me2 m p
= 1.2 × 10 29 m = 1.2 × 1013 light years
This is 3 orders of magnitude large than the universe, so a gravitationally bound hydrogen atom cannot exist in the imaginary universe. 5) Resisting the light (a) I used a 15W bulb, the measured resistance was about 85±1 ohms. (b) The resistance of the bulb when operating should be R=V2/Power=(120V)2/W=960 ohms. (c) The operating resistance is an order of magnitude higher than the measured room temperature resistance, so the the resistance of the filament must increase with temperature.
Page 4 of 4
2002-2003 Physics Olympiad Preparation Program – University of Toronto –
Problem Set 1: General Due 28 October 2002 Here are a few comments before starting your first problem set. • If you want to know what kind of physics you might need to solve these problems, look at the Physics Olympiad syllabus at http://www.jyu.fi/ipho/syllabus.html. You do not have to know the syllabus by heart to do the problems, but you should recognize what you need to know so you can look it up. • Don’t forget to look at the information given in the POPBits™ section at the end of the problem set. It sometimes has information helpful or necessary for particular problems. • Pay attention to words like “estimate” or “about”. They indicate that the expected answer is not exact because either the input data is not precisely known or approximations or simplifying assumptions are necessary. Much of real physics is learning how to turn insoluble exact problems into soluble approximations. • “Nothing ventured, nothing gained.” Whether you finish a problem or not, please make sure your reasoning and analysis are clear. If you write down nothing, we can only give you zero. Your basic ideas may be right even if you make a mistake or get stuck. 1. Two Alberts AE must throw a ball so that it reaches AM after hitting the ground one time. AM remains fixed at his position. The distances shown in the diagram are in metres. (a) What should AE do to achieve the above? Ignore air resistance. (b) Is there more than one way for AE to throw the ball so that it hits the ground only once before arriving at AM? AE
10
AM 5 5
5
5
Page 1 of 2
2. Two violins How much louder are two violins compared to one violin? Suppose that each violin produces a single pitch (frequency) wave f(t)=Asin(2πνt+φ), where A is the constant amplitude, ν is the frequency, and φ is a phase factor that might vary randomly in time for each instrument. You will need to consider several cases. 3. Temperature of the planet Venus All objects emit radiation. The energy flux density J of this radiation is the energy emitted per unit time and per unit area. The Stefan-Boltzmann law is J=eσΤ4, where e=1 for a perfect radiator, σ is a constant, and T is the temperature of the object. What is the equilibrium temperature of the planet Venus? Assume that the only radiation incident on Venus is from the Sun. Assume that Venus and the Sun are perfect radiators. If your calculated T is different from the measured temperature of Venus of about 700 Kelvin, speculate on why this is the case. 4. Can the can roll? Rest an empty aluminum pop can on its side and on a table. Bring a charged object close to the can, but do not touch the can. Can you get the can to roll? If it does start to roll, why does it do so? Can you halt the can’s motion by using the charged object, but without hitting the can? Describe the charged object you used in this experiment. 5. Circuits When the following circuit is connected, how bright are the bulbs A, B, C, D, and E relative to one another? The bulbs are identical. If bulb A is removed from the circuit, what happens to the brightness of B, C, D, and E? If bulb A is replaced by two new bulbs, what happens to the brightness of the remaining bulbs? You can try this question experimentally and/or theoretically.
A
battery
D
B
E
C
POPBits™ – Possibly useful bits of information Constants and units1,2 Stefan-Boltzmann radiation constant Mean Sun-Venus distance Temperature of the Sun Radius of the Sun
1 2
σ RSV TS RS
(5.670400±0.000040)×10-8 W/m2/K4 1.08×108 km 5800 K 6.96×105 km
http://physics.nist.gov/cuu/Constants/index.html http://pdg.lbl.gov/2000/contents_sports.html
Page 2 of 2
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http://mathhelpforum.com/differential-equations/132828-general-solution.html | ## General Solution
Find the general soution
(a) $\Delta^{2} U_{n+2} + 2U_{n}=0$
(b) $\Delta^{2} x_{n+1} - 2x_{n}=0$
I have these two problem and I keep getting imaginary number for the roots and I dont know what do with it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8173896670341492, "perplexity": 513.0557548928615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860124045.24/warc/CC-MAIN-20160428161524-00141-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://www.conservapedia.com/Angular_momentum | # Angular momentum
The angular momentum of a point mass about a point is defined as $\vec H = \vec r \times \vec p$ where r is the position vector of the point mass with respect to the point of reference and p is the linear momentum vector of the point mass.
The derivative of angular momentum with respect to time is equal to the sum of the external moments applied to the system. This relation is shown by the equation $\vec {\dot H} = \vec r \times \vec F$ From this, it can be concluded that in the absence of an external moment, angular momentum must be conserved. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9925951361656189, "perplexity": 59.01910383747666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704818711/warc/CC-MAIN-20130516114658-00091-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.neetprep.com/question/6595-rate-chemical-reaction-doubles-every-C-rise-temperature-Ifthe-temperature-raised-C-rate-reaction-increases--timesb--timesc--timesd--times?courseId=19 | • Subject:
...
• Topic:
...
The rate of a chemical reaction doubles for every 10°C rise of temperature. If the temperature is raised by 50°C, the rate of the reaction increases by about :
(a) 10 times
(b) 24 times
(c) 32 times
(d) 64 times | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9283995032310486, "perplexity": 4258.796978002741}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201882.11/warc/CC-MAIN-20190319012213-20190319034213-00471.warc.gz"} |
https://infoscience.epfl.ch/record/200996?ln=en | Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flow modifications
The question of optimal spanwise-periodic modification for the stabilisation of spanwise-invariant flows is addressed. A second-order sensitivity analysis is conducted for the linear temporal stability of parallel flows U-0 subject to small-amplitude spanwise-periodic modification epsilon U-1, epsilon << 1. It is known that spanwise-periodic flow modifications have a quadratic effect on stability properties, i.e. the first-order eigenvalue variation is zero, hence the need for a second-order analysis. A second-order sensitivity operator is computed from a one-dimensional calculation, which allows one to predict how eigenvalues are affected by any flow modification U-1, without actually solving for modified eigenvalues and eigenmodes. Comparisons with full two-dimensional stability calculations in a plane channel flow and in a mixing layer show excellent agreement. Next, optimisation is performed on the second-order sensitivity operator: for each eigenmode streamwise wavenumber alpha(0) and base flow modification spanwise wavenumber beta, the most stabilising/destabilising profiles U-1 are computed, together with lower/upper bounds for the variation in leading eigenvalue. These bounds increase like beta(-2) as beta goes to zero, thus yielding a large stabilising potential. However, three-dimensional modes with wavenumbers beta(0) = +/-beta, +/-beta/2 are destabilised, and therefore larger control wavenumbers should be preferred. The most stabilising U-1 optimised for the most unstable streamwise wavenumber alpha(0, max) has a stabilising effect on modes with other alpha(0) values too. Finally, the potential of transient growth to amplify perturbations and stabilise the flow is assessed with a combined optimisation. Assuming a separation of time scales between the fast unstable mode and the slow transient evolution of the optimal perturbations, combined optimal perturbations that achieve the best balance between transient linear amplification and stabilisation of the nominal shear flow are determined. In the mixing layer with beta <= 1.5, these combined optimal perturbations appear similar to transient growth-only optimal perturbations, and achieve a more efficient overall stabilisation than optimal spanwise-periodic and spanwise-invariant modifications computed for stabilisation only. These results are consistent with the efficiency of streak-based control strategies.
Published in:
Journal of Fluid Mechanics, 782, 491-514
Year:
2015
Publisher:
New York, Cambridge University Press
ISSN:
0022-1120
Keywords:
Note:
National Licences
Laboratories: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9128966927528381, "perplexity": 3333.980521087807}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737152.0/warc/CC-MAIN-20200807025719-20200807055719-00082.warc.gz"} |
http://math.stackexchange.com/users/17341/jose-garcia?tab=activity&sort=reviews | # Jose Garcia
less info
reputation
1414
bio website location Spain , Basque Country , Bilbao age member for 2 years, 10 months seen Aug 4 at 10:10 profile views 950
i got a degree on Physics and have several ideas on my own , unfortunately i do not have a physics sponsor :) in order to get an investigation grant :)
# 40 Reviews
Aug4 reviewed Approve suggested edit on How can I calculate $AB$ in a quadrilateral figure $(ABCD)$ Jun29 reviewed Approve suggested edit on expected value and variance of the difference of number of people in a row. Jun29 reviewed Approve suggested edit on An example of an $m$-primary ideal in noetherian local domain Jun10 reviewed Reject suggested edit on Splitting fields and isomorphic Jun10 reviewed Approve suggested edit on Hex game winning strategy Jun8 reviewed Approve suggested edit on Field homomorphism into itself Jun5 reviewed Approve suggested edit on How to find $x^2 - x$? Jun5 reviewed Approve suggested edit on Evaluate h(x) = x + 1/x Jun5 reviewed Approve suggested edit on Differentiability of a series Jun1 reviewed Approve suggested edit on How many eulerian circuits are in $K_{2,n}$ ? May31 reviewed Approve suggested edit on schur index in GAP May30 reviewed Approve suggested edit on Newton's Interpolation Formula: Difference between the forward and the backward formula May29 reviewed Approve suggested edit on Subgroup Lattices and Dimension May29 reviewed Approve suggested edit on Is there a fast algorithm for computing the $(2^n)+1$ th last digit of $3^{2^n}$ in base $2$? May29 reviewed Approve suggested edit on Chance of rolling at least five 3s on fifteen dice May28 reviewed Approve suggested edit on |x|+|y|+|z|=15. How many integer solutions do exist? May26 reviewed Approve suggested edit on Hölder-Zygmund Spaces on compact sets and for integer smoothness parameters May26 reviewed Approve suggested edit on $M_R$ is finitely generated iff Every submodule of $M_R$ is finitely generated May23 reviewed Approve suggested edit on A planar graph problem May23 reviewed Approve suggested edit on Straight-line embedding planar graph | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9109195470809937, "perplexity": 3782.015267468607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500824209.82/warc/CC-MAIN-20140820021344-00213-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/3090/packages-that-need-to-be-included-in-a-specific-order | # Packages that need to be included in a specific order
This isn't a real question, so I'm marking it community wiki.
When I was new to LaTeX, I thought that the `\usepackage` command worked much like the `#include` command of the C language: just put whatever you need in the preamble, in whatever order you like.
But then bugs began to bite me and I realised, with growing horror, that commutative packages seem to be the exception rather than the rule.
They appear so often that I think it would be useful to have a list here of problematic interactions between pairs of packages, and their correct order.
Also, a question for amusement:
I want to know if there exists three packages `A`, `B` and `C`, such that `A` has to be included before `B` and `B` needs to be included before `C`, but `C` needs to be included before `A`.
-
Is that last sentence a question if three such packages exist? – TH. Sep 15 '10 at 3:59
Hmm. Better reword it. – Mateus Araújo Sep 15 '10 at 4:11
`\usepackage` does work like the `#include` directive, and if you could overwrite existing definitions in C, you'd run into the same issues that LaTeX has. – Philipp Sep 15 '10 at 9:42
At least in theory, you could have just two packages that don't "play nice": If both package `foo` and package `bar` both redefine two commands, `\jedi` and `\sith`, and you want `foo`'s `\jedi` but `bar`'s `\sith`, then no order of package loading will get you both... – Seamus Sep 15 '10 at 10:51
As much as I like using math terminology for non-math situations, this question seemed to be a bit hard to find for non-math people. I changed the title to something more directly descriptive. – Caramdir Oct 24 '10 at 23:24
Of course you have less trouble with `inputenc` in XeLaTeX, simply because you never load `inputenc`. – Philipp Sep 15 '10 at 9:43
Regarding `hyperref`: it's not the hyperref package that is to be criticized, but rather the LaTeX2e kernel, which seems to combine every possible software development antipattern in a single package. – Philipp Sep 15 '10 at 9:44 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9084912538528442, "perplexity": 1706.5015972616086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929869.17/warc/CC-MAIN-20150521113209-00043-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/spring-resonance-frequency-and-a-bit-of-fun.376022/ | # Homework Help: Spring, resonance frequency and a bit of fun
1. Feb 6, 2010
### magwas
1. The problem statement, all variables and given/known data
The Hungarian national highway M7 is made of concrete blocks 4m long. The joints have a small gap.
Józsi have a mass of 80 kg.
He owns a Polski Fiat, which is lowered by 10 cm when he sits in.
At what speed would Józsi's car disintegrate?
2. Relevant equations
v = l/t
F = -k*dx
F = m*a
g = 9.81 kg*m/s
3. The attempt at a solution
I remember I have solved it once (maybe with other numbers), but the road got some asphalt coating since.
2. Feb 7, 2010
### kuruman
I have never heard of a car disintegrating on the highway, but then again I have never driven a Polski Fiat. You need to find the resonant frequency of the car, but for that I think that you will need the mass of the car in addition to Józsi's mass.
3. Feb 7, 2010
### magwas
hooke's law:
$$F = - dy k$$
resonance frequency:
$$\omega = \sqrt{\frac{k}{m}}$$
solve hooke's law for k and substitue gravitational force of Józsi
$$k = - \frac{g m_{j}}{dy}$$
substituing k for the resonance frequency
$$\omega = \sqrt{- \frac{g m_{j}}{dy m}}$$
m here is the combined mass of Józsi and the car:
$$\omega = \sqrt{- \frac{g m_{j}}{dy \left(m_{c} + m_{j}\right)}}$$
substituing the given numbers and 800 kg for the car:
$$\omega = 2.98633250295104 \ \ \frac{1}{s}$$
now $$v = l \omega = 11.9453300118042 \frac{m}{s} = 43.003188042495 \frac{km}{h}$$
If you would have driven on M7 at that time, you would not be surprised when your car disintegrated :)
4. Feb 7, 2010
### kuruman
One slight problem. The speed should be v/T where T is the period, i.e. the time elapsed from bump to bump. How is ω related to T?
5. Feb 8, 2010
### magwas
I would think $$\omega=\frac{1}{T}$$. Each bumps should reach the car at the same phase to increase the amplitude, I guess.
6. Feb 8, 2010
### ehild
$$\omega=\frac{2\pi}{T}$$
ehild
7. Feb 8, 2010
### magwas
I see. $$\omega$$ is the angular velocity, given in $$\frac{rad}{s}$$, and not the frequency.
So $$v = l \frac{\omega}{2\pi} = 1.9 \frac{m}{s} = 6.8 \frac{km}{h}$$
8. Feb 8, 2010
### kuruman
That speed is a bit low for a car to disintegrate, but you know M7 and the Polski Fiat (and what they can do to each other) better than I.
9. Feb 8, 2010
### kuruman
This is where the problem is
If m is the combined mass supported by the spring equivalent of the car when the car is loaded, then
$$k=\frac{mg}{dy}$$
$$\omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{g}{dy}}$$
10. Feb 8, 2010
### magwas
Well, our teacher back then have given numbers which lead to some 80 km/s, presumably dy has been very different, and the mass of the car a bit (I have looked it up, 905 kg is given by manufacturer).
I always thought of it as a funny example which raises the spirit in the classroom, making understanding resonance frequency easier. Now it also serves an example of how much one can forget about highschool physics if it isn't used:)
Tried to come up with a dy to get 80 km/h. I got something around 0.6 mm.
dy is the displacement of the car as Józsi sits in, so I believe that one should use only mass of Józsi in the numerator. However to find the resonance frequency of the whole system, one should use the combined mass, which is in the denominator.
11. Feb 8, 2010
### kuruman
You are correct, of course. I don't know what I was thinking.
12. Feb 8, 2010
### ideasrule
I guess what they say about Hungarians is true...
13. Feb 8, 2010
### magwas
I am curious. What do they say?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8452653288841248, "perplexity": 1950.2935497901246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823009.19/warc/CC-MAIN-20181209185547-20181209211547-00158.warc.gz"} |
https://arxiv.org/abs/1401.6154 | Full-text links:
hep-ph
(what is this?)
# Title: Constraints on muon-specific dark forces
Abstract: The recent measurement of the Lamb shift in muonic hydrogen allows for the most precise extraction of the charge radius of the proton which is currently in conflict with other determinations based on $e-p$ scattering and hydrogen spectroscopy. This discrepancy could be the result of some new muon-specific force with O(1-100) MeV force carrier---in this paper we concentrate on vector mediators. Such an explanation faces challenges from the constraints imposed by the $g-2$ of the muon and electron as well as precision spectroscopy of muonic atoms. In this work we complement the family of constraints by calculating the contribution of hypothetical forces to the muonium hyperfine structure. We also compute the two-loop contribution to the electron parity violating amplitude due to a muon loop, which is sensitive to the muon axial-vector coupling. Overall, we find that the combination of low-energy constraints favors the mass of the mediator to be below 10 MeV, and that a certain degree of tuning is required between vector and axial-vector couplings of new vector particles to muons in order to satisfy constraints from muon $g-2$. However, we also observe that in the absence of a consistent standard model embedding, high energy weak-charged processes accompanied by the emission of new vector particles are strongly enhanced by $(E/m_V)^2$, with $E$ a characteristic energy scale and $m_V$ the mass of the mediator. In particular, leptonic $W$ decays impose the strongest constraints on such models completely disfavoring the remainder of the parameter space.
Comments: 10 pages, 7 figures Subjects: High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Experiment (hep-ex); Nuclear Theory (nucl-th) Journal reference: Phys. Rev. D 90, 073004 (2014) DOI: 10.1103/PhysRevD.90.073004 Cite as: arXiv:1401.6154 [hep-ph] (or arXiv:1401.6154v1 [hep-ph] for this version)
## Submission history
From: David McKeen [view email]
[v1] Thu, 23 Jan 2014 20:30:59 GMT (152kb,D) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8892703652381897, "perplexity": 1252.2411082029193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218187227.84/warc/CC-MAIN-20170322212947-00158-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3064820/when-given-expected-value-is-finite | # When given expected value is finite?
I consider the stochastic process $$(X_t)_{t\geq 0}$$ that satisfies: $$dX_t = a(t, X_t) dt + b(t, X_t) dB_t.$$ We assume that solution of the above SDE is unique non-exploding solution, e.g. $$(X_t)_{t\geq 0}$$ can be a Brownian Motion $$\left(a(t, X_t)=0 \text{ and } b(t, X_t) = 1\right)$$ or geometric Brownian motion $$\left(a(t, X_t)=r X_t \text{ and } b(t, X_t) = \sigma X_t\right)$$.
Let us consider the expected value $$E[e^{-\int_t^T X_sds}].$$
I wonder when this expected value is finite. I suppose that for some functions $$a(t, X_t)$$ and $$b(t, X_t)$$ I can encounter inifite expected value (I suppose the problem arises when $$(X_t)_{t\geq 0}$$ will be 'very' negative). My question is: Are there any necessary conditions on $$a(t, X_t)$$ and $$b(t, X_t)$$ which guarantee that the above expectation is finite. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9961092472076416, "perplexity": 140.678799954447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256147.15/warc/CC-MAIN-20190520202108-20190520224108-00087.warc.gz"} |
https://www.physicsforums.com/threads/direction-of-electric-field-lines.400053/ | # Direction of electric field lines
1. May 1, 2010
I have an intuitive understanding of electric fields but when it comes to defining it on a test I wouldn't know what to say. Is the electric field defined as the force produced by a charged object on a positive test charge or any other charged object? Also if you were told to draw the electric field lines emanating from a charged object on its own (as opposed to acting on another charged object) would you just assume its in relation to a positive test i.e. lines pointing outward for a positively charged object lines pointing inward for a negatively charged object.
2. May 2, 2010
### Born2bwire
There is probably a more official, technical answer, but I would just say that the electric field is the vector force acting on a positive test charge. The Lorentz force for the electric field is simply qE (where E is a vector). So the field will point in the same direction as the force for positive charge. The value of your test charge is going to be dictated by your units. For example, we could just come up with a sytem of units so that the electric field has units of N/C. Thus, we would use a test charge of 1 C (or any positive charge in C but you divide the measured source by the charge's value) and directly get the electric field by measuring the force in Newtons.
Similar Discussions: Direction of electric field lines | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9069352746009827, "perplexity": 187.71809312119518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105451.99/warc/CC-MAIN-20170819124333-20170819144333-00027.warc.gz"} |
https://math.stackexchange.com/questions/3329066/rational-functions-from-integers-to-integers | # Rational functions from integers to integers
A polynomial with integer coefficients maps integers to integers. A rational function (which is a quotient of two polynomials) tends to map integers to rational numbers. Other than the trivial case where the denominator is a factor of the numerator, are there any rational functions that map all integers to integers? What is known about them and how can they be found?
Let $$r(z) = \frac{p(z)}{q(z)}$$ such that $$r:\mathbb{Z}\rightarrow\mathbb{Z}$$ and $$p,q$$ polynomials. Then $$q(z)|p(z)$$ for all $$z \in \mathbb{Z}$$ and $$\deg p \geq \deg q$$ (check what happens at $$\infty$$).
You can check here, for example, how the division between polynomials over a commutative ring (like $$\mathbb{Z}$$) works.
So if $$q$$ is monic, then you can divide the polynomials, so $$p(z)=f(z)q(z)+g(z)$$ with $$f,g$$ polynomials and $$\deg g < \deg q$$. Then if $$g \not \equiv 0$$, $$r(z)=f(z)+\frac{g(z)}{q(z)}$$, so $$\frac{g(z)}{q(z)}$$ is also an "integer" rational function, which is absurd. So $$g\equiv0$$ and $$r$$ is a integer polynomial.
If $$q$$ is non-monic, you can apply the theorem of the linked answer: call $$q_0$$ the leading coefficient of $$q$$, then $$q_0^kp(z) = f(z)q(z) + g(z)$$. Consider the integer rational function $$q_0^kr(z) = \frac{q_0^kp(z)}{q(z)} = f(z)+ \frac{g(z)}{q(z)}$$. As above, this implies that $$g(z) \equiv 0$$. So $$r(z)=\frac{p(z)}{q(z)}=\frac{q_0^kp(z)}{q_0^kq(z)}=\frac{f(z)q(z)}{q_0^kq(z)}=\frac{f(z)}{q_0^k}$$. Then again $$r$$ is a integer polynomial. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 25, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9614641070365906, "perplexity": 116.80715414673455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039626288.96/warc/CC-MAIN-20210423011010-20210423041010-00464.warc.gz"} |
https://puzzling.stackexchange.com/questions/57583/how-many-tries-to-roll-a-6/55008 | # How many tries to roll a 6?
Suppose you roll a (fair, 6-sided, perfectly ordinary) die repeatedly until you roll a 6. As is well known, the expected (i.e., long-term average over many trials) number of rolls required is 6.
Now suppose we ask this question conditional on never having rolled any odd numbers. That is, suppose you learn that someone has followed the procedure above, and that they rolled only even numbers up to the point where they rolled that 6. What is the expected number of rolls they took?
(That is: What is the expected number of rolls taken to roll a 6, conditional on having got no odd numbers in the process of rolling that 6? The question is not anything like "what's the expected number of rolls next time, given that you didn't get any odd numbers last time?". I've added this paragraph and slightly reworded the question above because it seems from comments that some readers interpreted the question in an unintended way.)
This question is interesting because there is an extremely tempting wrong answer.
Some history: The question seems to have first been asked by Elchanan Mossel while teaching an undergraduate probability course at the University of Pennsylvania in 2014-15. Gil Kalai posted it on his blog as part of his Test Your Intuition series, and then posted a followup. There is interesting discussion in the comments of both. And here is a nice little article about the question, in PDF form.
WARNING: All the links in the paragraph above may take you to places that give away some or all of the solution. If you want to solve the puzzle for yourself, don't follow those links until you've done it.
[See this Meta question for some relevant PSE-specific context, which in particular explains why this is community wiki. An older question has been merged into this one, and it's possible that some details of answers or comments may look odd as a result.]
• This is a poorly formulated puzzle. The problem is that we are first given a random number generator with specific properties. The formulation strongly implies that we have a fair six-sided die. We know the properties of such a die: when rolled it will give a result of 1 to 6, each with an equal 1/6 probability, and the result is independent of previous results. But then you go on to say that when rolling this particular die, the distribution of results will not be that, and you are not specific why the die will have a non-standard result distribution. (continued) – MichaelK Sep 11 '17 at 13:42
• In other words: we cannot be sure what the distribution function of this die actually is. Why are we getting only evens? Is the die unfair so that it only rolls evens? Is someone excluding results when odds are rolled? Do we reset the throw count when odds are rolled? Is there any other reason for the non-uniform result distribution? So it is a poorly formulated puzzle, and I hold the opinion that we do not have the information we need to answer it. – MichaelK Sep 11 '17 at 13:44
• @MichaelKarnerfors The problem is completely unambiguous as long as you are familiar with conditional probability. If X is a random variable and A is an event, then the expected value of X conditioned on A is $\sum_{n=0}^\infty nP(X=n\cap A)/P(A)$. Here, X is the number of rolls and A is the event that all rolls are even. This is equivalent to both your "exclude roll counts" and "reset throw count" interpretations. – Mike Earnest Sep 11 '17 at 15:40
• Yes. To define the event A and random variable X, you must first define the sample space, S, that they live in. Define S to be the set of all finite sequences of die rolls where the last roll is 6 and no previous rolls are 6. Then A is the subset of such sequences which are all even, and X is the function which returns the length of a sequence. This is how the stopping condition is taken into account. @MichaelKarnerfors – Mike Earnest Sep 11 '17 at 16:57
• The question does not imply any such thing. That isn't how conditional probability works. It may -- or may not -- be instructive to consider the following question. A fair die is rolled until it comes up 6. Conditional on its having come up 6 on the first roll, how many rolls did it take? Of course the answer to this question is 1, even though it's a fair die and only comes up 6 one-sixth of the time. And yes, coming up 6 on the first roll would be "just a stroke of luck". This is a question about probability, after all. – Gareth McCaughan Dec 3 '17 at 2:59
112 rolls
...because the question is equivalent to...
What is 1 more than the average number of consecutive rolls of only 2s and/or 4s before any other roll terminates the series?
One way to see this equivalence is to note that only 2/4 streaks terminated by 6 are considered. (A first-roll 6 terminates a 2/4 streak of length 0.) Yet the average length of those streaks is the same as of streaks terminated by 1, 3 or 5. Collect all such streaks, including those terminated by 6, and the result is all streaks of 2/4, each with the same expected length.
Calculations:
E = expected number of consecutive even rolls up to and including the first 6
= (expected number of consecutive 2s and/or 4s) + (one more roll of 6)
= T + 1
where
T = expected number of consecutive 2s and/or 4s before some other roll
= (probability of 2 or 4) × (1 for this roll + expected subsequent 2s and/or 4s)
= 13 (1 + T)
so
T = 12
thus
E = T + 1
= 112
• Note to possibly add in a future edit: The expected number of rolls is surprisingly small, but that is because most often it will include just 1 roll, a lone 6, as odd numbers are more likely than non-6 even numbers to precede any 6. – humn Sep 10 '17 at 18:09
• Why is the probability of a 2 or 4 only 1/3? Surely if you have been told that all throws were not odd that leave the chance of it being a 2 or 4 as 2/3? – Chris Sep 11 '17 at 14:38
• Ah... So I think in my interpretation rolls of 1,3,5 are strictly forbidden to occur whereas in your interpretation they may occur but cause the sequence to be discarded and discounted from the results? – Chris Sep 11 '17 at 15:22
• Sigh, @Chris, all too often the primary puzzle is to figure out what is the actual intended puzzle. Many solutions, too, only make sense to the solver and the puzzle's poser, until comments help point out how much was incorrectly assumed to have been clear. I'm looking forward to adding "discussion" to this solution thanks to your comments. – humn Sep 11 '17 at 15:30
• As Mike mentioned on the comments to the main question, @Chris, this is not at all ambiguous if you're familiar with conditional probability, but what it basically means is - someone rolled a fair die until they rolled a 6. If I give you the extra information that only evens were rolled, how does this change your expectation of the number of throws? (this does not change the fairness of the die, or the rolling procedure. it's just extra information about how this particular roll went) – ffao Sep 11 '17 at 21:31
This surprisingly beguiling puzzle may also be solved with a surprisingly unsophisticated approach. Symmetry, by itself, predicts the average length of evens-only sequences ending with 6 to be...
...1 12 throws.
Start with T many random throws: 2153664315121226553111444142566363625461525 . . 3644464461
Sift them into 4 groups that, due to symmetric criteria, are each expected to include T4 throws.
Every 1 and any preceding 2s/4s: 21......1.121......1114441.............1... . . .........1
Every 3 and any preceding 2s/4s: ...3..43..........3............3.3......... . . 3.........
Every 5 and any preceding 2s/4s: ..5......5......55........425......25...525 . . ..........
Every 6 and any preceding 2s/4s: ....66.......226.............66.6.6..46.... . . .64446446.
The last group, of 6s and their preceding 2s/4s, precisely contains every evens-ending-with-6 sequence considered by the puzzle. The number of such sequences is just the original number of 6s, as every 6 is in exactly one such sequence. From 6’s symmetry with the other 5 equally-likely throws —1, 2, 3, 4 and 5 — that number is expected to be T6 sequences.
Thus...
(Average length) = (number of throws in these sequences) / (number of sequences)
= T4 throws / T⁄ 6 sequences
= 1 12 throws /sequence.
Notes
The expected number of rolls seems suspiciously small but, as seen among the example throws above, it reflects the likelihood that most evens-ending-with-6 sequences will be just 1 roll long, a lone 6. This does make sense after all by noting that each sequence-ending 6 has 46 chance of following a 1, 3, 5 or 6(!) that allows no prior rolls to count in that 6’s sequence.
The condition “all throws gave even numbers” is a potent red herring that suggests 6s have equal status to 2s and 4s. Their status is not equal, however, after retaining all 6s originally rolled but discarding the 34 of 2s/4s that did not lead to 6s.
This solution’s approach does have a loose end, literally, as commented by Artur Kirkoryan. Any stretch of T throws could end with a streak of 2s/4s that might extend to become a qualifying sequence of any length if allowed to continue. That comment and Deusovi’s reminder to take density into account lead to the realization that possible 2s/4s loose ends would need an infinite expected length to make a difference because any bounded expectation would have a vanishing effect as T increases without limit. This further leads to acknowledgment that the evens-ending-with-6 sequences in question are only assumed, not proven, to have a finite expected length. Presumption of their nonetheless having a bounded expectation forces any 2s/4s loose ends to also have a bounded expectation and thus not undermine this solution’s approach.
This solution ensued from surprise at the match between the 4 kinds of non-2s/4s throws and the 14 fraction of undiscarded throws exhibited by a Lisp routine’s simulating a million throws at a time.
( defun RollEm (many)
( let ( (i many) (sixes 0) (streakSum 0) (streak 0) )
( while (>= (setq i (1- i)) 0)
( pcase (1+ (random 6))
( 2 ( setq streak (1+ streak) ) )
( 4 ( setq streak (1+ streak) ) )
( 6 ( setq sixes (1+ sixes) )
( setq streakSum (+ streakSum streak 1) )
( setq streak 0 ) )
( else ( setq streak 0 ) )
) )
( insert ( format "\n%d (%d%%) undiscarded throws / %d streaks = %.3f\n"
streakSum
( / ( + (* 200 streakSum) many) (* 2 many) )
sixes
( / (float streakSum) sixes )
) ) ) )
(RollEm 1000000) 249372 (25%) undiscarded throws / 166527 streaks = 1.497
"" 250234 (25%) undiscarded throws / 166972 streaks = 1.499
"" 249503 (25%) undiscarded throws / 166256 streaks = 1.501
"" 250777 (25%) undiscarded throws / 166953 streaks = 1.502
"" 249947 (25%) undiscarded throws / 166791 streaks = 1.499
• Wow, what a brilliant solution! – Deusovi Sep 12 '17 at 16:26
• This is very nice analysis and I upvoted the answer, but I am not convinced the argument is very rigorous. We still have to argue that the various halted 2-4 sequences are represented the same way in the long term, and justify taking T to infinity. I assume there is a theorem which states something like this, e.g. "if we have a sequence of experiments one after another, grouped by outcomes, then the statistical length of the experiment in group X, converges to the truth average, under certain conditions". Once again, I like this argument, but I think I like your first solution more:) – Puzzle Prime Sep 14 '17 at 19:10
• True about not treating infinity rigorously here, @Artur Kirkoryan. I wanted to disagree but remembered the simple example of comparing the population counts of primes versus whole numbers. Finite comparisons produce increasingly large disparities in favor of wholes whereas an infinite comparison comes out perfectly equal. – humn Sep 14 '17 at 19:23
• "An infinite comparison comes out perfectly equal"? Only in terms of cardinality. In terms of natural density (which I'd argue is more natural (pun completely intended) in this case), the natural density of the primes is 0, which seems perfectly consistent with disparities increasing as $n\to\infty$. – Deusovi Sep 17 '17 at 23:40
• I might understand your comments even better now, @Artur K and $\raise-.2ex\unicode{x40}$Deusovi, and have added a Note about them. – humn Sep 19 '17 at 20:06
Let $X_n$ be the event that the dice takes $n$ rolls to get the first 6, given all the rolls are even. Let $A_n$ be the event that it takes $n$ rolls to get the first 6, and let $B$ be the event that all rolls up to the first 6 are even.
$P(X_n)=P(A_n|B)=\dfrac{P(B|A_n)P(A_n)}{P(B)}$ (using Bayes' theorem)
Now:
$P(A_n)=\dfrac{1}{6}\cdot\left(\dfrac{5}{6}\right)^{n-1}$ (since the first $n-1$ rolls are not 6 and the last roll is)
$P(B)=\displaystyle\sum^\infty_{i=0}\dfrac{1}{6}\cdot\left(\dfrac{1}{3}\right)^{i}$ $=\dfrac{1}{6}\cdot\displaystyle\sum^\infty_{i=0}\left(\dfrac{1}{3}\right)^{i}$ $=\dfrac{1}{6}\cdot\dfrac{3}{2}=\dfrac{1}{4}$ (since at each stage there is a 1 in 3 chance that the roll is 2 or 4 and a 1 in 6 chance of getting a 6 at each roll)
$P(B|A_n)=\left(\dfrac{2}{5}\right)^{n-1}$ (since the last roll is even but for all the other rolls there are 2 even outcomes and 3 odd outcomes)
So $P(A_n|B)$ $=\dfrac{\dfrac{1}{6}\cdot\left(\dfrac{2}{5}\right)^{n-1}\cdot\left(\dfrac{5}{6}\right)^{n-1}}{\dfrac{1}{4}}$ $=\dfrac{2}{3}\cdot\dfrac{2^{n-1}\cdot5^{n-1}}{5^{n-1}\cdot6^{n-1}}$ $=\dfrac{2}{3}\cdot\left(\dfrac{1}{3}\right)^{n-1}$
Now, we wish to find
\begin{align} \sum^{\infty}_{i=1}i\cdot P(X_i) &=\sum^{\infty}_{i=1}i\cdot\dfrac{2}{3}\cdot\left(\dfrac{1}{3}\right)^{i-1} \\ &=2\cdot\sum^{\infty}_{i=1}i\left(\dfrac{1}{3}\right)^{i} \\ &=2\cdot\sum^{\infty}_{i=0}\left(\left(\dfrac{1}{3}\right)^{i}\cdot\sum^{\infty}_{j=1}\left(\dfrac{1}{3}\right)^{j}\right) \\ &=2\cdot\sum^{\infty}_{i=1}\left(\dfrac{1}{3}\right)^{i}\cdot\dfrac{3}{2} \\ &=3\cdot\sum^{\infty}_{i=1}\left(\dfrac{1}{3}\right)^{i} \\ &=\dfrac{3}{2} \end{align}
So it takes on average 1.5 rolls for the dice to get a 6, given that all the rolls are even.
Gamow wants me to keep my previous solution: if you want to see it, here it is. It is wrong because the definition of $B_n$ is faulty - for example, a sequence such as 26143 isn't counted in $B_5$ when it should be, and a sequence such as 24(16) is counted in $B_2$ when it shouldn't be, since we are given that all the rolls up to the first six are even, not that the rolls up to a certain point are even.
• Could you explain how ∑[1;∞]i(1/3)^i = ∑[0;∞]((1/3)^i⋅∑[1;∞](1/3)^j) ? thx – sousben Sep 10 '17 at 17:20
• @Sousben We are adding 1/3+2/9+3/27+4/81…=1/3+1/9+1/27+1/81…+1/9+1/27+1/81+1/243…+1/27+1/81+1/243+1/729…+1/81+1/243+1/729+1/2187……=1x(1/3+1/9+1/27+1/81…)+1/3x(1/3+1/9+1/27+1/81…)+1/9x(1/3+1/9+1/27+1/81…)+1/27x(1/3+1/9+1/27+1/81…)… – boboquack Sep 10 '17 at 21:17
• Can you explain P(B) a bit more? I'd have expected it to depend on N since you are considering the chances of all N rolls being even. ie it would just be 1/3*(2/3)^n-1. – Chris Sep 11 '17 at 14:45
• @Chris the reason why it's not the chance of all n rolls being even is because that counts configurations such as 26, 264, 2642 etc as part of $P(B)$ multiple times where in reality they should only be counted once as 26. – boboquack Sep 11 '17 at 21:16
• Great answer! I wish you would have kept the first version, where the result was "3 rolls on average". (That's also the answer I came up with, and with which most other people come up. It takes some time to accept that this is wrong.) – Gamow Sep 12 '17 at 6:24
$\frac{3}{2}$
This is computational calculation, so it is not statistical answer.
Here is the simple code
It is for the people who try to find it probabilistically.
Here is the probabilistic solution:
First of all, we know that the probability of getting 6 on the first roll is $\frac{1}{6}$, then getting 6 after an even roll is $\frac{2}{6}\frac{1}{6}$, and so on as the equation below:
$P_n=\left (\frac{1}{6} \right )\left (\frac{2}{6} \right )^{n-1}$
so if we calculate all possible solutions and sum them we will get the probability set among all other possibilities (such as getting odd after getting some even):
$\sum_{1}^{\infty }\left (\frac{1}{6} \right )\left (\frac{2}{6} \right )^{n-1}=0.25$
So if we multiply this by $4$ we will normalize the probability and find the actual expected value of getting $6$ given that all the rolls are even as below such as;
1. Getting $6$ in the first roll expected value:
$E_1=4\left (\frac{1}{6} \right )\left (\frac{2}{6} \right )^{1-1}=\frac{2}{3}$
1. Getting $6$ in the second roll (requires not getting $6$ in the first roll)
$E_2=\left [1-4\left (\frac{1}{6} \right )\left (\frac{2}{6} \right )^{1-1} \right ]+4\left (\frac{1}{6} \right )\left (\frac{2}{6} \right )^{2-1}=\frac{5}{9}$
In general, we can generalize the formula as below,
$E=\sum_{1}^{n}(1-A_{n-1})+B_n$
$A_{n}$ represents the sum of previous normalized probabilities and $B_{n}$ is the normalized probability of getting 6 at that roll.
firstly:
$A_n=\sum_{1}^{n}(4\times\frac{1}{6}\frac{2}{6}^{n-1})=1-3^{-n}$
secondly,
$B_n=4\times\frac{1}{6}\frac{2}{6}^{n-1})=2\times 3^{-n}$
so
$E_{2,\inf}=\sum_{2}^{\inf}(1-1-3^{1-n})+2\times 3^{-n}=\frac{5}{6}$
lastly, we are going to add the first result since the general sum will work after 1:
$E=E_1+E_{2,\inf}=\frac{2}{3}+\frac{5}{6}=\frac{3}{2}$
So if we sum all possible outcome expected values, we will get the actual expected value of getting $6$ given that all the rolls are even:
$E =\sum_{1}^{\infty}E_n=\frac{3}{2}$
• Whether your code is correct or not is hard to tell without comments, but it's easy to see that the output is 1 greater than the value you've given here. – Peter Taylor Sep 10 '17 at 8:22
• If all the rolls are even, the probability of getting 6 tends to 1, so I don't follow your argument. – boboquack Sep 10 '17 at 9:48
• @boboquack it is not actually ignoring getting odd rolls. just i normalize the probability. i will explain it more after i arrive home – Oray Sep 10 '17 at 9:52
• In any case, the probability of getting 6 tends to 1, and after normalising should be 4, then...? – boboquack Sep 10 '17 at 9:53
Somewhat related to Oray's solution.
$\frac{3}{2}$
Why?
The average chance to get a 6 can be modelled using an infinite sum:
$$\sum_{n=0}^\infty \frac{1}{3} \times \frac{2}{3} ^ r$$
Where $\frac{1}{3}$ is the chance to get a 6, $\frac{2}{3}$ is the chance to not get a 6 and $r$ is the number of fails.
This is equivalent to a geometric series with $a_0 = \frac{1}{3}$ and $r=\frac{2}{3}$
\begin{align}S_\infty &= \frac{ar}{1-r}\\ &= \frac{\frac{1}{3} \times \frac{2}{3}}{1-\frac{2}{3}}\\ &= \frac{\frac{1}{3} \times \frac{2}{3}}{\frac{1}{3}}\\ &= \frac{2}{3}\end{align}
The average number of rolls needed is just $\frac{1}{S_\infty} = \frac{1}{\frac{2}{3}} = \frac{3}{2}$
• Where do you get S∞=ar/(1−r) from? I read that lim N→∞ = a(1−r^(N+1))/(1−r) here: math.stackexchange.com/questions/29023/value-of-sum-limits-n-xn in which case the result would be 1 throw which is obviously wrong. – sousben Sep 10 '17 at 13:10
• @sousben wait never mind, this is not right - I need to divide by number of rolls too, I mean it's only logical the chance to get 6 is 100% in the long run – somebody Sep 10 '17 at 13:27
• Thanks it all makes sense now, I was making the same mistake as you and couldn't understand my result... – sousben Sep 10 '17 at 13:30
• This works out in this case, but I'm not sure that it works generally. If we get rid of the evenness condition (i.e., if we wanted the number of throws before seeing a 6 when throwing a dice normally and including all throws), I believe this approach would give 6/5? And it would give 2 if we discarded all throws that weren't, say, either 1 or 6. Seems like in the end it just inverts the probability to not get a 6 on one roll. – Matthew Read Sep 10 '17 at 18:00
Going by definition of what is the expected number we get
$$\sum_{n=1}^{\infty} n \cdot 1/3 \cdot (2/3)^{n-1} = 3$$
where 1/3 part is the probability to get "6" after reaching the n'th roll and $(2/3)^{n-1}$ - probability to reach that step.
Edit: well, it looks like the "reroll" [only the steps that produce odd number] and "reset" [the whole sequence] interpretations are not equal and the later is the intended meaning of the question. Under the later the probability to roll "6" on each step is in fact
not equal to 1/3 as in the first case.
Let $p_1$ be the conditional probability to roll "6" (equal for all steps because after roll of "2" or "4" all outcomes are still conditionally as probable as before the roll, just with the chain of rolls being longer by one). Conditional probability to roll "2" or "4" would be of course $1-p_1$.
Let "A" be the event that the first roll is "6", "B" be the event that the sequence ended on "6" and did not contain any odd numbers.
\begin{align} p_1 & = P(A|B) \\ & = P(A \& B)/P(B) \\ & = (1/6)/(P(B)) \end{align} $$P(B)= \sum_{n=1}^{\infty} 1/6 \cdot (2/6)^{n-1}=1/4$$ $$p_1=(1/6)/(1/4)=2/3$$ which is indeed rather counterintuitive.
The final answer in this approach:
$$\sum_{n=1}^{\infty} n \cdot p_1 \cdot (1-p_1)^{n-1}=\sum_{n=1}^{\infty} n \cdot 2/3 \cdot (1/3)^{n-1}=3/2$$
I was struggling to get my head around this so I decided to test it programatically to see if I had misunderstood the question. Here's the code:
#!/usr/bin/perl
use strict;
use warnings;
my $die_size = 6; my$number_of_runs = 100000;
my $total_count = 0; sub count_rolls { my$number_of_rolls = 0;
my $number_rolled = 0; while ($number_rolled != $die_size) {$number_of_rolls++;
$number_rolled = 1 + int rand($die_size);
if ($number_rolled % 2 == 1) { return count_rolls(); } } return$number_of_rolls;
}
for (1..$number_of_runs) { my$roll_count = count_rolls();
$total_count +=$roll_count;
}
my $mean_count =$total_count / $number_of_runs; print "$mean_count\n";
Try it online!
The result this gives is:
1.5
Hopefully this is helpful to anyone else who is a bit confused here. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8862106800079346, "perplexity": 834.0437446550504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703547333.68/warc/CC-MAIN-20210124044618-20210124074618-00007.warc.gz"} |
https://trigonometricfunctions.com/ | # Trigonometric Functions
Welcome to trigonometric functions. Here we discuss the mathematical functions of angles, often simply called trig functions.
Apart from similarity, the shape of a polygon with three edges and three vertices, a triangle, is completely determined by the angles, because similar triangles keep the same ratios between their sides.
As the angles of a triangle add up to 180°, in a triangle which has one angle of 90 degrees, the sum of the two acute angles is 90 degrees.
In other words, the third angle is fixed, provided that the two other angles are known.
Moreover, if we know the angles, then the ratios of the sides are determined, irrespective of the overall size of the triangle.
And in case the length of any single side is known, then the other two sides can be calculated.
It follows that all other angles and lengths are determined, once the value of a single angle and the length of a single side and are known.
The aforementioned ratios are given by the trigonometric functions which you can find below.
The following depiction of a triangle shows the hypotenuse h, the adjacent side a as well as the opposite side o:
Frequent calculations on trigonometricfunctions.com include, for example:
### Sine
The sine of an angle is defined as the ratio opposite/hypotenuse, and for the sine the abbreviation sin is the rule.
sin A = o / h
The inverse of the trigonometric function sin is the arcsine, and the reciprocal of the sin is the cosecant, both of which you can find by reading on this page.
### Cosine
The cosine of an angle is defined as the ratio adjacent/hypotenuse, and for the cosine the abbreviation cos is common.
cos A = a / h
The inverse of the cosine sin is the arccosine, and the reciprocal of the cos function is called secant. Keep reading this article to learn about both of them.
### Tangent
The tangent of an angle is defined as the ratio opposite/adjacent, and for the tangent the abbreviation tan is prevalent, but tg can also be seen.
tan A = o / a
The inverse of the trigonometric function known as tan is the arctangent, and the reciprocal of the tan is the cotangent. They, too, are explained further below.
### Cotangent
The cotangent of an angle is defined as adjacent/opposite, and for the cotangent the abbreviation cot is the most frequent. The short forms ctg, cotg and cotan can also be found.
cot A = a / o
The inverse of the cotangent is the arccotangent, and the reciprocal of the cot is the tangent function. Both, the tangent as well as the arccotangent are discussed on this page.
### Secant
The secant of an angle is defined as hypotenuse/adjacent, and for the secant the abbreviation sec is common.
sec A = h / a
The inverse function of the secant is the arcsecant, and the reciprocal of the sec is the cosine. You can find about the arcsecant in the section inverse trigonometric functions.
### Cosecant
The cosecant of an angle is defined as hypotenuse/opposite, and for the cosecant the abbreviation csc is the rule.
csc A = h / o
The inverse of the cosecant is the arccosecant, and the reciprocal of the csc is the sine function. To learn about the cosecant read our paragraph inverse trigonometric functions further down.
## Unit Circle Trigonometric Functions
As the above right-angled triangle definitions for trigonometric functions only work for angles between 0 and 90 degrees, we introduce the unit circle definition for trigonometric functions to account for all negative and positive arguments.
In a Cartesian coordinate system, x2 + y2 = r2 is the origin equation for a circle centered at the origin. If we make a line through the origin O of the circle and the point P, in the intersection of the circle and the line, P(x,y), x = r cos(θ) and y = r sin(θ); θ is the angle formed by OP and the x-axis.
With r = 1 we obtain x = cos(θ) and y = sin(θ):
So far, for the ease of the triangle definition of the trig functions, we have been measuring the plane angles in the unit degree, denoted by °. However, the standard unit of angular measure is the radian with the symbol rad.
90° are equal to π/2 for example. To change degrees to the unit radian multiply the angle by π / 180°, and if want to change radians to degrees multiply the argument by 180° / π.
## Period of Trigonometric Functions
All the six trigonometric functions are periodic, with a period of 360 degrees / 2π radians except for the tangent and cotangent functions of 180° / π radian. Colloquially, this is refereed to as period of trig functions.
In the next section we discuss the inverse trigonometric functions, which are not periodic in the strict sense. Rather, in order for sin, cos, tan etc having an inverse function, the domains of the original functions are restricted to a subset.
## Inverse Trigonometric Functions
The inverse trigonometric functions are the inverse of the functions discussed above with their domains suitably restricted domains. They are often called inverse trig functions, and used to obtain the angle from any of the angle’s trigonometric ratios sin, cos, tan, cot, sec, and csc.
### Arcsine
The arcsine is the inverse function of the sine, defined as x = sin(y), usually written as y = arcsin(x). Its domain of x for a real result of arcsin(x) is −1 ≤ x ≤ 1. The range of the usual principal value in degrees is: −90° ≤ y ≤ 90°; in radians −π/2 ≤ y ≤ π/2.
### Arccosine
The arccosine is the inverse function of the cosine, defined as x = cos(y), normally notated as y = arccos(x). Its domain of x for a real result of arccos(x) is −1 ≤ x ≤ 1. The range of the usual principal value in degrees is 0° ≤ y ≤ 180°; in radians 0 ≤ y ≤ π.
### Arctangent
The arctangent is the inverse function of the tangent, defined as x = tan(y), conventionally spelled as y = arctan(x). The domain of x for a real result of arctan(x) is $\mathbb{R}$, the set of real numbers. Its range of the usual principal value in degrees is −90° < y < 90°; in radians
### Arccotangent
The arccotangent is the inverse function of the cotangent, defined as x = cot(y), commonly written as y = arccot(x). Its domain of x for a real result of arccot(x) is $\mathbb{R}$. The range of the usual principal value in degrees is 0° < y < 180°; in radians −π/2 < y < π/2.
### Arcsecant
The arcsecant is the inverse function of the secant, defined as x = sec(y), typically notated as y = arcssec(x). The domain of x for a result in $\mathbb{R}$ is x ≤ −1 or 1 ≤ x. Its range of the usual principal value in degrees is 0° ≤ y < 90° or 90° < y ≤ 180°; in radians 0 < y < π.
### Arccosecant
The arccosecant is is the inverse function of the cosecant, defined as x = csc(y), generally spelled as y = arccsc(x). Its domain of x for a result in $\mathbb{R}$ is x ≤ −1 or 1 ≤ x. The range of the usual principal value in degrees is −90° ≤ y < 0° or 0° < y ≤ 90°; in radians −π/2 ≤ y < 0 or 0 < y ≤ π/2.
Make sure to understand the difference between the inverse trig functions, just discussed, and the reciprocal of sin cos tan, which is ahead.
## Reciprocal Trigonometric Functions
In trigonometry, the inverse functions, and the reciprocal trigonometric functions are not the same: f(x)-1 ≠ 1/f(x). The reciprocal trigonometric functions are as follows:
• f(x) = sin x, sin-1 x = 1 / csc x
• f(x) = cos x, cos-1 x = 1 / sec x
• f(x) = tan x, tan-1 x = 1 / cot x
• f(x) = cot x, cot-1 x = 1 / tan x
• f(x) = sec x, sec-1 x = 1 / cos x
• f(x) = csc x, csc-1 x = 1 / sin x
## Conclusion
You have made it to the end of our introduction to the trigonometric functions sin cos tan etc.
We plan to add an article for each trigonometric function, along with a calculator.
Until then, use our search form located in the sidebar to locate and calculate any trig functions and inverse trig functions. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9555309414863586, "perplexity": 709.8821994177075}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662594414.79/warc/CC-MAIN-20220525213545-20220526003545-00373.warc.gz"} |
http://mathhelpforum.com/calculus/220089-derivative-problem-print.html | # derivative problem
$t = {n^2} + 2n\$/extract_itex] $dt = 2ndn + 2dn\\$ $ddt = 2d{n^2}\\$ ${d^3}t = 0$ i know how dt, ${d^3}t$ is calculated but have no idea regarding ddt,..... what does it even mean • June 23rd 2013, 08:42 AM Plato Re: derivative problem Quote: Originally Posted by mpx86 $t = {n^2} + 2n\\$ $dt = 2ndn + 2dn\\$ $ddt = 2d{n^2}\\$ ${d^3}t = 0$ i know how dt, ${d^3}t$ is calculated but have no idea regarding ddt,..... what does it even mean Your notation is awful. $\\t=n^2+2n\\ \frac{dt}{dn}=2n+2\text{ first derivative} \\\frac{d^2t}{dn^2}=2\text{ second derivative}\\\frac{d^3t}{dn^3}=0\text{ third derivative}$ • June 23rd 2013, 09:01 AM mpx86 Re: derivative problem Quote: Originally Posted by Plato Your notation is awful. $\\t=n^2+2n\\ \frac{dt}{dn}=2n+2\text{ first derivative} \\\frac{d^2t}{dn^2}=2\text{ second derivative}\\\frac{d^3t}{dn^3}=0\text{ third derivative}$ i know this ...... but i saw $\[{\rm{ }}d(dt) = 2d{n^2}$$ in a book and m curious to know how the author solved it... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9504362940788269, "perplexity": 1511.3118525063248}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988311.72/warc/CC-MAIN-20150728002308-00251-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://mathhelpforum.com/discrete-math/164801-proofs-induction.html | # Thread: Proofs by Induction
1. ## Proofs by Induction[SOLVED]
Ok, I think I got most of the steps covered on these problems but can't seem to get the last part correct:
sorry, image from maple since i still have not grasp on doing the html for the symbols
i don't know if im doing the last steps correctly or not as the other problems without the summation and factorials worked fine.
2. In the second last line, you write $\displaystyle\sum_{i=1}^{k+2}i\cdot2^i=\sum_{i=1}^ {k+1}i\cdot2^i+k\cdot2^k$, whereas the right-hand side should be $\displaystyle\sum_{i=1}^{k+1}i\cdot2^i+(k+2)\cdot2 ^{k+2}$.
A style remark. In the beginning of the induction step, you write, "If p(k) is true for all integers k >= 2, then p(k+1) is true". First, it's helpful to write if this is your goal to prove or if you have already shown it. Second, this sentence can be interpreted as $(\forall k\ge 2.\;p(k))\to p(k+1)$, which is trivial. It is better to say, "For all k >= 2, if p(k), then p(k+1)" or "p(k) implies p(k+1) for all k >= 2".
3. Originally Posted by emakarov
In the second last line, you write $\displaystyle\sum_{i=1}^{k+2}i\cdot2^i=\sum_{i=1}^ {k+1}i\cdot2^i+k\cdot2^k$, whereas the right-hand side should be $\displaystyle\sum_{i=1}^{k+1}i\cdot2^i+(k+2)\cdot2 ^{k+2}$.
A style remark. In the beginning of the induction step, you write, "If p(k) is true for all integers k >= 2, then p(k+1) is true". First, it's helpful to write if this is your goal to prove or if you have already shown it. Second, this sentence can be interpreted as $(\forall k\ge 2.\;p(k))\to p(k+1)$, which is trivial. It is better to say, "For all k >= 2, if p(k), then p(k+1)" or "p(k) implies p(k+1) for all k >= 2".
ok i understand that part now. so the second problem should have be [(k+1)!-1]+(k+1)(k+1!). how do you simply this and the first problem to show that they equal p(k+1)?
4. so the second problem should have be [(k+1)!-1]+(k+1)(k+1!). how do you simply this and the first problem to show that they equal p(k+1)?
[(k+1)!-1]+(k+1)(k+1!) = (k + 1)! * (1 + k + 1) - 1 = (k+2)! - 1. In the first problem you similarly factor out $2^{k+2}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8980494737625122, "perplexity": 391.5633455918468}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187826840.85/warc/CC-MAIN-20171023221059-20171024001059-00343.warc.gz"} |
https://papers.nips.cc/paper/2021/hash/23e582ad8087f2c03a5a31c125123f9a-Abstract.html | #### Authors
Daiki Chijiwa, Shin'ya Yamaguchi, Yasutoshi Ida, Kenji Umakoshi, Tomohiro INOUE
#### Abstract
Pruning the weights of randomly initialized neural networks plays an important role in the context of lottery ticket hypothesis. Ramanujan et al. (2020) empirically showed that only pruning the weights can achieve remarkable performance instead of optimizing the weight values. However, to achieve the same level of performance as the weight optimization, the pruning approach requires more parameters in the networks before pruning and thus more memory space. To overcome this parameter inefficiency, we introduce a novel framework to prune randomly initialized neural networks with iteratively randomizing weight values (IteRand). Theoretically, we prove an approximation theorem in our framework, which indicates that the randomizing operations are provably effective to reduce the required number of the parameters. We also empirically demonstrate the parameter efficiency in multiple experiments on CIFAR-10 and ImageNet. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9553607106208801, "perplexity": 1309.6565378545167}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662636717.74/warc/CC-MAIN-20220527050925-20220527080925-00085.warc.gz"} |
http://physics.stackexchange.com/questions/64562/why-metric-system-uses-kilogram-as-a-basic-si-unit | # Why metric system uses kilogram as a basic SI unit?
SI system uses all (that I know) measurement basic units as 1 (single) instance: meter, second, ampere, etc, except the KILOgram. It already defined with 1000 multiplier (kilo).
It prevents from using usual multiplier prefixes: mega, giga, tera, ... Though we sometimes use miligrams or micrograms. All points to the "gram" as a basic unit. But for some reason it isn't the case.
Does anybody have explanation of this fact? Why kilogram and not gram was decided to be a basic unit?
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Fun fact: in the waste industry, Mg (Megagram) is sometimes used instead of t (ton). This is entirely logical, somewhat counterintuitive and done only to separate the insiders from the outsiders. – mart May 14 '13 at 13:59
@mart That would also help avoid confusion between short, long and metric tons. – Kinnisal Mountain Chicken May 14 '13 at 14:07
What is the physics in this question? This seems way too localized to me ... – Dilaton May 14 '13 at 14:33
This excellent video by Veritasium answers your question and goes on to discuss much more about the kilogram. – ejrb May 14 '13 at 14:35
@Dilaton Are you kidding? When the kilogram will be redefined in terms of the Planck constant—I don't know—later this decade, then that will be physics news, as SI will rid itself of the last artefact. Some of your favourite bloggers might write about it, including even why the New SI will still use kilograms instead of grams. See here for the draft (p. 7). – Kinnisal Mountain Chicken May 14 '13 at 15:17
## 3 Answers
Among the base units of the International System, the kilogram is the only one whose name and symbol, for historical reasons, include a prefix. Names and symbols for decimal multiples and submultiples of the unit of mass are formed by attaching prefix names to the unit name "gram", and prefix symbols to the unit symbol "g" (CIPM 1967, Recommendation 2).
BIPM
The reason why "kilogram" is the name of a base unit of the SI is an artefact of history.
Louis XVI charged a group of savants to develop a new system of measurement. Their work laid the foundation for the "decimal metric system", which has evolved into the modern SI. The original idea of the king's commission (which included such notables as Lavoisier) was to create a unit of mass that would be known as the "grave". By definition it would be the mass of a litre of water at the ice point (i.e. essentially 1 kg). The definition was to be embodied in an artefact mass standard.
After the Revolution, the new Republican government took over the idea of the metric system but made some significant changes. For example, since many mass measurements of the time concerned masses much smaller than the kilogram, they decided that the unit of mass should be the "gramme". However, since a one-gramme standard would have been difficult to use as well as to establish, they also decided that the new definition should be embodied in a one-kilogramme artefact. This artefact became known as the "kilogram of the archives". By 1875 the unit of mass had been redefined as the "kilogram", embodied by a new artefact whose mass was essentially the same as the kilogram of the archives.
The decision of the Republican government may have been politically motivated; after all, these were the same people who condemned Lavoisier to the guillotine. In any case, we are now stuck with the infelicity of a base unit whose name has a "prefix".
BIPM
The International Bureau of Weights and Measures (French: Bureau international des poids et mesures), is an international standards organisation, one of three such organisations established to maintain the International System of Units (SI) under the terms of the Metre Convention (Convention du Mètre). The organisation is usually referred to by its French initialism, BIPM.
Wikipedia
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Thanks for the answer. Looks like a truth. But from nowadays point of view I don't see the reason of not using small units. We still use bytes in computer science despite we don't measure anything in them - at least kilobytes or even gigabytes. I see no problem using teragrams. But that's more a question to BIMP that to you =) – Oleksandr Pshenychnyy May 14 '13 at 14:27
@OleksandrPshenychnyy Teragram (Tg) is actually the "rightest" way. Tonne is a non-SI unit accepted for use with the SI, so megatonne is also fine. – Kinnisal Mountain Chicken May 14 '13 at 14:45
This has some interesting history, but I don't think it's a correct answer to the question. The choice of a standard such as a 1 kg platinum-iridium bar rather than a 1 g bar is unrelated to the question of whether to do calculations in a system where the base unit is the kilogram or the gram. – Ben Crowell May 14 '13 at 15:21
@Gugg: you can distinguish because many derived units such as newton, joule, pascal, volt etc are products of kilogram and other basic units. Because of this, when you do calculations writing all numbers in SI measurements, you use kilogram so it matches values with other units, and to match expectations of other people. The gram only behaves like a basic unit when you write values with prefixed units and the unit starts with kilogram. – b_jonas May 14 '13 at 15:40
@b_jonas If the kilogram weren't the SI unit, then, counterfactually, the gram probably wouldn't be the SI unit either. Instead we would have an SI unit called somethingelsewithoutaprefix that happened to be equal to the kilogram. – Kinnisal Mountain Chicken May 14 '13 at 16:10
The kilogram is the base unit of mass because electrical engineers in the late 19th century chose a particular set of practical electrical units. These practical units were a success, and we are still using them today: ohm, volt, ampere, and the joule.
In 1874 the mechanical units cm, g, s ('CGS') were adopted as the coherent system of units for science. In 1881 the CGS was coherently extended with the “absolute" electrical units abampere, abvolt, abohm. Coherence in this case primarily means that electrical energy and mechanical energy have identical units: $V\ I\ t = F \ L$. Unfortunately, the abvolt, abohm, were inconveniently small. Another inconveniently small unit was the unit of mechanical energy, the erg (=1 g⋅cm/s2).
In 1881-1889 the 'practical' units electrical units ampere, volt, ohm, and joule (1 joule = 1 V⋅A⋅s = 10^7 erg) were introduced for practical use. Their magnitude was convenient, and they were decimally compatible with CGS, but they were not coherent with CGS. In 1901, Giorgi discovered that the practical units happen to be coherent with m, kg, s.
Derivation: $[E] = [F\cdot L] = [(M\ L\ t^{-2})\ L] = [M]\ [L]^{2}\ [t]^{-2}\rightarrow$ $\rightarrow [M] = [E]\ [t]^2\ [L]^{-2} = 1\ J s^{2} m^{-2} = 1\ kilogram$
Because of this, the scientific community adopted the SI system in 1960, with the kilogram as base unit.
In summary: the kilogram became the base unit because it is coherent with the joule, which is derived from the practical units volt and ampere.
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Note that the kilogram being a base unit has nothing to do with the kilogram being the calibration artefact. This is shown by the example of the CGS unit system. – jkien Aug 18 at 10:57
There are various systems of units that fall within the family colloquially referred to as the metric system. The SI (formerly known by the more descriptive term mks) is based on the meter, kilogram, and second. The cgs system is based on the centimeter, gram, and second. As far as I know, nobody actually uses an mgs (meter-gram-second) system, although it would be logical to use a system with no prefixes.
Whether to use mks or cgs is an arbitrary convention, and it depends on the field you're in. Astronomers and particle physicists often use cgs. Engineers tend to use mks. The cgs units for force and energy are inconveniently small for many mechanical applications.
Mks and cgs are also associated with different electrical units. This causes equations such as Maxwell's equations to have different forms in the two systems. WP has the gory details.
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## protected by Qmechanic♦Aug 18 at 17:37
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https://www.physicsforums.com/threads/an-ambiguity-in-the-definition-of-tensors.585238/ | # An ambiguity in the definition of tensors
1. Mar 9, 2012
### ShayanJ
One of the definitions of the tensors says that they are multidimensional arrays of numbers which transform in a certain form under coordinate transformations.No restriction is considered on the coordinate systems involved.So I thought they should transform as such not only under rotations but also under transformation from cartesian to plane polar coordinates,so I tried it on the contravariant tensor below:
$\left(\begin{array}{cc}-xy&-y^{2}\\x^{2}&xy\end{array}\right)$
But I got zero for all four elements.I got confused then I thought maybe curvilinear coordinates are somehow different from cartesian.Is it correct?If not,what's the reason?
thanks
2. Mar 12, 2012
### quasar987
Maybe you can show us your computations.
3. Mar 12, 2012
### ShayanJ
$r= \sqrt {x^2+y^2}$
$\theta=tan^{-1}{\frac{y}{x}}$
$\frac {\partial r} {\partial x} = \frac {x}{\sqrt {x^2+y^2}}$
$\frac {\partial r} {\partial y} = \frac {y}{\sqrt {x^2+y^2}}$
$\frac{\partial \theta}{\partial x}=\frac {-y}{x^2+y^2}$
$\frac{\partial \theta}{\partial y}=\frac {x}{x^2+y^2}$
And then I used the transformation rule below and the partial derivatives above:
$A^{' kl} = \frac {\partial x^{' k}} {\partial x^{i}} \frac {\partial x^{' l}} {\partial x^{j}} A^{ij}$
I calculate one of them here:
$A^{' 11}=( \frac {x}{\sqrt {x^2+y^2}} )^2 \times (-xy) + \frac {x}{\sqrt {x^2+y^2}} \times \frac {y}{\sqrt {x^2+y^2}} \times (-y^2) + \frac {x}{\sqrt {x^2+y^2}} \times \frac {y}{\sqrt {x^2+y^2}} \times x^2 + ( \frac {y}{\sqrt {x^2+y^2}} )^2 \times xy =0$
I did the calculations several times,I'm sure there was nothing wrong with them.
Last edited: Mar 12, 2012
4. Mar 16, 2012
### jimskea
There are different types of tensors. If you're only worried about how your objects transform between cartesian coordinate systems, then you'll define what you mean by a tensor in terms of orthogonal transformations (and you'll get Cartesian tensors). On the other hand, if you're interested in considering how objects transform under general coordinate transformations, you'll define a tensors (or general tensor) by the transformations of the components as you did above.
Now, not all collections of components will be tensors. Maybe the matrix you defined above doesn't represent the components of a general tensor?
5. Mar 16, 2012
### ShayanJ
Thanks
Now the question is,how can I understand that?
6. Mar 17, 2012
### genericusrnme
How do you mean understand?
Tensors are defined as things that transform like $T_{a,b}=\frac{ \partial x^n}{\partial x^a}\frac{ \partial x^m}{\partial x^b}T'_{n,m}$ or the other way if it's mixed or contravariant. If it doesn't transform like this then it isn't a tensor.
There's nothing to really 'understand' about it, it's just a definition, things that don't transform like tensors aren't tensors!
7. Mar 17, 2012
### ShayanJ
Ok.That's right.
But the matrix I've given in my first posts,transforms as a contravariant tensor under rotations but gives zero under transformation from cartesian to polar coordinates.
That's what I wanna know the reason.
8. Mar 17, 2012
### Alesak
I went through your calculation as well, your transformation rule seems correct...
Maybe the problem is that your coordinate matrix has determinant 0, but I don't see why it's wrong.
9. Mar 17, 2012
### ShayanJ
What you mean by coordinate matrix?
I just know one matrix relating this and that's the Jacobian which does not have zero determinant here.
I think if we analyze another tensor which works well here,we can find the wrong thing.
10. Mar 17, 2012
### Alesak
Well, $\left(\begin{array}{cc}-xy&-y^{2}\\x^{2}&xy\end{array}\right)$ has determinant zero. It probably doens't play any role here, but I´m not an expert.
If we took $\left(\begin{array}{cc}x&y\\y&x\end{array}\right)$, it would be
$A^{' 11}=( \frac {x}{\sqrt {x^2+y^2}} )^2 \times x + \frac {x}{\sqrt {x^2+y^2}} \times \frac {y}{\sqrt {x^2+y^2}} \times y + \frac {x}{\sqrt {x^2+y^2}} \times \frac {y}{\sqrt {x^2+y^2}} \times y + ( \frac {y}{\sqrt {x^2+y^2}} )^2 \times x$
which obviously is not zero. So it seems to be as unfortunate choice of tensor and coordinate system. But still, it is wierd. If we tried coordinate transformation to polar coordinates and back to cartesian, result would be zero. Which is wrong.
11. Mar 17, 2012
### HallsofIvy
Staff Emeritus
You have yet to give any reason why you think that is a tensor!
12. Mar 18, 2012
### ShayanJ
Very good point HallsofIvy.
Just the professor at university told that it is a tensor and has done the transformation for rotations and it proved to be a tensor under rotations.
Maybe its not a tensor because it does not work well under this kind of transformation.
So very unfortunate choice.
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http://www.newton.ac.uk/seminar/20050503163018002 | # Betti numbers of sets defined by quantifier-free formulas
Presented by:
A Gabrielov [Purdue]
Date:
Tuesday 3rd May 2005 -
16:30 to 18:00
Venue:
INI Seminar Room 2
Abstract:
(Joint work with N. Vorobjov)
Upper bounds for the Betti numbers of real algebraic sets were obtained by Oleinik-Petrovskii (1949), Milnor (1964) and Thom (1965). These bounds, based on Morse theory, were single exponential in the number of variables. Basu (1999) extended these results to real semialgebraic sets defined by equations and non-strict inequalities. However, the best previously known upper bounds for general semialgebraic sets were double exponential. Gabrielov and Vorobjov (2005) obtained a single exponential upper bound on the Betti numbers of a general semialgebraic set. Given a semialgebraic set X, another semialgebraic set Y defined by equations and non-strict inequalities is constructed, with the same Betti numbers as X. Basu's theorem applied to Y provides the upper bound for the Betti numbers of X. The method easily generalizes to non-algebraic functions, such as Pfaffian functions. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9774898886680603, "perplexity": 832.7300790878269}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541322.19/warc/CC-MAIN-20161202170901-00288-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://mathhelpforum.com/discrete-math/87191-help-particular-solution-non-homogeneous-recurrence-relation.html | # Math Help - HELP with Particular solution in the non-homogeneous recurrence relation
1. ## HELP with Particular solution in the non-homogeneous recurrence relation
Find the solution of the recurrence relation $a_{n} = 2a_{n-1} + 2n^2$ with initial condition a1 = 4.
As this is the linear non-homogeneous recurrence relation which has a general solution in the form of an = h(n) + p(n).
I tried solving the h(n), homogeneous part first $a_{n} = 2a_{n-1}$.
I got $h(n) = A2^n$ , for some constant A.
2. Hello, sanorita_belle!
Find the solution of the recurrence relation: . $a_n \:=\: 2a_{n-1} + 2n^2$
with initial condition $a_1 = 4$
This is a linear non-homogeneous recurrence which has a general solution
of the form: . $a_n \:=\: h(n) + p(n)$
I tried solving the $h(n)$, homogeneous part first: . $a_n \:=\: 2a_{n-1}$
I got: . $h(n) = A\!\cdot\!2^n$, for some constant $A.$ . . . . Good!
But now i am confused about the particular solution $p(n).$
How do i find that?
I don't know the procedure you were taught, but I can guess . . .
You might conjecture that the particular solution is a quadratic ...
. . of the form: . $p(n) \:=\:Bn^2 + Cn + D$
I was taught a different approach . . . quite long and tedious . . .
. . and came up with: . ${\color{blue}a_n \;=\;13\!\cdot\!2^n - 2(n^2 + 4n + 6)}$
. . which cranks out the first few terms: . $4,\: 16,\: 50,\: 132,\; \hdots$
3. Thanks Soroban, Can you please provide the way you learned it? It would be a great help!
4. Hello, sanorita_belle!
For the sake of anyone looking on, here it is.
I warned you that this long and tedious . . .
Find the solution of the recurrence relation: . $a_n \:=\: 2a_{n-1} + 2n^2,\;\;a_1 = 4$
We can reduce this to a homogeneous relation ... eventually.
$\begin{array}{ccccc}
\text{previous term:} & a_{n=1} &=& 2a_{n-2} + 2(n-1)^2 & [1] \\
n^{th}\text{ term:} & a_n &=& 2n_{n-1} + 2n^2 & [2] \end{array}$
Subtract [2] - [1]: . $a_n-a_{n-1} \:=\:2a_{n-1} - 2a_{n-2} + 2n^2 - 2(n-1)^2$
$\begin{array}{cccccc}\text{We have:} &a_n - 3a_{n-1} + 2a_{n-2} &=& 4n-2 & [3] \\ \text{Next term:} & a_{n+1} - 3a_n + 2a_{n-1} &=& 4(n+1)-2 & [4] \end{array}$
Subtract [4] - [3]: . $a_{n+1} - a_n - 3a_n + 3a_{n-1} + 2a_{n-1} - 2a_{n-2} \:=\:[4(n-2)-2] - (4n-2)$
$\begin{array}{cccccc}\text{We have:} & a_{n+1} - 4a_n + 5a_{n-1} - 2a_{n-2} &=& 4 & [5] \\
\text{Next term:} & a_{n+2} - 4a_{n+1} + 5a_n - 2a_{n-1} &=& 4 & [6] \end{array}$
Subtract [6] - [5]: . $a_{n+2} - 5a_{n+1} + 9a_n - 7a_{n-1} + 2a_{n-2} \:=\:0$ . . . There! .Homogeneous!
Let $X^n \:=\:a_n$
We have: . $X^{n+2} - 5X^{n+1} + 9X^n - 5X^{n-1} + 2X^{n-2} \:=\:0$
Divide by $X^{n-2}\!:\quad X^4 - 5X^3 + 9X^2 - 7X + 2 \:=\:0$
Lucky for us, this factors: . $(X-1)^3(X-2) \:=\:0$
. . which has roots: . $X \;=\;1,1,1,2$
So the function is: . $f(n) \;=\;A(1^n) + Bn(1^n) + Cn^2(1^n) + D(2^n)$
. . That is: . $f(n) \;=\;A + Bn + Cn^2 + D\cdot2^n$ .(a)
The first four terms of the sequence are: . $4,\:16,\:50,\:132$
. . Substitute these values into (a).
$\begin{array}{ccccc}f(1) = 4 & A + B + C + 2D &=& 4 & [1] \\
f(2) = 16 & A + 2B + 4C + 4D &=& 16 & [2] \\
f(3) = 50 & A + 3B + 9C + 8D &=& 50 & [3] \\
f(4) = 132 & A + 4B + 16C + 16D &=& 132 & [4] \end{array}$
$\begin{array}{ccccc}\text{Subtract [2] - [1]:} & B + 3C + 2D &=& 12 & [5] \\ \text{Subtract [3] - [2]:} & B + 5C + 4D &=& 34 & [6] \\
\text{Subtract [4] - [3]:} & B + 7C + 8D &=& 82 & [7] \end{array}$
$\begin{array}{ccccc}\text{Subtract[6] - [5]:} & 2C + 2D &=& 22 & [8] \\ \text{Subtract [7] - [6]:} & 2C + 4D &=& 48 & [9] \end{array}$
$\text{Subtract [9] - [8]: }\;\;2D \:=\:26 \quad\Rightarrow\quad\boxed{ D \:=\:13}$
Back-substitute and get: . $\boxed{C \:=\:\text{-}2} \quad\boxed{ B \:=\:\text{-}8} \quad \boxed{A \:=\:\text{-}12}$
Therefore: . $f(n) \;=\;-12 - 8n - 2n^2 + 13\!\cdot\!2^n$
I need a nap . . .
.
5. THANKSSSS ALOT SOROBAN for all the time you spent on this question, I really appreciate it. You are the BEST
This makes much more sense than what i learned in-class, thanks once again | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 35, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9739323258399963, "perplexity": 733.6013177658917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510268533.31/warc/CC-MAIN-20140728011748-00005-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://paperswithcode.com/conference/ieee-uemcon-2019-2019-10 | # Derivative-Free Optimization of Neural Networks using Local Search
LS is an algorithm where constrained noise is iteratively applied to subsets of the search space.
2 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9873964190483093, "perplexity": 1446.0742150329872}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347392142.20/warc/CC-MAIN-20200527075559-20200527105559-00015.warc.gz"} |
http://papers.nips.cc/paper/3696-boosting-with-spatial-regularization | # NIPS Proceedingsβ
## Boosting with Spatial Regularization
[PDF] [BibTeX] [Supplemental]
### Abstract
By adding a spatial regularization kernel to a standard loss function formulation of the boosting problem, we develop a framework for spatially informed boosting. From this regularized loss framework we derive an efficient boosting algorithm that uses additional weights/priors on the base classifiers. We prove that the proposed algorithm exhibits a grouping effect, which encourages the selection of all spatially local, discriminative base classifiers. The algorithms primary advantage is in applications where the trained classifier is used to identify the spatial pattern of discriminative information, e.g. the voxel selection problem in fMRI. We demonstrate the algorithms performance on various data sets. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8317539095878601, "perplexity": 1474.5906865621828}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123276.44/warc/CC-MAIN-20170423031203-00311-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-10-review-page-748/108 | ## Algebra: A Combined Approach (4th Edition)
$\dfrac{3\sqrt[4]{2x^2}}{2x^{3}}$
$\bf{\text{Solution Outline:}}$ To rationalize the denominator of the given expression, $\sqrt[4]{\dfrac{81}{8x^{10}}} ,$ simplify first the radicand by extracting the root of the factor that is a perfect power of the index. Then multiply by an expression equal to $1$ which will make the denominator a perfect power of the index. Then extract again the root of the factor that is a perfect power of the index. $\bf{\text{Solution Details:}}$ Extracting the root of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt[4]{\dfrac{81}{x^{8}}\cdot\dfrac{1}{8x^2}} \\\\= \sqrt[4]{\left( \dfrac{3}{x^{2}}\right)^4\cdot\dfrac{1}{8x^2}} \\\\= \dfrac{3}{x^{2}}\sqrt[4]{\dfrac{1}{8x^2}} .\end{array} Multiplying the given expression by an expression equal to $1$ which will make the denominator a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{3}{x^{2}}\sqrt[4]{\dfrac{1}{8x^2}\cdot\dfrac{2x^2}{2x^2}} \\\\= \dfrac{3}{x^{2}}\sqrt[4]{\dfrac{1}{16x^4}\cdot2x^2} .\end{array} Extracting the root of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{3}{x^{2}}\sqrt[4]{\left(\dfrac{1}{2x}\right)^4\cdot2x^2} \\\\= \dfrac{3}{x^{2}}\cdot\dfrac{1}{2x}\sqrt[4]{2x^2} \\\\= \dfrac{3}{2x^{3}}\sqrt[4]{2x^2} \\\\= \dfrac{3\sqrt[4]{2x^2}}{2x^{3}} .\end{array} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9813029170036316, "perplexity": 946.9467903793313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945855.61/warc/CC-MAIN-20180423070455-20180423090455-00632.warc.gz"} |
http://mathhelpforum.com/calculus/27148-gradient-question.html | # Math Help - Gradient Question
A and B are points (1,p) and (2,4) respectively on the curve with equation y = x³ - 4.
The gradient of the line AB is?
----
2. Originally Posted by James McNiven
A and B are points (1,p) and (2,4) respectively on the curve with equation y = x³ - 4.
The gradient of the line AB is?
----
The first thing to do is to find a value for p. I get that $y(1) = p = 1^3 - 4 = -3$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8415712118148804, "perplexity": 836.1236183574333}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990114.79/warc/CC-MAIN-20150728002310-00105-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://hal.in2p3.fr/in2p3-00212255 | # Density-induced suppression of the alpha-particle condensate in nuclear matter and the structure of alpha cluster states in nuclei
Abstract : At low densities, with decreasing temperatures, in symmetric nuclear matter alpha-particles are formed, which eventually give raise to a quantum condensate with four-nucleon alpha-like correlations (quartetting). Starting with a model of alpha-matter, where undistorted alpha particles interact via an effective interaction such as the Ali-Bodmer potential, the suppression of the condensate fraction at zero temperature with increasing density is considered. Using a Jastrow-Feenberg approach, it is found that the condensate fraction vanishes near saturation density. Additionally, the modification of the internal state of the alpha particle due to medium effects will further reduce the condensate. In finite systems, an enhancement of the S state wave function of the c.o.m. orbital of alpha particle motion is considered as the correspondence to the condensate. Wave functions have been constructed for self-conjugate 4n nuclei which describe the condensate state, but are fully antisymmetrized on the nucleonic level. These condensate-like cluster wave functions have been successfully applied to describe properties of low-density states near the n alpha threshold. Comparison with OCM calculations in 12C and 16O shows strong enhancement of the occupation of the S-state c.o.m. orbital of the alpha-particles. This enhancement is decreasing if the baryon density increases, similar to the density-induced suppression of the condensate fraction in alpha matter. The ground states of 12C and 16O show no enhancement at all, thus a quartetting condensate cannot be formed at saturation densities.
Document type :
Journal articles
http://hal.in2p3.fr/in2p3-00212255
Contributor : Suzanne Robert <>
Submitted on : Tuesday, January 22, 2008 - 3:46:11 PM
Last modification on : Wednesday, September 16, 2020 - 4:00:52 PM
### Citation
Y. Funaki, H. Horiuchi, G. Röpke, P. Schuck, A. Tohsaki, et al.. Density-induced suppression of the alpha-particle condensate in nuclear matter and the structure of alpha cluster states in nuclei. Physical Review C, American Physical Society, 2008, 77, pp.064312. ⟨10.1103/PhysRevC.77.064312⟩. ⟨in2p3-00212255⟩
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http://www.anthonysalib.com/solution-to-problem-3/ | Select Page
So this problem comes from the functional analysis course I took last semester. Although simple, I thought was pretty cool. In the actual exercise they had broken it down into three parts, which I did not show in the problem post. Here I will show the steps as claims.
Claim 1: The operator $latex {\Delta^2: A \rightarrow L^2(\Omega)}$ is a bijection
Proof: To show injectivity we show that this operator has trivial kernel. One inclusion is obvious, we just need to show that $latex {\Delta^2 u = 0~\implies u=0}$. To see this we recall from elliptic regularity theory for the Laplacian that if $latex {\Delta \phi = f}$ for some $latex {f\in H^k(\Omega)}$ then
$latex \displaystyle \norm{\phi}_{H^{k+2}}\leq C\norm{f}_{H^k}.$
Now back to our question which can be written as $latex {\Delta(\Delta u)=0}$, and so applying the estimate we can see that $latex {\norm{\Delta u}_{H^{2}}\leq 0}$ and so $latex {\Delta u = 0}$. Applying the estimate one more time shows that $latex {u=0}$. For surjectivity we begin taking some $latex {f\in L^2(\Omega)}$ and let $latex {u\in H^1_0(\Omega)}$ be the unique solution to $latex {\Delta u = f}$, and by the regularity theory $latex {u \in H^2(\Omega)}$. Then let $latex {v \in H^1_0(\Omega)}$ be the unique solution $latex {\Delta v = u}$. It follows that $latex {v\in A}$ and $latex {\Delta(\Delta v)=f}$. (Super cool right?!) $latex \Box$
Claim 2: If $latex {\Delta^2 u =f }$ then
$latex \displaystyle \forall \phi \in A: \int_{\Omega}u\Delta^2 \phi = \int_{\Omega}\phi f.$
Proof: Let $latex {\phi\in A}$, multiplying $latex {\Delta^2 u =f }$ by $latex {\phi}$ and integrating we get
$latex \displaystyle \int_{\Omega} \Delta^2 u \phi = \int_{\Omega} f\phi.$
Since $latex {\phi \in H^1_0(\Omega)}$ we can integrate by parts once with vanishing boundary terms on the left hand side (just replace $latex {\phi}$ with a sequence of compactly supported smooth functions, integrate by parts using the definition of weak derivative passing limits by dominated convergence) to get
$latex \displaystyle \int_{\Omega} \Delta(\Delta u) \phi =-\int_{\Omega} \nabla \Delta u \nabla \phi .$
Now, $latex {\Delta u \in H^1_0}$ (since it is the solution to $latex {\Delta v =f}$) and so we can integrate by parts once more, again with vanishing boundary terms and we get
$latex \displaystyle \int_{\Omega} f\phi=\int_{\Omega} \Delta u \Delta\phi. \ \ \ \ \ (1)$
Now if we try to integrate by parts again we will run into a dead end, try it! Instead we play the same game with $latex {\int_{\Omega} u \Delta^2 \phi }$ to get
$latex \displaystyle \int_{\Omega} u \Delta^2 \phi = \int_{\Omega} \Delta u \Delta\phi . \ \ \ \ \ (2)$
Equating (1) and (2) gives the claim. $latex \Box$
Now we are ready to finish the problem, which is to prove some regularity result on weak solutions to $latex {\Delta^2 u =f }$ for $latex {f\in L^2}$; that is $latex {u\in A}$. By surjectivity of $latex {\Delta^2}$ there exists some $latex {\tilde{u}\in A}$ such that $latex {\Delta^2 \tilde{u} =f}$. We just need to conclude now that $latex {\tilde{u}=u}$. By claim 2 we have that
$latex \displaystyle \forall \phi \in A: \int_{\Omega}(u-\tilde{u})\Delta^2 \phi = 0,$
and since $latex {\Delta^2}$ is bijective we have that this is equivalent to
$latex \displaystyle \forall g \in L^2: \int_{\Omega}(u-\tilde{u})g = 0.$
Setting $latex {g = u-\tilde{u}}$ we get that $latex {u-\tilde{u}=0}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9803216457366943, "perplexity": 69.46904210933857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141205147.57/warc/CC-MAIN-20201130035203-20201130065203-00398.warc.gz"} |
http://physics.stackexchange.com/questions/68783/line-shape-asymmetry-in-undergraduate-hydrogen-deuterium-experiment | # Line-shape asymmetry in undergraduate Hydrogen-Deuterium experiment
I'm working as an LA (undergraduate TA) for an undergraduate physics laboratory experiment where students test the Bohr model and use reduced mass to determine the approximate mass of the neutron.
In the experiment, a discharge tube is filled with a mixture of H and D gas, and the data that come from the spectrometer (Ocean Optics HR-2000 with an optical resolution of about 0.04 nm) look fairly defined. However there's an asymmetry in the peak (the peaks are skewed to the left slightly), and this is bothering me a little bit. The results that students get from fitting gaussians to the peaks are pretty good in terms of agreeing with the literature but the models themselves don't get great reduced chi-square values.
Where could the asymmetry come from? Doppler and natural lifetime broadening are symmetric and should yield a Voigt profile, which is symmetric. These contributions are also much smaller than the instrument resolution.
Using the optical resolution as the standard deviation of a gaussian instrument error function, I fit these peaks with a model taking fine-structure into account (fine-structure is about half the size of the instrument error). This, however yields some very poor confidence intervals in the fit and still does not fix the high chi-squared value.
The only things left that I can think of are instrument error (which I think could be asymmetric) or (grasping at straws here) something kind of complicated like a fano resonance.
What could this be? Is there any reason to think that the peak line shape of H$_\alpha$ should be skewed?
-
Now this is a good question. A picture or plot would be make it even better. I think the inherent shape of the peak is actually a Lorentzian rather than a Gaussian, but that won't give you the observed asymmetry. Another place to look would be in the behavior of your instruments. In any case, very interesting. – dmckee Jun 22 '13 at 17:05
Actually, poking around the link I gave for the Lorentzian suggests yoo may actually have a Voigt lineshape. Might be worth looking at. – dmckee Jun 22 '13 at 17:22
Thanks for the prompt reply! So my belief in using gaussians is based on the thought that the doppler broadening dominates the natural lifetime broadening. Doppler should have a contribution on the order of $\sqrt{k_BT/m_{H_2}} \lambda_0$, much larger than the natural lifetime broadening effect (of order $10^{-4}$ angstroms). In the limit where the Lorentzian looks like a delta relative to the doppler broadening, the voigt function approaches the gaussian. – Dawson Baker Jun 22 '13 at 19:46
add comment | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8339260816574097, "perplexity": 789.0786196064074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00148-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://testbook.com/question-answer/which-of-the-following-is-not-the-element-of-hydro--6022614b0345e887e5fb48bf | # Which of the following is not the element of hydroelectric power plant?
This question was previously asked in
RPSC ME Lecturer 2014: Official Paper
View all RPSC Lecturer Tech Edu Papers >
1. Catchment area
2. Dam
3. Draft tube
4. Condenser
Option 4 : Condenser
Free
ST 1: Engineering Mechanics
1029
20 Questions 20 Marks 16 Mins
## Detailed Solution
Explanation:
Hydroelectric Power Plant:
• The electricity produced from the kinetic energy of flowing water is called hydroelectricity and a plant that generates electricity on a large scale from flowing water is called a hydroelectric power plant.
• The requirements for a hydroelectric power plant are a strongfast flow of water and a significant drop down in which the water can fal
• In a hydroelectric power plant, more electrical power can be generated, if waterfalls from a greater height because the rise in water level causes the increase in potential energy of water.
• Thus, when it flows from a higher position more amount of kinetic energy is formed by the conversion of higher potential energy and this kinetic energy in the form of moving water can produce more electrical power.
Draft Tube: The draft tube is a conduit that connects the runner exit to the tailrace where the water is being finally discharged from the turbine.
The draft tube has two purposes as follows if
1. It permits a negative or suction head to be established at the runner exit, thus making it possible to install the turbine above the tailrace level without loss head.
2. It converts a large proportion of velocity energy rejected from the runner into useful pressure energy.
∴ A condenser is not used in a hydroelectric power plant. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8937809467315674, "perplexity": 1923.171138164937}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056856.4/warc/CC-MAIN-20210919095911-20210919125911-00286.warc.gz"} |
https://stacks.math.columbia.edu/tag/02IB | Lemma 5.20.3. Let $X$ be a topological space. Let $\delta$, $\delta '$ be two dimension functions on $X$. If $X$ is locally Noetherian and sober then $\delta - \delta '$ is locally constant on $X$.
Proof. Let $x \in X$ be a point. We will show that $\delta - \delta '$ is constant in a neighbourhood of $x$. We may replace $X$ by an open neighbourhood of $x$ in $X$ which is Noetherian. Hence we may assume $X$ is Noetherian and sober. Let $Z_1, \ldots , Z_ r$ be the irreducible components of $X$ passing through $x$. (There are finitely many as $X$ is Noetherian, see Lemma 5.9.2.) Let $\xi _ i \in Z_ i$ be the generic point. Note $Z_1 \cup \ldots \cup Z_ r$ is a neighbourhood of $x$ in $X$ (not necessarily closed). We claim that $\delta - \delta '$ is constant on $Z_1 \cup \ldots \cup Z_ r$. Namely, if $y \in Z_ i$, then
$\delta (x) - \delta (y) = \delta (x) - \delta (\xi _ i) + \delta (\xi _ i) - \delta (y) = - \text{codim}(\overline{\{ x\} }, Z_ i) + \text{codim}(\overline{\{ y\} }, Z_ i)$
by Lemma 5.20.2. Similarly for $\delta '$. Whence the result. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9970051050186157, "perplexity": 87.64077848346889}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573744.90/warc/CC-MAIN-20220819161440-20220819191440-00002.warc.gz"} |
http://math.wikia.com/wiki/Heaviside_function | FANDOM
1,022 Pages
The Heaviside function, also known as the unit step function or Heaviside step function and usually denoted as $u_c$ , is a discontinuous function that
$u_c(x)=\begin{cases}0&xc\end{cases}$
The Heaviside function can be defined as the integral of the Dirac delta function.
$u_c(t)=\int\limits_{-\infty}^t\delta(s-c)ds$
The Heaviside function is often used in differential equations to model non-continuous events such as force in a driven harmonic oscillator or voltage in a circuit. If the Dirac delta function represents force being applied on an object, the Heaviside function will represent its momentum.
The value at $x=c$ is usually of little importance, although it is sometimes defined as $x=\tfrac12$ .
It has the Laplace transform
$\mathcal{L}\{u_c(t)\}=\frac{e^{-cs}}{s}$
for $c>0$ . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9846580028533936, "perplexity": 232.2061113531812}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424287.86/warc/CC-MAIN-20170723062646-20170723082646-00008.warc.gz"} |
https://es.mathworks.com/help/pde/ug/clamped-square-isotropic-plate-with-a-uniform-pressure-load.html | # Clamped Square Isotropic Plate with Uniform Pressure Load
This example shows how to calculate the deflection of a structural plate under a pressure loading.
The partial differential equation for a thin isotropic plate with a pressure loading is
`${\nabla }^{2}\left(D{\nabla }^{2}w\right)=-p,$`
where $D$ is the bending stiffness of the plate given by
`$D=\frac{E{h}^{3}}{12\left(1-{\nu }^{2}\right)},$`
and $E$ is the modulus of elasticity, $\nu$ is Poisson's ratio, $h$ is the plate thickness, $w$ is the transverse deflection of the plate, and $p$ is the pressure load.
The boundary conditions for the clamped boundaries are $w=0$ and ${w}^{\prime }=0$, where ${w}^{\prime }$ is the derivative of $w$ in a direction normal to the boundary.
Partial Differential Equation Toolbox™ cannot directly solve this fourth-order plate equation. Convert the fourth-order equation to these two second-order partial differential equations, where $v$ is the new dependent variable.
`${\nabla }^{2}w=v$`
`$D{\nabla }^{2}v=-p$`
You cannot directly specify boundary conditions for both $w$ and ${w}^{\prime }$ in this second-order system. Instead, specify that ${w}^{\prime }$ is 0, and define ${v}^{\prime }$ so that $w$ also equals 0 on the boundary. To specify these conditions, use stiff "springs" distributed along the boundary. The springs apply a transverse shear force to the plate edge. Define the shear force along the boundary due to these springs as $n\cdot D\nabla v=-kw$, where $n$ is the normal to the boundary, and $k$ is the stiffness of the springs. This expression is a generalized Neumann boundary condition supported by the toolbox. The value of $k$ must be large enough so that $w$ is approximately 0 at all points on the boundary. It also must be small enough to avoid numerical errors due to an ill-conditioned stiffness matrix.
The toolbox uses the dependent variables ${\mathit{u}}_{1}$ and ${\mathit{u}}_{2}$ instead of $w$ and $v$. Rewrite the two second-order partial differential equations using variables ${\mathit{u}}_{1}$ and ${\mathit{u}}_{2}$:
`$-{\nabla }^{2}{u}_{1}+{u}_{2}=0$`
`$-D{\nabla }^{2}{u}_{2}=p$`
Create a PDE model for a system of two equations.
`model = createpde(2);`
Create a square geometry and include it in the model.
```len = 10; gdm = [3 4 0 len len 0 0 0 len len]'; g = decsg(gdm,'S1',('S1')'); geometryFromEdges(model,g);```
Plot the geometry with the edge labels.
```figure pdegplot(model,"EdgeLabels","on") ylim([-1,11]) axis equal title("Geometry With Edge Labels Displayed")```
PDE coefficients must be specified using the format required by the toolbox. For details, see
The c coefficient in this example is a tensor, which can be represented as a 2-by-2 matrix of 2-by-2 blocks:
`$\left[\begin{array}{cccc}c\left(1\right)& c\left(2\right)& \cdot & \cdot \\ \cdot & c\left(3\right)& \cdot & \cdot \\ \cdot & \cdot & c\left(4\right)& c\left(5\right)\\ \cdot & \cdot & \cdot & c\left(6\right)\end{array}\right]$`
This matrix is further flattened into a column vector of six elements. The entries in the full 2-by-2 matrix (defining the coefficient a) and the 2-by-1 vector (defining the coefficient f) follow directly from the definition of the two-equation system.
```E = 1.0e6; % Modulus of elasticity nu = 0.3; % Poisson's ratio thick = 0.1; % Plate thickness pres = 2; % External pressure D = E*thick^3/(12*(1 - nu^2)); c = [1 0 1 D 0 D]'; a = [0 0 1 0]'; f = [0 pres]'; specifyCoefficients(model,"m",0,"d",0,"c",c,"a",a,"f",f);```
To define boundary conditions, first specify spring stiffness.
`k = 1e7;`
Define distributed springs on all four edges.
```bOuter = applyBoundaryCondition(model,"neumann","Edge",(1:4),... "g",[0 0],"q",[0 0; k 0]);```
Generate a mesh.
`generateMesh(model);`
Solve the model.
`res = solvepde(model);`
Access the solution at the nodal locations.
`u = res.NodalSolution;`
Plot the transverse deflection.
```numNodes = size(model.Mesh.Nodes,2); figure pdeplot(model,"XYData",u(:,1),"Contour","on") title("Transverse Deflection")```
Find the transverse deflection at the plate center.
```numNodes = size(model.Mesh.Nodes,2); wMax = min(u(1:numNodes,1))```
```wMax = -0.2763 ```
Compare the result with the deflection at the plate center computed analytically.
`wMax = -.0138*pres*len^4/(E*thick^3)`
```wMax = -0.2760 ``` | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 34, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9717361927032471, "perplexity": 409.08714296507617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948976.45/warc/CC-MAIN-20230329120545-20230329150545-00503.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/145991-isomorphism-question.html | # Math Help - Isomorphism question
1. ## Isomorphism question
Let G=<x, y|x^4=y^4=e, xyxy^–1=e>. Show that |G|≤16. Assuming |G|=16, show G/<y^2> is isomorphic to D4.
2. Originally Posted by aabsdr
Let G=<x, y|x^4=y^4=e, xyxy^–1=e>. Show that |G|≤16. Assuming |G|=16, show G/<y^2> is isomorphic to D4.
$D_{2n}$ of order 2n have presentations
1. $$,
2. $$.
2 is denoted as $I_2(n)$, implying that $D_{2n}$ can also be generated by two reflections instead of one rotation and one reflection. You can find this idea in the context of Coxeter groups.
If you just play a little bit of your question, you will reduce your presentation to $$, which is $D_4$.
I'll leave it to you to covert the presentation 1 into 2 and verify that they are indeed equivalent.
3. Here is what I have:
since xyxy^-1=e, we know that yxy^-1=x^-1=x^3, so we know that <x) is normal. So G= <x> union y<x> union y^2<x> union y^3<x> and |G|<= 16.
Lets assume that |G|=16.
So |G|/|<y^2>|=8, thus |G|/|<y^2>|>= |D4|.
D4=<a,b|a^4=b^2=(ab)^2=e, or ab=ba^3>.
Let w=x<y^2> , z=y<y^2> and q=e<y^2> (where q is are identity element in G/<y^2>.
Need to show that G/<y^2>= <w,z|w^4=z^2=(wz)^2=e>.
1. Since x and y are generates in G, then w and z are generates in G/<y^2>. I am not sure if this is right, could someone explain this to me? (I need to show that w and z generate G/<y^2>).
2. w^4=e since (x<y^2>)^4=x^4<y^2>=e<y^2>=q.
3. z^2=e since (y<y^2>)^2=y^2<y^2>=e<y^2>=q.
4. (wz)^2=(wz)(wz)=e. Since wz=(z^-1)(w^-1), then wz=xy<y^2> and z^-1w^-1=y^3x^3<y^2>=yx^3<y^2>. Thus xy<y^2>=yx^3<y^2>.
Therefore, G/<y^2> is isomorphic to D4.
4. how would I find the center for this group if the groups order was 16? I know that since xyxy^-1=e that xy=yx^3, so x=yx^3y^-1, so than x^2y=xyx^3=yx^6=yx^2. So x^2 and e is in the center. How would I show that nothing else is in the center besides <x^2>?
5. Originally Posted by aabsdr
how would I find the center for this group if the groups order was 16? I know that since xyxy^-1=e that xy=yx^3, so x=yx^3y^-1, so than x^2y=xyx^3=yx^6=yx^2. So x^2 and e is in the center. How would I show that nothing else is in the center besides <x^2>?
Looking at the presentation you give, the third relation gives you something which looks like commutativity. Essentially, it allows you to write every element in the form $x^iy^j$ for $i, j \in \{0, 1, 2, 3\}$. What you need to do is find the values of $i,j$ such that $x^iy^jx^ay^b = x^ay^ba^iy^j$ for all $a, b$. So just plug it in...
Alternatively, noting that you are in $D_{2n}$ you can do the same trick. And yes, your group will have a center of order 2 in this case as $D_{2n}$ has a center of order 2 if and only if $n$ is even. Otherwise, it has trivial center.
6. Originally Posted by aabsdr
Here is what I have:
since xyxy^-1=e, we know that yxy^-1=x^-1=x^3, so we know that <x) is normal. So G= <x> union y<x> union y^2<x> union y^3<x> and |G|<= 16.
Lets assume that |G|=16.
So |G|/|<y^2>|=8, thus |G|/|<y^2>|>= |D4|.
D4=<a,b|a^4=b^2=(ab)^2=e, or ab=ba^3>.
Let w=x<y^2> , z=y<y^2> and q=e<y^2> (where q is are identity element in G/<y^2>.
Need to show that G/<y^2>= <w,z|w^4=z^2=(wz)^2=e>.
1. Since x and y are generates in G, then w and z are generates in G/<y^2>. I am not sure if this is right, could someone explain this to me? (I need to show that w and z generate G/<y^2>).
2. w^4=e since (x<y^2>)^4=x^4<y^2>=e<y^2>=q.
3. z^2=e since (y<y^2>)^2=y^2<y^2>=e<y^2>=q.
4. (wz)^2=(wz)(wz)=e. Since wz=(z^-1)(w^-1), then wz=xy<y^2> and z^-1w^-1=y^3x^3<y^2>=yx^3<y^2>. Thus xy<y^2>=yx^3<y^2>.
Therefore, G/<y^2> is isomorphic to D4.
In $G/$, simply plug $y^2=1$ into your presentation of G.
Then, the original presentation reduces to $=$, where $y=y^{-1}$ since $y^2=1$. It is exactly the presentation of D_4 of order 4 (See my previous reply).
To convert the presentation 1 into presentation 2 in my previous reply, substitue s for x and sr for y. Then you will see that both are equivalent presentations.
7. Originally Posted by TheArtofSymmetry
In $G/$, simply plug $y^2=1$ into your presentation of G.
(It should perhaps be pointed out that relations in a presentation, homomorphic images and quotients are all equivalent. Thus, putting a relation into your presentation is the same as taking a quotient.
This is because, formally, $\langle X ; R=1 \rangle :\cong F(X)/<>$ where $F(X)$ is the free group over the alphabet $X$ and $<>$ is the normal subgroup generated by the elements of $R$.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 27, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9942409992218018, "perplexity": 4340.867916576724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447547818.113/warc/CC-MAIN-20141224185907-00069-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://www.reference.com/browse/Pulsar | Definitions
# pulsar
[puhl-sahr]
pulsar, in astronomy, a neutron star that emits brief, sharp pulses of energy instead of the steady radiation associated with other natural sources. The study of pulsars began when Antony Hewish and his students at Cambridge built a primitive radio telescope to study a scintillation effect on radio sources caused by clouds of electrons in the solar wind. Because this telescope was specially designed to record rapid variations in signals, in 1967 it readily recorded a signal from a totally unexpected source. Jocelyn Bell Burnell noticed a strong scintillation effect opposite the sun, where the effect should have been weak. After an improved recorder was installed, the signals were received again as a series of sharp pulses with intervals of about a second. By the end of 1968 it was clear that the team had discovered a rapidly spinning neutron star, a remnant of a supernova.
In 1974 the first binary pulsar—two stars, at least one of which is a neutron star, that orbit each other—was discovered by Russell A. Hulse and Joseph H. Taylor, for which they shared the 1993 Nobel Prize in Physics. Using this binary system, they observed indirect evidence of gravitational waves and also tested the general theory of relativity. Several dozen binary pulsars are now known. In 1995 the orbiting Compton Gamma Ray Observatory detected the first object that bursts and pulses at the same time. This bursting pulsar, another class of pulsars, is currently the strongest source of X rays and gamma rays in the sky. Fewer than a dozen bursting pulsars are known to exist.
The intense magnetic field and plasma that are believed to surround a neutron star provide an effective source of radio waves. The high-energy electrons of the plasma spiral around the magnetic field and emit radio waves and other forms of electromagnetic radiation. This synchrotron radiation is highly directional, like a flashlight beam. If the neutron star is rotating, it will act like a revolving beacon and produce the observed pulses. The pulses recur at precise intervals, but successive pulses differ considerably in strength. Since 1968 more than 700 pulsars have been observed, with pulse rates from 4 seconds to 1.5 milliseconds; the very rapid ones are called millisecond pulsars. The interval between pulses decreases ever so slightly with the passage of time, and it is believed that the slower pulsers are the older stars while the rapid pulsers are the younger. Pulsars in the Crab Nebula and at the site of the Vela supernova can be detected optically as well as at X-ray and gamma-ray frequencies.
A pulsar emits two beams of electromagnetic radiation along its magnetic axis. If the magnetic axis elipsis
Any of a class of cosmic objects that appear to emit extremely regular pulses of radio waves. A few give off short rhythmic bursts of visible light, X rays, and gamma radiation as well. Thought to be rapidly spinning neutron stars, they were discovered by Antony Hewish and Jocelyn Bell Burnell in 1967 with a specially designed radio telescope. More than 550 have been detected since. All behave similarly, but the intervals between pulses (and thus their rotation periods) range from one-thousandth of a second to four seconds. Charged particles from the surface enter the star's magnetic field, which accelerates them so that they give off radiation, released as intense beams from the magnetic poles. These do not coincide with the pulsar's own axis of rotation, so as the star spins, the radiation beams swing around like lighthouse beams and are seen as pulses. Pulsars have been shown to be slowing down, typically by a millionth of a second per year. It has been calculated that pulsars “switch off” after about 10 million years, when their magnetic fields weaken enough.
Pulsars are highly magnetized rotating neutron stars that emit a beam of electromagnetic radiation in the form of radio waves. Their observed periods range from 1.4 ms to 8.5 s. The radiation can only be observed when the beam of emission is pointing towards the Earth. This is called the lighthouse effect and gives rise to the pulsed nature that gives pulsars their name. Because neutron stars are very dense objects, the rotation period and thus the interval between observed pulses are very regular. For some pulsars, the regularity of pulsation is as precise as an atomic clock. Pulsars are known to have planets orbiting them, as in the case of PSR B1257+12. Werner Becker of the Max-Planck-Institut für extraterrestrische Physik said in 2006, "The theory of how pulsars emit their radiation is still in its infancy, even after nearly forty years of work.
## History
### Discovery
The first pulsar was observed in July 1967 by Jocelyn Bell Burnell and Antony Hewish. Initially baffled as to the seemingly unnatural regularity of its emissions, they dubbed their discovery LGM-1, for "little green men" (a name for intelligent beings of extraterrestrial origin). The hypothesis that pulsars were beacons from extraterrestrial civilizations was never serious, but some discussed the the far-reaching implications if it turned out to be true. Their pulsar was later dubbed CP 1919, and is now known by a number of designators including PSR 1919+21, PSR B1919+21 and PSR J1921+2153.
Although CP 1919 emits in radio wavelengths, pulsars have, subsequently, been found to emit in the X-ray and/or gamma ray wavelengths.
The word pulsar is a contraction of "pulsating star", and first appeared in print in 1968:
"An entirely novel kind of star came to light on Aug. 6 last year and was referred to, by astronomers, as LGM (Little Green Men). Now it is thought to be a novel type between a white dwarf and a neutron [sic]. The name Pulsar is likely to be given to it. Dr. A. Hewish told me yesterday: "… I am sure that today every radio telescope is looking at the Pulsars."
The suggestion that pulsars were rotating neutron stars was put forth independently by Thomas Gold and Franco Pacini in 1968, and was soon proven beyond doubt by the discovery of a pulsar with a very short (33-millisecond) pulse period in the Crab nebula.
In 1974, Antony Hewish became the first astronomer to be awarded the Nobel Prize in physics. Considerable controversy is associated with the fact that Professor Hewish was awarded the prize while Bell, who made the initial discovery while she was his Ph.D student, was not.
### Subsequent history
In 1974, Joseph Hooton Taylor, Jr. and Russell Hulse discovered, for the first time, a pulsar in a binary system, PSR B1913+16. This pulsar orbits another neutron star with an orbital period of just eight hours. Einstein's theory of general relativity predicts that this system should emit strong gravitational radiation, causing the orbit to continually contract as it loses orbital energy. Observations of the pulsar soon confirmed this prediction, providing the first ever evidence of the existence of gravitational waves. As of 2004, observations of this pulsar continue to agree with general relativity. In 1993, the Nobel prize in physics was awarded to Taylor and Hulse for the discovery of this pulsar.
In 1982, a pulsar with a rotation period of just 1.6 milliseconds was discovered, by Shri Kulkarni and Don Backer. Observations soon revealed that its magnetic field was much weaker than ordinary pulsars, while further discoveries cemented the idea that a new class of object, the "millisecond pulsars" (MSPs) had been found. MSPs are believed to be the end product of X-ray binaries. Owing to their extraordinarily rapid and stable rotation, MSPs can be used by astronomers as clocks rivalling the stability of the best atomic clocks on Earth. Factors affecting the arrival time of pulses at the Earth by more than a few hundred nanoseconds can be easily detected and used to make precise measurements. Physical parameters accessible through pulsar timing include the 3D position of the pulsar, its proper motion, the electron content of the interstellar medium along the propagation path, the orbital parameters of any binary companion, the pulsar rotation period and its evolution with time. (These are computed from the raw timing data by Tempo, a computer program specialized for this task.) After these factors have been taken into account, deviations between the observed arrival times and predictions made using these parameters can be found and attributed to one of three possibilities: intrinsic variations in the spin period of the pulsar, errors in the realization of Terrestrial Time against which arrival times were measured, or the presence of background gravitational waves. Scientists are currently attempting to resolve these possibilities by comparing the deviations seen amongst several different pulsars, forming what is known as a Pulsar Timing Array. With luck, these efforts may lead to a time scale a factor of ten or better than currently available, and the first ever direct detection of gravitational waves.
The first ever extrasolar planets were found orbiting a MSP, by Aleksander Wolszczan. This discovery presented important evidence concerning the widespread existence of planets outside the solar system, although it is very unlikely that any life form could survive in the environment of intense radiation near a pulsar.
## Pulsar classes
Three distinct classes of pulsars are currently known to astronomers, according to the source of energy that powers the radiation:
Although all three classes of objects are neutron stars, their observable behaviour and the underlying physics are quite different. There are, however, connections. For example, X-ray pulsars are probably old rotation-powered pulsars that have already lost most of their energy, and have only become visible again after their binary companions expanded and began transferring matter on to the neutron star. The process of accretion can in turn transfer enough angular momentum to the neutron star to "recycle" it as a rotation-powered millisecond pulsar.
## Naming
Initially pulsars were named with letters of the discovering observatory followed by their right ascension (e.g. CP 1919). As more pulsars were discovered, the letter code became unwieldy and so the convention was then superseded by the letters PSR (Pulsating Source of Radio) followed by the pulsar's right ascension and degrees of declination (e.g. PSR 0531+21) and sometimes declination to a tenth of a degree (e.g. PSR 1913+167). Pulsars that are very close together sometimes have letters appended (e.g. PSR 0021-72C and PSR 0021-72D).
The modern convention is to prefix the older numbers with a B (e.g. PSR B1919+21) with the B meaning the coordinates are for the 1950.0 epoch. All new pulsars have a J indicating 2000.0 coordinates and also have declination including minutes (e.g. PSR J1921+2153). Pulsars that were discovered before 1993 tend to retain their B names rather than use their J names (e.g. PSR J1921+2153 is more commonly known as PSR B1919+21). Recently discovered pulsars only have a J name (e.g. PSR J0437-4715). All pulsars have a J name that provides more precise coordinates of its location in the sky.
## Glitch prediction
In June 2006, astronomer John Middleditch and his team at LANL announced the first prediction of glitches with observational data from the Rossi X-ray Timing Explorer. They used observations of the pulsar PSR J0537-6910.
## Applications
The study of pulsars has resulted in many applications in physics and astronomy. Striking examples include the confirmation of the existence of gravitational radiation as predicted by general relativity and the first detection of an extrasolar planetary system.
### As probes of the interstellar medium
The radiation from pulsars passes through the interstellar medium (ISM) before reaching Earth. Free electrons in the warm (8000 K), ionized component of the ISM and H II regions affect the radiation in two primary ways. The resulting changes to the pulsar's radiation provide an important probe of the ISM itself.
Due to the dispersive nature of the interstellar plasma, lower-frequency radio waves travel through the medium slower than higher-frequency radio waves. The resulting delay in the arrival of pulses at a range of frequencies is directly measurable as the dispersion measure of the pulsar. The dispersion measure is the total column density of free electrons between the observer and the pulsar,
$mathrm\left\{DM\right\} = int_0^D n_e\left(s\right) ds,$
where $D$ is the distance from the pulsar to the observer and $n_e$ is the electron density of the ISM. The dispersion measure is used to construct models of the free electron distribution in the Milky Way Galaxy.
Additionally, turbulence in the interstellar gas causes density inhomogeneities in the ISM which cause scattering of the radio waves from the pulsar. The resulting scintillation of the radio waves—the same effect as the twinkling of a star in visible light due to density variations in the Earth's atmosphere—can be used to reconstruct information about the small scale variations in the ISM. Due to the high velocity (up to several hundred km/sec) of many pulsars, a single pulsar scans the ISM rapidly, which results in changing scintillation patterns over timescales of a few minutes.
## Significant pulsars
• The first radio pulsar CP 1919 (now known as PSR 1919+21), with a pulse period of 1.337 seconds and a pulse width of 0.04 second, was discovered in 1967 (Nature 217:709-713, 1968).
• The first binary pulsar, PSR 1913+16, whose orbit is decaying at the exact rate predicted due to the emission of gravitational radiation by general relativity
• The first millisecond pulsar, PSR B1937+21
• The brightest millisecond pulsar, PSR J0437-4715
• The first X-ray pulsar, Cen X-3
• The first accreting millisecond X-ray pulsar, SAX J1808.4-3658
• The first extrasolar planets to be discovered orbit the pulsar PSR B1257+12
• The first double pulsar binary system, PSR J0737−3039
• The magnetar SGR 1806-20 produced the largest burst of energy in the Galaxy ever experimentally recorded on 27 December 2004
• PSR B1931+24 "... appears as a normal pulsar for about a week and then 'switches off' for about one month before emitting pulses again. [..] this pulsar slows down more rapidly when the pulsar is on than when it is off. [.. the] braking mechanism must be related to the radio emission and the processes creating it and the additional slow-down can be explained by a wind of particles leaving the pulsar's magnetosphere and carrying away rotational energy.
• PSR J1748-2446ad, at 716 Hz, the pulsar with the highest rotation speed.
• PSR J0108-1431, the closest known pulsar to the Earth. It lies in the direction of the constellation Cetus, at a distance of about 85 parsecs (280 light years). Nevertheless, it was not discovered until 1993 due to its extremely low luminosity. It was discovered by the Danish astronomer Thomas Tauris in collaboration with a team of Australian and European astronomers using the Parkes 64-meter radio telescope. The pulsar is 1000 times weaker than an average radio pulsar and thus this pulsar may represent the tip of an iceberg of a population of more than half a million such dim pulsars crowding our Milky Way.
• PSR J1903+0327, a ~2.15 ms pulsar discovered to be in a highly eccentric binary star system with a sun-like star.
## References
• Duncan R. Lorimer, "Binary and Millisecond Pulsars at the New Millennium", Living Rev. Relativity 4, (2001), http://www.livingreviews.org/lrr-2001-5
• D. R. Lorimer & M. Kramer; Handbook of Pulsar Astronomy; Cambridge Observing Handbooks for Research Astronomers, 2004
• Ingrid H. Stairs, "Testing General Relativity with Pulsar Timing", Living Rev. Relativity 6, (2003): http://www.livingreviews.org/lrr-2003-5
• Peter A. Sturrock; The UFO Enigma: A New Review of the Physical Evidence; Warner Books, 1999; ISBN 0-446-52565-0 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8371626734733582, "perplexity": 1618.9265975807868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345762908/warc/CC-MAIN-20131218054922-00069-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/the-independance-of-horizontal-and-vertical-motion.79968/ | # The independance of horizontal and vertical motion
1. Jun 22, 2005
### Cheman
The independance of horizontal and vertical motion....
Obviously it is possible to prove the independance of horizontal and vertical motion empirically - we only need look at projectile motion following a parabolic path. However, I have never found a convincing algebraic proof the for the independence of these 2 types of motion.
I have asked my physics teacher and he says that it can apparetly be proved by treating Gravity as a Vector, and then assessing the overall motion of a body - apparently things like "gcos90" appears, which obviously equal "0", and this can be used to show the independance of horizontal and vertical motion.
If anyone could supply me with a convincing algebriac proof i would be really greatful! :tongue2:
2. Jun 22, 2005
### ZapperZ
Staff Emeritus
What do you mean by "algebraic proof"? If you accept the mathematical concept of a vector, then if you have the vector oriented along the x-axis, can you find the component of the vector along the y-axis?
Zz.
3. Jun 22, 2005
### Claude Bile
The x and y unit vectors are orthogonal, thus the x and y components can be treated seperately. A ZapperZ pointed out, there really is no algebraic proof to consider.
Claude.
4. Jun 22, 2005
### pervect
Staff Emeritus
You also need to define how vectors multiply before you can say that$$\hat{x}$$ and $$\hat{y}$$ are orthagonal. A metric such as
ds^2 = dx^2 + dy^2
is one way of giving the necessary defintion of the vector product $$\hat{x} \cdot \hat{y}$$, and a diagonal metric such as the specific example above is necessary and sufficient to make these two vectors orthagonal.
5. Jun 23, 2005
### HallsofIvy
Staff Emeritus
You can't prove a physical fact mathematically! They can only be proven by experiment (what you called "empirically").
(If you assume that physical velocity can be represented by mathematical vectors, then you can use the properties of vectors. Of course, you would have to base that "assumption" on experiment.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9797219634056091, "perplexity": 708.5895949404968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279379.41/warc/CC-MAIN-20170116095119-00007-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://mathhelpforum.com/math-software/194886-plotting-fourier-series-approximation-square-wave-using-matlab.html | # Math Help - Plotting Fourier series approximation for a square wave using MATLAB.
1. ## Plotting Fourier series approximation for a square wave using MATLAB.
Suppose $f(x)$ is a square wave of period $2\pi$ such that
$f(x)=1$ for $0
$f(x)=-1$ for $\pi
The Fourier series for $f(x)$ is
$f(x)=\frac{4}{\pi}\left[\frac{\sin x}{1}+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}+\ldots\right]$
I tried plotting this using MATLAB (using the first 10 terms):
Code:
x=linspace(-4*pi,4*pi,100);
y=0;
k=1;
for n=1:10
y=y+4*sin(k*x)/pi*k;
k=k+2;
end
plot(x,y)
axis([-4*pi 4*pi -2 2])
xlabel('x')
ylabel('Square wave approximation using first 10 terms')
I have attached my output but it doesn't look anything like a square wave. What am I doing wrong?
Attached Thumbnails
2. ## Re: Plotting Fourier series approximation for a square wave using MATLAB.
Hi alexmahone!
Perhaps you should write
y=y+4*sin(k*x)/(pi*k);
y=y+4*sin(k*x)/pi*k;
3. ## Re: Plotting Fourier series approximation for a square wave using MATLAB.
Originally Posted by ILikeSerena
Hi alexmahone!
Perhaps you should write
y=y+4*sin(k*x)/(pi*k); | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9950501322746277, "perplexity": 2885.5440147781032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395546.12/warc/CC-MAIN-20160624154955-00089-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/135129-what-variance-division-2-random-variables.html | # Thread: What is the variance of the division of 2 random variables?
1. ## What is the variance of the division of 2 random variables?
More specifically, if X and Y are Poisson-distributed, what is the variance of X/Y (assuming that X and Y are independent)?
Thanks!
2. Originally Posted by floats
More specifically, if X and Y are Poisson-distributed, what is the variance of X/Y (assuming that X and Y are independent)?
Thanks!
As far as I know, Poisson random variables can be 0 (with probability $e^{-\lambda}$), so why would the ratio $\frac{X}{Y}$ make sense?
3. I see, that is a good point.
What if we consider X/(Y+1) instead? What would the variance of that be?
4. Originally Posted by floats
What if we consider X/(Y+1) instead? What would the variance of that be?
To get the variance, you need the expectation, and the expectation of the square. Since the variables are independent, you need $E[X]E\left[\frac{1}{Y+1}\right]$ and $E[X^2]E\left[\frac{1}{(1+Y)^2}\right]$. I guess you know how to get $E[X]$ and $E[X^2]$ ( $\lambda$ and $\lambda+\lambda^2$ since I know the variance is $\lambda$). So let's consider the other ones.
$E\left[\frac{1}{Y+1}\right]=e^{-\lambda}\sum_{n=0}^\infty \frac{1}{n+1}\frac{\lambda^n}{n!}=e^{-\lambda}\sum_{n=0}^\infty \frac{\lambda^n}{(n+1)!}$ $=\frac{1}{\lambda}e^{-\lambda}\sum_{n=0}^\infty \frac{\lambda^{n+1}}{(n+1)!}$ ; then let $m=n+1$ in the sum to see that it equals $e^{\lambda}-1$.
$E\left[\frac{1}{(Y+1)^2}\right]=e^{-\lambda}\sum_{n=0}^\infty\frac{1}{(n+1)^2}\frac{\l ambda^n}{n!}$ is more tricky. I'd let $f(\lambda)=\sum_{n=0}^\infty \frac{1}{(n+1)^2}\frac{\lambda^{n+1}}{n!}$ and compute $f'(\lambda)=\sum_{n=0}^\infty \frac{1}{n+1}\frac{\lambda^n}{n!}$, which we have computed above ; now integrate to get $f(\lambda)$, and multiply by $e^{-\lambda}\frac 1\lambda$ to get $E[\frac{1}{(Y+1)^2}]$.
5. I got the $E[\frac{1}{Y+1}]$ part but was struggling with the $E[\frac{1}{(Y+1)^2}]$ part.
Thanks for the help! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.997850775718689, "perplexity": 260.6428082661724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891808539.63/warc/CC-MAIN-20180217224905-20180218004905-00214.warc.gz"} |
https://apboardsolutions.in/ap-board-7th-class-maths-solutions-chapter-5-intext-questions-new-syllabus/ | # AP Board 7th Class Maths Solutions Chapter 5 Triangles InText Questions
SCERT AP 7th Class Maths Solutions Pdf Chapter 5 Triangles InText Questions and Answers.
## AP State Syllabus 7th Class Maths Solutions 5th Lesson Triangles InText Questions
Check Your Progress [Page No. 92]
Question 1.
Classify the following triangles according to the length of their sides:
[Page No. 93]
Question 2.
Classify the following triangles according to the measure of their angles.
Let’s Think [Page No. 97]
Question 1.
Can you find a triangle in which each angle is less than 60°?
Sum of angles in a triangle is 180°.
If each angle is less than 60°, then the sum of angles of a triangle is < 180°. So, triangle cannot form. So, we cannot find a triangle in which each angle is less than 60°.
Check Your Progress [Page No. 99]
Question 1.
Find the value of ‘x’ in the given figure.
In the given figure,
∠HAC = ∠TAE = a (∵ Vertically opposite angles)
In ∆AHC,
we know, ∠H + ∠C + ∠HAC 180°
⇒ 60° + 80° + a = 180°
⇒ 140° + a – 140° = 180°- 140° … a = 40°.
∴ ∠HAC = ∠TAE = a = 40°
In ∆ATE, we know,
∠T + ∠TAE + ∠E = 180°
⇒ x + a + 70° = 180°
⇒ x + 40° + 70° = 180°
⇒ x + 110°- 110° = 180°- 110°
∴ x = 70°
Let’s Do Activity [Page No. 101]
Question 1.
Measure the length of the sides of ∆ABC and fill the following table.
From the above table, we can conclude that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Question 2.
Take the same measurements of the previous activity and note the results as given below.
From the above table we can conclude that the difference of the lengths of any two sides of a triangle is less than the length of the third side.
Question 3.
Fill the following table as per measurements given in the above triangles:
What do you observe from the last four columns in above table ?
• In any triangle the opposite side to the biggest angle is bigger than the other two sides.
• In any triangle the opposite side to the smallest angle is smaller than the other two sides.
Let’s Think [Page No. 102]
Question 1.
What are the measurements of angles of an equilateral triangle?
AD = DI – AI = 6 cm (All angles are equal)
∠A = ∠D = ∠I = 60° (All angles are equal)
So, in an equilateral triangle,
All Sides are equal in length. Each angle is equal to 60°.
Puzzle Time [Page No. 104]
Question 1.
Find the interior angles of all triangles by using given dues.
Hints:
Each angle in an equilateral triangle is 60°.
Straight angle = 180°
In an isosceles triangle, two angles are equal.
In a scalene triangle all angles are unequal.
Right angle = 90°.
Obtuse angle > 90°.
Examples:
Question 1.
In a triangle, two angles are 43° and 57°. Find the third angle.
Given two angles of a triangle are 43° and 57°.
Sum of these two angles = 43° + 57° = 100°
In a triangle the sum of the interior angles is 180°.
∴ Third angle = 180° – 100° = 80°
Question 2.
Find the value of ‘x’ in the following triangles :
From the figure
∠A + ∠B + ∠C = 180° (v Sum of three angles in triangle is 180°)
⇒ 65° + 60° + x = 180°
⇒ 125° + x = 180°
∴ x =180°- 125° = 55°
From the figure
∠P + ∠Q + ∠R = 180°
x + x + 80° = 180°
⇒ 2x + 80° = 180°
⇒ 2x = 180°-80°
⇒ 2x = 100°
⇒ x = $$\frac{100^{\circ}}{2}$$ = 50°
∴ x = 50°
From the figure
∠X + ∠Y + ∠Z – 180°
⇒ x + 90° + 42° = 180°
⇒ x + 132° = 180°
⇒ x = 180°- 132°
∴ x = 48°
Question 3.
Find the values of x and y in the given triangle.
In the ∆ABC
∠ACD + ∠ACB = 180° (linear pair of angles)
110° + y = 180°
⇒ y = 180°- 110°
∴ y = 70° ……………….(i)
∠BAC + ∠ACB + ∠CBA = 180°
⇒ x + y + 60° = 180°
⇒ x + 70° + 60° = 180° (From (i))
⇒ x + 130° = 180°
∴ x = 50°
Question 4.
In a right angled triangle one acute angle is 44°, then find the other acute angle.
We know that, sum of the two acute angles in a right angled triangle is 90°.
Given that, in the right angled tri-angle one of the acute angles = 44°
Other acute angle in right angled triangle = 90° – 44° = 46°
Question 5.
The angles of a triangle are (x + 10)°, (x – 20)° and (x + 40)°. Find the value of x and the measure of the angles.
Given that the angles of the triangle are
(x + 10)°, (x – 20)° and (x + 40)°
(x + 10)° + (x – 20)° + (x + 40)° = 180°
⇒ x + 10° + x – 20° + x + 40° = 180°
⇒ 3x + 30° = 180°
⇒ 3x – 180° – 30°
⇒ 3x = 150°
⇒ x = $$\frac{150^{\circ}}{3}$$ = 50°
The angles are,
x + 10° = 50° + 10° = 60°
x – 20° = 50° – 20° = 30°
x + 40° = 50° + 40° = 90°
∴ Measure of the angles are 60°, 30° and 90°.
Question 6.
In the ∆ABC, exterior angle at ∠C = 105° and ∠A=65°. Find the other interior opposite angle.
Given that, exterior angle at ∠C = 105°.
One of the interior opposite angle ∠A = 65°.
The other interior opposite angle is ∠B .
∠A + ∠B =105°(∵ Exterior angle property of a triangle)
65° + ∠B = 105°
⇒ ∠B = 105° – 65° = 40°
Question 7.
Find the exterior angle of the given triangle.
From the figure, ∠P = 30°, ∠Q = 35°
Exterior angle at R = ∠P + ∠Q (∵ Exterior angle property of a triangle)
= 30° + 35° = 65°
Question 8.
Find the values of ‘x’ and ‘y’ in the following figure. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8251051902770996, "perplexity": 2658.263382684524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104248623.69/warc/CC-MAIN-20220703164826-20220703194826-00668.warc.gz"} |
https://hyperelectronic.net/learn-electronics/lesson-voltage-divider/ | # Lesson : Voltage Divider
In the past chapter, we looked at series and parallel circuit. One things you may have notice is that the series circuit can be used as voltage divider. We have already seen a small portion of the theory about the voltage divider in the series circuit lessons but we will now study in depth the voltage divider. We will also take a look at some real life applications of the voltage divider.
The voltage divider is actually quite simple. It is two resistors in series used to reduce the input voltage (in this case VDC) to a smaller voltage (Vout).
Below you will find all the equations for the voltage divider. All these formulas are found with Ohm’s law and the equivalent circuit:
$V_{out} = R_{2} \cdot I$
$I = \cfrac{VDC}{(R_{1}+R_{2})}$
If we replace I in the first formula, we get the formula for the voltage divider:
$V_{out} = VDC \cdot \cfrac{R_{2}}{(R_{1}+R_{2})}$
The voltage divider has multiple applications in real life. Below you can find one of the popular applications for it:
Voltage monitoring with microcontroller:
Often in personal computer and server, you have a microcontroller monitoring voltage, current, etc. These controllers often works at low voltage (< 3.3V) but needs to monitor voltage higher than this (12V, 5V). Sometimes the integrated analog to digital converter (ADC) works at even lower voltage than the microcontroller since they have internal reference. If you want to monitor the 12V rail, you will need to reduce the 12V to the appropriate voltage for your ADC. The voltage divider is perfect for this but you will need to use resistors which have a pretty low tolerance to get accurate result (<1%).
Example :
In the image below, we have 12V from the D.C voltage source. R1 = 100 ohms and R2 = 50 ohms. We want to calculate Vout. (Vout = V2 in the image below)
$Vout=Vin*\cfrac{R2}{R1+R2}$
$Vout=12V*\cfrac{50\Omega}{50\Omega+100\Omega}$
$Vout=4V$
We can also calculate V1 with the equation above. We only need to modify the equation for V1. We replace Vout with V1 and instead of R2, we have R1 :
$V1=Vin*\cfrac{R1}{R1+R2}$
$V1=12V*\cfrac{100\Omega}{50\Omega+100\Omega}$
$V1=8V$
The equation of the voltage divider could be modify for more than two resistors too if needed as long as the resistors are in series :
$Vx=Vin*\cfrac{Rx}{Rx+Rn1+Rn2+\dots}$
where Vx is the voltage across the resistor Rx and Vin is the voltage across all resistors in series. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 10, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8877478837966919, "perplexity": 636.6499029710058}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178381989.92/warc/CC-MAIN-20210308052217-20210308082217-00209.warc.gz"} |
https://www.physicsforums.com/threads/even-more-induction.241711/ | # Homework Help: Even More Induction
1. Jun 23, 2008
### Moridin
1. The problem statement, all variables and given/known data
Show that
$$\forall n \in \mathbb{N}: 3^{n} \geq n^{3}$$
3. The attempt at a solution
(1) Show that it is true for n = 1:
$$3^{1} \geq 1^{3}$$
(2) Show that if it is true for n = p, then it is true for n = p + 1:
Assume that $$3^{p} \geq p^{3}$$
Now,
$$3^{p+1} = 3 \cdot 3^{p} = 3^{p} + 3^{p} + 3^{p}$$
$$(p+1)^{3} = p^{3} + 3p^{2} + 3p + 1$$
Given our assumption, we know that if it could be demonstrated that
$$3^{p} + 3^{p} \geq 3p^{2} + 3p + 1$$
then we are done. From here, I'm not sure how to proceed. Should I pull some moves from analysis and argue that certain functions grow faster than others above a certain n? The last inequality is also a stronger criteria, but does not apply to p = 1 or p = 2, since 3^{p} was larger than p^{3}.
2. Jun 23, 2008
### dynamicsolo
Why not just compare
$$3^{p} + 3^{p} + 3^{p}$$
term-by-term with
$$p^{3} + 3p^{2} + (3p + 1)$$ ?
You've assumed
$$3^{p} \geq p^{3}$$
and, beyond some low value of p,
$$3^{p} \geq 3p^{2}$$
and
$$3^{p} \geq 3p+1$$.
EDIT: the last two inequalities only fail for p = 1, but you've already shown that the proposed inequality works there. As for showing that the inequalities work for p>= 2,
how about taking log base 3 of both sides?
Last edited: Jun 23, 2008 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8172643184661865, "perplexity": 598.9203101472942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867559.54/warc/CC-MAIN-20180526151207-20180526171207-00453.warc.gz"} |
https://homework.cpm.org/category/CON_FOUND/textbook/mc2/chapter/2/lesson/2.3.4/problem/2-130 | ### Home > MC2 > Chapter 2 > Lesson 2.3.4 > Problem2-130
2-130.
Simplify the expressions below.
1. $8 - 13 - (-4)$
1. $4 - (-7) + (-2)$
1. $-7 + 3 - 20$
$−1$
If you start at $0$, move to left $7$, right $3$, and left $20$, where do you end up? | {"extraction_info": {"found_math": true, "script_math_tex": 8, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9535096883773804, "perplexity": 2517.2157710857764}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057524.58/warc/CC-MAIN-20210924110455-20210924140455-00545.warc.gz"} |
https://www.physicsforums.com/threads/probability-problem-urgend.103869/ | Probability Problem(Urgend)
1. Dec 12, 2005
Mathman23
Hi Guys,
I have this probability Problem where I have become stuck, therefore I pray that there is somebody in here who can give me some advice :)
It goes a something like that:
The two dimensional discrete Stochastic vector (X,Y) has the probability function $$P_{X,Y}$$ which is
$$P(X=x,Y=y) = \left\{ \begin{array}{ll}\frac{{c e^{- \lambda}{\lambda ^{y}}}}{{y!}} & \textrm{where} \ x \in (-2,-1,0,1) \ \textrm{and} \ \ y \in (0,1,\ldots)&\\0 & \textrm{other.}&\\\end{array} \right.$$
where $$\lambda > 0$$ and c > 0
(a) The support supp P_{X,Y} = {-2,-1,0,1, \ldots}
(b)
The Probability functions $$P_X$$ and $$P_Y$$ for X and Y are
$$P(X=x) = \left\{ \begin{array}{ll}c & \textrm{where} \ x \in (-2,-1,0,1) \\0 & \textrm{other.}&\\\end{array} \right.$$
$$P(Y=y) = \left\{ \begin{array}{ll}\frac{{c e^{- \lambda}{\lambda ^{y}}}}{{y!}} & \textrm{and} \ \ y \in (0,1,\ldots)&\\0 & \textrm{other.}&\\\end{array} \right.$$
This is done by showing that $$Y \ \~{} pol(\lambda)$$,
$$\sum_{y=0} ^{\infty} \frac{{e^{- \lambda}{\lambda ^{y}}}}{{y!}} =\sum_{y=1} ^{\infty} \frac{{ e^{- \lambda}{\lambda ^{y}}}}{{y-1!}} = \lambda \sum_{y=1} ^{\infty} \frac{{ e^{- \lambda}{\lambda ^{(y-1)}}}}{{(y-1)!}} = \lambda \sum_{v=0} ^{\infty} \frac{{ e^{- \lambda}{\lambda ^{(v)}}}}{{(v)!}} = \lambda$$
where v = (y-1)
I have two question
(c) I need to find the constant 'c'. How do I go about doing that?
(d)
If $$\lambda = 1$$, then $$P(X=Y) = 1/2 e^ -1$$. Any hits on how I show that ?
Sincerely and God bless You
Fred
Last edited: Dec 12, 2005
2. Dec 12, 2005
EnumaElish
Is this homework?
3. Dec 12, 2005
Mathman23
Yes,
I there is somebody out there who culd please give me a hit on how to solve question (c) and (d)??
Many thanks in advance and God bless You :)
/Fred
4. Dec 13, 2005
EnumaElish
5. Dec 14, 2005
EnumaElish
Similar Discussions: Probability Problem(Urgend) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.909018337726593, "perplexity": 1808.662688948255}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891811352.60/warc/CC-MAIN-20180218023321-20180218043321-00769.warc.gz"} |
https://gmatclub.com/forum/an-experiment-conducted-aboard-space-lab-in-1983-was-the-154071.html | It is currently 25 Jun 2017, 20:50
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An experiment conducted aboard Space Lab in 1983 was the
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An experiment conducted aboard Space Lab in 1983 was the first attempt to grow protein crystals in the low-gravity environment of space. That experiment is still cited as evidence that growing crystals in microgravity can increase crystal size: the authors reported that they grew lysozyme protein crystals 1,000 times larger than crystals grown in the same device on Earth. Unfortunately, the authors did not point out that their crystals were no larger than the average crystal grown using other, more standard techniques in an Earth laboratory.
No research has yet produced results that could justify the enormous costs of producing crystals on a large scale in space. To get an unbiased view of the usefulness of microgravity crystal growth, crystals grown in space must be compared with the best crystals that have been grown with standard techniques on Earth. Given the great expense of conducting such experiments with proper controls, and the limited promise of experiments performed thus far, it is questionable whether further experiments in this area should even be conducted.
1. According to the passage, which of the following is true about the Space Lab experiment conducted in 1983?
(A) It was the first experiment to take place in the microgravity environment of space.
(B) It was the first experiment in which researchers in space were able to grow lysozyme protein crystals greater in size than those grown on Earth.
(C) Its results have been superseded by subsequent research in the field of microgravity protein crystal growth.
(D) Its results are still considered by some to be evidence for the advantages of microgravity protein crystal growth.
(E) Its results are considered by many to be invalid because nonstandard techniques were employed.
[Reveal] Spoiler:
D
2. It can be inferred from the passage that the author would find the Space Lab experiment more impressive if which of the following were true?
(A) The results of the Space Lab experiment could be replicated in producing other kinds of crystals in addition to lysozyme protein.
(B) The device used in the experiment produced larger crystals on Earth than it did in space.
(C) The size of the crystals produced in the experiment exceeded the size of crystals grown in Earth laboratories using standard techniques.
(D) The cost of producing the crystals in space exceeded that of producing them using standard laboratory techniques.
(E) The standard techniques used in Earth laboratories were modified in the Space Lab experiment due to the effects of microgravity.
[Reveal] Spoiler:
C
3. Which of the following can be inferred from the passage about the device used to grow crystals in the Space Lab experiment?
(A) The device is more expensive to manufacture than are the devices used in standard techniques in an Earth laboratory.
(B) The device has not been used to grow crystals in space since the Space Lab experiment of 1983.
(C) Crystals grown in the device on Earth tend to be much smaller than crystals grown in it in space.
(D) Crystals grown in the device in space have been exceeded in size by crystals grown in subsequent experiments in space using other devices.
(E) The experiments in which the device was used were conducted with proper controls.
[Reveal] Spoiler:
C
4. The passage suggests that the author would most probably agree with which of the following assessments of the results of the Space Lab experiment?
(A) Although the results of the experiment are impressive, the experiment was too limited in scope to allow for definitive conclusions.
(B) The results of the experiment are impressive on the surface but the report is misleading.
(C) The results of the experiment convincingly confirm what researchers have long suspected.
(D) Because of design flaws, the experiment did not yield any results relevant to the issue under investigation.
(E) The results of the experiment are too contradictory to allow for easy interpretation.
[Reveal] Spoiler:
A
[Reveal] Spoiler: Question #1 OA
[Reveal] Spoiler: Question #2 OA
[Reveal] Spoiler: Question #3 OA
[Reveal] Spoiler: Question #4 OA
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Re: An experiment conducted aboard Space Lab in 1983 was the [#permalink]
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08 Aug 2013, 01:42
an easy passage
got one wrong.
overall time: 0643
post your doubts if need help.
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08 Aug 2013, 15:05
The answer to question 4 seems incorrect, it should be "B", no? The fact that the author points out that the report failed to say that the crystals grown in the space lab tended to be the same size as those grown by different means on earth would point to "misleading"...
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11 Aug 2013, 06:24
Allie8238 wrote:
The answer to question 4 seems incorrect, it should be "B", no? The fact that the author points out that the report failed to say that the crystals grown in the space lab tended to be the same size as those grown by different means on earth would point to "misleading"...
agreed ... the report is misleading ... i go with B
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11 Aug 2013, 10:46
Total time:aprox 6 mins
DCCA
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Re: An experiment conducted aboard Space Lab in 1983 was the [#permalink]
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23 Aug 2013, 08:37
1
KUDOS
Completed within 6mins with 1 wrong.
Can some one explain Question 1 and its answer choice?!
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24 Aug 2013, 12:30
vabhs192003 wrote:
Completed within 6mins with 1 wrong.
Can some one explain Question 1 and its answer choice?!
Ans option D reflect the geography or the main idea of the 1st paragraph.
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24 Aug 2013, 22:29
vabhs192003 wrote:
Completed within 6mins with 1 wrong.
Can some one explain Question 1 and its answer choice?!
1. According to the passage, which of the following is true about the Space Lab experiment conducted in 1983?
Here the author is asking which of the following statements is true about the space lab experiment conducted in 1983.....
(A) It was the first experiment to take place in the microgravity environment of space.we dont know if this is the first experiment in space environment. Passage mentions first attempt to grow protein crystals in the low-gravity environment of space.
(B) It was the first experiment in which researchers in space were able to grow lysozyme protein crystals greater in size than those grown on Earth. This choice says this was the first time the researchers were able to grow protein in space and have failed other times when the experiment was carried out...
(C) Its results have been superseded by subsequent research in the field of microgravity protein crystal growth.The second paragraph last line says no further experiment in this was carried out.
(D) Its results are still considered by some to be evidence for the advantages of microgravity protein crystal growth.Line 2 in passage "That experiment is still cited as evidence that growing crystals in microgravity" exactly implies this.
(E) Its results are considered by many to be invalid because nonstandard techniques were employed.Not mentioned in passage that results were considered invalid.
Hope this explains....
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Re: An experiment conducted aboard Space Lab in 1983 was the [#permalink]
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24 Aug 2013, 22:41
mehtakaustubh wrote:
vabhs192003 wrote:
Completed within 6mins with 1 wrong.
Can some one explain Question 1 and its answer choice?!
1. According to the passage, which of the following is true about the Space Lab experiment conducted in 1983?
Here the author is asking which of the following statements is true about the space lab experiment conducted in 1983.....
(A) It was the first experiment to take place in the microgravity environment of space.we dont know if this is the first experiment in space environment. Passage mentions first attempt to grow protein crystals in the low-gravity environment of space.
(B) It was the first experiment in which researchers in space were able to grow lysozyme protein crystals greater in size than those grown on Earth. This choice says this was the first time the researchers were able to grow protein in space and have failed other times when the experiment was carried out...
(C) Its results have been superseded by subsequent research in the field of microgravity protein crystal growth.The second paragraph last line says no further experiment in this was carried out.
(D) Its results are still considered by some to be evidence for the advantages of microgravity protein crystal growth.Line 2 in passage "That experiment is still cited as evidence that growing crystals in microgravity" exactly implies this.
(E) Its results are considered by many to be invalid because nonstandard techniques were employed.Not mentioned in passage that results were considered invalid.
Hope this explains....
I got thrown off by the word "Advantage" in option D. The passage does not indicate any kinda advantage from the experiment. The passage though indicates that evidences from the experiment are used to decide whether such experiments are practical or not.
But I did get lured to option B and ended up selecting it. I guess option D is best among the worst.
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24 Aug 2013, 22:47
vabhs192003 wrote:
mehtakaustubh wrote:
vabhs192003 wrote:
Completed within 6mins with 1 wrong.
Can some one explain Question 1 and its answer choice?!
1. According to the passage, which of the following is true about the Space Lab experiment conducted in 1983?
Here the author is asking which of the following statements is true about the space lab experiment conducted in 1983.....
(A) It was the first experiment to take place in the microgravity environment of space.we dont know if this is the first experiment in space environment. Passage mentions first attempt to grow protein crystals in the low-gravity environment of space.
(B) It was the first experiment in which researchers in space were able to grow lysozyme protein crystals greater in size than those grown on Earth. This choice says this was the first time the researchers were able to grow protein in space and have failed other times when the experiment was carried out...
(C) Its results have been superseded by subsequent research in the field of microgravity protein crystal growth.The second paragraph last line says no further experiment in this was carried out.
(D) Its results are still considered by some to be evidence for the advantages of microgravity protein crystal growth.Line 2 in passage "That experiment is still cited as evidence that growing crystals in microgravity" exactly implies this.
(E) Its results are considered by many to be invalid because nonstandard techniques were employed.Not mentioned in passage that results were considered invalid.
Hope this explains....
I got thrown off by the word "Advantage" in option D. The passage does not indicate any kinda advantage from the experiment. The passage though indicates that evidences from the experiment are used to decide whether such experiments are practical or not.
But I did get lured to option B and ended up selecting it. I guess option D is best among the worst.
The passage Line 2 says "growing crystals in microgravity can increase crystal size......grew lysozyme protein crystals 1,000 times larger than crystals grown in the same device on Earth" which is a advantage of microgravity.
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25 Aug 2013, 11:22
HI,
Can someone explain Q3 and Q4 analysis.
THanks,
Jai
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26 Aug 2013, 10:09
I am not getting logic to question 4. can anyone explain?
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27 Aug 2013, 05:32
aryabhatta wrote:
I am not getting logic to question 4. can anyone explain?
Para 1 Line 2 states: growing crystals in microgravity can increase crystal size: the authors reported that they grew lysozyme protein crystals 1,000 times larger than crystals grown in the same device on Earth.
Para1 line 3 states: the authors of exp did not point out that their crystals were no larger than the average crystal grown using other, more standard techniques in an Earth laboratory
Para2 line 2 states:To get an unbiased view of the usefulness of microgravity crystal growth, crystals grown in space must be compared with the best crystals that have been grown with standard techniques on Earth.
now lets analyze the choices considering the above three points:
(A) Although the results of the experiment are impressive, the experiment was too limited in scope to allow for definitive conclusions. This is the correct one...
(B) The results of the experiment are impressive on the surface but the report is misleading.Refering to Para1 line 3 we can say results were drawn without comparing the output of other methods(limited scope) that does not mean the report is misleading...
(C) The results of the experiment convincingly confirm what researchers have long suspected.It is not mentioned in passage what researchers suspected b4 the experiment
(D) Because of design flaws, the experiment did not yield any results relevant to the issue under investigation.No design flaws are discussed in experiment.
(E) The results of the experiment are too contradictory to allow for easy interpretation.Its never mentioned that the results of exp are contradictory
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Re: An experiment conducted aboard Space Lab in 1983 was the [#permalink] 27 Aug 2013, 05:32
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https://cs.stackexchange.com/questions/73840/rice-shapiro-theorem-example?noredirect=1 | # Rice–Shapiro theorem example
Rice-Shapiro theorem is:
Let $Γ$ be a set of computable functions such that the set $R_Γ$ is recursively enumerable. We have $f ∈ Γ$ if and only if there exists a finite function $θ ∈ Γ$ such that $θ ⊆ f$.
If $Γ$ is singleton set, for example $Γ = \lbrace ϕ_{10}\rbrace$ then $R_Γ = \lbrace 10 \rbrace$ and is recursively enumerable, but $Γ$ does not satisfy:
$f ∈ Γ$ if and only if there exists a finite function $θ ∈ Γ$ such that $θ ⊆ f$.
Where I have made a mistake?
The flaw is that $R_\Gamma$ includes not just the natural number $10$, but any natural number which gives rise to the same partial recursive function. It will be infinite, and not recursively enumerable.
In general, for any partial recursive function $f$ you can think of, there will be many Godel numbers which describe it. In other words, many partial recursive function descriptions result in the same actual function. If you are more familiar with Turing machines, there will always be many different Turing machines which compute the same abstract function or language.
In summary: $\Gamma$ is any set of functions. However, $R_\Gamma$ is the set of natural numbers which encode that function. In your case, for $\Gamma = \{\phi_{10}\}$, $R_\Gamma$ is the set of natural numbers which encode the same function as $10$, and this set is not recursively enumerable.
• I think you are right, but can you describe for example halting problem ,as a RE set, by this theorem? I mean for halting problem I can not construct $\Gamma$... – Hossein Apr 13 '17 at 7:44
• @Hossein It is not true that any RE set is equal to $R_\Gamma$ for some $\Gamma$. For the halting problem, though, you can do it. Take $\Gamma$ to be the set of functions which are defined on input $0$. Then $R_\Gamma$ is the set of descriptions of such functions, which is the set of descriptions (programs) which halt on input $0$. – 6005 Apr 13 '17 at 21:00
• You can also see that the theorem holds for thtis $\Gamma$. We have that $f \in \Gamma$ if and only if $f(0)$ is defined, so equivalently, $f \in \Gamma$ if and only if the function just sending $0$ to $a$ is a subset of $f$ for some $a \in \mathbb{N}$. – 6005 Apr 13 '17 at 21:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8593047857284546, "perplexity": 110.1484195571696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038087714.38/warc/CC-MAIN-20210415160727-20210415190727-00062.warc.gz"} |
http://mathhelpforum.com/calculus/212981-simplify-ax-ax-axb-dotc.html | # Math Help - simplify ax[ax(axb)]dotc
1. ## simplify ax[ax(axb)]dotc
Hello, we were given a few practice problems in class the other day and one of them asks...
Simplify a x [a x (a x b)] dot c (where the x refers to cross product)
I have tried to simplify it but have ended up with about a page and a half of computation that I don't know what to do with.
I imagine there is a trick to this but I am clueless. If anyone can help I would appreciate it.
2. ## Re: simplify ax[ax(axb)]dotc
Hey lytwynk.
Hint: Use the fact that the cross product is associative and re-group [a x (a x b)] = (a x a) x b.
3. ## Re: simplify ax[ax(axb)]dotc
But I thought the cross product was not associative for example a x (b x c) does not equal (a x b) x c... At least that's what my text says.
4. ## Re: simplify ax[ax(axb)]dotc
Sorry you are right: it is not associative.
Try using the Jacobi identity and you should get some cancellations:
Jacobi identity - Wikipedia, the free encyclopedia
5. ## Re: simplify ax[ax(axb)]dotc
You could also use the identity $A \times (B \times C) = (A \cdot C)B - (A \cdot B)C$.
- Hollywood | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9623387455940247, "perplexity": 971.7394014258136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257828010.65/warc/CC-MAIN-20160723071028-00198-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/188114/could-there-be-an-exact-formula-for-the-ramsey-numbers/188197#188197 | # Could there be an exact formula for the Ramsey numbers?
Let $R(k)$ denote the diagonal Ramsey number, i.e. the minimal $n$ such that every red-blue colouring of the edges of $K_n$ produces at least one monochromatic $K_k$.
Is it possible that there exists an absolute constant $C>0$ and polynomial $p\in \mathbb{Q}[x]$ such that for all $k$ (or perhaps for all sufficiently large $k$) $$R(k) = C^kp(k)?$$
Even the asymptotic behaviour of $R(k$) is not known (it is known that $\sqrt{2}^{k(1+o(1))}\leq R(k)\leq 4^{k(1+o(1)}$.) So a proof of such an exact result is certainly out of reach. But showing that there cannot be such a formula might be possible.
There are exact formulas known for related Ramsey-type functions (e.g. fractional Ramsey numbers [2] and Ramsey numbers restricted to various classes of graphs, such as planar graphs, line graphs, and perfect graphs [1,3].) As far as I can see, however, the question of an exact formula for the classical Ramsey numbers has not been considered.
[1] Belmonte, Heggernes, van't Hof and Saei, Ramsey numbers for line graphs and perfect graphs, Computing and combinatorics, 204–215, Lecture Notes in Comput. Sci., 7434, Springer, Heidelberg, 2012. http://link.springer.com/chapter/10.1007%2F978-3-642-32241-9_18
[2] Brown and Hoshino, Proof of a conjecture on fractional Ramsey numbers, J. Graph Theory 63 (2010), no. 2, 164–178. http://onlinelibrary.wiley.com/doi/10.1002/jgt.20416/pdf
[3] Steinberg and Tovey, Planar Ramsey numbers, J. Combin. Theory Ser. B 59 (1993), no. 2, 288–296
• Somehow, this makes me think of pattern avoidance: en.wikipedia.org/wiki/Permutation_pattern The permutations that avoid the pattern 1324 is still unknown, and finding a formula for these is according to some, a really tough question (I recall Zeilberger compares this to RH).... Nov 26, 2014 at 9:14
• Here it's mandatory to quote Paul Erdős: ”[...] imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5,5) or they will destroy our planet. In that case, [...], we should marshal all our computers and our mathematicians and attempt to find the value. Suppose, instead, that they ask for R(6,6). In that case, [...], we should attempt to destroy the aliens.” Nov 26, 2014 at 10:23
• I see no reason why there can't be an exact formula, but no reason why it should be of the given form. Something involving binomial coefficients is more likely, but of course this is pure speculation. Nov 26, 2014 at 10:43
Obviously I cannot prove it, but I think the answer is almost surely 'no'. In fact, I don't think any combination of polynomials, exponentials, and factorials will do the job: I believe you should need (at least) to have involvement of the parity mod $2$ function.
My reason for this is of course quite trivial. When you try to find a lower bound example for $R(3,4)$, you observe that neighbourhoods are independent hence the maximum degree of the graph you are constructing is three. Then you note that every vertex has at most five non-neighbours, otherwise you use the value of $R(3,3)$ in the non-neighbourhood. So you think: OK, I should have a $3$-regular graph on $9$ vertices. Unfortunately this doesn't exist because $3$ and $9$ are both odd, and you are forced to use $8$ vertices. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8701032400131226, "perplexity": 267.1104921194133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711003.56/warc/CC-MAIN-20221205032447-20221205062447-00206.warc.gz"} |
https://www.physicsforums.com/threads/check-a-h-k-are-correct-or.345345/ | # Check a , h , k are correct or ?
1. Oct 13, 2009
### r-soy
check a , h , k are correct or ?
If no correct can put the solve in paper to understand more
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2. Oct 13, 2009
### Staff: Mentor
You'll get a quicker reply if you post your problem directly in the message box. Attachments can sometimes take many hours to be approved, and they can't be opened until they're approved.
3. Oct 14, 2009
### Staff: Mentor
So it's now 12 hours later, and your attachment is still pending. That's a lot of time wasted if you were looking for an answer.
4. Oct 14, 2009
### HallsofIvy
Staff Emeritus
First your original equation is written wrong. What you have written is "4x^2+ 12- 9" when I am sure you meant 4x^2+ 12 x+ 9.
After that, everything looks okay until part (3).
You have
Where did you get that "-24"? 4(-3/2)^2= 4(9/4)= 9 so your equation should be 9= 9+ k and that leads to k= 9- 9= 0
I had a bit of difficulty deciphering you handwriting (for a while, I thought your "k" was "15" or "19"!). That often happens when handwritten materials are scanned into the computer. Also, as Mark44 said, it can take a long time for attachments to be approved. If you expect people to take the trouble to help you, at least take the trouble to type something that simple in yourself. It would be best to learn LaTex but I refrained from using using LaTex to show how easy it is to type in something like a polynomial.
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http://www.thespectrumofriemannium.com/tag/motion/ | ## LOG#179. Extra dimensions: Kepler 3rd law.
Hi, everyone! Long time since the last time I wrote here. I am sorry. Life is complex and complicated…But I know you are there, dear followers. Today, Kepler motion and Kepler third law with extra dimensions! Are you prepared? Let … Continue reading
## LOG#053. Derivatives of position.
Position or displacement and its various derivatives define an ordered hierarchy of meaningful concepts. There are special names for the derivatives of position (first derivative is called velocity, second derivative is called acceleration, and some other derivatives with proper name), … Continue reading | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9398706555366516, "perplexity": 3417.279365306186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370500482.27/warc/CC-MAIN-20200331115844-20200331145844-00184.warc.gz"} |
http://sepwww.stanford.edu/data/media/public/docs/sep123/huazhong1/paper_html/node13.html | Next: (3) Least-squares migration/inversion Up: (A) Non-iterative linearized migration/inversion Previous: (1) Wave equation prestack
### (2) Wave theory tomography
(a) Fourier Diffraction Tomography for constant background Wu and Toksoz (1987) gave the plane-wave response in the direction from an incident wave :
(39)
where is the 3D Fourier transform of the object function . is some kind of projection. Comparing this to linear Radon transform, we know that the object function can be accurately restored if the angles of the plane waves continuously change around the object. Fig.3 shows the projection from the real plane wave source and from the virtual plane wave source.
planewave_tomography Figure 3 The geometry of plane wave propagation.
(b) Inverse Generalized Radon Transform for variable background The scattered wavefield after Bron and WKBJ approximation is of the following form:
(40)
where and .It is known that the diffraction-time surface in the data space is a counterpart of the isochron surface in the model space. These dual geometric associations naturally give rise to a corresponding pair of projection operators. Equation (40) can be written as
(41)
The diffraction curve in the data space is a projection of an isochron in the model space. This is a kind of Radon transform Hubral et al. (1996); Miller et al. (1987). The standard Radon transform and inverse Radon transform in three dimensions are given by
(42)
and
(43)
respectively, where p is the distance from the origin to a plane which cuts through the object body, is the unity direction vector which is normal to the plane, and is a point on the plane.
GRT_fig Figure 4 The geometry between the incident and scattering rays near the scattering point, or imaging point .
Comparing this with the classical Radon transform and its inverse, the final 3D inversion formula can be given as
(44)
In equation (44), the angle variable near the imaging point is used, rather than the measurement configuration at the surface. Fig.4 illustrates this. The angle variable is related to the measurement configuration and reflects the seismic wave illumination aperture. Only if the aperture is large can a high resolution image be obtained. The relative true-amplitude imaging is severely affected by the angle variable. Bleistein and Stockwell (2001); Zhang (2004) gave some similar true-amplitude migration/inversion formulas.
Next: (3) Least-squares migration/inversion Up: (A) Non-iterative linearized migration/inversion Previous: (1) Wave equation prestack
Stanford Exploration Project
11/1/2005 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9412201642990112, "perplexity": 1066.7020044178457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187820466.2/warc/CC-MAIN-20171016214209-20171016234209-00639.warc.gz"} |
https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/376483 | # Examples of common false beliefs in mathematics
The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while they are learning mathematics, but quickly abandon when their mistake is pointed out -- and also in why they have these beliefs. So in a sense I am interested in commonplace mathematical mistakes.
Let me give a couple of examples to show the kind of thing I mean. When teaching complex analysis, I often come across people who do not realize that they have four incompatible beliefs in their heads simultaneously. These are
(i) a bounded entire function is constant;
(ii) $$\sin z$$ is a bounded function;
(iii) $$\sin z$$ is defined and analytic everywhere on $$\mathbb{C}$$;
(iv) $$\sin z$$ is not a constant function.
Obviously, it is (ii) that is false. I think probably many people visualize the extension of $$\sin z$$ to the complex plane as a doubly periodic function, until someone points out that that is complete nonsense.
A second example is the statement that an open dense subset $$U$$ of $$\mathbb{R}$$ must be the whole of $$\mathbb{R}$$. The "proof" of this statement is that every point $$x$$ is arbitrarily close to a point $$u$$ in $$U$$, so when you put a small neighbourhood about $$u$$ it must contain $$x$$.
Since I'm asking for a good list of examples, and since it's more like a psychological question than a mathematical one, I think I'd better make it community wiki. The properties I'd most like from examples are that they are from reasonably advanced mathematics (so I'm less interested in very elementary false statements like $$(x+y)^2=x^2+y^2$$, even if they are widely believed) and that the reasons they are found plausible are quite varied.
• I have to say this is proving to be one of the more useful CW big-list questions on the site... – Qiaochu Yuan May 6 '10 at 0:55
• The answers below are truly informative. Big thanks for your question. I have always loved your post here in MO and wordpress. – Unknown May 22 '10 at 9:04
• wouldn't it be great to compile all the nice examples (and some of the most relevant discussion / comments) presented below into a little writeup? that would make for a highly educative and entertaining read. – Suvrit Sep 20 '10 at 12:39
• It's a thought -- I might consider it. – gowers Oct 4 '10 at 20:13
• Meta created tea.mathoverflow.net/discussion/1165/… – user9072 Oct 8 '11 at 14:27
False belief: It is obvious how to prove that $\sin'=\cos$.
Not so much... if $\cos$ and $\sin$ are defined geometrically. You need to prove geometrically that $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ and a (non-circular) proof of that is not obvious (see here).
Personally I'm aware of that just today! (thanks to a remedial course given to my niece).
• The string of comments below the Math.SE link shows that really pursuing this gets one down a rabbit hole of rigorous discussions of arc length and such. On the other hand, one can prove that if $S$ (think "sine") and $C$ (think "cosine") are continuous functions which satisfy the standard addition formulas and the Pythagorean theorem $C^2 + S^2 \equiv 1$, then $S'(0)$ exists and $S'(x) = S'(0)C(x)$. There is a whole family of sine-like functions $S_a: x \mapsto S(ax)$; adjusting the parameter $a$ so that $S_a^\prime(0) = 1$, you can define the standard sine to be this $S_a$: a neat finesse. – Todd Trimble Oct 8 '17 at 3:14
• @ToddTrimble: Should we interpret your comment as for or against this false belief? Or else, is it just a neutral complement? For a full proof, is your way really easier? – Sebastien Palcoux Oct 10 '17 at 11:56
• It's mainly in agreement with you, with a shade of neutral. I don't claim that my suggestion really makes it easier, but only that one can prove things rigorously without getting into considerations of arc length, with (theoretically anyway) no prerequisites past introductory differential calculus. Mainly it's based on convexity arguments; I have a write-up here: ncatlab.org/toddtrimble/published/…; see theorem 3.1 and the crucial lemma 3.4. The "finesse" is akin to how we adjust parameter $a$ in $f: x \mapsto a^x$ to $a = e$ to get $f'(0) = 1$. – Todd Trimble Oct 10 '17 at 13:22
Maybe I am a bit late to the party, but here is something that I falsely believed in for a while:
False Belief: If $$f$$ is a continuously differentiable function with a horizontal asymptote, i.e. $$\lim_{x\to +\infty}f(x)=L$$, then $$f'(x)\to 0$$ as $$x$$ goes to infinity.
Of course, this is incorrect even for smooth functions; take $$g$$ to be a smooth $$L^1$$ function with oscillatory behavior so that $$\lim_{x\to +\infty}g(x)$$ does not exist and then set $$f(x)=\int_{0}^xg(s)ds$$.
I came to this misconception during an ODE course I took in my undergrad. In the study of the asymptotic behavior of solutions of an autonomous equation $$\frac{dy}{dx}=F(y)$$ one argues that if a solution $$y=y(x)$$ tends to a limit $$L$$ as $$x\to +\infty$$, that limit must be a zero of the vector field. We then proceeds with determining which zeros of $$F$$ are asymptotically stable to find $$L$$. The point is that here not only $$\lim_{x\to +\infty}y$$, but also $$\lim_{x\to +\infty}\frac{dy}{dx}$$ exists due to the ODE. Hence:
Correct Statement: If $$f$$ is a continuously differentiable function with a horizontal asymptote, and if $$f'$$ admits a limit as $$x$$ goes to infinity, that limit must be zero.
This readily follows from the mean value theorem.
• A related old answer. I have found that people usually need some prompting to think of an $L^1$ function that does not converge to $0$ at infinity, even though all one needs is a sequence of spikes of increasing height, but sufficiently rapidly decreasing width so that the sum of the areas under them converges. – Robert Furber Mar 3 '20 at 17:02
Maybe that's a cute one:
If $$f$$ is a continuous real function, then $$f^{-1}(x)$$ is at most countable unless $$f$$ is constant on some interval.
• How would you stumble upon this belief? The most basic example of continuous functions (constant ones) fail it, so it seems like an easy belief to correct. – Mark Nov 14 '20 at 21:00
• Well, thanks, clarified what was left as obvious. – Alessandro Della Corte Nov 14 '20 at 21:04
• Hmm, with the clarification added, really not obvious. What is an example of a counterexample? – JoshuaZ Nov 14 '20 at 23:21
• Any component of a Peano curve does the job. – Alessandro Della Corte Nov 14 '20 at 23:59
• You don't need that fancy Peano curve. Every closed set in a metric space is the zero set of a real valued function. – user20948 Feb 8 at 22:43
For some reason, I believed until recently that every compact operator on a Hilbert space must have eigenvalues. It's obviously true for finite-rank operators and Hermitian operators, so I must have subconsciously generalized.
This was despite being well aware of (say) the Volterra operator, say; I simply had two contradictory ideas in my head.
• The example I had heard of for this was the operator on $\ell^2$ you get by multiplying pointwise by $\frac{1}{n}$ after doing a unilateral right shift. I realize now that this is the Volterra operator in disguise. – Robert Furber Jun 2 at 2:31
• Yes, your example was what I first thought of as well, and only afterwards did someone bring up the Volterra operator. – Yonah Borns-Weil Jun 2 at 19:30
These are 2 instances which i have seen to happen with my friends. If $A$ and $B$ are 2 matrices, then they believe that $(A+B)^{2}=A^{2}+ 2 \cdot A \cdot B +B^{2}$.
Another mistake is if one i asked to solve this equation, $\displaystyle\frac{\sqrt{x}}{2}=-1$, people generally square both the sides and do get $x$ as $4$.
• What "people"? Non-mathematicians? – Todd Trimble May 4 '11 at 0:03
• @Todd: No i was talking of high school students. – crskhr May 4 '11 at 4:08
• @S.C.:if squarring both sides will not give the solution then how can second problem be solved? – Styles Oct 25 '17 at 16:08
"It cannot be shown without some form of AC that the union (or disjoint union) of countably many countable sets is countable. I have a countably infinite set X of countably infinite sets. Therefore, the union of X cannot be shown to be countable without Choice."
The fallacy is that in many cases of interest, it is possible to exhibit an explicit counting of every element of X. In such a case a counting of X by antidiagonals is easily constructed. The usual counting of the rationals is an example of this.
I think this may even be an example of a more general phenomenon of "people think AC is necessary for a certain construction, but in fact it turns out not to be necessary for the example they have in mind". For example, AC is necessary to find a maximal ideal in an arbitrary ring ... but it isn't if you're prepared to assume the ring is Noetherian.
• If "Noetherian" is defined by the ascending chain condition or by requiring all ideals to be finitely generated, then in order to deduce the existence of maximal ideals, you still need a weak form of the axiom of choice. The usual argument uses the axiom of dependent choice. (Of course, if you define "Noetherian" to mean that every set of ideals has a maximal element, then deducing the existence of maximal ideals is a choiceless triviality.) A good reference is "Six impossible rings" by Wilfrid Hodges (J. Algebra 31 (1974) 218-244). – Andreas Blass Oct 22 '10 at 15:29
• Thanks Andreas! I had a feeling there was a technicality somewhere there, but couldn't remember what it was. As a philosophical point I personally think that of course in the absence of AC you want to define Noetherian so that my original statement is true, but admittedly that's a harder sell than my countable-sets example. – Karol Nov 16 '10 at 21:06
• @AndreasBlass's reference, clickably: Hodges - 6 impossible rings. – LSpice Feb 5 '19 at 1:09
Hopefully this isn't a repeat answer. False belief: a matrix is positive definite if its determinant is positive.
• Is this really a common(!) false belief? – Martin Brandenburg Oct 3 '11 at 7:23
A set is compact iff it is closed and bounded.
• This is perhaps a common false belief among undergraduates, but one that is dispelled by just a superficial acquaintance with functional analysis. – Todd Trimble Dec 9 '13 at 2:45
• @ Todd Trimble: true, but then also the belief about $sin$ suggested by the OP is only common among people who have not completed a course in complex analysis. – Delio Mugnolo Dec 13 '13 at 8:34
• I thought "bounded" is only defined on metric spaces, and this is true on metric spaces. Is that wrong? – Akiva Weinberger Sep 1 '15 at 2:48
• I have seen analysis textbooks take this as a definition. I hope they realize that they are contributing to future confusion in their readers once they move on to topology or even metric spaces. @AkivaWeinberger, The Heine-Borel theorem stated in this way makes sense for arbitrary metric spaces, but it is only true for complete metric spaces for which balls are totally bounded. The correct statement of H-B for general metric spaces is "a metric space is compact iff it is complete and totally bounded". – Mario Carneiro Oct 20 '15 at 21:27
• @AkivaWeinberger: Yes, it is wrong. The closed unit ball of an normed vector space is compact if and only if the space is finite dimensional. – ACL Apr 21 '16 at 13:43
In ${\mathbb F}_p^\times$, the non-squares are the opposite of the squares. In other words, $a$ is square iff $-a$ is not a square.
This is a confusion with the facts that the kernel of $x\mapsto x^2$ is $\{1,-1\}$ and the subgroup of squares has index $2$.
Anytime I wanted to write an answer to this question, I doubted maybe it is not as common as worthy of mentioning here. In fact, I am also not sure how common is the false belief that I observed today in a PDE class. I didn't observe that in many years of teaching calculus, but today four or five students in a small PDE class when calculating a definite integral by parts only applied the limits of the integral to the "second" integral, that is:
$$\int_{a}^b{f(x) g'(x) dx}=f(x) g(x) - \int_{a}^b{f'(x) g(x) dx}$$
Haven't I observed well enough in my calculus classes?
• Definitely integrals are numbers and $f(x)g(x)$ is a function of variable $x$. Formula as written is something very strange. – Fedor Petrov Apr 20 '16 at 18:58
• @FedorPetrov More strange is that most students don't see such a very strange something :) – Amir Asghari Apr 20 '16 at 19:50
I'm seven years late to the game, but here is mine:
False belief: The irrational numbers, in their usual topology as a subset of $\mathbb{R}$, are not a complete metric space.
• Could you write that more carefully? You mean the false belief to be that the irrationals, as a topological space, can't be complete for some metric. What you wrote can be easily confused with saying the irrational numbers are not complete for the usual metric coming from $\mathbf R$, which is true rather than false. Consider the "false belief" that $(-1,1)$ with its topology from $\mathbf R$ can't be made into a complete metric space for some metric. Certainly it's not complete for the usual metric, but it is if we use $\tan(\pi x/2)$ to identify $(-1,1)$ with $\mathbf R$ topologically. – KConrad Jul 15 '17 at 3:26
• "are not completely metrizable" is the wording you want. – Andrés E. Caicedo Jul 15 '17 at 3:28
• @KConrad and Andres: The wording is part of what made that false belief so believable! At the time I didn't think about the fact that there could be multiple metrics, much less that the completeness of those metrics wasn't a topological property. I only realized my mistake when I was introduced to the ideas contained in your two comments. (That, and picturing the irrationals as complete is HARD!) – Pace Nielsen Jul 15 '17 at 4:15
• Pace, you probably realize by now that continued fractions make the picture a lot easier (whereby the space of irrationals between $0$ and $1$ can be identified with a product space $\mathbb{N}^\mathbb{N}$). – Todd Trimble Jul 23 '17 at 21:07
• @DuchampGérardH.E. A topological subspace of a completely metrizable topological space is completely metrizable if and only if it is a $G_\delta$, that is a countable intersection of open sets. One can use Baire's category theorem to show that $\mathbb{Q}$ is not a $G_\delta$. All this can be found at: en.wikipedia.org/wiki/G%CE%B4_set – Michael Greinecker Aug 11 '17 at 14:41
People seem to believe that conventional computation (for example, running a chaotic irreversible cellular automaton) can be as efficient as one wants simply with good engineering, but this is not the case. Landauer's principle states that erasing a bit of information always takes $\ln(2)\cdot k\cdot T$ energy where $k$ is Boltzmann's constant ($k=1.38065\cdot 10^{-23}$ Joules/Kelvin) and $T$ is the temperature. Landauer's principle is a consequence of the second law of thermodynamics since if Landauer's principle were violated, then entropy would decrease. Landauer's principle means that conventional irreversible computation always must take $\ln(2)\cdot k\cdot T$ energy per bit erased (and one can erase data just by running it through AND and OR gates, so every irreversible gate must take a minimum amount of energy by Landauer's principle). However, Landauer's principle does not apply to reversible computation since reversible computers are not allowed to erase data.
• Okay, I see what you are aiming at with this last edit. The idea that ordinary computation can be made arbitrarily efficient is a reasonably common false belief about our physical world (and may even have a somewhat solid mathematical interpretation). I withdraw my initial objection. – S. Carnahan Aug 11 '17 at 14:04
I once misunderstood the definition of monads, and thought that for a monad $(T,\eta,\mu)$, we have $T\eta_X = \eta_{TX}$ (or fmap return == return in Haskell). Of course this is not the case (in case of $T=$[], fmap return [1,2] is [[1],[2]], whereas return [1,2] is [[1,2]]).
This might not be common, but I once believed the following.
Let $A, B$ be integers, and define a sequence by the linear recurrence $s_n = A s_{n-1} + B s_{n-2}$ with the base case $s_0 = 0$, $s_1 = 1$. Two important special cases are the Fibonacci sequence ($A = B = 1$) and the sequence $s_n = 2^n - 1$ (where $A = 3$, $B = -2$). Then, for any integers $n$ and $k$, $\gcd(s_n, s_k) = s_{\gcd(n,k)}$.
This is true in the two mentioned special cases, so it's tempting to believe it's true in general. But there's a counterexample: $A = B = k = 2$, $n = 3$.
Update: corrected the powers of two minus one example from $B = 2$ to $B = -2$. Thanks to Harry Altman.
• Quick correction, that should be A=3, B=-2 for 2^n-1. – Harry Altman Apr 7 '11 at 21:23
• Hmm, this raises an obvious question of whether it is true whenever (A,B)=1. – JoshuaZ Apr 22 '20 at 23:09
Here are two beliefs. I think everybody will agree that one of them, at least, is false. I adhere to the second one.
Belief 1. The simplest way to compute the exponential $e^A$ of a complex square matrix $A$ is to use the Jordan decomposition.
Belief 2. It's simpler and more efficient to use the following fact.
Let $f(z)$ be the minimal polynomial of $A$, let $g(z)$ be $f(z)$ times the singular part of $e^z/f(z)$, and observe $e^A=g(A)$.
(By abuse of notation $z$ is at the same time an indeterminate and a complex variable.) (The problems of computing the exponential of $A$ and that of computing the Jordan decomposition of $A$ have the same difficulty level. But, to solve one of them, there is no need to refer to the other.) Here are two references
http://www.iecl.univ-lorraine.fr/~Pierre-Yves.Gaillard/DIVERS/Constant_coefficients/
Jordan decomposition is often mentioned in relation with matrix exponentials. I'm convinced (rightly or wrongly) that the association of these notions in this context is purely irrational. I think somebody once made this association by accident, and then many people repeated it mechanically.
Here is another attempt to describe the situation.
Put $B:=\mathbb C[A]$. This is a Banach algebra, and also a $\mathbb C[X]$-algebra ($X$ being an indeterminate). Let $$\mu=\prod_{s\in S}\ (X-s)^{m(s)}$$ be the minimal polynomial of $A$, and identify $B$ to $\mathbb C[X]/(\mu)$. The Chinese Remainder Theorem says that the canonical $\mathbb C[X]$-algebra morphism $$\Phi:B\to C:=\prod_{s\in S}\ \mathbb C[X]/(X-s)^{m(s)}$$ is bijective. Computing exponentials in $C$ is trivial, so the only missing piece in our puzzle is the explicit inversion of $\Phi$. Fix $s$ in $S$ and let $e_s$ be the element of $C$ which has a one at the $s$ place and zeros elsewhere. It suffices to compute $\Phi^{-1}(e_s)$. This element will be of the form $$f=g\ \frac{\mu}{(X-s)^{m(s)}}\mbox{ mod }\mu$$ with $f,g\in\mathbb C[X]$, the only requirement being $$g\equiv\frac{(X-s)^{m(s)}}{\mu}\mbox{ mod }(X-s)^{m(s)}$$ (the congruence taking place in the ring of rational fractions defined at $s$). So $g$ is given by Taylor's Formula.
This can be summarized as follows:
There is a unique polynomial $E$ such that $\deg E<\deg\mu$ and $e^A=E(A)$. Moreover $E$ can be uniquely written as $$E=\sum_{s\in S}\\ E_s\\ \frac{\mu}{(X-s)^{m(s)}}$$ with (for all $s$) $\deg E_s < m(s)$ and $$E_s\equiv e^s\ e^{X-s}\\ \frac{(X-s)^{m(s)}}{\mu}\mbox{ mod }(X-s)^{m(s)},$$ the congruence taking place in $\mathbb C[[X-s]]$.
• Dear Johannes, please reread my post. – Pierre-Yves Gaillard May 12 '10 at 15:18
• Even a cursory examination of Nick Higham's book amazon.co.uk/Functions-Matrices-Computation-Nicholas-Higham/dp/… will show that both these opinions on the evaluation of matrix exponentials are hopelessly naive. – Robin Chapman May 15 '10 at 9:17
• Dear Robin, Thanks for your answer. I don't have Nick Higham's book. I was wondering if you could be more precise. Your comment is very surprising to me: I thought I was stating a triviality. Here are two references en.wikipedia.org/wiki/Matrix_exponential#Alternative iecn.u-nancy.fr/~gaillard/DIVERS/Constant_coefficients Looking forward to hearing from you. – Pierre-Yves Gaillard May 15 '10 at 10:51
• Your opinions are normative statements: "one should" and "it is better". It is naive to suppose that there is one best method that one should use to compute the matrix exponential. – Robin Chapman May 15 '10 at 14:07
• I don't think the OP wants examples of normative statements. As I read it, the question is about conceptual errors regarding non-normative mathematical statements. – Qiaochu Yuan May 17 '10 at 6:19
False belief: relativization is well-defined and the corresponding notation $C^A$ is unambiguous. Which is not quite true because $P=NP$ would not imply $P^A=NP^A$.
• Maybe some more explanation would be useful. If decision problems and oracles are subsets of $\mathbb{N}$, and complexity classes are subsets of $P(\mathbb{N})$, then there is in general no such operation as relativization. I'm not sure how common of a false belief this is, but once I settled on my preference for the set point of view and saw what was going on here I lost some interest in the idea of relativization. – Dan Brumleve Jan 13 '18 at 5:16
I used to think that the subset of even norm vectors in an integral lattice is a sub-lattice. This is true for the "classically integral" lattice defined by $<u,v> \in \mathbb{Z}$ for $u,v$ in the lattice because the even vectors is the kernel of the group homomorphism $v \rightarrow <v,v>$ mod 2. However this fails for the more general notion of "integer norm" lattice where we only require the quadratic form is integer valued (ie. the coefficients are all integral or that the off diagonal entries in the Gram matrix may be half integral eg $x^2+xy+2y^2$). For the hexagonal lattice $x^2+xy+y^2$ which is not classically integral, the even vectors is a sub-lattice but for a different reason that it is the lattice scaled by 2.
If $t = 3$ mod 4, the lattice $L_t$ with quadratic form $x^2+ty^2$ is classically integral. Its even sublattice $L_{t0}$ has quadratic form $4(x^2+xy+(t+1)y^2/4)$ which clearly equals 2$W_t$ where $W_t$ is the lattice with quadratic form $(x^2+xy+(t+1)y^2/4)$. If $t=7$ mod 8, the coefficients of the form is [odd,odd,even], the even vectors is not a subgroup since for example $[0,1]$ and $[1,1]$ has even norm but $[0,1]+[1,1]=[1,2]$ has odd norm. If $t$ is 3 mod 8, the form $(x^2+xy+(t+1)y^2/4)$ can only be even only if both $x,y$ are even since all coefficients are odd. So the even vectors in $W_t$ turn out to be $2W_t$. It is a sub-lattice and it is the subset of even vectors but it is index 4 in $W_t$. It is the 2-scaled sub-lattice.
If $v \in L_t$, $2v \in 2L_t \subset L_{t0}=2W_t$, so $L_t \subset W_t$. So the picture is $2W_t=L_{t0} \subset L_t \subset W_t$ with each containment is index 2.
Just today, I realised that I had been mis-interpreting the FTFGAG. One can speak of the $2^\infty$-torsion subgroup or the $2'$-torsion subgroup, the $3^\infty$-torsion or the $3'$-torsion subgroup, or even just the torsion subgroup of a FGAG or the maximal torsion-free quotient … so surely one can speak of the torsion-free subgroup, right? In fact, when I was corrected on this, my first thought was to reply: "just take the subgroup consisting of all infinite-order elements", and it only occured to me as I was saying it to wonder how the identity element would squeeze its way into this so-called 'subgroup'.
• A good way to think of this is as a natural, exact sequence of abelian groups $1\to tor(G)\to G\to G/tor(G)\to 1$ (and similar sequences). If $G$ is f.g., the sequence happens to split, but not in a natural way. Thus there is not such thing as "the" complement of the torsion subgroup. – Johannes Hahn Mar 16 '18 at 23:08
• Consider the unit group $\mathbf Z[\sqrt{2}]^\times = \{\pm 1\} \times (1+\sqrt{2})^{\mathbf Z}$. Applying conjugation turns $(1+\sqrt{2})^{\mathbf Z}$ into $(1-\sqrt{2})^{\mathbf Z}$ and these are not the same subgroup complementing the torsion subgroup. – KConrad Mar 16 '18 at 23:39
• To be clear, I am aware that this is false! – LSpice Mar 17 '18 at 0:46
• @JohannesHahn, what was misleading me was the fact that, if $\pi$ is any set of primes and $G$ is torsion (as well as finitely generated Abelian), then $1 \to G[\pi^{\mathbb Z}] \to G \to G/G[\pi^{\mathbb Z}] \to 1$ does split canonically; so it seems so tempting to think that a similar result should hold without the torsion hypothesis …. – LSpice Mar 17 '18 at 1:52
It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior
• Can you give a concrete example why it is a "mistake"? – Willie Wong Apr 25 '20 at 13:15
• If you mean set theoretic surjective then in every concrete category a morphism is surjective then the morphism is epi and in all the category that I know epi are something very reasonable for example in Hausdorff spaces epi are exactly the map with dense image ... – Amos Kaminski Sep 16 '20 at 13:00
• I think the best example where this constitutes a mistake is the category of rings. For example the inclusion $\mathbb{Z}\to \mathbb{Q}$ is an epimorphism. – Louis Hainaut Oct 16 '20 at 11:39
I already thought that the following two sets of matrices are one. $$M(\color{blue}{\Bbb R},2n)\qquad \text{and}\qquad M(\color{red}{\Bbb C},n).$$
• ... matrices, presumably? – AlexArvanitakis Nov 5 '20 at 4:17
This is more sort of a convention issue than an outright false belief (connected to the usual $$\emptyset$$ vs $$\{\emptyset\}$$ stuff), but I find it funny. I guess a fair share of mathematicians believe that: $$$$\bigcap\emptyset=\emptyset\label{eq}$$$$ while retaining the standard definition for intersection: $$\bigcap S:=\{x\ \text{such that}\ \forall Y(Y\in S\implies x\in Y)\}$$ according to which in fact: $$\bigcap\emptyset=V$$ where $$V$$ is the universal class. The condition in round brackets is of course vacuously true. So in a way - this is what I find funny - the former is the worst possible tentative solution of an equation ever.
A common misconception in analysis is that if two $$C^1$$-functions are linearly independent then their Wronskian is non-zero at some point.
This is the result of carelessly reversing a true implication. The mistake was first made in 1882 by Thomas Muir (who is responsible for the name "Wronskian") in his "Treatise on the Theory of Determinants" and Giuseppe Peano pointed out a counterexample as early as 1889 (it suffices to take $$y=x^2$$ and $$y=x|x|$$).
For more information on the history of this mistake see "Peano on Wronskians: A Translation". by Susannah M. Engdahl and Adam E. Parker.
I don't think I've seen it in here:
Every vector space has a non-trivial dual space ($L^p$ for $0 < p < 1$ was a counter-example only mentioned during one of the classes in measure theory)
And of course there's the common false belief of people outside of mathematics that "mathematicians work with numbers and formulae all day long" :)
• Well, it is true that every vector space has a dual space, even $L^{1/2}$... and it is even true that every topological vector space has a continuous dual space... What you mean is that it is not true that every topological vector space has a non-trivial continuous dual space (or, that the continuous dual of a topological vector space does not necessarily separate points) – Mariano Suárez-Álvarez Jul 7 '10 at 18:54
• You are indeed correct. I'll do better not to dismiss the trivial case the next time. – Asaf Karagila Jul 7 '10 at 19:31
• The existence of a nonzero functional on a locally convex space is guarenteed by the Hahn-Banach theorem. $p$-Banach spaces are in general not locally convex if $p<1$. – user20948 Feb 8 at 22:45
False Belief: "The suspension spectrum map from spaces to (edit: symmetric) spectra preserves smash-products"
The facts that one denotes the smash product of spectra and the smash product of a space with a spectrum (levelwise) with the same $\wedge$ and tends to leave away the $\Sigma^\infty$ when one embeds a space into spectra are also not helpful in getting used to the harsh reality that the above is wrong.
• Yay! 100th answer! – Peter Arndt Oct 4 '10 at 21:33
• I don't see that this qualifies as a false belief. In order for the question of whether it is true or false to even be meaningful, you have to first commit yourself to one of the many different notions of spectrum, not to mention smash product of spectra. – Tom Goodwillie Oct 5 '10 at 0:35
• True. I meant symmetric spectra with the smash product coming from their description as modules over the symmetric sequence of spheres. – Peter Arndt Oct 5 '10 at 10:52
The assumption that a cubic surface expressed as a foliation of Weierstrass curves cannot be rational, because a general Weierstrass curve is not rational.
I've seen this false assumption more than once on sci.math over the years. But there are simple counterexamples, such as:
$(x + y) (x^2 + y^2) = z^2$
On defining $u = x/y$ and $v = z/y$ one obtains $y (u + 1) (u^2 + 1) = v^2$, and hence x, y, z as rational functions of u, v.
I'd love to have a reference to a procedure for calculating the geometric genus and algebraic genus of surfaces like this, because they are rational if and only if both these quantities are zero, and for other cubic surfaces that interest me it would save a lot of fruitless hacking around trying to find a rational solution that probably doesn't exist! Are there any symbolic algebra packages that can do this?
I mean for example is $x y (x y + 1) (x + y) = z^2$ rational? I'm almost sure it isn't; but how can one be sure?
• Algebraic genus in software: Seems to be available in Singular and Maple and may be available in Mathematica by now (from mathematica.stackexchange.com/a/5453/16237 ). (Aside: I don't know anything about the software mentioned.) – Eric Towers Aug 13 '17 at 2:06
If a matrix $A$ is self-adjoint/skew-self-adjoint with respect to a symmetric bilinear form, then it is diagonalizable.
True for matrices over $\mathbb{R}$, with respect to a positive definite inner product.
False over other fields. For example, over $\mathbb{C}$, $\left( \begin{smallmatrix} 1 & i \\ i & -1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & 1 & i \\ -1 & 0 & 0 \\ -i & 0 & 0 \end{smallmatrix} \right)$ are nilpotent, but self-adjoint and skew self-adjoint respectively with respect to the standard inner product.
False for other nondegenerate symmetric bilinear forms: $\left( \begin{smallmatrix} 1 & 1 \\ -1 & -1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & -1 & -1 \\ 1 & 0 & 0 \\ -1 & 0 & 0 \end{smallmatrix} \right)$ are nilpotent, but self-adjoint and skew self-adjoint respectively with respect to $\left( \begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{smallmatrix} \right)$.
You can exponentiate the skew-self-adjoint matrices to get examples of matrices preserving a nondegenerate symmetric bilinear form, with Jordan blocks of the form $\left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)$.
• You seem to have a different definition of "the standard inner product on $\mathbb{C}^n$" than I do. I think that phrase normally refers to the familiar positive definite sesquilinear form, with respect to which self-adjoint matrices are indeed diagonalizable. – Mark Meckes Jan 28 '11 at 16:46
• But that's not a bilinear form. And it has no generalization to other fields (what is it on $\overline{\mathbb{F}_p}$?). How can it be standard? :) I certainly agree that people should know that matrices which are self-adjoint with respect to the standard sesquilinear form are diagonalizable. – David E Speyer Jan 28 '11 at 17:51
• Of course it's not bilinear -- an "inner product" on a complex vector space is defined to be sesquilinear, not bilinear -- I've spent a lot of time trying to get my linear algebra students to remember that. The failure of such a form to generalize to other fields is indeed sad, but I think the richness of Hilbert space theory helps to make up for that disappointment. :) – Mark Meckes Jan 28 '11 at 21:24
If $a$ is a real zero of a cubic polynomial with rational coefficients then $a$ can be written as a combination of cube roots of rational numbers.
More generally if $a$ is a real zero of an irreducible polynomial with rational coefficients that is solvable by radicals then students expect the following:
1. Any expression inside a radical evaluates to a real number
2. Any sub-expression of the expression for $a$ evaluates to an algebraic number of order less than or equal to the order of $a$
Of course the problem is that from Cardan's solution to the cubic we can have negative rational numbers inside a square root. Let $c$ = $4*(-1 + \sqrt{-3})$.
$a$ = $\frac{\sqrt[3]{c}}{4} + \frac{1}{\sqrt[3]{c}}$
$f(x) = 4x^3 - 3x + \frac{1}{2}$.
So while $a$ is an algebraic number of degree three, it can not be written as combination of cube roots of rational numbers. Indeed, it is counter-intuitive that $\sqrt[3]{c}$ has degree 6 over the rational numbers yet we can use this number and simple arithmetic to produce an algebraic number of degree 3.
Also $a$ = $\sin(50^{\circ})$. For many values of $\theta$, $\sin \theta$ is a radical number. See also radical values for sine and cosine
If every collection of disjoint open sets in a topological space is at most countable, then the space is separable
Something I was sure about until earlier today:
Suppose $\kappa$ is an $\aleph$ number, then $AC_\kappa$ is equivalent to $W_\kappa$, namely the universe holds that the product of $\kappa$ many sets is non-empty if and only if every cardinality is either of size less than $\kappa$ or has a subset of cardinality $\kappa$.
In fact this is only true if you assume full $AC$, and $(\forall \kappa) AC_\kappa$ doesn't even imply $W_{\aleph_1}$, I was truly shocked.
Furthermore, $W_\kappa$ doesn't even imply $AC_\kappa$ in most cases.
The strongest psychological implication is that most people actually think of the well-ordering principle as a the "correct form" of choice, when it is actually Dependent Choice (limited to $\kappa$, or unbounded) which is the "proper" form, that is $DC_\kappa$ implies both $AC_\kappa$ and $W_\kappa$.
• How common is this misconception? – Thierry Zell Apr 17 '11 at 3:08
• @Thierry: For the past couple of weeks I spent a lot time considering models without choice, not only I held that misconception but not once anyone corrected me about it - grad students and professors alike. – Asaf Karagila Apr 17 '11 at 6:09
Coordinates on a manifold do not have an immediate metric meaning. Until becoming familiar with differential geometry one tends to think they do. (Einstein wrote that he took seven years to free himself from this idea.)
For example, linear control theory is for the most part metric with variables in $R^n$. When moving away from linear control theory, variables are represented as coordinates on a manifold. Nevertheless, much of the literature tends to either abandon metric notions altogether, or to keep using an Euclidean metric though it is no longer very useful. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 46, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8845847249031067, "perplexity": 351.06959278807665}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487622113.11/warc/CC-MAIN-20210625054501-20210625084501-00044.warc.gz"} |
http://eccc.hpi-web.de/report/2012/111/ | REPORTS > DETAIL:
### Paper:
TR12-111 | 5th September 2012 15:17
#### Faster SDP hierarchy solvers for local rounding algorithms
TR12-111
Authors: Venkatesan Guruswami, Ali Kemal Sinop
Publication: 5th September 2012 15:18
Keywords:
Abstract:
Convex relaxations based on different hierarchies of
linear/semi-definite programs have been used recently to devise
approximation algorithms for various optimization problems. The
approximation guarantee of these algorithms improves with the number
of {\em rounds} $r$ in the hierarchy, though the complexity of solving
(or even writing down the solution for) the $r$'th level program grows
as $n^{\Omega(r)}$ where $n$ is the input size.
In this work, we observe that many of these algorithms are based on
{\em local} rounding procedures that only use a small part of the SDP
solution (of size $n^{O(1)} 2^{O(r)}$ instead of $n^{\Omega(r)}$). We
give an algorithm to find the requisite portion in time polynomial in
its size. The challenge in achieving this is that the required portion
of the solution is not fixed a priori but depends on other parts of
the solution, sometimes in a complicated iterative manner.
Our solver leads to $n^{O(1)} 2^{O(r)}$ time algorithms to obtain the
same guarantees in many cases as the earlier $n^{O(r)}$ time
algorithms based on $r$ rounds of the Lasserre hierarchy. In
particular, guarantees based on $O(\log n)$ rounds can be realized in
polynomial time. For instance, one can (i) get $O(1/\lambda_r)$
approximations for graph partitioning problems such as minimum
bisection and small set expansion in $n^{O(1)} 2^{O(r)}$ time, where $\lambda_r$ is the $r$'th smallest eigenvalue of
the graph's normalized Laplacian; (ii) a similar guarantee in
$n^{O(1)} k^{O(r)}$ for Unique Games where $k$ is the number of labels (the polynomial dependence on $k$ is new); and (iii) find an independent set of size $\Omega(n)$ in $3$-colorable graphs in $(n 2^r)^{O(1)}$ time provided $\lambda_{n-r} \le 1.11$.
We develop and describe our algorithm in a fairly general abstract
framework. The main technical tool in our work, which might be of
independent interest in convex optimization, is an efficient ellipsoid
algorithm based separation oracle for convex programs that can output
a {\em certificate of infeasibility with restricted support}. This is
used in a recursive manner to find a sequence of consistent points in
nested convex bodies that fools'' local rounding algorithms.
ISSN 1433-8092 | Imprint | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8570240139961243, "perplexity": 1058.841643607888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507447819.35/warc/CC-MAIN-20141017005727-00130-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Kolmogorov_continuity_theorem | # Kolmogorov continuity theorem
In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.
## Statement of the theorem
Let ${\displaystyle (S,d)}$ be some complete metric space such that d is ${\displaystyle {\mathcal {B}}(S)\otimes {\mathcal {B}}(S)}$ measurable, and let ${\displaystyle X:[0,+\infty )\times \Omega \to S}$ be a stochastic process. Suppose that for all times ${\displaystyle T>0}$, there exist positive constants ${\displaystyle \alpha ,\beta ,K}$ such that
${\displaystyle \mathbb {E} [d(X_{t},X_{s})^{\alpha }]\leq K|t-s|^{1+\beta }}$
for all ${\displaystyle 0\leq s,t\leq T}$. Then there exists a modification ${\displaystyle {\tilde {X}}}$ of ${\displaystyle X}$ that is a continuous process, i.e. a process ${\displaystyle {\tilde {X}}:[0,+\infty )\times \Omega \to S}$ such that
• ${\displaystyle {\tilde {X}}}$ is sample-continuous;
• for every time ${\displaystyle t\geq 0}$, ${\displaystyle \mathbb {P} (X_{t}={\tilde {X}}_{t})=1.}$
Furthermore, the paths of ${\displaystyle {\tilde {X}}}$ are locally ${\displaystyle \gamma }$-Hölder-continuous for every ${\displaystyle 0<\gamma <{\tfrac {\beta }{\alpha }}}$.
## Example
In the case of Brownian motion on ${\displaystyle \mathbb {R} ^{n}}$, the choice of constants ${\displaystyle \alpha =4}$, ${\displaystyle \beta =1}$, ${\displaystyle K=n(n+2)}$ will work in the Kolmogorov continuity theorem. Moreover for any positive integer ${\displaystyle m}$, the constants ${\displaystyle \alpha =2m}$, ${\displaystyle \beta =m-1}$ will work, for some positive value of ${\displaystyle K}$ that depends on ${\displaystyle n}$ and ${\displaystyle m}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 26, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9916749596595764, "perplexity": 116.8062438955798}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593142.83/warc/CC-MAIN-20180722080925-20180722100925-00221.warc.gz"} |
http://www-old.newton.ac.uk/programmes/BNR/seminars/2007081015301.html | BNR
Seminar
Posterior consistency of species sampling priors
Lee, J (Seoul)
Friday 10 August 2007, 15:30-16:30
Seminar Room 1, Newton Institute
Abstract
Recently there have been increasing interests on species sampling priors, the nonparametric priors defined as the directing random probability measures of the species sampling sequences. In this paper, we show that not all of the species sampling priors produce consistent posteriors. In particular, in the class of Pitman-Yor process priors, the only priors rendering posterior consistency are essentially the Dirichlet process priors. For the general species sampling priors, we also give a necessary and sufficient condition for the posterior consistency. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9057096242904663, "perplexity": 1139.1195625614794}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541950.13/warc/CC-MAIN-20161202170901-00219-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://www.math-only-math.com/rational-expressions-involving-addition-and-subtraction.html | # Rational Expressions Involving Addition and Subtraction
How to simplify rational expressions involving addition and subtraction of three or more rational numbers?
1. Evaluate 3/5 + 7/3 + -11/5 + -2/3
Solution:
Using the commutative and associative law, it follows that we may arrange the terms in any manner suitably. Using this rearrangement property, we have:
3/5 + 7/3 + -11/5 + -2/3
= (3/5 + -11/5) + (7/3 + -2/5)
= {(3 + (-11)}/5 + {7 + (-2)}/3
= -8/5 + 5/3
= (-24 + 25)/15
= 1/15
2. Simplify: (4/7 + -8/9 + -5/21 + 1/3)
Solution:
Using the rearrangement property, we have:
4/7 + -8/9 + -5/21 + 1/3 = (4/7 + -5/21) + (-8/9 + 1/3)
= {12 + (-5)}/21 + {-8 + 3}/9
= (7/21 + -5/9)
= {21 + (-35)}/63
= -14/63
= -2/9
3. What should be subtracted from -5/7 to get -1?
Solution:
Let the required number be x. Then,
-5/7 – x = -1
-5/7 = x - 1
x = (-5/7 + 1)
= (-5 + 7)/7
= 2/7
Hence, the required number is 2/7.
Rational Numbers
What is Rational Numbers?
Is Every Rational Number a Natural Number?
Is Zero a Rational Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
Positive Rational Number
Negative Rational Number
Equivalent Rational Numbers
Equivalent form of Rational Numbers
Rational Number in Different Forms
Properties of Rational Numbers
Lowest form of a Rational Number
Standard form of a Rational Number
Equality of Rational Numbers using Standard Form
Equality of Rational Numbers with Common Denominator
Equality of Rational Numbers using Cross Multiplication
Comparison of Rational Numbers
Rational Numbers in Ascending Order
Representation of Rational Numbers on the Number Line
Addition of Rational Number with Same Denominator
Addition of Rational Number with Different Denominator
Properties of Addition of Rational Numbers
Subtraction of Rational Number with Different Denominator
Subtraction of Rational Numbers
Properties of Subtraction of Rational Numbers
Simplify Rational Expressions Involving the Sum or Difference
Multiplication of Rational Numbers
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Division of Rational Numbers
Rational Expressions Involving Division
Properties of Division of Rational Numbers
To Find Rational Numbers
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8602299690246582, "perplexity": 3861.816407652298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541518.17/warc/CC-MAIN-20161202170901-00376-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://asmedigitalcollection.asme.org/PVP/proceedings-abstract/PVP2012/55003/631/282667 | The Rules on Fitness-for-Service for Nuclear Power Plants of the Japan Society of Mechanical Engineers (JSME Code) has the reference fatigue crack growth curve for austenitic stainless steels in BWR environment. This reference curve was determined as the upper bound of crack growth data excluding the outlier data. However, the other reference curves for fatigue crack growth rate such as austenitic stainless steels and ferritic steels in air environment and ferritic steels in water environment in the ASME Boiler and Pressure Vessel Code, Section XI and the JSME Code, austenitic stainless steels in PWR environment in the JSME Code and Ni-base alloys in PWR environment in the JSME Code Case are determined based on the 95% upper confidential limit by statistic data treatment. In the present study, the fatigue crack growth data of austenitic stainless steels in BWR environment were re-evaluated statistically. It was found that the current reference curve almost coincides with 95% upper confidential limit of fatigue crack growth data in the Paris region. Consequently, the current reference fatigue crack growth curve for austenitic stainless steels in BWR environment in the JSME Code can be regarded to stand on the same technical bases with other reference fatigue crack growth curves. Furthermore, the authors proposed to extend applicable upper bound of load rising time tr from 1000 s to 32000 s.
This content is only available via PDF. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9510710835456848, "perplexity": 2438.3463893686776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319915.98/warc/CC-MAIN-20190824063359-20190824085359-00357.warc.gz"} |
http://notesonmathematics.wordpress.com/2012/09/19/the-axiom-of-choice/ | # The axiom of choice
This post is about the axiom of choice, arguably the most famous and the most controversial of the axioms underpinning the set theoretic foundations of mathematics today. The axiom of choice is about making choices: choosing an element each from an arbitrary collection of nonempty sets. The axiom basically says that this can always be done, although it provides no recipe for doing so.
Before formally presenting the axiom we state some definitions. The treatment is essentially taken from here.
Definition: A function $I$ from a set $\Lambda$ onto a set $X$ is said to index the set $X$ by $\Lambda$. The set $\Lambda$ is called the index and $X$ is the indexed set. If $I(\lambda)=x$, we write $x_\lambda$ for $I(\lambda)$.
Definition: Let $X$ be a nonempty set indexed by a set $\Lambda$. The cartesian product of $X$ is defined to be the set $\Pi _{\lambda\in\Lambda}x_\lambda$ of all functions $f$ with domain $\Lambda$ and codomain $\cup X=\cup_{x\in X}x$, satisfying the condition $f(\lambda)\in x_\lambda$.
Let us consider an example before we go further. We let $X=\{\{a,b\},\{c,d\}\}$ be indexed by $\Lambda=\{1,2\}$ where $1\to \{a,b\}, 2\to \{c,d\}$. Now there are four functions possible from $\Lambda$ to $\{a,b,c,d\}$ which satisfy the definition:
The function $f_1$ for which $f(1)=a, f(2)=c$.
The function $f_2$ for which $f(1)=b, f(2)=c$.
The function $f_3$ for which $f(1)=a, f(2)=d$.
The function $f_4$ for which $f(1)=b, f(2)=d$.
The collection $\{f_1,f_2,f_3,f_4\}$ is the cartesian product. Now intuitively the behavior of the function $f_1$ can be captured by simply writing $(a,c)$ since the fact that the first coordinate corresponds to $a$ and the second to $c$ indicates the function $1\to a,2\to c$. Similarly the behavior of the function $f_2,f_3,f_4$ are also captured by $(b,c),(a,d),(b,d)$ respectively. Note that a particular ordered pair always corresponds to a unique function $f_i$. So the cartesian product may be simply represented as $\{(a,c),(b,c),(a,d),(b,d)\}$.
In fact if $\Lambda$ has at most countable number of elements within it, we can always consider it as an subset of the form $\{1,2,3\cdots\}$ (which may or may not terminate), and index appropriately. Then the cartesian product obtained can be equivalently thought of as a collection of n-tuples or infinite sequences where the ith coordinate corresponds to the image of $i$ in the indexing. For example if $X=\{\{0,1\}\}$ and $\Lambda=\mathbb{N}$ then all zero-one sequences correspond to the requisite functions and their collection forms the cartesian product.
Definition: Any function $f$ which satisfies the conditions required to make it a member of a cartesian product is called a choice function.
We now give three formulations of the axiom of choice:
Axiom of choice:
1. For every nonempty set whose elements are nonempty sets and are indexed by a set $I$ there exists a choice function.
2. If $\{a_i\}$ is a family of nonempty sets, indexed by a nonempty set $I$, then there exists a family $\{x_i\}$ with $i\in I$ such that $x_i\in a_i$ for each $i\in I$. (This corresponds roughly to our informal comment about choices in the first paragraph.)
3. The cartesian product of a nonempty collection of nonempty sets is nonempty.
Theorem: The three formulations are equivalent.
Proof: $1\Rightarrow 2$: Let $\{a_i\}_{i\in I}$ be a family of sets, indexed by the nonempty set $I$. Let $S=\{a_i\mid i\in I\}$. (Note that the elements of $S$ are themselves sets.) We index $S$ by itself, and consider the cartesian product of $S$. By the assumption and by $I\ne \emptyset$, the set $S$ is a nonempty set of nonempty sets. Hence owing to (1) there exists a choice function, i.e. a function $f\colon S\to \bigcup S$ such that $f(x)\in x$ for all $x \in S$. For $i\in I$ let $x_i=f(a_i)$. Then $\{x_i\}_{i\in I}$ is a family with the required properties.
$2\Rightarrow 1$: Let $S=\{a_i:i\in I\}$ be a nonempty set whose elements are themselves nonempty where $S$ is indexed by $I$. By (2) we have a family $x_i$ with $x_i\in a_i$. We define our choice function to be $f:I\to\cup S$ where $f(i)=x_i$.
$1\Leftrightarrow 3$: This is obvious by the definitions.$\Box$
Due to this result any of these three statements may be referred to as the axiom of choice. In fact, there are many other equivalent statements. The term ZFC is usually used to denote the Zermelo Fraenkel axioms together with the axiom of choice, and these axioms form a basis of most of the mathematics done today.
Why is this axiom so relevant? Firstly if one refuses to accept this axiom, one loses a lot of interesting results. Secondly, many times assuming the axiom of choice is so much easier. Many results such as the Cantor-Berstein-Schroeder theorem were proved originally using the axiom of choice, although later a “choice-free” proof was discovered. Similarly the axiom of choice makes undergraduate analysis easier by enabling one to say that $f(x)$ is continuous at $x=a$ if $f(a_n)$ tends to $f(a)$ for each sequence $(a_n)$ that tends to $a$.
Are there any negative issues with the axiom of choice? Well, firstly it is not constructive and doesn’t actually tell us the choices to make. It just guarantees that some choice exists. In one formulation it tells us that the real numbers can be well ordered but it gives us no recipe of doing so. In other words, it doesn’t actually construct a well order of the reals for us. Secondly, some counter intuitive results such as the Banach Tarsky paradox are true in ZFC and many mathematicians find this troubling and “fishy”.
Thirdly, and probably most importantly, there is history. After it was introduced (in another form) in 1904 by Zermelo, many mathematicians wondered whether or not the axiom was consistent with ZF. In other words people asked whether accepting it would lead to contradictions. Many decades passed before in 1938 Godel showed that ZFC is indeed consistent. Another doubt was whether the axiom of choice was independent of ZF. It was only by 1963 that Cohen proved that the axiom of choice cannot derived from ZF and so together with Godel’s result this implied that the axiom of choice is independent of ZF. During this long period from 1904 to 1963, many bizzare results such as the Banach Tarsky paradox, existence of non-Lebesgue measurable sets etc were found. All this sowed a suspicion in people’s minds as to whether this axiom was “false” at some fundamental level. As a result mathematicians began to pay close attention to results in which the axiom was used, and to try and device proofs which avoided the axiom as much as possible. This habit somewhat lessened after the independence results, but it has not been dropped by everyone even today.
The fact of the matter is that while rejecting the axiom of choice leads to many weird results, so does accepting it. See the chapters entitled “Disasters without Choice” and “Disasters with Choice” in this book for details. However in the long run, the advantages of accepting the axiom outweigh the advantages of rejecting it primarily because the sheer quantity of beautiful and aesthetic results flowing from the axiom of choice provide substance to mathematics as a whole. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 85, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9708744883537292, "perplexity": 160.53498131878237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770742.121/warc/CC-MAIN-20141217075250-00008-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://electronics.stackexchange.com/questions/125123/what-is-the-transfer-function-for-a-first-order-active-high-pass-filter | # What is the transfer function for a first order active high-pass filter
What is the transfer function for the below circuit?
simulate this circuit – Schematic created using CircuitLab
Since the op-amp has unity gain, the transfer function should be the same as a passive high pass RC filter.
This can be found by calculating the voltage across $R$ using the potential divider rule, in the $s$ domain:
The impedance of a capacitor in the S domain is $\frac{1}{sC}$, so the transfer function is:
$$H(s) = \frac{R}{R+\frac{1}{sC}}$$
In standard form this is:
$$H(s) = \frac{RCs}{RCs+1}$$
$$H(s) = RC * s * \frac{1}{RCs+1}$$
DC Gain: $20log(RC)$ dB
Gain due to single zero at origin: $20log(\omega)$ dB ; Argument: $90^o$
Gain due to pole:
• at high frequencies: $-20log(RC\omega) = -20log(RC)-20log(\omega)$ dB
• at low frequencies: $20log(1) = 0$ dB
• with corner frequency: $\frac{1}{RC}$
Argument (phase) due to pole: $-tan^{-1}(RC\omega)$
So the complete gain is:
• $for \:\omega >> \frac{1}{RC}$
$|H(j\omega)| = 0$ dB
• $for \:\omega << \frac{1}{RC}$
$|H(j\omega)| = 20log(\omega) + 20log(RC)$ dB
Complete phase response:
$\angle H(j\omega) = 90-tan^{-1}(RC\omega)$
However, in my lecture notes it says that the transfer function for the above circuit is:
$$H(s) = \frac{s}{1+sCR}$$
Without any derivation.
Which is correct?
-
The last expression cannot be correct because the numerator has the dimension rad/s. The whole function H(s) must be dimensionless. (Perhaps the author assumes a factor T=1 second, however should be mentioned in this case). But your DC gain is not correct. Instead, we have H(s=0)=0. – LvW Aug 14 at 12:47
Your expression for $H(s)$ is correct:
$$H(s)=\frac{RCs}{1+RCs}$$
where $\tau=RC$ is the time constant of the high pass filter. The most important features of this transfer function are the location of the pole ($s_{\infty}=-1/RC$) and the location of the zero ($s_0=0$). Note that the location of the pole and the zero is of course the same for the transfer function in your lecture notes. However, the gain factor in your notes is wrong. The consequence of this is that for $s\rightarrow\infty$ (i.e. for very high frequencies) your transfer function converges to unity gain, whereas the one in your notes converges to $1/RC$.
Thanks for this answer. In my notes there is also a transfer function for a similar circuit, but with the op-amp in a non inverting configuation, given as $\frac{Ks}{1+sCR}$, where $K=1+\frac{R_1}{R_2}$. Im guessing this is also wrong, and there should be an $RC$ in the numerator of this as well? – Blue7 Aug 14 at 14:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.932497501373291, "perplexity": 608.159269397141}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009751.93/warc/CC-MAIN-20141125155649-00181-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://clay6.com/qa/23357/the-equation-of-the-plane-that-meets-the-coordinate-axes-at-the-points-a-b- | # The equation of the plane that meets the coordinate axes at the points $A,B,C$ respectively and the centroid of the $\Delta\:ABC$ is $(2,1,5)$ is ?
$(a)\:5x+10y+2z-30=0\:\:\qquad\:(b)\:2x+y+5z-30=0\:\:\qquad\:(c)\:5x+10y+2z+30=0\:\:\qquad\:(d)\:2x+y+5z+30=0$
Let the points on the coordinate axes, at which the plane meets be $A(a,0,0),\:\:B(0,b,0)\:\:and\:\:C(0,0,c)$
$\therefore$ The equation of the plane is $\large\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
Given that the centroid of the triangle $ABC=(2,1,5)$
$\Rightarrow\:\large\frac{a}{3}$$=2,\:\:\large\frac{b}{3}$$=1,\:\:\large\frac{c}{3}$$=5$
$\Rightarrow\:a=6,\:\:b=3\:\:and\:\:c=15$
$\Rightarrow\:$ The equation of the plane is $\large\frac{x}{6}+\frac{y}{3}+\frac{z}{15}=1$
$i.e.,\:\:5x+10y+2z=30$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8163305521011353, "perplexity": 136.39721390992068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864256.26/warc/CC-MAIN-20180621172638-20180621192638-00417.warc.gz"} |
https://www.physicsforums.com/threads/angle-theta-baseball-problem.370199/ | # Angle theta baseball problem
1. Jan 17, 2010
### Larrytsai
1. The problem statement, all variables and given/known data
A boy throws a ball upward with a speed of vo= 12m/s. The wind imparts a horizontal acceleration of 0.4m/s^2 to the left. At what angle theta must the ball be thrown so that it returns to the point of release. (assuming wind does not affect vertical motion.
2. Relevant equations
All physics equation
3. The attempt at a solution
I have written down all the formulas in the x and y directions, but it doesnt work out. I have also tried to look for what distance Vy=0 but i still end up having 2 unknowns.
2. Jan 17, 2010
### tiny-tim
Hi Larrytsai!
(have a theta: θ and try using the X2 tag just above the Reply box )
You should have only two unknowns, θ and t, so two equations should be enough.
Show us your full calculations, and then we can see what went wrong, and we'll know how to help!
3. Jan 17, 2010
### Larrytsai
Re: Physics
ok so what i have so far is,
for y:
y= (1/2 a t^2) + (Vo cosθ t)
(Vcosθ)^2 = (Vocosθ)^2 + 2ay
for x:
x= (1/2 a t^2) + (Vo sinθ t)
(Vsinθ)^2 = (Vosinθ)^2 + 2ax
so far how i see it is im going to substitute my 2 x equations and my 2 y equations, then ill have 2 equations left to subtitute which are θ and time.
4. Jan 17, 2010
### tiny-tim
(hmm … the second and fourth equations are just V = V0 … that doesn't help much)
What's the difficulty?
Put all the numbers in, including x = y = 0, and what do you get?
5. Jan 17, 2010
### Larrytsai
Re: Physics
ohh ok, i got the answer, thnx so much, but im a little confused as to why x is 0, im thinking that when the height and horizontal distance is 0 , the ball has landed back in the boys hand, therefore the time elapsed is the same so we can subsitute for time?
btw thanx alot
6. Jan 17, 2010
### tiny-tim
Hi Larrytsai!
Yes, you're getting confused.
When the ball returns to the hand, both x and y are 0, and the time elapsed is t (say).
So the same t appears in both the equation for x and the equation for y.
7. Feb 12, 2010
Re: Physics | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8170745372772217, "perplexity": 1583.997207587474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257650685.77/warc/CC-MAIN-20180324132337-20180324152337-00567.warc.gz"} |
http://cms.math.ca/Reunions/ete14/abs/ps.html | Réunion d'été SMC 2014
Université du Manitoba, 6 - 9 juin 2014
Conférences plénières
[PDF]
BELA BOLLOBAS, Cambridge and Memphis
Percolation, Bootstrap Percolation, and Random Monotone Cellular Automata [PDF]
Percolation theory, founded by Broadbent and Hammersley in 1957, and $r$-neighbour bootstrap percolation, introduced by Chalupa, Leath and Reich in 1979, have much in common. Thus, on the infinite lattice ${\mathbb Z}^d$ we may start the same way, by selecting vertices of ${\mathbb Z}^d$ with the same probability $p$ to obtain a random set $A_0$. In percolation a basic question is whether $A_0$ percolates, i.e. the subgraph induced by it has an infinite component; the infimum of those probabilities is the critical probability $p_c$. In $r$-neighbour bootstrap percolation the set $A_0$ is only the beginning: considering the vertices in it infected, the infection spreads in a very simple way: infected vertices remain infected for ever, and if at any stage a vertex has at least $r$ infected neighbours then itself becomes infected. This system percolates if each site is infected eventually. The critical probability $p_c$ is defined as in percolation.
In percolation much work has been done on the critical probability, since it is strictly between $0$ and $1$ in every non-trivial model; however, in $r$-neighbour bootstrap percolation the critical probability is of no interest, since every lattice has critical probability $0$ or $1$.
Recently, Smith, Uzzell and the lecturer introduced a wide-ranging extension of $r$-neighbour bootstrap percolation: they initiated the study of completely general monotone, local, and homogeneous cellular automata in a random environment. Among other results, they classified these $\mathcal U$-percolation models into three types, and proved results about the phase transition in two of them. The phase transition in the third type has been clarified by Balister, Przykucki, Smith and the lecturer: in particular, they have shown that in ${\mathbb Z}^2$ these processes have non-trivial critical probabilities. These results have reopened the study of critical probabilities in generalized bootstrap processes' on ${\mathbb Z}^2$.
In the lecture I shall survey some classical results in percolation and bootstrap percolation, and will sketch some of the recent results on monotone cellular automata obtained by Balister, Duminil-Copin, Gunderson, Holmgren, Morris, Przykucki, Smith, Uzzell, and myself.
MARK LEWIS, University of Alberta
The mathematics behind biological invasion processes [PDF]
Models for invasions track the front of an expanding wave of population density. They take the form of parabolic partial differential equations and related integral formulations. These models can be used to address questions ranging from the rate of spread of introduced invaders and diseases to the ability of vegetation to shift in response to climate change.
In this talk I will focus on scientific questions that have led to new mathematics and on mathematics that have led to new biological insights. I will investigate the mathematical and empirical basis for multispecies invasions, for accelerating invasion waves, and for nonlinear stochastic interactions that can determine spread rates.
JAMES MAYNARD, CRM, Universite de Montreal
Small Gaps between primes [PDF]
It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer $m$ there should be infinitely many sets of $m$ primes, with each set contained in an interval of size roughly $m\log{m}$. Although proving these conjectures seems to be beyond our current techniques, recent progress has enabled us to obtain some partial results. We will introduce a refinement of the GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty$ for any integer $m$, and so there are infinitely many bounded length intervals containing $m$ primes.
THOMAS RANSFORD, Université Laval
Computation of analytic capacity and the subadditivity problem [PDF]
Analytic capacity measures the size of compact plane sets from the point of view of certain aspects of complex analysis. It was first introduced in the 1940’s in connection with the so-called Painlevé problem of characterizing removable singularities of bounded holomorphic functions. Later, in the 1960’s, it played a key role in the solution of fundamental problems in rational approximation. The last twenty years have seen significant advances in the understanding of analytic capacity, one of the most striking breakthroughs being the positive solution to the long-standing conjecture that analytic capacity is semi-additive (the capacity of the union of two sets is bounded by C times the sum of their individual capacities, where C is a universal constant). However, whether analytic capacity is subadditive (can we take C=1?) still remains an open problem.
The plan for the talk is (1) a brief history of analytic capacity and its applications, (2) a practical method for rigorous computation of analytic capacity, and (3) the hunt for a counterexample to the subadditivity problem. Parts (2) and (3) are based on joint work with Malik Younsi. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8823694586753845, "perplexity": 520.0420649120618}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410665301782.61/warc/CC-MAIN-20140914032821-00097-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://mathoverflow.net/feeds/question/106479 | What is wrong with the proof of Theorem 1.39 of "Additive Combinatorics"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:46:59Z http://mathoverflow.net/feeds/question/106479 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106479/what-is-wrong-with-the-proof-of-theorem-1-39-of-additive-combinatorics What is wrong with the proof of Theorem 1.39 of "Additive Combinatorics"? unknown (google) 2012-09-06T05:50:54Z 2012-09-06T05:50:54Z <p>According to <a href="http://terrytao.wordpress.com/books/additive-combinatorics/" rel="nofollow">http://terrytao.wordpress.com/books/additive-combinatorics/</a></p> <p>we have:</p> <p>p. 36-37: The proof of Theorem 1.39 requires a number of significant changes. After the first paragraph, add “We will show first that with probability 1, that any natural number has at most a bounded number of representations as the sum of $k$ elements of $A$ between $n^\epsilon$ and $n$; the treatment of the remaining sums in which at least one term is less than $n^\epsilon$ is left as an exercise.” ...</p> <p>Now, this is referring to Theorem 1.39 of Additive Combinatorics:</p> <p>For any $h \geq 1$ and $\epsilon > 0$, there exists a set $A \subset Z^+$ with $|A \cap [0,n]| = \Omega_h(n^{1/h-\epsilon})$ for all large $n$, which is a $B_h[g]$ set for some $g=g_{h, \epsilon}$.</p> <p>Now, the part of the proof which I think the errata above mentions is:</p> <p>$E(Y_m) = \sum_{n_1 \leq ... \leq n_h: n_1 + ... +n_h = m} n_1^{1/h - \epsilon} ... n_h^{1/h-\epsilon}$= ...</p> <p>Now, supposedly there is an error here (and thus the need for the errata). What is wrong with the proof of Theorem 1.39?</p> | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9790122509002686, "perplexity": 524.6087512933434}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704433753/warc/CC-MAIN-20130516114033-00061-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://www.computer.org/csdl/trans/tb/2012/04/06152087-abs.html | Subscribe
Issue No.04 - July-Aug. (2012 vol.9)
pp: 1203-1211
Hazem N. Nounou , Electr. Eng. Dept., Texas A&M Univ. at Qatar, Doha, Qatar
ABSTRACT
This paper considers the problem of learning the structure of gene regulatory networks from gene expression time series data. A more realistic scenario when the state space model representing a gene network evolves nonlinearly is considered while a linear model is assumed for the microarray data. To capture the nonlinearity, a particle filter-based state estimation algorithm is considered instead of the contemporary linear approximation-based approaches. The parameters characterizing the regulatory relations among various genes are estimated online using a Kalman filter. Since a particular gene interacts with a few other genes only, the parameter vector is expected to be sparse. The state estimates delivered by the particle filter and the observed microarray data are then subjected to a LASSO-based least squares regression operation which yields a parsimonious and efficient description of the regulatory network by setting the irrelevant coefficients to zero. The performance of the aforementioned algorithm is compared with the extended Kalman filter (EKF) and Unscented Kalman Filter (UKF) employing the Mean Square Error (MSE) as the fidelity criterion in recovering the parameters of gene regulatory networks from synthetic data and real biological data. Extensive computer simulations illustrate that the proposed particle filter-based network inference algorithm outperforms EKF and UKF, and therefore, it can serve as a natural framework for modeling gene regulatory networks with nonlinear and sparse structure.
INDEX TERMS
time series, bioinformatics, cellular biophysics, genetics, Kalman filters, least squares approximations, sparse structure, gene regulatory network, nonlinear state space models, sparsity exploitation, gene expression time series data, particle filter-based state estimation algorithm, parameter vector, microarray data, LASSO-based least squares regression, extended Kalman filter, unscented Kalman filter, mean square error, fidelity criterion, particle filter-based network inference algorithm, Kalman filters, Mathematical model, Data models, Estimation, Approximation algorithms, Gene expression, Noise, LASSO., Gene regulatory network, particle filter, Kalman filter, parameter estimation
CITATION
Hazem N. Nounou, "Inferring Gene Regulatory Networks via Nonlinear State-Space Models and Exploiting Sparsity", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.9, no. 4, pp. 1203-1211, July-Aug. 2012, doi:10.1109/TCBB.2012.32 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8730397820472717, "perplexity": 1480.349150269537}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246651873.94/warc/CC-MAIN-20150417045731-00291-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/speed-of-light.184199/ | # Speed of light
1. Sep 12, 2007
### Killer slug
is it possible to catch light particals and have them push you along in some kind of space ship, and if you could would it be the speed of light
2. Sep 12, 2007
### Morga
No, but apparently swans make good propulsion for early space craft. Light particles or "Photons" are packets of energy that you can't catch. If you were to try and "grab" one with your hand and catch a ride you may be disappointed. Traveling at light speed is impossible for such a large mass as yourself, but seriously swans all the way...
3. Sep 12, 2007
### Staff: Mentor
Actually, every time a photon hits you and you absorb it, you have "caught" it. And when you do, it transfers its momentum to you. So yes, you could propel a spaceship that way (actually, reflecting them is better...), but you'd need an enormous solar sail to catch enough photons to be useful. And no, you could never reach the speed of light. You could get arbitrarily close to it, though.
Anyway, look here: http://en.wikipedia.org/wiki/Solar_sail
4. Sep 12, 2007
### HallsofIvy
"swans"? Is that a technical term.
"Light Sails" have long been a staple of science fiction. I don't know if NASA is doing any research but it would certainly been possible.
5. Sep 12, 2007
### jobyts
Why is the speed of light limited to 3*10^8 m/s? What is blocking it from going faster than that?
6. Sep 12, 2007
### lanman
The speed of lights is an absolute maximum to speed, because it is the rate at which time flows. The digits it is expressed as are based on our units of measurement.
7. Sep 12, 2007
### Staff: Mentor
The speed of light is not the rate at which time flows.
The speed of light is a dictated by the permeability and permittivity of space, as a consequence of Maxwell's equations.
8. Sep 13, 2007
### prasannapakkiam
Well Watters has explained it clearly...
9. Sep 13, 2007
### lanman
Not sure where I heard that...
10. Sep 13, 2007
### genneth
In a sense, c is the conversion rate between space and time. In many theoretical papers, "natural" units are adopted, where c = 1, \hbar = 1, etc. So that mass, length, time are all inter-convertible without changing the number.
11. Sep 13, 2007
### jobyts
From the speed of light, did we derive the permeability and permittivity of space, or is it the other way?
Can anyone explain permeability and permittivity of space in simpler terms? I googled those keywords, but gave me pages with lots of greek symbols, which I used to understand in the past, for a short period in my life.
12. Sep 13, 2007
### Loren Booda
I would think that the definition of c, the speed of light, would rely somewhat on the Higgs field - that which endows mass upon particles. Massless particles travel at the speed of light, massive ones at less than c.
13. Sep 13, 2007
### Staff: Mentor
If I understand the history correctly, I think they were measured about the same time the first accurate measurements of the speed of light were being taken (by Michelson). So the speed of light was separately measured and calculated and found to be the same value.
Basically, the resistance to passing magnetic and electric fields.
Last edited: Sep 13, 2007 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.887677788734436, "perplexity": 1383.9419114509396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812978.31/warc/CC-MAIN-20180220145713-20180220165713-00147.warc.gz"} |
http://entiaetnomina.blogspot.com/2010/10/busting-myth-about-lesniewski-and.html | ### Busting a myth about Lesniewski and definitions
Some time ago I run into a few logicians chatting at a conference about how Lesniewski formulated the standard theory of definitions. For some reason I felt obliged to say a few words about this. One thing let to another, and this ended up in a footnote to Gupta's entry on definitions at the Stanford Encyclopedia (see footnote three). A few more coincidences and I started writing a paper about this with Severi Hamari. After a freakish number of revisions (around 20) we finally agreed on a version, and I posted it on academia.
Many thanks for their comments to John MacFarlane, Nuel Belnap, Wilfrid Hodges, Paolo Mancosu, Oystein Linnebo and Jan von Plato (I hope I didn't forget anyone).
Abstract:
A theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to S. Lesniewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Lesniewski’s published orunpublished work is known where the standard conditions are discussed. Second, Lesniewski’s own logical theories allow for creative definitions. Third, Lesniewski’s celebrated ‘rules of definition’ lay merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 30s, a study of these meta-theoretical conditions is more readily found in the works of J. Lukasiewicz and K. Ajdukiewicz. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8191142678260803, "perplexity": 2532.0570149601044}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867977.85/warc/CC-MAIN-20180527004958-20180527024958-00016.warc.gz"} |
http://orbi.ulg.ac.be/browse?type=author&value=Aerts%2C+C | Browse ORBi by ORBi project The Open Access movement
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References of "Aerts, C" in Complete repository Arts & humanities Archaeology Art & art history Classical & oriental studies History Languages & linguistics Literature Performing arts Philosophy & ethics Religion & theology Multidisciplinary, general & others Business & economic sciences Accounting & auditing Production, distribution & supply chain management Finance General management & organizational theory Human resources management Management information systems Marketing Strategy & innovation Quantitative methods in economics & management General economics & history of economic thought International economics Macroeconomics & monetary economics Microeconomics Economic systems & public economics Social economics Special economic topics (health, labor, transportation…) Multidisciplinary, general & others Engineering, computing & technology Aerospace & aeronautics engineering Architecture Chemical engineering Civil engineering Computer science Electrical & electronics engineering Energy Geological, petroleum & mining engineering Materials science & engineering Mechanical engineering Multidisciplinary, general & others Human health sciences Alternative medicine Anesthesia & intensive care Cardiovascular & respiratory systems Dentistry & oral medicine Dermatology Endocrinology, metabolism & nutrition Forensic medicine Gastroenterology & hepatology General & internal medicine Geriatrics Hematology Immunology & infectious disease Laboratory medicine & medical technology Neurology Oncology Ophthalmology Orthopedics, rehabilitation & sports medicine Otolaryngology Pediatrics Pharmacy, pharmacology & toxicology Psychiatry Public health, health care sciences & services Radiology, nuclear medicine & imaging Reproductive medicine (gynecology, andrology, obstetrics) Rheumatology Surgery Urology & nephrology Multidisciplinary, general & others Law, criminology & political science Civil law Criminal law & procedure Criminology Economic & commercial law European & international law Judicial law Metalaw, Roman law, history of law & comparative law Political science, public administration & international relations Public law Social law Tax law Multidisciplinary, general & others Life sciences Agriculture & agronomy Anatomy (cytology, histology, embryology...) & physiology Animal production & animal husbandry Aquatic sciences & oceanology Biochemistry, biophysics & molecular biology Biotechnology Entomology & pest control Environmental sciences & ecology Food science Genetics & genetic processes Microbiology Phytobiology (plant sciences, forestry, mycology...) Veterinary medicine & animal health Zoology Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences Chemistry Earth sciences & physical geography Mathematics Physics Space science, astronomy & astrophysics Multidisciplinary, general & others Social & behavioral sciences, psychology Animal psychology, ethology & psychobiology Anthropology Communication & mass media Education & instruction Human geography & demography Library & information sciences Neurosciences & behavior Regional & inter-regional studies Social work & social policy Sociology & social sciences Social, industrial & organizational psychology Theoretical & cognitive psychology Treatment & clinical psychology Multidisciplinary, general & others Showing results 1 to 20 of 109 1 2 3 4 5 6 Pulsating B-type Stars in the Open Cluster NGC 884: Frequencies, Mode Identification, and AsteroseismologySaesen, S.; Briquet, Maryline ; Aerts, C. et alin Astronomical Journal (2013), 146Recent progress in the seismic interpretation of field β Cep stars has resulted in improvements of the physical description in the stellar structure and evolution model computations of massive stars ... [more ▼]Recent progress in the seismic interpretation of field β Cep stars has resulted in improvements of the physical description in the stellar structure and evolution model computations of massive stars. Further asteroseismic constraints can be obtained from studying ensembles of stars in a young open cluster, which all have similar age, distance, and chemical composition. We present an observational asteroseismology study based on the discovery of numerous multi-periodic and mono-periodic B stars in the open cluster NGC 884. We describe a thorough investigation of the pulsational properties of all B-type stars in the cluster. Overall, our detailed frequency analysis resulted in 115 detected frequencies in 65 stars. We found 36 mono-periodic, 16 bi-periodic, 10 tri-periodic, and 2 quadru-periodic stars and one star with nine independent frequencies. We also derived the amplitudes and phases of all detected frequencies in the U, B, V, and I filter, if available. We achieved unambiguous identifications of the mode degree for 12 of the detected frequencies in nine of the pulsators. Imposing the identified degrees and measured frequencies of the radial, dipole, and quadrupole modes of five pulsators led to a seismic cluster age estimate of log (age/yr) = 7.12-7.28 from a comparison with stellar models. Our study is a proof-of-concept for and illustrates the current status of ensemble asteroseismology of a young open cluster. [less ▲]Detailed reference viewed: 6 (1 ULg) An observational asteroseismic study of the pulsating B-stars in the open cluster NGC 884Saesen, S.; Briquet, Maryline ; Aerts, C. et alin IAU Symposium No. 301 (2013), "Precision Asteroseismology" (Wroclaw), W. Chaplin, J. Guzik, G. Handler & A. Pigulski (eds.) (2013, October 01)Recent progress in the seismic interpretation of field {\beta} Cep stars has resulted in improvements of the physical description in the stellar structure and evolution model computations of massive stars ... [more ▼]Recent progress in the seismic interpretation of field {\beta} Cep stars has resulted in improvements of the physical description in the stellar structure and evolution model computations of massive stars. Further asteroseismic constraints can be obtained from studying ensembles of stars in a young open cluster, which all have similar age, distance and chemical composition. We present an observational asteroseismic study based on the discovery of numerous multi-periodic and mono-periodic B-stars in the open cluster NGC 884. Our study illustrates the current status of ensemble asteroseismology of a young open cluster. [less ▲]Detailed reference viewed: 5 (0 ULg) Low-amplitude rotational modulation rather than pulsations in the CoRoT B-type supergiant HD 46769Aerts, C.; Simón-Díaz, S.; Catala, C. et alin Astronomy and Astrophysics (2013), 557
Aims: We aim to detect and interpret photometric and spectroscopic variability of the bright CoRoT B-type supergiant target HD 46769 (V = 5.79). We also attempt to detect a magnetic field in the ... [more ▼]
Aims: We aim to detect and interpret photometric and spectroscopic variability of the bright CoRoT B-type supergiant target HD 46769 (V = 5.79). We also attempt to detect a magnetic field in the target.
Methods: We analyse a 23-day oversampled CoRoT light curve after detrending and spectroscopic follow-up data using standard Fourier analysis and phase dispersion minimization methods. We determine the fundamental parameters of the star, as well as its abundances from the most prominent spectral lines. We perform a Monte Carlo analysis of spectropolarimetric data to obtain an upper limit of the polar magnetic field, assuming a dipole field.
Results: In the CoRoT data, we detect a dominant period of 4.84 d with an amplitude of 87 ppm and some of its (sub-)multiples. Given the shape of the phase-folded light curve and the absence of binary motion, we interpret the dominant variability in terms of rotational modulation, with a rotation period of 9.69 d. Subtraction of the rotational modulation signal does not reveal any sign of pulsations. Our results are consistent with the absence of variability in the Hipparcos light curve. The spectroscopy leads to a projected rotational velocity of 72 ± 2 km s[SUP]-1[/SUP] and does not reveal periodic variability or the need to invoke macroturbulent line broadening. No signature of a magnetic field is detected in our data. A field stronger than ~500 G at the poles can be excluded, unless the possible non-detected field were more complex than dipolar.
Aims: We deduce and summarise observational constraints for the hybrid ... [more ▼]Context. B-type stars are promising targets for asteroseismic modelling, since their frequency spectrum is relatively simple.
Aims: We deduce and summarise observational constraints for the hybrid pulsator, HD 50230, earlier reported to have deviations from a uniform period spacing of its gravity modes. The combination of spectra and a high-quality light curve measured by the CoRoT satellite allow a combined approach to fix the position of HD 50230 in the HR diagram.
Methods: To describe the observed pulsations, classical Fourier analysis was combined with short-time Fourier transformations and frequency spacing analysis techniques. Visual spectra were used to constrain the projected rotation rate of the star and the fundamental parameters of the target. In a first approximation, the combined information was used to interpret multiplets and spacings to infer the true surface rotation rate and a rough estimate of the inclination angle.
Results: We identify HD 50230 as a spectroscopic binary and characterise the two components. We detect the simultaneous presence of high-order g modes and low-order p and g-modes in the CoRoT light curve, but were unable to link them to line profile variations in the spectroscopic time series. We extract the relevant information from the frequency spectrum, which can be used for seismic modelling, and explore possible interpretations of the pressure mode spectrum. The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA's RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain. Based on observations made with the ESO telescopes at La Silla Observatory under the ESO Large Programme LP182.D-0356, and on observations made with the Mercator Telescope, operated on the island of La Palma by the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, and on observations obtained with the HERMES spectrograph, which is supported by the Fund for Scientific Research of Flanders (FWO), Belgium, the Research Council of K.U. Leuven, Belgium, the Fonds National de la Recherche Scientifique (FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Genève, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany.Appendix A is available in electronic form at http://www.aanda.org [less ▲]Detailed reference viewed: 12 (2 ULg) Gravito-inertial and pressure modes detected in the B3 IV CoRoT target HD 43317Pápics, P. I.; Briquet, Maryline ; Baglin, A. et alin Astronomy and Astrophysics (2012), 542Context. OB stars are important building blocks of the Universe, but we have only a limited sample of them well understood enough from an asteroseismological point of view to provide feedback on the ... [more ▼]Context. OB stars are important building blocks of the Universe, but we have only a limited sample of them well understood enough from an asteroseismological point of view to provide feedback on the current evolutionary models. Our study adds one special case to this sample, with more observational constraints than for most of these stars.
Aims: Our goal is to analyse and interpret the pulsational behaviour of the B3 IV star HD 43317 using the CoRoT light curve along with the ground-based spectroscopy gathered by the HARPS instrument. This way we continue our efforts to map the β Cep and SPB instability strips.
Methods: We used different techniques to reveal the abundances and fundamental stellar parameters from the newly-obtained high-resolution spectra. We used various time-series analysis tools to explore the nature of variations present in the light curve. We calculated the moments and used the pixel-by-pixel method to look for line profile variations in the high-resolution spectra.
Results: We find that HD 43317 is a single fast rotator (v[SUB]rot[/SUB] ≈ 50% v[SUB]crit[/SUB]) and hybrid SPB/β Cep-type pulsator with Solar metal abundances. We interpret the variations in photometry and spectroscopy as a result of rotational modulation connected to surface inhomogeneities, combined with the presence of both g and p mode pulsations. We detect a series of ten consecutive frequencies with an almost constant period spacing of 6339 s as well as a second shorter sequence consisting of seven frequencies with a spacing of 6380 s. The dominant frequencies fall in the regime of gravito-inertial modes. The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA's RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain.Based on data gathered with HARPS installed on the 3.6 m ESO telescope (ESO Large Programme 182.D-0356) at La Silla, Chile.Table A.1 is only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/542/A55 [less ▲]Detailed reference viewed: 16 (3 ULg) CoRoT's view on variable B8/9 stars: spots versus pulsations. Evidence for differential rotation in HD 174648Degroote, P.; Acke, B.; Samadi, R. et alin Astronomy and Astrophysics (2011), 536Context. There exist few variability studies of stars in the region in the Hertzsprung-Russell diagram between the A and B-star pulsational instability strips. With the aid of the high precision ... [more ▼]Context. There exist few variability studies of stars in the region in the Hertzsprung-Russell diagram between the A and B-star pulsational instability strips. With the aid of the high precision continuous measurements of the CoRoT space satellite, low amplitudes are more easily detected, making a study of this neglected region worthwhile.
Aims: We collected a small sample of B stars observed by CoRoT to determine the origin of the different types of variability observed.
Methods: We combine literature photometry and spectroscopy to measure the fundamental parameters of the stars in the sample, and compare asteroseismic modelling of the light curves with (differentially rotating) spotted star models.
Results: We found strong evidence for the existence of spots and differential rotation in HD 174648, and formulated hypotheses for their origin. We show that the distinction between pulsations and rotational modulation is difficult to make solely based on the light curve, especially in slowly rotating stars. The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA's RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain.Based on observations made with the ESO telescopes at La Silla Observatory under the ESO Large Programme LP182.D-0356.Based on observations made with the Mercator Telescope, operated on the island of La Palma by the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.Appendix A is available in electronic form at http://www.aanda.org [less ▲]Detailed reference viewed: 8 (0 ULg) Seismic modelling of the β Cephei star HD 180642 (V1449 Aquilae)Aerts, C.; Briquet, Maryline ; Degroote, P. et alin Astronomy and Astrophysics (2011), 534Context. We present modelling of the β Cep star HD 180642 based on its observational properties deduced from CoRoT and ground-based photometry as well as from time-resolved spectroscopy.
Aims: We ... [more ▼]Context. We present modelling of the β Cep star HD 180642 based on its observational properties deduced from CoRoT and ground-based photometry as well as from time-resolved spectroscopy.
Aims: We investigate whether present-day state-of-the-art models are able to explain the full seismic behaviour of this star, which has extended observational constraints for this type of pulsator.
Methods: We constructed a dedicated database of stellar models and their oscillation modes tuned to fit the dominant radial mode frequency of HD 180642, by means of varying the hydrogen content, metallicity, mass, age, and core overshooting parameter. We compared the seismic properties of these models with those observed.
Results: We find models that are able to explain the numerous observed oscillation properties of the star, for a narrow range in mass of 11.4-11.8 M[SUB]&sun;[/SUB] and no or very mild overshooting (with up to 0.05 local pressure scale heights), except for an excitation problem of the ℓ = 3, p[SUB]1[/SUB] mode. We deduce a rotation period of about 13 d, which is fully compatible with recent magnetic field measurements. The seismic models do not support the earlier claim of solar-like oscillations in the star. We instead ascribe the power excess at high frequency to non-linear resonant mode coupling between the high-amplitude radial fundamental mode and several of the low-order pressure modes. We report a discrepancy between the seismic and spectroscopic gravity at the 2.5σ level. [less ▲]Detailed reference viewed: 20 (13 ULg) Variability in the CoRoT photometry of three hot O-type stars. HD 46223, HD 46150, and HD 46966Blomme, R.; Mahy, Laurent ; Catala, C. et alin Astronomy and Astrophysics (2011), 533Context. The detection of pulsational frequencies in stellar photometry is required as input for asteroseismological modelling. The second short run (SRa02) of the CoRoT mission has provided photometric ... [more ▼]Context. The detection of pulsational frequencies in stellar photometry is required as input for asteroseismological modelling. The second short run (SRa02) of the CoRoT mission has provided photometric data of unprecedented quality and time-coverage for a number of O-type stars.
Aims: We analyse the CoRoT data corresponding to three hot O-type stars, describing the properties of their light curves and search for pulsational frequencies, which we then compare to theoretical model predictions.
Methods: We determine the amplitude spectrum of the data, using the Lomb-Scargle and a multifrequency HMM-like technique. Frequencies are extracted by prewhitening, and their significance is evaluated under the assumption that the light curve is dominated by red noise. We search for harmonics, linear combinations, and regular spacings among these frequencies. We use simulations with the same time sampling as the data as a powerful tool to judge the significance of our results. From the theoretical point of view, we use the MAD non-adiabatic pulsation code to determine the expected frequencies of excited modes.
Results: A substantial number of frequencies is listed, but none can be convincingly identified as being connected to pulsations. The amplitude spectrum is dominated by red noise. Theoretical modelling shows that all three O-type stars can have excited modes, but the relation between the theoretical frequencies and the observed spectrum is not obvious.
Conclusions: The dominant red noise component in the hot O-type stars studied here clearly points to a different origin than the pulsations seen in cooler O stars. The physical cause of this red noise is unclear, but we speculate on the possibility of sub-surface convection, granulation, or stellar wind inhomogeneities being responsible. The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA's RSSD and Science Programmes, Austria, Belgium, Brazil, Germany and Spain.Tables 2-4 are available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/533/A4 [less ▲]Detailed reference viewed: 23 (2 ULg) CoRoT high-precision photometry of the B0.5 IV star HD 51756Pápics, P. I.; Briquet, Maryline ; Auvergne, M. et alin Astronomy and Astrophysics (2011), 528Context. OB stars are important constituents for the ecology of the Universe, and there are only a few studies on their pulsational properties detailed enough to provide important feedback on current ... [more ▼]Context. OB stars are important constituents for the ecology of the Universe, and there are only a few studies on their pulsational properties detailed enough to provide important feedback on current evolutionary models.
Aims: Our goal is to analyse and interpret the behaviour present in the CoRoT light curve of the B0.5 IV star HD 51756 observed during the second long run of the space mission and to determine the fundamental stellar parameters from ground-based spectroscopy gathered with the Coralie and Harps instruments after checking for signs of variability and binarity, thus making a step further in mapping the top of the β Cep instability strip.
Methods: We compared the newly obtained high-resolution spectra with synthetic spectra of late O-type and early B-type stars computed on a grid of stellar parameters. We matched the results with evolutionary tracks to estimate stellar parameters. We used various time series analysis tools to explore the nature of the variations present in the light curve. Additional calculations were carried out based on distance and historical position measurements of the components to impose constraints on the binary orbit.
Results: We find that HD 51756 is a wide binary with both a slow (vsini ≈ 28 km s[SUP]-1[/SUP]) and a fast (vsini ≈ 170 km s[SUP]-1[/SUP]) early-B rotator whose atmospheric parameters are similar (T[SUB]eff[/SUB] ≈ 30 000 K and log g ≈ 3.75). We are unable to detect pulsation in any of the components, and we interpret the harmonic structure in the frequency spectrum as a sign of rotational modulation, which is compatible with the observed and deduced stellar parameters of both components.
Conclusions: The non-detection of pulsation modes provides a feedback on the theoretical treatment, given that non-adiabatic computations applied to appropriate stellar models predict the excitation of both pressure and gravity modes for the fundamental parameters of this star. The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA's RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain.Based on data gathered with Coralie installed on the 1.2 m Euler telescope at La Silla, Chile; and Harps installed on the 3.6 m ESO telescope (ESO Large Programme 182.D-0356) at La Silla, Chile.Appendix A is only available in electronic form at http://www.aanda.org [less ▲]Detailed reference viewed: 14 (8 ULg) Kepler Detected Gravity-Mode Period Spacings in a Red Giant StarBeck, P. G.; Bedding, T. R.; Mosser, B. et alin Science (2011), 332Stellar interiors are inaccessible through direct observations. For this reason, helioseismologists made use of the Sun’s acoustic oscillation modes to tune models of its structure. The quest to detect ... [more ▼]Stellar interiors are inaccessible through direct observations. For this reason, helioseismologists made use of the Sun’s acoustic oscillation modes to tune models of its structure. The quest to detect modes that probe the solar core has been ongoing for decades. We report the detection of mixed modes penetrating all the way to the core of an evolved star from 320 days of observations with the Kepler satellite. The period spacings of these mixed modes are directly dependent on the density gradient between the core region and the convective envelope. [less ▲]Detailed reference viewed: 17 (1 ULg) An asteroseismic study of the O9V star HD 46202 from CoRoT space-based photometryBriquet, Maryline ; Aerts, C.; Baglin, A. et alin Astronomy and Astrophysics (2011), 527The O9V star HD 46202, which is a member of the young open cluster NGC 2244, was observed by the CoRoT satellite in October/November 2008 during a short run of 34 days. From the very high-precision light ... [more ▼]The O9V star HD 46202, which is a member of the young open cluster NGC 2244, was observed by the CoRoT satellite in October/November 2008 during a short run of 34 days. From the very high-precision light curve, we clearly detect β Cep-like pulsation frequencies with amplitudes of ~0.1 mmag and below. A comparison with stellar models was performed using a χ[SUP]2[/SUP] as a measure for the goodness-of-fit between the observed and theoretically computed frequencies. The physical parameters of our best-fitting models are compatible with the ones deduced spectroscopically. A core overshooting parameter α[SUB]ov[/SUB] = 0.10 ± 0.05 pressure scale height is required. None of the observed frequencies are theoretically excited with the input physics used in our study. More theoretical work is thus needed to overcome this shortcoming in how we understand the excitation mechanism of pulsation modes in such a massive star. A similar excitation problem has also been encountered for certain pulsation modes in β Cep stars recently modelled asteroseismically. The CoRoT space mission was developed and is operated by the French space agency CNES, with the participation of ESA's RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain. [less ▲]Detailed reference viewed: 14 (7 ULg) Plaskett's star: analysis of the CoRoT photometric dataMahy, Laurent ; Gosset, Eric ; Baudin, F. et alin Astronomy and Astrophysics (2011), 525Context. The second short run (SRa02) of the CoRoT space mission for asteroseismology was partly devoted to stars belonging to the Mon OB2 association. An intense monitoring has been performed on Plaskett ... [more ▼]Context. The second short run (SRa02) of the CoRoT space mission for asteroseismology was partly devoted to stars belonging to the Mon OB2 association. An intense monitoring has been performed on Plaskett's star (HD 47129) and the unprecedented quality of the light curve allows us to shed new light on this very massive, non-eclipsing binary system.
Aims: We particularly aimed at detecting periodic variability that might be associated with pulsations or interactions between both components. We also searched for variations related to the orbital cycle that could help to constrain the inclination and the morphology of the binary system.
Methods: We applied an iterative Fourier-based prewhitening and a multiperiodic fitting procedure to analyse the time series and extract the frequencies of variations from the CoRoT light curve. We describe the noise properties to tentatively define an appropriate significance criterion and, in consequence, to only point out the peaks at a certain significance level. We also detect the variations related to the orbital motion and study them with the NIGHTFALL programme.
Results: The periodogram computed from Plaskett's star CoRoT light curve mainly exhibits a majority of peaks at low frequencies. Among these peaks, we highlight a list of 43 values, notably including two different sets of harmonic frequencies whose fundamental peaks are located at about 0.07 and 0.82 d[SUP]-1[/SUP]. The former represents the orbital frequency of the binary system, whilst the latter could probably be associated with non-radial pulsations. The study of the 0.07 d[SUP]-1[/SUP] variations reveals a hot spot most probably situated on the primary star and facing the secondary. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8194818496704102, "perplexity": 4623.577719095599}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00193-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.rdmag.com/news/2012/10/satellites-track-rapid-changes-earths-core | Annual to decadal changes in the earth's magnetic field in a region that stretches from the Atlantic to the Indian Ocean have a close relationship with variations of gravity in this area. From this it can be concluded that outer core processes are reflected in gravity data. This is the result presented by a German-French group of geophysicists in the latest issue of Proceedings of the National Academy of Sciences.
The main field of the Earth's magnetic field is generated by flows of liquid iron in the outer core. The Earth's magnetic field protects us from cosmic radiation particles. Therefore, understanding the processes in the outer core is important to understand the terrestrial shield. Key to this are measurements of the geomagnetic field itself. A second, independent access could be represented by the measurement of minute changes in gravity caused by the fact that the flow in the liquid Earth's core is associated with mass displacements. The research group has now succeeded to provide the first evidence of such a connection of fluctuations in the Earth's gravity and magnetic field.
They used magnetic field measurements of the GFZ-satellite CHAMP and extremely accurate measurements of the Earth's gravity field derived from the GRACE mission, which is also under the auspices of the GFZ.
"The main problem was the separation of the individual components of the gravity data from the total signal," explains Vincent Lesur from the GFZ German Research Centre for Geosciences, who is involved in the study.
A satellite only measures the total gravity, which consists of the mass fractions of Earth's body, water and ice on the ground and in the air. To determine the mass redistribution by flows in the outer core, the thus attained share of the total gravity needs to be filtered out.
"Similarly, in order to capture the smaller changes in the outer core, the proportion of the magnetic crust and the proportion of the ionosphere and magnetosphere need to be filtered out from the total magnetic field signal measured by the satellite," Vincent Lesur explains.
The data records of the GFZ-satellite missions CHAMP and GRACE enabled this for the first time.
During the investigation, the team focused on an area between the Atlantic and the Indian Ocean, as the determined currents flows were the highest here. Extremely fast changes (so-called magnetic jerks) were observed in the year 2007 at the Earth's surface. These are an indication for sudden changes of liquid flows in the upper outer core and are important for understanding the magneto-hydrodynamics in the Earth's core. Using the satellite data, a clear signal of gravity data from the Earth's core could be received for the first time.
This results in consequences for the existing conceptual models. Until now, for example, it was assumed that the differences in the density of the molten iron in the earth's core are not large enough to generate a measurable signal in the earth's gravitational field. The newly determined mass flows in the upper outer core allow a new approach to Earth's core hydrodynamics.
Recent changes of the Earth's core derived from satellite observations of magnetic and gravity fields
Source: Helmholtz Association of German Research Centres | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8722484111785889, "perplexity": 560.6906560129439}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721347.98/warc/CC-MAIN-20161020183841-00244-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/mass-ratios-and-mutiple-proportions.90297/ | # Mass Ratios and Mutiple Proportions
1. Sep 22, 2005
### Soaring Crane
The "Law of Multiple proportions" states that if two elements combine in different ways to form different substances, the mass ratios are small, whole number multiples of each other. Suppose that you have 66g of carbon combined with oxygen in carbon dioxide (CO2), how many grams of oxygen would you expect to be involved?
Well, I looked at the atomic mass ratios of oxygen to carbon, or 32.00 to 12.01. By dividing this I get 8/3, so I multiply (8/3)*66 g for 176 g of O? Is this correct?
Oh, exactly how would I express this in significant figures (or the proper form)? 1.8*10^2 g (if this is correct)?
Thanks.
Last edited: Sep 22, 2005
2. Sep 22, 2005
### bomba923
"66 grams" implies two significant figures. The ratio of carbon to oxygen in pure carbon dioxide is 1 atom carbon : 2 atoms oxygen . So,
$$\left( {\frac{{66g\;{\text{C}}}}{1}} \right)\left( {\frac{{1{\text{ mole C}}}}{{12g\;{\text{C}}}}} \right)\left( {\frac{{{\text{2}}\;{\text{moles O}}}}{{1{\text{ mole C}}}}} \right)\left( {\frac{{{\text{16}}g\;{\text{O}}}}{{1{\text{ mole O}}}}} \right) = 176g\;{\text{O}} \Rightarrow \boxed{{\text{180g}}\;{\text{O}}}$$
Just 180 g is fine I think
Last edited: Sep 23, 2005 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9595487117767334, "perplexity": 2388.546903124814}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376831933.96/warc/CC-MAIN-20181219090209-20181219112209-00446.warc.gz"} |
http://sepwww.stanford.edu/data/media/public/docs/sep94/biondo1/paper_html/node3.html | Next: COMMON-AZIMUTH IMAGING OF A Up: Biondi: Common-Azimuth Imaging Previous: AZIMUTH-MOVEOUT TRANSFORMATION
# COMMON-AZIMUTH MIGRATION
Common-azimuth migration is the second component of the common-azimuth imaging procedure. Common-azimuth migration Biondi and Palacharla (1996) is based on a downward-continuation operator derived from the full 3-D prestack downward-continuation operator. Common-azimuth data sets are recursively evaluated at increasing depth levels, starting from the common-azimuth data set recorded at the surface. Common-azimuth data has zero cross-line offset, and thus it is only four-dimensional. Consequently, the common-azimuth operator is also only four dimensional, whereas the full 3-D prestack downward continuation operator is five dimensional. This reduction in dimensionality of the continuation operator results in a substantial reduction of computational and storage requirements, though at the cost of some potential loss in accuracy.
The full prestack downward continuation operator is expressed in the frequency-wavenumber domain by the Double Square Root (DSR) dispersion relation
(2)
where is the temporal frequency, kx and ky are the midpoint wavenumbers, and khx and khy are the offset wavenumbers; and are respectively the propagation velocities at the source and receiver location.
A stationary-phase approximation of the full DSR yields the common-azimuth dispersion relation. This new dispersion relation can be expressed as the cascade of two 2-D relations. The first relation is that for 2-D prestack downward-continuation along the in-line direction,
(3)
and the second relation is the one for 2-D zero-offset downward continuation along the cross-line axis,
(4)
The common-azimuth downward continuation operator is not as a general operator as the full DSR, and consequently it introduces some approximations. However, the components of the wavefield that contribute significantly to the final image are correctly extrapolated Biondi and Palacharla (1996).
Next: COMMON-AZIMUTH IMAGING OF A Up: Biondi: Common-Azimuth Imaging Previous: AZIMUTH-MOVEOUT TRANSFORMATION
Stanford Exploration Project
11/11/1997 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9126748442649841, "perplexity": 4609.460898599545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824357.3/warc/CC-MAIN-20171020211313-20171020231313-00256.warc.gz"} |
https://byjus.com/questions/find-the-median-of-the-first-ten-prime-numbers/ | # Find the median of the first ten prime numbers?
Solution:
The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Here number of terms = 10.
So median is the mean of (n/2)th term and [(n/2)+1] th term.
Here it is mean of 5th and 6th term.
Median = (11 + 13)/2
= 24/2
= 12
Hence the median of the first ten prime numbers is 12. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8205987811088562, "perplexity": 703.4610196119959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585231.62/warc/CC-MAIN-20211019012407-20211019042407-00290.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-slope-and-y-intercept-of-the-line-y-x-10 | Algebra
Topics
# How do you find the slope and y-intercept of the line y=x+10?
Slope : 1
Intercept : 10
#### Explanation:
Write the Equation in The Slope Intercept Form, i.e. $y = m x + c$
$y = 1 \cdot x + 10$
Comparing this to the Slope Intercept Form(Gradient Intercept Form),
$m = 1$ (which is the slope)
& $c = 10$ (which is the intercept on $y$-axis)
##### Impact of this question
274 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8784821033477783, "perplexity": 2670.4925988003783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986717235.56/warc/CC-MAIN-20191020160500-20191020184000-00134.warc.gz"} |
https://www.jiskha.com/questions/1012594/a-square-field-has-an-area-of-22-500-square-feet-to-the-nearest-foot-what-is-the | # Math
A square field has an area of 22,500 square feet. To the nearest foot, what is the diagonal distance across the field?
F. 150 feet
G. 178 feet
H. 191 feet
I. 212 feet
J. 260 feet
1. 👍
2. 👎
3. 👁
1. AREA=SIDE X SIDE
THEREFORE FIND SQUARE ROOT OF 22,500
THEN USE PYTHAGORAS THEOREM Z^2=X^2+Y^2
FIND SQUARE ROOT OF YOUR VALUE.THAT GIVES THE ANSWER TO Z= DIAGONAL
1. 👍
2. 👎
2. sqrt(22,500) = 150
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2. 👎
3. c^2 = a^2 + b^2
c^2 = 150^2 + 150^2
c^2 = 22,500 + 22,500
c^2 = 45,000
sqrt(c^2) = sqrt(45,000)
c = 212
I. 212 feet
1. 👍
2. 👎
4. What are the steps to finding the answer to the question: A square field has an area of 22,500 square feet. To the nearest foot, what is the diagonal distance across the field?
1. 👍
2. 👎
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For any circle, which ratio is equal to the number π? Select all that apply. ***A. Circumference ------------------- Diameter B. Circumference ------------------- Radius ***C. Circumference ------------------- 2 Times Radius D.
1. ### math Help!!!!!!
George has a square sandbox in the backyard. The sandbox has an area of 160 square feet. To the nearest foot, what is the length of one side of the sanbox?(Hint: A=s^2 A. 11 ft B. 12 ft C. 40 ft D. 80 ft. Is the answer B? Thanks
2. ### Math
Julie is digging a square area in her backyard for a vegetable garden. If the area is 52 square feet, what is the approximate length of one side of her garden? 6 square feet 7 square feet 8 square feet 9 square feet 7 square feet
3. ### Pre-Algebra
8. Mark needs to cut a piece of glass to replace a broken window. He has four pieces of glass: 6 feet long, 5 feet long, 3 feet long, and 7 feet long. If the length of the glass he needs to cut is square root 20 feet long, which
4. ### Math help
Paul is painting a square wall with an area of 115 square feet. To the nearest foot, what is the length of the wall? (1 point) 11 29 ** 42 58
1. ### Math
Shelly sews a square blanket that has an area of 144 square ft. It has 36 square blocks each of the same size, what is the approximate length of each side of a block. A.1 foot B.2 feet C.3 feet D.4 feet Please I need help with
2. ### Math plz help someone.
Which of the following is a possible measure of the area of a square rug? • 18 square feet • 24 square feet • 36 square feet • 48 square feet Based on your answer to the previous question, what would the length of one side
3. ### Math Help or I fail
1. What is the solution to negative Start Root 75 End Root to the nearest integer? (1 point) 9 8 -8 - My answer -9 2. Shelly sews a blanket that has an area of 170 square feet. It has 30 square blocks, each the same size. What is
4. ### Math
The volume of a cube is 8000 cubic feet, what is the area of its faces A 40 square feet B 80 square feet C 200 square feet D 400 square feet | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8826039433479309, "perplexity": 1807.8587851431635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703518201.29/warc/CC-MAIN-20210119072933-20210119102933-00403.warc.gz"} |
http://mathhelpforum.com/pre-calculus/168316-find-polynomials-p-x.html | # Math Help - Find the polynomials p (x)
1. ## Find the polynomials p (x)
Find all polynomials p(x) where xp(x -1) = (x-5)p(x) and p(6) = 5 factorials
2. Originally Posted by rcs
Find all polynomials p(x) where xp(x -1) = (x-5)p(x) and p(6) = 5 factorials
Hint :
If $p(x)$ satisfies $xp(x-1)=(x-5)p(x)$ then,
For $x=0$ , $0=-5p(0)$ that is, $p(0)=0$ .
For $x=1$ , $1p(0)=-4p(1)$ that is, $p(1)=0$ .
...
$x=0,1,2,3,4$ are roots of $p(x)$
Fernando Revilla
3. Didn't we just recently have this?
Here's how I would do it (rather different from the way FernandoRevilla did it). Taking x= 5, 5p(4)= 0 so that p(4)= 0. Since p is a polynomial, that means that p has a factor of x- 4: we can write p(x)= (x- 4)q(x) for some polynomial q.
Putting that into the given equation, xp(x- 1)= x((x- 1)- 4)q(x-1)= x(x- 5)q(x-1)= (x- 5)(x- 4)q(x). Since this is true for all x, we can cancel the "x- 5" terms an see that q satisfies
xq(x- 1)= (x- 4)q(x).
Talking x= 4, we have 4q(3)= 0 so that q(3)= 0 and we can write q(x)= (x- 3)r(x) for some polynomial r.
Now xq(x-1)= x(x-1- 3)r(x-1)= (x- 4)(x- 3)r(x). Since this is true for all x, we cancel the "x-4" terms and have
xr(x-1)= (x- 3)r(x). Taking x= 3, we have 3r(2)= 0 so that r(x)= (x- 2)s(x) for some polynomial s.
Get the point? Keep doing that until it terminates (it will).
4. Originally Posted by HallsofIvy
Didn't we just recently have this?
Yes.
5. ## answer from the book without solution
hello everybody Dear Mathematicians... i saw the answer from where i got that problem i posted above... this is the answer but i don't know how they reach to this
6. Originally Posted by rcs
hello everybody Dear Mathematicians... i saw the answer from where i got that problem i posted above... this is the answer but i don't know how they reach to this
We have proved that $x=0,1,2,3,4$ are roots of $p(x)$ so, all the polynomials satisfying
$xp(x-1)=(x-5)p(x)\quad[*]$
have the form:
$p(x)=x(x-1)(x-2)(x-3)(x-4)q(x)$
On the other hand, $[*]$ implies:
$x(x-1)(x-2)(x-3)(x-4)(x-5)q(x-1)=$
$(x-5)x(x-1)(x-2)(x-3)(x-4)q(x)$
that is, $q(x)=q(x-1)$ and necessarily $q(x)=a$ constant ( Why ?).
So, all polynomials $p(x)$ satisfying $[*]$ are:
$p(x)=ax(x-1)(x-2)(x-3)(x-4)$
and the condition $p(6)=5!$ implies $a=1/6$ .
Fernando Revilla | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 24, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9781444668769836, "perplexity": 2618.9844118743617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775225.37/warc/CC-MAIN-20141217075255-00030-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/users/3795/fendi?tab=summary | # Fendi
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# 7 Questions
6 Can someone explain how water from a garden hose can propagate in a sine/cosine wave? 3 Conical Pendulum — Can it rotate at 90 degrees? 2 When are Maximum Velocity and Acceleration acheived in Simple Harmonic Motion? 2 2D - Kinematics - Linkage System using Vector Algebra [closed] 1 Kinematics Problem
# 189 Reputation
+33 Can someone explain how water from a garden hose can propagate in a sine/cosine wave? +18 Conical Pendulum — Can it rotate at 90 degrees? +10 2D - Kinematics - Linkage System using Vector Algebra +5 Kinematics Problem
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https://www.physicsforums.com/threads/help-fluids-problem.51826/ | # Homework Help: Help - FLUIDS problem
1. Nov 7, 2004
### chattkis3
A pump and its horizontal intake pipe are located 11.5 m beneath the surface of a reservoir. The speed of the water in the intake pipe causes the pressure there to decrease, in accord with Bernoulli's principle. Assuming nonviscous flow, what is the maximum speed with which water can flow through the intake pipe?
--Im getting confused because they give you so few numbers to work with. Any ideas? Thanks, --
2. Nov 7, 2004
### Pyrrhus
Bernoulli's Equation
$$P + \frac{1}{2} \rho v^2 + \rho gh = constant$$
Well you know the density of the water and atmospheric pressure.
3. Nov 7, 2004
### chattkis3
Ok so this is what I have: (P1) + ((1/2)density*v1^2) + (density*g*h1) = (P2) + ((1/2)density*v2^2) + (density*g*h2)
So for this question I can ignore the left side of this longer version of Bernoulli's? Do I substitute in a value of 1 or 0 so that I can solve for v ?
4. Nov 7, 2004
### Staff: Mentor
Ignore the speed of the water at the surface. (Assume the surface area is much greater than the cross-sectional area of the pipe.)
5. Nov 7, 2004
### chattkis3
Ok am I on the right track with this? :
1.013 x 10^5 Pa = (1.013E5Pa + (1000 kg/m3*9.8m/s2*11.5m)) + .5(1000 kg/m3)* v^2 + (1000 kg/m3 *9.8m/s2 *11.5m)
Im trying to plug in what I know with Bernoulli's...
6. Nov 8, 2004
### Pyrrhus
It should be:
$$P_{o} + \rho g h = P_{pipe} + \frac{1}{2} \rho v^2$$
Last edited: Nov 8, 2004
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8327701091766357, "perplexity": 2313.123635931236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156513.14/warc/CC-MAIN-20180920140359-20180920160759-00086.warc.gz"} |
https://cs.stackexchange.com/questions/124550/data-compression-entropy | # Data compression - Entropy
Let's say I have an alphabet $$\Sigma = \{A, B, C, D, E\}$$
with probabilities $$P(A) = P(B) = P(C) = 0.25 \text{ and } P(D)=P(E) = 0.125.$$
I know that the entropy then is: $$H(\Sigma) = 3 \cdot 0.25 \cdot \log 4 + 2 \cdot 0.125 \cdot \log 8 = 2.25.$$
My question now is: What does this mean in relation to the lower limit of compression? How many bits will I at least need to compress a text that consists of the above alphabet?
• According to Shannon's source coding theorem, the entropy of a source is the optimal rate of compression. The theorem gives both an upper bound and a lower bound. – Yuval Filmus Apr 22 at 12:04
• So I would need min. 2.25 bits per character? – Philipp Wilhelm Apr 22 at 12:05
• en.wikipedia.org/wiki/Shannon%27s_source_coding_theorem – Yuval Filmus Apr 22 at 12:07
• (Keep in mind that Shannon presumes independent distribution: not given in many applications.) – greybeard Apr 22 at 16:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8857683539390564, "perplexity": 672.0450739527986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655908294.32/warc/CC-MAIN-20200710113143-20200710143143-00128.warc.gz"} |
https://www.groundai.com/project/blockwise-svd-with-error-in-the-operator-and-application-to-blind-deconvolution/ | Blockwise SVD with error in the operator and application to blind deconvolution
# Blockwise SVD with error in the operator and application to blind deconvolution
S. Delattre 111Université Denis Diderot Paris 7 and CNRS-UMR 7099, 175 rue du Chevaleret 75013 Paris, France. E-mail: [email protected], [email protected], [email protected] M. Hoffmann222ENSAE and CNRS-UMR 8050, 3 avenue Pierre Larousse, 92245 Malakoff Cedex, France. E-mail: [email protected] D. Picard footnotemark: and T. Vareschi footnotemark:
###### Abstract
We consider linear inverse problems in a nonparametric statistical framework. Both the signal and the operator are unknown and subject to error measurements. We establish minimax rates of convergence under squared error loss when the operator admits a blockwise singular value decomposition (blockwise SVD) and the smoothness of the signal is measured in a Sobolev sense. We construct a nonlinear procedure adapting simultaneously to the unknown smoothness of both the signal and the operator and achieving the optimal rate of convergence to within logarithmic terms. When the noise level in the operator is dominant, by taking full advantage of the blockwise SVD property, we demonstrate that the block SVD procedure overperforms classical methods based on Galerkin projection [14] or nonlinear wavelet thresholding [18]. We subsequently apply our abstract framework to the specific case of blind deconvolution on the torus and on the sphere.
Keywords: Blind deconvolution; blockwise SVD; circular and spherical deconvolution; nonparametric adaptive estimation; linear inverse problems; error in the operator.
Mathematical Subject Classification: 62G05, 62G99, 65J20, 65J22.
## 1 Introduction
### 1.1 Motivation
Consider the following idealised statistical problem: estimate a function (a signal, an image) from data
yn=Kf+n−1/2˙W, (1.1)
where
K:H→G
is a linear operator between two Hilbert spaces and . The observation of the unknown is challenged by the action of the linear degradation as well as contaminated by an experimental Gaussian white noise on with vanishing noise level as . Alternatively, in a density estimation setting, we observe a random sample drawn from a probability distribution333In that setting, must therefore also be a probability density . with density . In each case, we do not know the operator exactly, but we have access to
Kδ=K+δ˙B, (1.2)
where is a Gaussian white noise on thanks to preliminary experiments or calibration through trial functions. This setting has been discussed in details in [14, 18]. In this paper, we are interested in operators admitting a singular value decomposition (SVD) or a blockwise SVD. In essence, we know the typical eigenfunctions of but not the eigenvalues. We cover two specific examples of interest: spherical and circular deconvolution.
###### Spherical deconvolution.
Used for the analysis of data distributed on the celestial sphere, see Section 4.1 below. One observes a random sample with
Zi=εiXi,i=1,…,n
where the are random elements in , the group of rotation matrices, and the are independent and identically distributed on the sphere , with common density with respect to the uniform probability distribution on . In this setting, if the have common density with respect to the Haar measure on , we have
Kf(x)=g⋆f(x)=∫SO(3)g(u)f(u−1x)du,x∈S2.
We are interested in the case where the exact form is unknown. However, is block-diagonal in the spherical harmonic basis. ∎
###### Circular deconvolution.
Used for restoring signal or images, see Section 4.2 below. We take the space of square integrable functions on the torus (or ) appended with periodic boundary conditions. We have
Kf(x)=g⋆f(x)=∫Tg(u)f(x−u)du,x∈T.
The degradation process is characterised by the impulse response function which we do not know exactly. However, is diagonal in the Fourier basis. ∎
Although the problem of estimating is fairly classical and well understood when is known (a selected literature is [32, 8, 12, 1, 17, 29, 28] and the references therein), only moderate attention has been paid in the case of an unknown despite its relevance in practice. When the eigenfunctions of are known solely, we have the results of Cavalier and Hengartner [5], Cavalier and Raimondo [6] but they are confined to the case where the error in the operator is negligible . In a general setting with error in the operator, Efromovitch and Kolchinskii [14] and later Hoffmann and Reiß [18] studied the recovery of when the eigenfunctions and eigenvalues of are unknown. In both contributions, a marginal attention is paid to the case of sparse or diagonal operators, but it is showed in both papers that unusual rates of convergence can be obtained when . In a univariate setting, Neumann [24] and Comte and Lacour [9] consider the case of deconvolution with an error density, known only through an auxiliary set of learning data. This formally corresponds to having in our setting. Minimax rates and adaptive estimators are derived in both regimes and . We address in the paper the following program:
• Construction of a feasible procedure estimating from data (1.1) and (1.2) that achieves optimal rates of convergence (up to inessential logarithmic terms). We require to be adaptive with respect to smoothness constraints on and .
• Identification of best achievable accuracy for under smoothness constraints on and so that the interplay between and can be explicitly related in the asymptotic and ; this includes the comparison with earlier results of [24, 14, 18] in the context of blockwise SVD.
• Application to spherical deconvolution on or circular deconvolution on the torus; this includes the discussion of our findings in terms of the existing literature on the topic [7, 25, 22] and some practical aspects of numerical implementation.
### 1.2 Main results and organisation of the paper
In Section 2, we present an abstract framework that allows for operators to admit a so-called blockwise SVD. This property is simply turned into the existence of pairs of increasing finite dimensional spaces ( that are stable under the action of . The blockwise SVD property is further appended with a smoothness condition quantified by the arithmetic decay of the operator norm of and its inverse on (resp. ) (the so-called ordinary smooth assumption, see e.g. [32]). By means of a reconstruction formula, we obtain in Section 2.2 an estimator of by first inverting on with a thresholding tuned with and then filter the resulting signal by a block thresholding tuned with . As for i) and ii), we establish in Theorems 3.1 and 3.4 of Section 3 the minimax rates of convergence for Sobolev constraints on under squared error loss and we demonstrate that is optimal and adaptive to within logarithmic terms. The explicit interplay between and is revealed and discussed in the case of sparse operator when , completing earlier findings in [14, 18] and to some extent [24] in the univariate case for density deconvolution. In particular, we demonstrate that a certain parametric regime dominates when the smoothness of the signal dominates the smoothing properties of the operator. Concerning iii), the method is applied to the case of spherical and circular deconvolution in Section 4 where harmonic Fourier analysis enables to provide explicit blockwise SVD for the convolution operator. We illustrate the numerical feasability of and the phenomena that appear in the case . Section 5 is devoted to the proofs.
We choose to state and prove our results in the white Gaussian model generated by the observation of and defined by (1.1) and (1.2). The extension to the case of density estimation, when is replaced by the observation of a random sample of size drawn from the distribution , like for instance in [24, 9] is a bit more involved, due to the intrinsic heteroscedasticity that appears when enforcing a formal analogy with the Gaussian setting (1.1). It is briefly addressed in the discussion Section 3.2
## 2 Estimation by blockwise SVD
### 2.1 The blockwise SVD property
Let denote a family of linear operators
K:H→G
between two Hilbert spaces and that shall represent our parameter set of unknown .
A fundamental property (Assumption 2.1 below) is that an explicit singular value decomposition (SVD) or blockwise SVD is known for all simultaneously. More specifically, we suppose that there exist two explicitly known bases of and of , as well as a partition of with if , and a constant such that:
ℓd−1≲|Λℓ|≲ℓd−1,
where means inequality up to a multiplicative constant that does not depend on . Here .
It is worthwhile to notice that in our examples as well as in the rates of convergence that we will exhibit later, plays the role of a dimension. In particular, corresponds to a ’standard SVD’, whereas creates blocks and deserves the name of ’blockwise’ SVD. However, there is no need in the paper to assume that is in . Set
Hℓ=Span{eλ,λ∈Λℓ}andGℓ=Span{gλ,λ∈Λℓ}.
The Galerkin projection of an operator onto is defined by , where is the orthogonal projector onto .
###### Assumption 2.1 (Blockwise SVD).
K|Hℓ=Kℓfor everyK∈G,ℓ≥1.
We further need to quantify the action of on . We denote by the operator norm of .
###### Assumption 2.2 (Spectral behaviour of K|Hℓ).
For every , is invertible and there exists such that
Q1(K)=supℓ≥1ℓ−ν∥(Kℓ)−1∥Gℓ→Hℓ<∞
and
Q2(K)=supℓ≥1ℓν∥Kℓ∥Hℓ→Gℓ<∞
for every .
We associate with the bases and the following decompositions
f=∑ℓ≥1∑λ∈Λℓ⟨f,eλ⟩eλ,g=∑ℓ≥1∑λ∈Λℓ⟨g,gλ⟩gλ for everyf∈H,g∈G,
where denotes the inner product either in or and the scale of Sobolev spaces
Ws ={f∈H,∥f∥2Ws=∑ℓ≥1ℓ2s∑λ∈Λℓ⟨f,eλ⟩2<∞},s∈R, (2.1) ˜Ws ={g∈G,∥g∥2˜Ws=∑ℓ≥1ℓ2s∑λ∈Λℓ⟨g,gλ⟩2<∞},s∈R.
For , Assumption 2.2 implies that is continuous. In particular, when , the operator is ill-posed with degree , see for instance [26].
### 2.2 Blockwise SVD reconstruction with noisy data
Under Assumption 2.1 and 2.2, we have the reconstruction formula
f=∑ℓ≥0(Kℓ)−1∑λ∈Λℓ⟨Kf,gλ⟩eλ. (2.2)
By the observed blurred version of in (1.2), we obtain a family of estimators of from data (1.2) by considering the operator
1{∥(Kδ,ℓ)−1∥Gℓ→Hℓ≤κ}(Kδ,ℓ)−1, (2.3)
where is a cutoff level, possibly depending on . Likewise, the coefficient can be estimated by
zn,λ:=⟨yn,gλ⟩. (2.4)
Mimicking the reconstruction formula (2.2) with the estimates (2.4) and (2.3), we finally obtain a (family of) estimator(s) of by setting
ˆfn,δ=∑0≤ℓ≤L(Kδ,ℓ)−1(∑λ∈Λℓzn,λeλ1{∑λ∈Λℓz2n,λ≥τ2ℓ})1Eδ,ℓ(κℓ)
where
Eδ,ℓ(κℓ)={∥(Kδ,ℓ)−1∥Gℓ→Hℓ≤κℓ}.
The procedure is specified by the maximal frequency level and the threshold levels
κℓ=(λ0|Λℓ|−1/2(δ2|logδ|)−1/2)⋀n1/2 (2.5)
and
τℓ=μ0|Λℓ|1/2(n−1logn)1/2, (2.6)
for some prefactors . The threshold rule we introduce in both the signal (with level ) and the operator (with level ) is inspired by classical block thresholding [21, 4, 3] and will enable to adapt with respect to the smoothness properties of both the signal and the operator , see below.
## 3 Main results
### 3.1 Minimax rates of convergence
We assess the performance of the estimator defined in Section 2.2 over the Sobolev spaces linked to the basis defined in (2.1). Define the Sobolev balls for and let
Gν(Q)={K∈G,Qi(K)≤Qi,i=1,2}. (3.1)
for with , , where the mapping constants are defined in Assumption 2.2.
###### Theorem 3.1 (Upper bounds).
Let be a class of operators satisfying Assumptions 2.1 and 2.2. Assume we observe given by (1.1) and (1.2), with and . Specify with
L=⌊(δ2)−1/(2ν+d−1)⌋⋀⌊n1/(2ν+d)⌋ (3.2)
and as in (2.5) and (2.6). For sufficiently small and sufficiently large , for every , with and such that , we have
supf∈Ws(M),K∈Gν(Q)E[∥∥ˆfn,δ−f∥∥2H] ≲ (δ2|logδ|)1⋀2s/(2ν+d−1)⋁(n−1logn)2s/(2(s+ν)+d) (3.3)
where means inequality up to a multiplicative constant that depends on and only.
The bounds for and are explicitly computable. In the model generated by in (1.1) and in (1.2), they depend on the dimension and on the absolute constants and of the concentration lemmas 5.3 and 5.6 below. However, they are in practice much too conservative, as is well known in the signal detection case [13] or the classical inverse problem case [1], see the numerical implementation Section 4.
Our next result shows that the rate achieved by is indeed optimal, up to logarithmic terms. The lower bound in the case is classical (Nussbaum and Pereverzev [26]) and will not decrease for increasing noise levels or whence it suffices to provide the case which formally corresponds to observing without noise while remains unknown.
###### Theorem 3.2 (Lower bounds).
In the same setting as Theorem 3.1, with in addition , assume we observe exactly and given by (1.2). For sufficiently small , we have
infˆfsupf∈Ws(M),K∈Gν(Q)E[∥∥ˆf−f∥∥2H]≳(δ2)1⋀2s/(2ν+d−1) (3.4)
where means inequality up to a positive multiplicative constant that depends on and only.
Combining (3.3) together with (3.4) and the results of [26], we conclude that is minimax over to within logarithmic terms in and , and that this result is uniform over the nuisance parameter .
### 3.2 Discussion
#### The case of diagonal operators
It is interesting to notice that the condition in Theorem 3.2 excludes the case where is diagonal. In this particular case, considered especially in the deconvolution example of Section 4.2 below, a closer inspection of the proof of the upper bound shows that the rate
n−s/(2(s+ν)+d)⋁δ1⋀s/ν
can be obtained (up to some extra logarithmic factors) as in the case where , which improves on the rate
n−s/(2(s+ν)+d)⋁δ1⋀2s/(2ν+d−1)
provided by Theorem 3.1. This sheds some light on the role of the number . It is in fact twofolds: it acts as a ’dimension’ in the term ; in the term involving error in the operator , it reflects the distance to the diagonal case expanding from in the diagonal case, to in the case . It is very plausible, though beyond the scope of this paper, to express conditions on leading to rates of the form , with continuously varying from 0 to . Note that in the case , we recover the minimax rate of density deconvolution with unknown error as proved by Neumann [24], see also [9].
#### Relation to other works in the case of sparse operators
For an unknown signal with smoothness and unknown operator with degree of ill-posedness , the optimal rates of convergence are
n−α(s,ν)/2⋁δβ(s,ν), (3.5)
up to inessential logarithmic terms. The exponents and are linked respectively to the error in the signal and the error in the operator . Efromovitch and Kolchinskii [14] established that under fairly general conditions on the operator , the optimal exponents are given by
α(s,ν)=β(s,ν)=2s2(s+ν)+d.
They noted however that if certain sparsity properties on are moreover assumed (and that we shall not describe here, for instance if is diagonal in an appropriate basis) then the exponent is no longer optimal, while remains unchanged.
In the related context of operators acting on Besov spaces of functions with smoothness measured in -norm, Hoffmann and Reiß [18] introduce an ad hoc hypothesis on the sparsity of the unknown operator (that we shall not describe here either), expressed in terms of the wavelet discretization of . They subsequently obtain new rates of convergence for a certain nonlinear wavelet procedure, and these rates overperform (3.5) as expected from the results by [14]. In particular, if one considers the estimation of , in the extreme case where the operator is diagonal in a wavelet basis, the procedure in [18] achieves the rate
n−α(s,ν)/2⋁(δ2)1⋀(s−d/2)/ν (3.6)
up to extra logarithmic terms. We may compare our results with the rate (3.6). In our setting, if we pick as the Fourier basis described in Section 4.2, then we have . Assuming to be diagonal in the basis which is the exact counterpart of the approach of Hoffmann and Reiß with being diagonal in a wavelet basis, then by Theorem 3.1, our estimator (nearly) achieves the rate
n−α(s,ν)/2⋁(δ2)1⋀2s/(2ν+d−1)
which already outperforms the rate (3.6) whenever the error in the signal is dominated by the error in the operator and is small compared to , as follows from the elementary inequality
2s/(2ν+d−1)>(s−d/2)/νfor2ν+d−1≥2s.
The superiority of the blockwise SVD in this setting is explained by the fact that the wavelet procedure in [18] is agnostic to the diagonal structure of in the wavelet basis, in contrast to that takes full advantage of the block structure of . As already explained in the preceding section, one could actually improve further this result in the specific case of being diagonal in and show that (nearly) achieves the rate , thus deleting the ‘dimensional effect’ of for the error in the operator.
#### Adaptation over the scales {Ws,s>0} and {Gν,ν≥0}
The estimator is fully adaptive over the family of Sobolev balls (in the sense that does not require the knowledge of nor ). However, the knowledge of the degree of ill-posedness of is required through the choice of the maximal frequency in (3.2). This restriction can actually be relaxed further in dimension . Indeed, setting formally in (3.2), one readily checks that becomes adaptive over and simultaneously. In dimension however, setting in (3.2) is forbidden, but an alternative adaptivity result can be obtained by taking for some , in which case is fully adaptive over the scale and the restricted family .
#### Extension to density estimation
We briefly show a line of proof for extending Theorem 3.1 to the framework of density estimation. Suppose that instead of we observe a random sample drawn from assumed to a probability density. By analogy to (2.4), we have an estimator of replacing with
Pn(gλ)=n−1n∑i=1gλ(Zi).
Writing
Pn(gλ)=⟨Kf,gλ⟩+n−1/2ηn,λ,
with , an inspection of the proof of Theorem 3.1 reveals that an extension to the density estimation setting carries over as soon as the vector satisfies a concentration inequality, namely
∃β1>0,c1>0,∀β≥β0,P(|Λλ|−1∑λ∈Λη2n,λ⩾β2)≤exp(−cβ2|Λℓ|),
see (5.6) in Lemma 5.2. To that end, we may apply a concentration inequality by Bousquet [2] as developed for instance in Massart [23], Eq (5.51) p. 171. The precise control of this extension requires further properties on the basis and on the density via the behaviour of , see Eq. (5.52) p. 171 in [23]. We do not pursue that here.
## 4 Application to blind deconvolution
### 4.1 Spherical deconvolution
###### Scientific context.
A common challenge in astrophysics is the analysis of complex data sets consisting of a number of objects or events such as galaxies of a particular type or ultra high energy cosmic rays (UHECR) and that are genuinely distributed over the celestial sphere. Such objects or events are distributed according to a probability density distribution on the sphere, depending itself on the physics that governs the production of these objects or events. For instance, UHECR are particles of unknown nature arriving at the earth from apparently random directions of the sky. They could originate from long-lived relic particles from the Big Bang. Alternatively, they could be generated by the acceleration of standard particles, such as protons, in extremely violent astrophysical phenomena. They could also originate from Active Galactic Nuclei (AGN), or from neutron stars surrounded by extremely high magnetic fields. As a consequence, in some hypotheses, the underlying probability distribution for observed UHECRs would be a finite sum of point-like sources. In other hypotheses, the distribution could be uniform, or smooth and correlated with the local distribution of matter in the universe. The distribution could also be a superposition of the above. Identifying between these hypotheses is of primordial importance for understanding the origin and mechanism of production of UHECRs. The observations, denoted by , are often perturbated by an experimental noise, say , that lead to the deconvolution problem described in Section 1.1. Following van Rooj and Ruymgart [33], Healy et al. [16], Kim and Koo [22] and Kerkyacharian et al. [25], we assume the following model: we observe an -sample with
Zi=εiXi,i=1,…,n
where the are distributed on the sphere , with common density with respect to the uniform probability distribution on and independent of the that have a common density with respect to the Haar probability measure on the group of rotation matrices. One proves in [16, 22] that the density of the is
Kf(ω)=g⋆f(ω):=∫SO(3)g(r)f(r−1x)dr,ω∈S2 (4.1)
and we are interested in the case where the exact form of the convolution operator is unknown, due for instance to insufficient knowledge of the device that is used to measure the observations. However, we assume approximate knowledge of through as defined in (1.2). ∎
###### Checking the blockwise SVD Assumptions 2.1 and 2.1.
We closely follow the exposition of [16, 22, 25] for an overview of Fourier theory on and in order to establish rigorously the connection to Theorem 3.1 and 3.4. Define
u(φ)=⎛⎜⎝cosφ−sinφ0sinφcosφ0001⎞⎟⎠anda(θ)=⎛⎜⎝cosθ0sinθ010−sinθ0cosθ⎞⎟⎠
where . Every rotation has representation for some . Define the rotational harmonics
Dlmn(r)=Dlmn(φ,θ,ψ)=e−i(mφ+nψ)Plmn(cos(θ))
for where are the second type Legendre functions described in details in [34]. The are the eigenfunctions of the Laplace-Beltrami operator on hence the family forms a complete orthonormal basis of on , where is the Haar probability measure. Every has a rotational Fourier transform
F(h)lmn=∫SO(3)h(u)Dlmn(u)du,l∈N,−l⩽m,n⩽l,
and for every we have a reconstruction formula
h =∑l∈N∑−l≤m,n≤lF(h)lmn¯¯¯¯¯¯¯¯¯¯Dlmn =∑l∈N∑−l≤m,n≤lF(h)lmnDlmn(∙−1)
An analogous analysis is available on . Any point is determined by its spherical coordinates for some . Define
Ylm(ω)=Yml(θ,φ)=(−1)m√2l+14π(l−m)!(l+m)!Plm(cos(θ))eimφ (4.2)
for where are the Legendre functions. We have and the constitute an orthonormal basis of on , generally referred to as the spherical harmonic basis. Any has a spherical Fourier transform
F(f)lm=∫S2f(ω)¯¯¯¯¯¯¯¯¯¯¯¯¯¯Ylm(ω)μ(dω)
and a reconstruction formula
f=∑ℓ∈N∑−l≤m≤lF(f)lmYlm.
If the spherical convolution operator defined in (4.1) satisfies
F(g⋆f)lm=l∑n=−lF(g)lmnF(f)ln (4.3)
and we retrieve the blockwise SVD formalism of Section 2.1 in dimension by setting , where the probability Haar measure on and
eλ=gλ=Yℓmwithλ=(m,ℓ),Λℓ={(m,ℓ),−ℓ⩽m⩽ℓ}.
We have and by (4.3), is the finite dimension operator stable on with matrix having entries
(Kℓ)mn=F(g)ℓmn.
Hence Assumption 2.1 is satisfied. Notice also that in this case is generally not diagonal. Assumption 2.2 is satisfied as we assume that is ordinary smooth in the terminology of Kim and Koo [22]. Our Assumption 2.2 exactly matches the constraint (3.6) in their paper with examples given by the Laplace distribution on the sphere () or the Rosenthal distribution ( arbitrary). ∎
###### Numerical implementation.
Following Kerkyacharian, Pham Ngoc and Picard [25] in their Example 2, we take with and . We have .
is the density of a Laplace distribution on , defined through . Hence, the matrices are homotheties whose ratios behave as . We have .
We plot in Figures 1 a -sample of with density on the sphere, and the action by on the , where the are distributed according to in Figure 2. Note that for the estimation of , we have access to a noisy version of with noise level only.
We display below the (renormalised) empirical squared error of (oracle choice ) for Monte-Carlo for several values of . The noise level is to to be compared with the noise level . The latter is chosen non-negative, in order to show the interaction between the two types of error, and sufficiently small to emphasize the influence of on the process of estimation.
Noise level
Mean error 0.0466 0.0542 0.1732 0.2784 0.4335
Standard dev. 0.0011 0.0022 0.0126 0.0355 0.0466
Finally, on a specific sample of data, we plot the target density (Figure 3) and its reconstruction for data with (Figure 4) and (Figure 5). At a visual level, we oversimplify the representation by plotting and its reconstruction with a view from above the sphere through the axis. We see that the contour in Figure 5 is not well recovered in the regions where is small (on the right side of the graph in Figure 5). The choice of remains unchanged.
### 4.2 Circular deconvolution
###### Scientific context.
In many engineering problems, the observation of a signal or image is distorted by the action of a linear operator . We assume for simplicity that lives on the torus (or ) appended with periodic boundary conditions. In many instances, the restoration of from the noisy observation of is challenged by the additional uncertainty about the operator . This is the case for instance in electronic microscopy [27] for the restoration of fluorescence Confocal Laser Scanning Microscope (CLSM) images. In other words, the quality of the image suffers from two physical limitations: error measurements or limited accuracy, and the fact that the exact PSF (the incoherent point spread function) that accounts for the blurring of (mathematically the action of ) is not precisely known. This is a classical issue that goes back to [31, 15]. An idealised additive Gaussian model for the noise contamination yields the observation (1.1) with
Kf(x)=g⋆f(x):=∫Tdg(u)f(x−u)du,x∈D=Td.
The degradation process is characterised by the impulse response function . In most cases of interest, we do not know the exact form of . In a condensed idealised statistical setup, we have access to
gδ=g+δ˙W′, (4.4)
where is another Gaussian white noise defined on and independent of . Experimental approaches that justify representation (4.4) are described in [10, 20, 30]. ∎
###### Checking Assumptions 2.1 and 2.1.
We obviously have and the bases and will coincide with the -dimensional extension of the circular trigonometric basis if we set:
eλ(x1,…,xd)=d∏j=1e2iπkjxj,(x1,…,xd)∈Td,
where we put
λ=(k1,…,kd),ℓ=|λ|=1+d∑j=1|kj|,andℓ≥1.
Any has a Fourier transform and moreover, if , we have
F(f⋆g)λ=Fλ(f)Fλ(g).
Therefore, is diagonal in the basis henceforth stable. Moreover, with , we have . Moreover and Assumption 2.1 follows. Assuming that satisfies for some and constants , we readily obtain Assumption 2.2. Note also that since is diagonal in the basis observing in the representation (4.4) is equivalent to observing in (1.2). ∎
###### Numerical implementation.
We numerically implement in dimension in the case where there is no noise in the signal (formally ) in order to illustrate the parametric effect that dominates in the optimal rate of convergence in Theorems 3.1 and 3.4 that becomes in that case. We take as target function belonging to for all and defined by its Fourier coefficients
F(f)λ=|λ|−5,λ∈{−1000,…,1000}.
We pick a family of blurring functions defined in the same manner by the formula
F(gν)λ=|λ|−ν,λ∈{−1000,…,1000},ν∈{1,4,5,6,8}.
We show in Figure 6 in a log-log plot the mean-squared error of for the oracle choice over 1000 Monte-Carlo simulations for and . For small values of the predicted slope of the curve gives a rough estimate of the rate of convergence. We visually see that for the critical case with and below, the slope is close to confirming the parametric rate that is obtained whenever . ∎
## 5 Proofs
### 5.1 Preliminary estimates
###### Preparation.
Recall that , and that (resp. ) denotes the orthogonal projector onto (resp. ). For , we have
PℓKh=PℓKQℓh+PℓK(Id−Qℓ)h.
Using Assumption 2.1, we have and therefore . As a consequence
PℓK=KℓQℓ. (5.1)
In turn, we have a convenient description of the observation defined in (1.2) and defined in (1.1) and in terms of a sequence space model that we shall now describe. ∎
###### Notation.
If , we denote by the (column) vector of coordinates of in the basis . If is a linear operator, we write for the matrix of the Galerkin projection of . ∎
###### Sequence model for error in the operator.
The observation of in (1.2) leads to the representation , or equivalently, in matrix notation
Kδ,ℓ=Kℓ+δ˙Bℓ,ℓ≥1, (5.2)
where is a matrix with entries that are independent centred Gaussian random variables, with unit variance. The following estimate is a classical concentration property of random matrices. For , denotes the operator norm for matrices (we shall skip the dependence upon in the notation).
###### Lemma 5.1 ([11], Theorem II.4).
There are positive constants such that
For allβ≥β0, P(|Λℓ|−1/2∥˙Bℓ∥op≥β)≤exp(−c0β2|Λℓ|2). (5.3)
An immediate consequence of Lemma 5.1 is the following moment bound:
For everyp>0,E[∥˙Bℓ∥pop]≲|Λℓ|p/2. (5.4)
###### Sequence model for error in the signal.
From (1.1), we observe the Gaussian measure , or equivalently, thanks to (5.1)
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http://www.wikinfo.org/English/index.php/Radiocarbon_dating?rdfrom=http%3A%2F%2Fwww.wikinfo.org%2Findex.php%3Ftitle%3DRadiocarbon_dating%26redirect%3Dno | File:Carbon 14 formation and decay.svg
1: Formation of Carbon-14
2: Decay of Carbon-14
3: The equation is for living organisms, and the inequality is for dead organisms, in which the C-14 then decays (See 2).
Radiocarbon dating (sometimes simply known as carbon dating) is a radiometric dating method that uses the naturally occurring radioisotope carbon-14 (14C) to estimate the age of carbon-bearing materials up to about 58,000 to 62,000 years.[1] Raw, i.e. uncalibrated, radiocarbon ages are usually reported in radiocarbon years "Before Present" (BP), "Present" being defined as 1950. Such raw ages can be calibrated to give calendar dates. One of the most frequent uses of radiocarbon dating is to estimate the age of organic remains from archaeological sites. When plants fix atmospheric carbon dioxide (CO2) into organic material during photosynthesis they incorporate a quantity of 14C that approximately matches the level of this isotope in the atmosphere (a small difference occurs because of isotope fractionation, but this is corrected after laboratory analysis[cn]). After plants die or they are consumed by other organisms (for example, by humans or other animals) the 14C fraction of this organic material declines at a fixed exponential rate due to the radioactive decay of 14C. Comparing the remaining 14C fraction of a sample to that expected from atmospheric 14C allows the age of the sample to be estimated.
The technique of radiocarbon dating was developed by Willard Libby and his colleagues at the University of Chicago in 1949. Emilio Segrè asserted in his autobiography that Enrico Fermi suggested the concept to Libby in a seminar at Chicago that year. Libby estimated that the steady state radioactivity concentration of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram. In 1960, he was awarded the Nobel Prize in chemistry for this work. He demonstrated the accuracy of radiocarbon dating by accurately estimating the age of wood from a series of samples for which the age was known, including an ancient Egyptian royal barge of 1850 BC.[2][3]
## Basic physics
Carbon has two stable, nonradioactive isotopes: carbon-12 (12C), and carbon-13 (13C). In addition, there are trace amounts of the unstable isotope carbon-14 (14C) on Earth. Carbon-14 has a relatively short half-life of 5730 years, meaning that the amount of carbon-14 in a sample is halved over the course of 5730 years due to radioactive decay. Carbon-14 would have long ago vanished from Earth were it not for the unremitting cosmic ray flux interactions with the Earth's atmosphere, which create more of the isotope. The neutrons resulting from the cosmic ray interactions participate in the following nuclear reaction on the atoms of nitrogen molecules (N2) in the atmosphere:
$^{14}_{7}N} \rightarrow \mathrm{^{14}_{6}C$
The highest rate of carbon-14 production takes place at altitudes of 9 to 15 km (30,000 to 50,000 ft), and at high geomagnetic latitudes, but the carbon-14 spreads evenly throughout the atmosphere and reacts with oxygen to form carbon dioxide. Carbon dioxide also permeates the oceans, dissolving in the water. For approximate analysis it is assumed that the cosmic ray flux is constant over long periods of time; thus carbon-14 is produced at a constant rate and the proportion of radioactive to non-radioactive carbon is constant: ca. 1 part per trillion (600 billion atoms/mole). In 1958 Hessel de Vries showed that the concentration of carbon-14 in the atmosphere varies with time and locality.[4] For the most accurate work, these variations are compensated by means of calibration curves. When these curves are used, their accuracy and shape are the factors that determine the accuracy of the age obtained for a given sample.[5] 14C can also be produced at ground level at a rate of 1 x 10−4 g−1s−1, which is not considered significant enough to impact on dating without a known other source of neutrons.[6]
Plants take up atmospheric carbon dioxide by photosynthesis, and are ingested by animals, so every living thing is constantly exchanging carbon-14 with its environment as long as it lives. Once it dies, however, this exchange stops, and the amount of carbon-14 gradually decreases through radioactive beta decay with a half-life of 5,730 ± 40 years.[7] Carbon-14 is stored in different amounts in different reservoirs because there is a dynamic equilibrium between 14C production and decay.[6]
$~^{14}_{6}C}\rightarrow\mathrm{~^{14}_{7}N}+ e^{-} + \bar{\nu$
Carbon-14 was discovered on February 27, 1940, by Martin Kamen and Sam Ruben at the University of California Radiation Laboratory at Berkeley.
## Computation of ages and dates
The number of decays per time is proportional to the current number of radioactive atoms. This is expressed by the following differential equation, where N is the number of radioactive atoms and λ is a positive number called the decay constant:
$dN}{dt$
As the solution to this equation, the number of radioactive atoms N can be written as a function of time:
$-\lambda t$
, which describes an exponential decay over a timespan t with an initial condition of N0 radioactive atoms at t = 0. Canonically, t is 0 when the decay started. In this case, N0 is the initial number of 14C atoms when the decay started.
For radiocarbon dating a once living organism, the initial ratio of 14C atoms to the sum of all other carbon atoms at the point of the organism's death and hence the point when the decay started, is approximately the ratio in the atmosphere.
Two characteristic times can be defined:
• mean- or average-life: mean or average time each radiocarbon atom spends in a given sample until it decays.
• half-life: time lapsed for half the number of radiocarbon atoms in a given sample, to decay,
It can be shown that:
$avg$
= $1}{\lambda$
= radiocarbon mean- or average-life = 8033 years (Libby value)
$1}{2$
= $avg$
= radiocarbon half-life = 5568 years (Libby value)
Notice that dates are customarily given in years BP which implies t(BP) = –t because the time arrow for dates runs in reverse direction from the time arrow for the corresponding ages. From these considerations and the above equation, it results:
$1}{\lambda} {\ln \frac{N}{N_0}$
and for a raw radiocarbon age:
$1}{\lambda} {\ln \frac{N}{N_0}$
After replacing values, the raw radiocarbon age becomes any of the following equivalent formulae:
using logs base e and the average life:
$avg}\cdot \ln{\frac{N}{N_0}$
and
using logs base 2 and the half-life:
$1}{2}\cdot \log_2 \frac{N}{N_0$
Wiggle matching uses the non-linear relationship between the 14C age and calendar age to match the shape of a series of closely sequentially spaced 14C dates with the 14C calibration curve.
## Measurements and scales
File:1 MV accelerator mass spectrometer.jpg
The use of accelerator mass spectrometers can improve the sensitivity of radiocarbon dating.
Measurements are traditionally made by counting the radioactive decay of individual carbon atoms by gas proportional counting or by liquid scintillation counting. For samples of sufficient size (several grams of carbon) this method is still widely used in the 2000s. Among others, all the tree ring samples used for the calibration curves (see below) were determined by these counting techniques. Such decay counting, however, is relatively insensitive and subject to large statistical uncertainties for small samples. When there is little carbon-14 to begin with, the long radiocarbon half-life means that very few of the carbon-14 atoms will decay during the time allotted for their detection, resulting in few disintegrations per minute.
The sensitivity of the method has been greatly increased by the use of accelerator mass spectrometry (AMS). With this technique 14C atoms can be detected and counted directly vs only detecting those atoms that decay during the time interval allotted for an analysis. AMS allows dating samples containing only a few milligrams of carbon.
Raw radiocarbon ages (i.e., those not calibrated) are usually reported in "years Before Present" (BP). This is the number of radiocarbon years before 1950, based on a nominal (and assumed constant – see "calibration" below) level of carbon-14 in the atmosphere equal to the 1950 level. These raw dates are also based on a slightly-off historic value for the radiocarbon half-life. Such value is used for consistency with earlier published dates (see "Radiocarbon half-life" below). See the section on computation for the basis of the calculations.
Radiocarbon dating laboratories generally report an uncertainty for each date. For example, 3000 ± 30 BP indicates a standard deviation of 30 radiocarbon years. Traditionally this included only the statistical counting uncertainty. However, some laboratories supplied an "error multiplier" that could be multiplied by the uncertainty to account for other sources of error in the measuring process. More recently, the laboratories try to quote the overall uncertainty, which is determined from control samples of known age and verified by international intercomparison exercises.[8] In 2008, a typical uncertainty better than ±40 radiocarbon years can be expected for samples younger than 10,000 years. This, however, is only a small part of the uncertainty of the final age determination (see section Calibration below).
Samples older than the upper age-limit cannot be dated because the small number of remaining intrinsic 14C atoms will be obscured by 14C background atoms introduced into the samples while they still resided in the environment, during sample preparation, or in the detection instrument. As of 2007, the limiting age for a 1 milligram sample of graphite is about ten half-lives, approximately 60,000 years.[9] This age is derived from that of the calibration blanks used in an analysis, whose 14C content is assumed to be the result of contamination during processing (as a result of this, some facilities[9] will not report an age greater than 60,000 years for any sample).
A variety of sample processing and instrument-based constraints have been postulated to explain the upper age-limit. To examine instrument-based background activities in the AMS instrument of the W. M. Keck Carbon Cycle Accelerator Mass Spectrometry Laboratory of the University of California, a set of natural diamonds were dated. Natural diamond samples from different sources within rock formations with standard geological ages in excess of 100 Ma yielded14C apparent ages 64,920 ± 430 BP to 80,000 ± 1100 BP as reported in 2007.[10]
### Calibration
#### The need for calibration
Calibration curve for the radiocarbon dating scale. Data sources: Stuiver et al. (1998).[11] Samples with a real date more recent than AD 1950 are dated and/or tracked using the N- & S-Hemisphere graphs. See following figure.
Atmospheric 14C, New Zealand[12] and Austria.[13] The New Zealand curve is representative for the Southern Hemisphere, the Austrian curve is representative for the Northern Hemisphere. Atmospheric nuclear weapon tests almost doubled the concentration of 14C in the Northern Hemisphere.[14]
Dates may be expressed as either uncalibrated or calibrated years. A raw BP date cannot be used directly as a calendar date, because the level of atmospheric 14C has not been strictly constant during the span of time that can be radiocarbon dated. The level is affected by variations in the cosmic ray intensity which is in turn affected by variations in the Earth's magnetosphere.[15] In addition, there are substantial reservoirs of carbon in organic matter, the ocean, ocean sediments (see methane hydrate), and sedimentary rocks. Changes in the Earth's climate can affect the carbon flows between these reservoirs and the atmosphere, leading to changes in the atmosphere's 14C fraction. As the graph to the right shows, there is an overstatement of the age of the sample of nearly 1000 years in an uncalibrated dating of 7000 BP.
Aside from these changes due to natural processes, the level has also been affected by human activities. From the beginning of the industrial revolution in the 18th century to the 1950s, the fractional level of 14C decreased because of the admixture of large quantities of CO2 into the atmosphere, due to the excavated oil reserves and combustion production of fossil fuel. This decline is known as the Suess effect, and also affects the 13C isotope. However, atmospheric 14C was almost doubled during the 1950s and 1960s due to atmospheric atomic bomb tests.[16]
#### Calibration methods
The raw radiocarbon dates, in BP years, are calibrated to give calendar dates. Standard calibration curves are available, based on comparison of radiocarbon dates of samples that can be dated independently by other methods such as examination of tree growth rings (dendrochronology), deep ocean sediment cores, lake sediment varves, coral samples, and speleothems (cave deposits).
The calibration curves can vary significantly from a straight line, so comparison of uncalibrated radiocarbon dates (e.g., plotting them on a graph or subtracting dates to give elapsed time) is likely to give misleading results. There are also significant plateaus in the curves, such as the one from 11,000 to 10,000 radiocarbon years BP, which is believed to be associated with changing ocean circulation during the Younger Dryas period. Over the historical period from 0 to 10,000 years BP, the average width of the uncertainty of calibrated dates was found to be 335 years, although in well-behaved regions of the calibration curve the width decreased to about 113 years while in ill-behaved regions it increased to a maximum of 801 years. Significantly, in the ill-behaved regions of the calibration curve, increasing the precision of the measurements does not have a significant effect on increasing the accuracy of the dates.[17]
The 2004 version of the calibration curve extends back quite accurately to 26,000 years BP. Any errors in the calibration curve do not contribute more than ±16 years to the measurement error during the historic and late prehistoric periods (0–6,000 yrs BP) and no more than ±163 years over the entire 26,000 years of the curve, although its shape can reduce the accuracy as mentioned above.[18]
In late 2009, the journal Radiocarbon announced agreement on the INTCAL09 standard, which extends a more accurate calibration curve to 50,000 years.[19][20]
## History
Carbon dating was developed by American scientist Willard Libby and his team at the University of Chicago. Libby calculated the half-life of carbon-14 as 5568 ± 30 years, a figure now known as the Libby half-life. Following a conference at the University of Cambridge in 1962, a more accurate figure of 5730 ± 40 years was agreed upon based on more recent experimental data (this figure is now known as the Cambridge half-life).
The chairman of the Cambridge conference, Harry Godwin, wrote to the scientific journal Nature, recommending that the Libby half-life continue to be used for the time being, as the Cambridge figure might itself be improved by future experiments.[21] Laboratories today continue to use the Libby figure to avoid inconsistencies with earlier publications, although the Cambridge half-life is still the most accurate figure that is widely known and accepted. However, the inaccuracy of the Libby half-life is not relevant if calibration is applied: the mathematical term representing the half-life is canceled out as long as the same value is used throughout a calculation.
## Carbon exchange reservoir
Libby's original exchange reservoir hypothesis assumes that the exchange reservoir is constant all over the world. The calibration method also assumes that the temporal variation in 14C level is global, such that a small number of samples from a specific year are sufficient for calibration.[22] However, since Libby's early work was published (1950 to 1958), latitudinal and continental variations in the carbon exchange reservoir have been observed by Hessel de Vries (1958;[4] as reviewed by Lerman et al.[23]). Subsequently, methods have been developed that allow the correction of these so-called reservoir effects, including:
• When CO2 is transferred from the atmosphere to the oceans, it initially shares the 14C concentration of the atmosphere. However, turnaround times of CO2 in the ocean are similar to the half-life of 14C (making 14C also a dating tool for ocean water).[24] Marine organisms feed on this "old" carbon, and thus their radiocarbon age reflects the time of CO2 uptake by the ocean rather than the time of death of the organism. This marine reservoir effect is partly handled by a special marine calibration curve,[25] but local deviations of several hundred years exist.
• Erosion and immersion of carbonate rocks (which are generally older than 80,000 years and so shouldn't contain measurable 14C) causes an increase in 12C and 13C in the exchange reservoir, which depends on local weather conditions and can vary the ratio of carbon that living organisms incorporate. This is believed negligible for the atmosphere and atmosphere-derived carbon since most erosion will flow into the sea.[26] The atmospheric 14C concentration may differ substantially from the concentration in local water reservoirs. Eroded from CaCO3 or organic deposits, old carbon may be assimilated easily and provide diluted 14C carbon into trophic chains. So the method is less reliable for such materials as well as for samples derived from animals with such plants in their food chain.
• Volcanic eruptions eject large amount of carbon into the air, causing an increase in 12C and 13C in the exchange reservoir and can vary the exchange ratio locally. This explains the often irregular dating achieved in volcanic areas.[26]
• The earth is not affected evenly by cosmic radiation, the magnitude of the radiation depends on land altitude and earth's magnetic field strength at any given location, causing minor variation in the local 14C production. This is accounted for by having calibration curves for different locations of the globe. However this could not always be performed, as tree rings for calibration were only recoverable from certain locations in 1958.[27] The rebuttals by Münnich et al.[4] and by Barker[28] both maintain that while variations of carbon-14 exist, they are about an order of magnitude smaller than those implied by Crowe's calculations.
These effects were first confirmed when samples of wood from around the world, which all had the same age (based on tree ring analysis), showed deviations from the dendrochronological age. Calibration techniques based on tree-ring samples have contributed to increased accuracy since 1962, when they were accurate to 700 years at worst.[29]
## Speleothem studies extend 14C calibration
Speleothems (such as stalagmites) are calcium carbonate deposits that form from drips in limestone caves. Individual speleothems can be tens of thousands of years old.[30] Scientists are attempting to extend the record of atmospheric carbon-14 by measuring radiocarbon in speleothems which have been independently dated using uranium-thorium dating.[31][32] These results are improving the calibration for the radiocarbon technique and extending its usefulness to 45,000 years into the past.[33] Initial results from a cave in the Bahamas suggested a peak in the amount of carbon-14 which was twice as high as modern levels.[34] A recent study does not reproduce this extreme shift and suggests that analytical problems may have produced the anomalous result.[32]
## References
1. Plastino, W.; Kaihola, L.; Bartolomei, P.; Bella, F. (2001). "Cosmic Background Reduction In The Radiocarbon Measurement By Scintillation Spectrometry At The Underground Laboratory Of Gran Sasso". Radiocarbon 43 (2A): 157–161.
2. Arnold, J. R.; Libby, W. F. (1949). "Age Determinations by Radiocarbon Content: Checks with Samples of Known Age". Science 110 (2869): 678–680. Bibcode 1949Sci...110..678A. doi:10.1126/science.110.2869.678. PMID 15407879.
3. Willard Frank Libby
4. Münnich KO, Östlund HG, de Vries H (1958). "Carbon-14 Activity during the past 5,000 Years". Nature 182 (4647): 1432–3. Bibcode 1958Natur.182.1432M. doi:10.1038/1821432a0.
5. "The shape of a distribution of calibrated 14C dates displays spurious peaks and troughs, brought about by changes in the slope of the calibration curve interacting with the spreading effect of the stochastic distribution of counting errors"
6. doi:10.1111/j.1475-4754.2008.00394.x
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7. http://hypertextbook.com/facts/1997/MargaretKong.shtml
8. Scott, EM (2003). "The Fourth International Radiocarbon Intercomparison (FIRI).". Radiocarbon 45: 135–285.
9. Taylor RE, Southon J (2007). "Use of natural diamonds to monitor 14C AMS instrument backgrounds". Nuclear Instruments and Methods in Physics Research B 259: 282–28. Bibcode 2007NIMPB.259..282T. doi:10.1016/j.nimb.2007.01.239.
10. Stuiver M, Reimer PJ, Braziunas TF (1998). "High-precision radiocarbon age calibration for terrestrial and marine samples". Radiocarbon 40: 1127–51.
11. 14CO2 record from Vermunt". Carbon Dioxide Information Analysis Center. Retrieved 1 May 2008.
12. "Radiocarbon dating". Utrecht University. Retrieved 1 May 2008.
13. Kudela K. and Bobik P. (2004). "Long-Term Variations of Geomagnetic Rigidity Cutoffs". Solar Physics 224: 423–431. Bibcode 2004SoPh..224..423K. doi:10.1007/s11207-005-6498-9.
14. Reimer, Paula J.; Brown, Thomas A.; Reimer, Ron W. (2004). "Discussion: Reporting and Calibration of Post-Bomb 14C Data". Radiocarbon 46 (3): 1299–1304
15. These results were obtained from a Monte Carlo analysis calibrating simulated measurements of varying precision using the 1993 version of the calibration curve. The width of the uncertainty represents a 2σ uncertainty (that is, a likelihood of 95% that the date appears between these limits). Niklaus TR, Bonani G, Suter M, Wölfli W (1994). "Systematic investigation of uncertainties in radiocarbon dating due to fluctuations in the calibration curve". Nuclear Instruments and Methods in Physics Research 92: 194–200. Bibcode 1994NIMPB..92..194N. doi:10.1016/0168-583X(94)96004-6.
16. Reimer Paula J et al. (2004). "INTCAL04 Terrestrial Radiocarbon Age Calibration, 0–26 Cal Kyr BP". Radiocarbon 46 (3): 1029–1058. A web interface is here.
17. Reimer, P.J.; et. al. (2009). "IntCal09 and Marine09 Radiocarbon Age Calibration Curves, 0–50,000 Years cal BP". Radiocarbon 51 (4): 1111–1150.
18. Balter, Michael (15 Jan 2010). "Radiocarbon Daters Tune Up Their Time Machine". ScienceNOW Daily News.
19. Godwin, H. (1962). "Half-life of Radiocarbon". Nature 195 (4845): 984. Bibcode 1962Natur.195..984G. doi:10.1038/195984a0.
20. Libby WF (1955). Radiocarbon dating (2nd ed.). Chicago: University of Chicago Press.
21. Lerman, J. C.; Mook, W. G.; Vogel, J. C.; de Waard, H. (1969). "Carbon-14 in Patagonian Tree Rings". Science 165 (3898): 1123–1125. Bibcode 1969Sci...165.1123L. doi:10.1126/science.165.3898.1123. PMID 17779805.
22. McNichol AP, Schneider RJ, von Reden KF, Gagnon AR, Elder KL, NOSAMS, Key RM, Quay PD (October 2000). "Ten years after - The WOCE AMS radiocarbon program". Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms 172 (1–4): 479–84. Bibcode 2000NIMPB.172..479M. doi:10.1016/S0168-583X(00)00093-8.
23. Stuiver M, Braziunas TF (1993). "Modelling atmospheric 14C influences and 14C ages of marine samples to 10,000 BC". Radiocarbon 35 (1): 137.
24. Kolchin BA, Shez YA (1972). Absolute archaeological datings and their problems. Moscow: Nauka.
25. Crowe C (1958). "Carbon-14 activity during the past 5000 years". Nature 182 (4633): 470–1. Bibcode 1958Natur.182..470C. doi:10.1038/182470a0.
26. Barker H (1958). "Carbon-14 Activity during the past 5,000 Years". Nature 182 (4647): 1433. Bibcode 1958Natur.182.1433B. doi:10.1038/1821433a0.
27. Libby WF (1962). "Radiocarbon; an atomic clock". Annual Science and Humanity Journal.
28. Wang YJ; Cheng, H; Edwards, RL; An, ZS; Wu, JY; Shen, CC; Dorale, JA (2001). "A High-Resolution Absolute-Dated Late Pleistocene Monsoon Record from Hulu Cave, China". Science 294 (5550): 2345–2348. Bibcode 2001Sci...294.2345W. doi:10.1126/science.1064618. PMID 11743199.
29. Beck JW; Richards, DA; Edwards, RL; Silverman, BW; Smart, PL; Donahue, DJ; Hererra-Osterheld, S; Burr, GS et al. (2001). "Extremely large variations of atmospheric C-14 concentration during the last glacial period". Science 292 (5526): 2453–2458. Bibcode 2001Sci...292.2453B. doi:10.1126/science.1056649. PMID 11349137.
30. Hoffmann DL; Beck, J. Warren; Richards, David A.; Smart, Peter L.; Singarayer, Joy S.; Ketchmark, Tricia; Hawkesworth, Chris J. (2010). "Towards radiocarbon calibration beyond 28 ka using speleothems from the Bahamas". Earth and Planetary Science Letters 289: 1–10. Bibcode 2010E&PSL.289....1H. doi:10.1016/j.epsl.2009.10.004.
31. Jensen MN (2001). "Peering deep into the past". University of Arizona, Department of Physics.
32. Pennicott K (10 May 2001). "Carbon clock could show the wrong time". PhysicsWeb. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 12, "math_score": 0.8511555790901184, "perplexity": 3699.438526391588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424559.25/warc/CC-MAIN-20170723122722-20170723142722-00508.warc.gz"} |
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