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Archimedes principle and buoyant force

https://www.youtube.com/watch?v=vzID7ds600c

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https://www.youtube.com/api/timedtext?v=vzID7ds600c&ei=F2WUZcPdGdfoxN8PvMOl4AM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249223&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2A08D99C677DAE634860B1E2411F72A6FB53E776.9F54C322AA0A4E971893FE43B54427FA5378F6D1&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.620 --> 00:00:03.080 Let's say we have a cup of water. 00:00:03.080 --> 00:00:04.330 Let me draw the cup. 00:00:06.800 --> 00:00:11.530 This is one side of the cup, this is the bottom of the cup, 00:00:11.530 --> 00:00:15.130 and this is the other side of the cup. 00:00:15.130 --> 00:00:16.325 Let me say that it's some liquid. 00:00:16.325 --> 00:00:19.920 It doesn't have to be water, but some arbitrary liquid. 00:00:19.920 --> 00:00:21.260 It could be water. 00:00:21.260 --> 00:00:23.850 That's the surface of it. 00:00:23.850 --> 00:00:26.570 We've already learned that the pressure at any point within 00:00:26.570 --> 00:00:30.970 this liquid is dependent on how deep 00:00:30.970 --> 00:00:32.580 we go into the liquid. 00:00:32.580 --> 00:00:35.450 One point I want to make before we move on, and I 00:00:35.450 --> 00:00:39.730 touched on this a little bit before, is that the pressure 00:00:39.730 --> 00:00:43.460 at some point isn't just acting downwards, or it isn't 00:00:43.460 --> 00:00:44.480 just acting in one direction. 00:00:44.480 --> 00:00:46.975 It's acting in all directions on that point. 00:00:46.975 --> 00:00:50.620 So although how far we go down determines how much pressure 00:00:50.620 --> 00:00:53.000 there is, the pressure is actually acting in all 00:00:53.000 --> 00:00:54.850 directions, including up. 00:00:54.850 --> 00:00:58.330 The reason why that makes sense is because I'm assuming 00:00:58.330 --> 00:01:02.950 that this is a static system, or that the fluids in this 00:01:02.950 --> 00:01:05.500 liquid are stationary, or you even could imagine an object 00:01:05.500 --> 00:01:07.360 down here, and it's stationary. 00:01:07.360 --> 00:01:09.800 The fact that it's stationary tells us that the pressure in 00:01:09.800 --> 00:01:12.000 every direction must be equal. 00:01:12.000 --> 00:01:13.970 Let's think about a molecule of water. 00:01:13.970 --> 00:01:15.870 A molecule of water, let's say it's roughly a sphere. 00:01:20.920 --> 00:01:24.440 If the pressure were different in one direction or if the 00:01:24.440 --> 00:01:28.070 pressure down were greater than the pressure up, then the 00:01:28.070 --> 00:01:30.310 object would start accelerating downwards, 00:01:30.310 --> 00:01:32.950 because its surface area pointing upwards is the same 00:01:32.950 --> 00:01:36.240 as the surface area pointing downwards, so the force 00:01:36.240 --> 00:01:37.410 upwards would be more. 00:01:37.410 --> 00:01:39.550 It would start accelerating downwards. 00:01:39.550 --> 00:01:43.520 Even though the pressure is a function of how far down we 00:01:43.520 --> 00:01:45.900 go, at that point, the pressure is 00:01:45.900 --> 00:01:48.150 acting in every direction. 00:01:48.150 --> 00:01:51.790 Let's remember that, and now let's keep that in mind to 00:01:51.790 --> 00:01:54.530 learn a little bit about Archimedes' principle. 00:01:54.530 --> 00:02:01.840 Let's say I submerge a cube into this liquid, and let's 00:02:01.840 --> 00:02:13.370 say this cube has dimensions d, so every side is d. 00:02:17.180 --> 00:02:19.560 What I want to do is I want to figure out if there's any 00:02:19.560 --> 00:02:22.870 force or what is the net force acting on this 00:02:22.870 --> 00:02:25.070 cube due to the water? 00:02:25.070 --> 00:02:27.770 Let's think about what the pressure on this cube is at 00:02:27.770 --> 00:02:29.980 different points. 00:02:29.980 --> 00:02:32.760 At the depths along the side of the cube, we know that the 00:02:32.760 --> 00:02:35.310 pressures are equal, because we know at this depth right 00:02:35.310 --> 00:02:38.350 here, the pressure is going to be the same as at that depth, 00:02:38.350 --> 00:02:40.110 and they're going to offset each other, and so these are 00:02:40.110 --> 00:02:41.900 going to be the same. 00:02:41.900 --> 00:02:44.080 But one thing we do know, just based on the fact that 00:02:44.080 --> 00:02:47.480 pressure is a function of depth, is that at this point 00:02:47.480 --> 00:02:50.510 the pressure is going to be higher-- I don't know how much 00:02:50.510 --> 00:02:54.010 higher-- than at this point, because this point is deeper 00:02:54.010 --> 00:02:55.330 into the water. 00:02:55.330 --> 00:02:59.275 Let's call this P1. 00:02:59.275 --> 00:03:03.210 Let's call that pressure on top, PT, and let's call this 00:03:03.210 --> 00:03:06.260 point down here PD. 00:03:06.260 --> 00:03:07.910 No, pressure on the bottom, PB. 00:03:11.330 --> 00:03:17.220 What's going to be the net force on this cube? 00:03:17.220 --> 00:03:22.330 The net force-- let's call that F sub N-- is going to be 00:03:22.330 --> 00:03:27.420 equal to the force acting upwards on this object. 00:03:27.420 --> 00:03:29.280 What's the force acting upwards on the object? 00:03:29.280 --> 00:03:35.830 It's going to be this pressure at the bottom of the object 00:03:35.830 --> 00:03:38.730 times the surface area at the bottom of the object. 00:03:38.730 --> 00:03:41.530 What's the surface area at the bottom of the object? 00:03:41.530 --> 00:03:42.930 That's just d squared. 00:03:42.930 --> 00:03:45.770 Any surface of a cube is d squared, so the bottom is 00:03:45.770 --> 00:03:54.440 going to be d squared minus-- I'm doing this because I 00:03:54.440 --> 00:03:56.850 actually know that the pressure down here is higher 00:03:56.850 --> 00:03:58.630 than the pressure here, so this is going to be a larger 00:03:58.630 --> 00:04:01.090 quantity, and that the net force is actually going to be 00:04:01.090 --> 00:04:04.430 upwards, so that's why I can do the minus confidently up 00:04:04.430 --> 00:04:07.570 here-- the pressure at the top. 00:04:07.570 --> 00:04:09.590 What's the force at the top? 00:04:09.590 --> 00:04:14.620 The force at the top is going to be the pressure on the top 00:04:14.620 --> 00:04:16.750 times the surface area of the top of the cube, 00:04:16.750 --> 00:04:20.700 right, times d squared. 00:04:20.700 --> 00:04:24.540 We can even separate out the d squared already at that point, 00:04:24.540 --> 00:04:30.980 so the net force is equal to the pressure of the bottom 00:04:30.980 --> 00:04:33.920 minus the pressure of the top, or the difference in pressure 00:04:33.920 --> 00:04:37.150 times the surface area of either the top or the bottom 00:04:37.150 --> 00:04:39.840 or really any of the sides of the cube. 00:04:39.840 --> 00:04:41.460 Let's see if we can figure what these are. 00:04:41.460 --> 00:04:45.770 Let's say the cube is submerged h units or h meters 00:04:45.770 --> 00:04:48.560 into the water. 00:04:48.560 --> 00:04:51.780 So what's the pressure at the top? 00:04:51.780 --> 00:04:55.600 The pressure at the top is going to be equal to the 00:04:55.600 --> 00:04:58.240 density of the liquid-- I keep saying water, but it could be 00:04:58.240 --> 00:05:02.610 any liquid-- times how far down we are. 00:05:02.610 --> 00:05:08.080 So we're h units down, or maybe h meters, times gravity. 00:05:08.080 --> 00:05:10.700 And what's the pressure the bottom? 00:05:10.700 --> 00:05:15.060 The pressure at the bottom similarly would be the density 00:05:15.060 --> 00:05:18.600 of the liquid times the depth, so what's the depth? 00:05:18.600 --> 00:05:21.270 It would be this h and then we're another d down. 00:05:24.226 --> 00:05:29.480 It's h plus d-- that's our total depth-- times gravity. 00:05:29.480 --> 00:05:31.946 Let's just substitute both of those back into our net force. 00:05:31.946 --> 00:05:35.750 Let me switch colors to keep from getting monotonous. 00:05:35.750 --> 00:05:40.360 I get the net force is equal to the pressure at the bottom, 00:05:40.360 --> 00:05:42.120 which is this. 00:05:42.120 --> 00:05:52.730 Let's just multiply it out, so we get p times h times g plus 00:05:52.730 --> 00:05:54.480 d times p times g. 00:05:58.580 --> 00:06:00.910 I just distributed this out, multiplied this out. 00:06:00.910 --> 00:06:06.230 That's the pressure at the bottom, then minus the 00:06:06.230 --> 00:06:13.950 pressure at the top, minus phg, and then we learned it's 00:06:13.950 --> 00:06:17.860 all of that times d squared. 00:06:17.860 --> 00:06:20.350 Immediately, we see something cancels out. 00:06:20.350 --> 00:06:23.905 phg, phg subtract. 00:06:23.905 --> 00:06:25.950 It cancels out, so we're just left with-- 00:06:25.950 --> 00:06:27.100 what's the net force? 00:06:27.100 --> 00:06:36.000 The net force is equal to dpg times d squared, or that 00:06:36.000 --> 00:06:42.490 equals d cubed times the density of the 00:06:42.490 --> 00:06:45.740 liquid times gravity. 00:06:45.740 --> 00:06:50.000 Let me ask you a question: What is d cubed? 00:06:50.000 --> 00:06:51.830 d cubed is the volume of this cube. 00:06:51.830 --> 00:06:54.780 And what else is it? 00:06:54.780 --> 00:06:56.800 It's also the volume of the water displaced. 00:06:56.800 --> 00:06:59.680 If I stick this cube into the water, and the cube isn't 00:06:59.680 --> 00:07:03.050 shrinking or anything-- you can even imagine it being 00:07:03.050 --> 00:07:05.860 empty, but it doesn't have to be empty-- but that amount of 00:07:05.860 --> 00:07:08.640 water has to be moved out of the way in order for 00:07:08.640 --> 00:07:11.040 that cube to go in. 00:07:11.040 --> 00:07:15.360 This is the volume of the water displaced. 00:07:15.360 --> 00:07:16.680 It's also the volume of the cube. 00:07:24.200 --> 00:07:28.010 This is the density-- I keep saying water, but it could be 00:07:28.010 --> 00:07:30.770 any liquid-- of the liquid. 00:07:30.770 --> 00:07:32.560 This is the gravity. 00:07:32.560 --> 00:07:33.190 So what is this? 00:07:33.190 --> 00:07:38.800 Volume times density is the mass of the liquid displaced, 00:07:38.800 --> 00:07:42.240 so the net force is also equal to the 00:07:42.240 --> 00:07:47.120 mass of liquid displaced. 00:07:47.120 --> 00:07:52.260 Let's just say mass times gravity, or we could say that 00:07:52.260 --> 00:07:55.660 the net force acting on this object is-- what's the mass of 00:07:55.660 --> 00:07:57.120 the liquid displaced times gravity? 00:07:57.120 --> 00:08:05.750 That's just the weight of liquid displaced. 00:08:05.750 --> 00:08:08.030 That's a pretty interesting thing. 00:08:08.030 --> 00:08:12.990 If I submerge anything, the net force acting upwards on 00:08:12.990 --> 00:08:15.630 it, or the amount that I'm lighter by, is equal to the 00:08:15.630 --> 00:08:18.370 weight of the water being displaced. 00:08:18.370 --> 00:08:20.960 That's actually called Archimedes' principle. 00:08:20.960 --> 00:08:24.260 That net upward force due to the fact that there's more 00:08:24.260 --> 00:08:27.090 pressure on the bottom than there is on the top, that's 00:08:27.090 --> 00:08:28.280 called the buoyant force. 00:08:28.280 --> 00:08:30.950 That's what makes things float. 00:08:30.950 --> 00:08:35.280 I'll leave you there to just to ponder that, and we'll use 00:08:35.280 --> 00:08:37.919 this concept in the next couple of videos to actually 00:08:37.919 --> 00:08:40.789 solve some problems. I'll see you soon.

Pressure at a depth in a fluid

https://www.youtube.com/watch?v=5EWjlpc0S00

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https://www.youtube.com/api/timedtext?v=5EWjlpc0S00&ei=GGWUZdPEJICFp-oPkISwkAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249224&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=0D9B8B4B5AD6C51D61237CBE6EF12A85201B9173.BF77DB14C60448AECFDE6DE741F46FF0E52F6B5E&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.860 --> 00:00:04.740 In the last video, we showed that any external pressure on 00:00:04.740 --> 00:00:08.490 a liquid in a container is distributed 00:00:08.490 --> 00:00:10.000 evenly through the liquid. 00:00:10.000 --> 00:00:13.340 But that only applied to-- and that was called Pascal's 00:00:13.340 --> 00:00:16.950 principle-- external pressure. 00:00:16.950 --> 00:00:19.670 Let's think a little bit about what the internal pressure is 00:00:19.670 --> 00:00:20.620 within a liquid. 00:00:20.620 --> 00:00:23.760 We're all familiar, I think, with the notion of the deeper 00:00:23.760 --> 00:00:26.930 you go into a fluid or the deeper you dive into the 00:00:26.930 --> 00:00:29.350 ocean, the higher the pressure is on you. 00:00:29.350 --> 00:00:32.619 Let's see if we can think about that a little bit more 00:00:32.619 --> 00:00:35.420 analytically, and get a framework for what the 00:00:35.420 --> 00:00:38.270 pressure is at any depth under the water, or 00:00:38.270 --> 00:00:39.730 really in any fluid. 00:00:39.730 --> 00:00:44.680 Here I've drawn a cylinder, and in that cylinder I have 00:00:44.680 --> 00:00:47.620 some fluid-- let's not assume that it's water, but some 00:00:47.620 --> 00:00:50.160 fluid, and that's the blue stuff. 00:00:50.160 --> 00:00:52.250 I'm also assuming that I'm doing this on a planet that 00:00:52.250 --> 00:00:55.670 has the same mass as Earth, but it has no atmosphere, so 00:00:55.670 --> 00:00:57.630 there's a vacuum up here-- there's no air. 00:00:57.630 --> 00:00:59.590 We'll see later that the atmosphere actually adds 00:00:59.590 --> 00:01:00.940 pressure on top of this. 00:01:00.940 --> 00:01:04.800 Let's assume that there's no air, but it's on a planet of 00:01:04.800 --> 00:01:07.770 the same mass, so the gravity is the same. 00:01:07.770 --> 00:01:10.530 There is gravity, so the liquid will fill this 00:01:10.530 --> 00:01:12.810 container on the bottom part of it. 00:01:12.810 --> 00:01:15.840 Also, the gravitational constant would be the same as 00:01:15.840 --> 00:01:19.040 Earth, so we can imagine this is a horrible situation where 00:01:19.040 --> 00:01:22.020 Earth has lost its magnetic field and the solar winds have 00:01:22.020 --> 00:01:23.820 gotten rid of Earth's atmosphere. 00:01:23.820 --> 00:01:26.300 That's very negative, so we won't think about that, but 00:01:26.300 --> 00:01:28.000 anyway-- let's go back to the problem. 00:01:28.000 --> 00:01:35.640 Let's say within this cylinder, I have a thin piece 00:01:35.640 --> 00:01:40.340 of foil or something that takes up the entire 00:01:40.340 --> 00:01:43.000 cross-sectional area of the cylinder. 00:01:43.000 --> 00:01:46.290 I did that just because I want that to be an indicator of 00:01:46.290 --> 00:01:49.150 whether the fluid is moving up or down or not. 00:01:49.150 --> 00:01:52.760 Let's say I have that in the fluid at some depth, h, and 00:01:52.760 --> 00:01:56.650 since the fluid is completely static-- nothing's moving-- 00:01:56.650 --> 00:02:00.335 that object that's floating right at that level, at a 00:02:00.335 --> 00:02:02.440 depth of h, will also be static. 00:02:02.440 --> 00:02:05.060 In order for something to be static, where it's not 00:02:05.060 --> 00:02:07.490 moving-- what do we know about it? 00:02:07.490 --> 00:02:11.060 We know that the net forces on it must be zero-- in fact, 00:02:11.060 --> 00:02:13.050 that tells that it's not accelerating. 00:02:13.050 --> 00:02:15.470 Obviously, if something's not moving, it has a velocity of 00:02:15.470 --> 00:02:18.170 zero, and that's a constant velocity-- it's not 00:02:18.170 --> 00:02:20.000 accelerating in any direction, and so its net 00:02:20.000 --> 00:02:22.830 forces must be zero. 00:02:22.830 --> 00:02:32.810 This force down must be equal to the force up. 00:02:32.810 --> 00:02:38.050 So what is the force down acting on this cylinder? 00:02:38.050 --> 00:02:41.860 It's going to be the weight of the water above it, because 00:02:41.860 --> 00:02:45.780 we're in a gravitational environment, and so this water 00:02:45.780 --> 00:02:47.030 has some mass. 00:02:49.310 --> 00:02:53.470 Whatever that mass is, times the gravitational constant, 00:02:53.470 --> 00:02:56.170 will equal the force down. 00:02:56.170 --> 00:02:57.240 Let's figure out what that is. 00:02:57.240 --> 00:03:00.960 The force down, which is the same thing is the force up, is 00:03:00.960 --> 00:03:12.860 going to equal the mass of this water, times the 00:03:12.860 --> 00:03:15.490 gravitational constant. 00:03:15.490 --> 00:03:19.020 Actually, I shouldn't say water-- let me change this, 00:03:19.020 --> 00:03:20.760 because I said that this is going to be some random 00:03:20.760 --> 00:03:23.140 liquid, and the mass is a liquid. 00:03:23.140 --> 00:03:27.360 The force down is going to be equal to the mass of the 00:03:27.360 --> 00:03:31.500 liquid times gravity. 00:03:31.500 --> 00:03:33.220 What is that mass of the liquid? 00:03:33.220 --> 00:03:35.565 Well, now I'll introduce you to a concept called density, 00:03:35.565 --> 00:03:38.460 and I think you understand what density is-- it's how 00:03:38.460 --> 00:03:41.150 much there is of something in a given amount of volume, or 00:03:41.150 --> 00:03:42.290 how much mass per volume. 00:03:42.290 --> 00:03:44.560 That's the definition of density. 00:03:44.560 --> 00:03:47.670 The letter people use for density is rho-- let me do 00:03:47.670 --> 00:03:50.440 that in a different color down here. 00:03:50.440 --> 00:04:01.476 rho, which looks like a p to me, equals mass per volume, 00:04:01.476 --> 00:04:03.940 and that's the density. 00:04:03.940 --> 00:04:12.380 The units are kilograms per meter cubed-- that is density. 00:04:12.380 --> 00:04:15.170 I think you might have an intuition that if I have a 00:04:15.170 --> 00:04:24.930 cubic meter of lead-- lead is more dense than marshmallows. 00:04:24.930 --> 00:04:28.690 Because of that, if I have a cubic meter of lead, it will 00:04:28.690 --> 00:04:32.320 have a lot more mass, and in a gravitational field, weigh a 00:04:32.320 --> 00:04:36.100 lot more than a cubic meter of marshmallows. 00:04:36.100 --> 00:04:39.050 Of course, there's always that trick people say, what weighs 00:04:39.050 --> 00:04:42.740 more-- a pound of feathers, or a pound of lead? 00:04:42.740 --> 00:04:46.600 Those, obviously, weigh the same-- the key is the volume. 00:04:46.600 --> 00:04:50.010 A cubic meter of lead is going to weigh a lot more than a 00:04:50.010 --> 00:04:52.180 cubic meter of feathers. 00:04:52.180 --> 00:04:54.890 Making sure that we now know what the density is, let's go 00:04:54.890 --> 00:04:57.360 back to what we were doing before. 00:04:57.360 --> 00:05:00.330 We said that the downward force is equal to the mass of 00:05:00.330 --> 00:05:04.300 the liquid times the gravitational force, and so 00:05:04.300 --> 00:05:06.390 what is the mass of the liquid? 00:05:06.390 --> 00:05:09.250 We could use this formula right here-- density is equal 00:05:09.250 --> 00:05:12.490 to mass times volume, so we could also say that mass is 00:05:12.490 --> 00:05:15.660 equal to density times volume. 00:05:15.660 --> 00:05:18.130 I just multiply both sides of this equation times volume. 00:05:21.240 --> 00:05:26.570 In this situation, force down is equal to-- let's substitute 00:05:26.570 --> 00:05:28.430 this with this. 00:05:28.430 --> 00:05:31.300 The mass of the liquid is equal to the density of the 00:05:31.300 --> 00:05:35.910 liquid times the volume of the liquid-- I could get rid of 00:05:35.910 --> 00:05:39.110 these l's-- times gravity. 00:05:39.110 --> 00:05:43.110 What's the volume of the liquid? 00:05:43.110 --> 00:05:45.150 The volume of the liquid is going to be the 00:05:45.150 --> 00:05:49.685 cross-sectional area of the cylinder times the height. 00:05:49.685 --> 00:05:53.110 So let's call this cross-sectional area A. 00:05:53.110 --> 00:05:58.290 A for area-- that's the area of the cylinder or the foil 00:05:58.290 --> 00:06:01.290 that's floating within the water. 00:06:01.290 --> 00:06:05.820 We could write down that the downward force is equal to the 00:06:05.820 --> 00:06:09.460 density of the fluid-- I'll stop writing the l or f, or 00:06:09.460 --> 00:06:13.800 whatever I was doing there-- times the 00:06:13.800 --> 00:06:15.310 volume of the liquid. 00:06:15.310 --> 00:06:19.870 The volume of the liquid is just the height times the area 00:06:19.870 --> 00:06:21.390 of the liquid. 00:06:21.390 --> 00:06:27.110 So that is just times the height times the area and then 00:06:27.110 --> 00:06:28.360 times gravity. 00:06:35.920 --> 00:06:39.560 We've now figured out if we knew the density, this height, 00:06:39.560 --> 00:06:42.840 the cross-sectional area, and the gravitational constant, we 00:06:42.840 --> 00:06:44.100 would know the force coming down. 00:06:44.100 --> 00:06:46.800 That's kind of vaguely interesting, but let's try to 00:06:46.800 --> 00:06:48.610 figure out what the pressure is, because that's what 00:06:48.610 --> 00:06:50.180 started this whole discussion. 00:06:50.180 --> 00:06:54.720 What is the pressure when you go to deep parts of the ocean? 00:06:54.720 --> 00:06:59.903 This is the force-- what is the pressure on this foil that 00:06:59.903 --> 00:07:01.190 I have floating? 00:07:01.190 --> 00:07:05.060 It's the force divided by the area of pressure on this foil. 00:07:05.060 --> 00:07:08.400 So I would take the force and divide it by the area, which 00:07:08.400 --> 00:07:11.780 is the same thing as A, so let's do that. 00:07:11.780 --> 00:07:15.470 Let's divide both sides of this equation by area, so the 00:07:15.470 --> 00:07:19.850 pressure coming down-- so that's P sub d. 00:07:25.610 --> 00:07:28.900 The downward pressure at that point is going to be equal 00:07:28.900 --> 00:07:31.460 to-- keep in mind, that's going to be the same thing as 00:07:31.460 --> 00:07:33.950 the upward pressure, because the upward force is the same. 00:07:33.950 --> 00:07:36.680 The area of whether you're going upwards or downwards is 00:07:36.680 --> 00:07:37.380 going to be the same thing. 00:07:37.380 --> 00:07:40.130 The downward pressure is going to be equal to the downward 00:07:40.130 --> 00:07:44.430 force divided by area, which is going to be equal to this 00:07:44.430 --> 00:07:46.260 expression divided by area. 00:07:46.260 --> 00:07:49.630 Essentially, we can just get rid of the area here, so it 00:07:49.630 --> 00:07:57.550 equals PhAg divided by A-- we get rid of the A's in both 00:07:57.550 --> 00:08:02.580 situations-- so the downward pressure is equal to the 00:08:02.580 --> 00:08:07.480 density of the fluid, times the depth of the fluid, or the 00:08:07.480 --> 00:08:11.040 height of the fluid above it, times the gravitational 00:08:11.040 --> 00:08:13.400 constant Phg. 00:08:13.400 --> 00:08:15.430 As I said, the downward pressure is equal to the 00:08:15.430 --> 00:08:16.910 upward pressure-- how do we know that? 00:08:16.910 --> 00:08:19.360 Because we knew that the upward force is the same as 00:08:19.360 --> 00:08:20.230 the downward force. 00:08:20.230 --> 00:08:26.170 If the upward force were less, this little piece of foil 00:08:26.170 --> 00:08:28.480 would actually accelerate downwards. 00:08:28.480 --> 00:08:31.050 The fact that it's static-- it's in one place-- lets us 00:08:31.050 --> 00:08:34.250 know that the upward force is equal to the downward force, 00:08:34.250 --> 00:08:35.390 so the upward pressure is equal to 00:08:35.390 --> 00:08:37.520 the downward pressure. 00:08:37.520 --> 00:08:41.640 Let's use that in an example. 00:08:41.640 --> 00:08:46.650 If I were on the same planet, and this is water, and so the 00:08:46.650 --> 00:08:53.330 density of water-- and this is something good to memorize-- 00:08:53.330 --> 00:08:58.805 is 1,000 kilograms per meter cubed. 00:09:02.710 --> 00:09:06.400 Let's say that we have no atmosphere, but I were to go 00:09:06.400 --> 00:09:09.230 10 meters under the water-- roughly 30 00:09:09.230 --> 00:09:10.000 feet under the water. 00:09:10.000 --> 00:09:11.980 What would be the pressure on me? 00:09:11.980 --> 00:09:16.840 My pressure would be the density of water, which is 00:09:16.840 --> 00:09:19.760 1,000 kilograms per meter cubed-- make sure your units 00:09:19.760 --> 00:09:21.750 are right, and I'm running out of space, so I don't have the 00:09:21.750 --> 00:09:25.110 units-- times the height, 10 meters, times the 00:09:25.110 --> 00:09:28.970 gravitational acceleration, 9.8 meters per second squared. 00:09:28.970 --> 00:09:30.440 It's a good exercise for you to make sure 00:09:30.440 --> 00:09:32.210 the units work out. 00:09:32.210 --> 00:09:36.000 It's 10,000 times 9.8, so the pressure is going to be equal 00:09:36.000 --> 00:09:41.200 to 98,000 pascals. 00:09:41.200 --> 00:09:42.500 This actually isn't that much-- it just 00:09:42.500 --> 00:09:43.960 sounds like a lot. 00:09:43.960 --> 00:09:47.100 We'll actually see that this is almost one atmosphere, 00:09:47.100 --> 00:09:50.450 which is the pressure at sea level in France, I think. 00:09:50.450 --> 00:09:53.380 Anyway, I'll see you in the next video.

Finding height of fluid in a barometer

https://www.youtube.com/watch?v=i6gz9VFyYks

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https://www.youtube.com/api/timedtext?v=i6gz9VFyYks&ei=GWWUZeL-MN-up-oPisy2wAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249225&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8173BC1E455FEDE4F248E4B5B8297892D7DE7150.CFF756DB528FCE1786CBEA28338C116F2E547C33&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:01.220 --> 00:00:04.070 In the last video, we learned that the pressure at some 00:00:04.070 --> 00:00:10.120 depth in a fluid is equal to the density of the fluid times 00:00:10.120 --> 00:00:13.670 how deep we are in the fluid, or how high is the column of 00:00:13.670 --> 00:00:16.990 fluid above us times gravity. 00:00:16.990 --> 00:00:19.870 Let's see if we can use that to solve a fairly typical 00:00:19.870 --> 00:00:22.750 problem that you'll see in your physics class, or even on 00:00:22.750 --> 00:00:24.660 an AP physics test. 00:00:24.660 --> 00:00:26.000 Let's say that I have a bowl. 00:00:37.900 --> 00:00:41.740 And in that bowl, I have mercury, and then I also have 00:00:41.740 --> 00:00:45.000 this kind of inverted test tube that I stick in the 00:00:45.000 --> 00:00:47.480 middle of-- this is the side view of the bowl, and I'll 00:00:47.480 --> 00:00:48.950 draw everything shortly. 00:00:48.950 --> 00:00:50.960 Let's say my test tube looks something like this. 00:00:58.320 --> 00:01:00.090 Let's say I have no air in this test tube-- there's a 00:01:00.090 --> 00:01:03.120 vacuum here-- but the outside of the bowl, this whole area 00:01:03.120 --> 00:01:04.950 out here, this is exposed to the air. 00:01:04.950 --> 00:01:08.530 We are actually on Earth, or actually in Paris, France, at 00:01:08.530 --> 00:01:14.550 sea level, because that's what an atmosphere is defined as-- 00:01:14.550 --> 00:01:15.940 the atmospheric pressure. 00:01:15.940 --> 00:01:17.970 Essentially, the way you could think about it-- the weight of 00:01:17.970 --> 00:01:21.830 all of the air above us is pushing down on the surface of 00:01:21.830 --> 00:01:24.270 this bowl at one atmosphere. 00:01:24.270 --> 00:01:27.530 An atmosphere is just the pressure of all of the air 00:01:27.530 --> 00:01:31.000 above you at sea level in Paris, France. 00:01:31.000 --> 00:01:32.650 And in the bowl, I have mercury. 00:02:05.380 --> 00:02:08.536 Let's say that that mercury-- there's no air in here, and it 00:02:08.536 --> 00:02:11.000 is actually going to go up this column a little bit. 00:02:11.000 --> 00:02:14.200 We're going to do the math as far as-- one, we'll see why 00:02:14.200 --> 00:02:16.830 it's going up, and then we'll do the math to figure out how 00:02:16.830 --> 00:02:19.110 high up does it go. 00:02:19.110 --> 00:02:24.030 Say the mercury goes up some distance-- this 00:02:24.030 --> 00:02:25.280 is all still mercury. 00:02:29.498 --> 00:02:31.820 And this is actually how a barometer works; this is 00:02:31.820 --> 00:02:34.860 something that measures pressure. 00:02:34.860 --> 00:02:39.730 Over here at this part, above the mercury, but still within 00:02:39.730 --> 00:02:45.040 our little test tube, we have a vacuum-- there is no air. 00:02:45.040 --> 00:02:47.680 Vacuum is one of my favorite words, because it has 00:02:47.680 --> 00:02:48.930 two u's in a row. 00:02:52.400 --> 00:02:55.740 We have this set up, and so my question to you is-- how high 00:02:55.740 --> 00:02:58.430 is this column of mercury going to go? 00:03:02.210 --> 00:03:05.270 First of all, let's just have the intuition as to why this 00:03:05.270 --> 00:03:07.660 thing is going up to begin with. 00:03:07.660 --> 00:03:09.780 We have all this pressure from all of the air above us-- I 00:03:09.780 --> 00:03:12.320 know it's a little un-intuitive for us, because 00:03:12.320 --> 00:03:15.140 we're used to all of that pressure on our shoulders all 00:03:15.140 --> 00:03:18.520 of the time, so we don't really imagine it, but there 00:03:18.520 --> 00:03:21.340 is literally the weight of the atmosphere above us. 00:03:21.340 --> 00:03:25.830 That's going to be pushing down on the surface of the 00:03:25.830 --> 00:03:28.420 mercury on the outside of the test tube. 00:03:28.420 --> 00:03:31.630 Since there's no pressure here, the mercury is going to 00:03:31.630 --> 00:03:34.220 go upwards here. 00:03:34.220 --> 00:03:36.810 This state that I've drawn is a static state-- we have 00:03:36.810 --> 00:03:40.260 assumed that all the motion has stopped. 00:03:40.260 --> 00:03:41.267 So let's try to solve this problem. 00:03:41.267 --> 00:03:44.300 Oh, and there are a couple of things we have to know before 00:03:44.300 --> 00:03:46.140 we do this problem. 00:03:46.140 --> 00:03:50.710 It's mercury, and we know the specific gravity-- I'm using 00:03:50.710 --> 00:03:53.000 terminology, because a lot of these problems, the hardest 00:03:53.000 --> 00:04:09.350 part is the terminology-- of mercury is 13.6. 00:04:09.350 --> 00:04:12.230 That's often a daunting statement on a test-- you know 00:04:12.230 --> 00:04:13.860 how to do all the math, and all of a sudden you go, what 00:04:13.860 --> 00:04:15.060 is specific gravity? 00:04:15.060 --> 00:04:19.589 All specific gravity is, is the ratio of how dense that 00:04:19.589 --> 00:04:21.300 substance is to water. 00:04:21.300 --> 00:04:29.460 All that means is that mercury is 13.6 00:04:29.460 --> 00:04:37.365 times as dense as water. 00:04:40.260 --> 00:04:42.070 Hopefully, after the last video-- because I told you 00:04:42.070 --> 00:04:44.240 to-- you should have memorized the density of water. 00:04:44.240 --> 00:04:48.870 It's 1,000 kilograms per meter cubed, so the density of 00:04:48.870 --> 00:04:53.210 mercury-- let's write that down, and that's the rho, or 00:04:53.210 --> 00:04:56.400 little p, depending on how you want to do it-- is going to be 00:04:56.400 --> 00:05:01.240 equal to 13.6 times the density of water, or times 00:05:01.240 --> 00:05:11.960 1,000 kilograms per meter cubed. 00:05:11.960 --> 00:05:14.830 Let's go back to the problem. 00:05:14.830 --> 00:05:17.060 What we want to know is how high this 00:05:17.060 --> 00:05:19.350 column of mercury is. 00:05:19.350 --> 00:05:22.290 We know that the pressure-- let's consider this point 00:05:22.290 --> 00:05:25.700 right here, which is essentially the base of this 00:05:25.700 --> 00:05:27.270 column of mercury. 00:05:27.270 --> 00:05:30.030 What we're saying is the pressure on the base of this 00:05:30.030 --> 00:05:33.630 column of mercury right here, or the pressure at this point 00:05:33.630 --> 00:05:39.800 down, has to be the same thing as the pressure up, because 00:05:39.800 --> 00:05:42.180 the mercury isn't moving-- we're in a static state. 00:05:42.180 --> 00:05:45.180 We learned several videos ago that the pressure in is equal 00:05:45.180 --> 00:05:50.350 to the pressure out on a liquid system. 00:05:50.350 --> 00:05:53.740 Essentially, I have one atmosphere pushing down here 00:05:53.740 --> 00:05:56.380 on the outside of the surface, so I must have one atmosphere 00:05:56.380 --> 00:05:57.630 pushing up here. 00:05:59.820 --> 00:06:03.590 The pressure pushing up at this point right here-- we 00:06:03.590 --> 00:06:05.520 could imagine that we have that aluminum foil there 00:06:05.520 --> 00:06:09.850 again, and just imagine where the pressure is hitting-- is 00:06:09.850 --> 00:06:14.450 one atmosphere, so the pressure down right here must 00:06:14.450 --> 00:06:18.090 be one atmosphere. 00:06:18.090 --> 00:06:20.840 What's creating the pressure down right there? 00:06:20.840 --> 00:06:25.330 It's essentially this column of water, or it's this 00:06:25.330 --> 00:06:28.380 formula, which we learned in the last video. 00:06:28.380 --> 00:06:31.000 What we now know is that the density of the mercury, times 00:06:31.000 --> 00:06:33.880 the height of the column of water, times the acceleration 00:06:33.880 --> 00:06:37.300 of gravity on Earth-- which is where we are-- has to equal 00:06:37.300 --> 00:06:40.810 one atmosphere, because it has to offset the atmosphere 00:06:40.810 --> 00:06:43.820 that's pushing on the outside and pushing up here. 00:06:43.820 --> 00:06:52.880 The density of mercury is this: 13.6 thousand, so 13,600 00:06:52.880 --> 00:06:58.550 kilogram meters per meter cubed. 00:06:58.550 --> 00:07:01.060 That's the density times the height-- we don't know what 00:07:01.060 --> 00:07:03.695 the height is, that's going to be in meters-- times the 00:07:03.695 --> 00:07:06.460 acceleration of gravity, which is 9.8 00:07:06.460 --> 00:07:09.340 meters per second squared. 00:07:09.340 --> 00:07:12.350 It's going to be equal to one atmosphere. 00:07:12.350 --> 00:07:14.020 Now you're saying-- Sal, this is strange. 00:07:14.020 --> 00:07:15.850 I've never seen this atmosphere before-- we've 00:07:15.850 --> 00:07:18.600 talked a lot about it, but how does an atmosphere relate to 00:07:18.600 --> 00:07:21.330 pascals or newtons? 00:07:21.330 --> 00:07:23.320 This is something else you should memorize: one 00:07:23.320 --> 00:07:32.120 atmosphere is equal to 103,000 pascals, and that also equals 00:07:32.120 --> 00:07:38.660 103,000 newtons per meter squared. 00:07:38.660 --> 00:07:41.350 One atmosphere is how much we're pushing down out here. 00:07:41.350 --> 00:07:43.230 So it's how much we're pushing up here, and that's going to 00:07:43.230 --> 00:07:45.560 be equal to the amount of pressure at this point from 00:07:45.560 --> 00:07:47.650 this column of mercury. 00:07:47.650 --> 00:07:54.380 One atmosphere is exactly this much, which equals 103,000 00:07:54.380 --> 00:07:56.800 newtons per meters squared. 00:07:59.430 --> 00:08:07.660 If we divide both sides by 13,609.8, we get that the 00:08:07.660 --> 00:08:21.130 height is equal to 103,000 newtons per meter cubed, over 00:08:21.130 --> 00:08:36.659 13,600 kilograms per meter cubed times 9.8 meters per 00:08:36.659 --> 00:08:37.909 second squared. 00:08:40.480 --> 00:08:41.940 Make sure you always have the units right-- that's the 00:08:41.940 --> 00:08:44.600 hardest thing about these problems, just to know that an 00:08:44.600 --> 00:08:49.210 atmosphere is 103,000 pascals, which is also the same as 00:08:49.210 --> 00:08:52.410 newtons per meter squared. 00:08:52.410 --> 00:09:01.810 Let's just do the math, so let me type this in-- 103,000 00:09:01.810 --> 00:09:17.000 divided by 13,600 divided by 9.8 equals 0.77. 00:09:17.000 --> 00:09:19.250 We were dealing with newtons, so height is 00:09:19.250 --> 00:09:22.870 equal to 0.77 meters. 00:09:22.870 --> 00:09:24.720 And you should see that the units actually work, because 00:09:24.720 --> 00:09:27.030 we have a meters cubed in the denominator up here, we have a 00:09:27.030 --> 00:09:29.060 meters cubed in the denominator down here, and 00:09:29.060 --> 00:09:32.260 then we have kilogram meters per second squared here. 00:09:32.260 --> 00:09:35.780 We have newtons up here, but what's a newton? 00:09:35.780 --> 00:09:39.720 A newton is a kilogram meter squared per second, so when 00:09:39.720 --> 00:09:42.700 you divide you have kilogram meters squared per second 00:09:42.700 --> 00:09:44.060 squared, and here you have kilogram 00:09:44.060 --> 00:09:45.390 meter per second squared. 00:09:45.390 --> 00:09:47.080 When you do all the division of the units, all you're left 00:09:47.080 --> 00:09:51.450 with is meters, so we have 0.77 meters, or roughly 77 00:09:51.450 --> 00:09:54.850 centimeters-- is how high this column of mercury is. 00:09:54.850 --> 00:09:57.255 And you can make a barometer out of it-- you can say, let 00:09:57.255 --> 00:09:59.760 me make a little notch on this test tube, and that represents 00:09:59.760 --> 00:10:01.550 one atmosphere. 00:10:01.550 --> 00:10:04.540 You can go around and figure out how many atmospheres 00:10:04.540 --> 00:10:05.780 different parts of the globe are. 00:10:05.780 --> 00:10:07.200 Anyway, I've run out of time. 00:10:07.200 --> 00:10:08.940 See you in the next video.

Pressure and Pascal's principle (part 2)

https://www.youtube.com/watch?v=lWDtFHDVqqk

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en

WEBVTT Kind: captions Language: en 00:00:00.630 --> 00:00:01.250 Welcome back. 00:00:01.250 --> 00:00:02.990 To just review what I was doing on the last video before 00:00:02.990 --> 00:00:07.050 I ran out of time, I said that conservation of energy tells 00:00:07.050 --> 00:00:09.980 us that the work I've put into the system or the energy that 00:00:09.980 --> 00:00:11.300 I've put into the system-- because they're really the 00:00:11.300 --> 00:00:14.740 same thing-- is equal to the work that I get out of the 00:00:14.740 --> 00:00:17.440 system, or the energy that I get out of the system. 00:00:17.440 --> 00:00:20.480 That means that the input work is equal to the output work, 00:00:20.480 --> 00:00:23.540 or that the input force times the input distance is equal to 00:00:23.540 --> 00:00:25.610 the output force times the output distance-- that's just 00:00:25.610 --> 00:00:27.020 the definition of work. 00:00:27.020 --> 00:00:30.040 Let me just rewrite this equation here. 00:00:30.040 --> 00:00:33.730 If I could just rewrite this exact equation, I could say-- 00:00:33.730 --> 00:00:42.780 the input force, and let me just divide it by this area. 00:00:42.780 --> 00:00:44.900 The input here-- I'm pressing down this piston that's 00:00:44.900 --> 00:00:47.580 pressing down on this area of water. 00:00:47.580 --> 00:00:54.560 So this input force-- times the input area. 00:00:54.560 --> 00:00:58.380 Let's call the input 1, and call the output 2 for 00:00:58.380 --> 00:00:59.630 simplicity. 00:01:01.590 --> 00:01:04.069 Let's say I have a piston on the top here. 00:01:04.069 --> 00:01:09.320 Let me do this in a good color-- brown is good color. 00:01:09.320 --> 00:01:15.060 I have another piston here, and there's going to be some 00:01:15.060 --> 00:01:18.140 outward force F2. 00:01:18.140 --> 00:01:20.500 The general notion is that I'm pushing on this water, the 00:01:20.500 --> 00:01:22.740 water can't be compressed, so the water's going to push up 00:01:22.740 --> 00:01:25.470 on this end. 00:01:25.470 --> 00:01:28.880 The input force times the input distance is going to be 00:01:28.880 --> 00:01:32.540 equal to the output force times the output distance 00:01:32.540 --> 00:01:34.560 right-- this is just the law of conservation of energy and 00:01:34.560 --> 00:01:37.590 everything we did with work, et cetera. 00:01:37.590 --> 00:01:40.060 I'm rewriting this equation, so if I take the input force 00:01:40.060 --> 00:01:46.610 and divide by the input area-- let me switch back to green-- 00:01:46.610 --> 00:01:50.660 then I multiply by the area, and then I just 00:01:50.660 --> 00:01:53.222 multiply times D1. 00:01:53.222 --> 00:01:55.730 You see what I did here-- I just multiplied and divided by 00:01:55.730 --> 00:01:56.710 A1, which you can do. 00:01:56.710 --> 00:01:59.230 You can multiply and divide by any number, and these two 00:01:59.230 --> 00:02:00.260 cancel out. 00:02:00.260 --> 00:02:03.160 It's equal to the same thing on the other side, which is 00:02:03.160 --> 00:02:07.420 F2-- I'm not good at managing my space on my whiteboard-- 00:02:07.420 --> 00:02:13.790 over A2 times A2 times D2. 00:02:13.790 --> 00:02:15.270 Hopefully that makes sense. 00:02:15.270 --> 00:02:20.890 What's this quantity right here, this F1 divided by A1? 00:02:20.890 --> 00:02:24.640 Force divided by area, if you haven't been familiar with it 00:02:24.640 --> 00:02:26.780 already, and if you're just watching my videos there's no 00:02:26.780 --> 00:02:29.450 reason for you to be, is defined as pressure. 00:02:29.450 --> 00:02:34.260 Pressure is force in a given area, so this is pressure-- 00:02:34.260 --> 00:02:35.620 we'll call this the pressure that I'm 00:02:35.620 --> 00:02:38.165 inputting into the system. 00:02:40.700 --> 00:02:43.360 What's area 1 times distance 1? 00:02:43.360 --> 00:02:46.950 That's the area of the tube at this point, the 00:02:46.950 --> 00:02:48.840 cross-sectional area, times this distance. 00:02:48.840 --> 00:02:51.750 That's equal to this volume that I calculated in the 00:02:51.750 --> 00:02:53.520 previous video-- we could say that's the 00:02:53.520 --> 00:02:56.410 input volume, or V1. 00:02:56.410 --> 00:03:02.420 Pressure times V1 is equal to the output pressure-- force 2 00:03:02.420 --> 00:03:06.115 divided by area 2 is the output pressure that the water 00:03:06.115 --> 00:03:07.670 is exerting on this piston. 00:03:07.670 --> 00:03:11.560 So that's the output pressure, P2. 00:03:11.560 --> 00:03:14.700 And what's area 2 times D2? 00:03:14.700 --> 00:03:18.110 The cross sectional area, times the height at which how 00:03:18.110 --> 00:03:20.490 much the water's being displaced upward, that is 00:03:20.490 --> 00:03:21.740 equal to volume 2. 00:03:24.460 --> 00:03:26.680 But what do we know about these two volumes? 00:03:26.680 --> 00:03:29.420 I went over it probably redundantly in the previous 00:03:29.420 --> 00:03:33.690 video-- those two volumes are equal, V1 is equal to V2, so 00:03:33.690 --> 00:03:36.170 we could just divide both sides by that equation. 00:03:36.170 --> 00:03:42.040 You get the pressure input is equal to the pressure output, 00:03:42.040 --> 00:03:43.380 so P1 is equal to P2. 00:03:52.480 --> 00:03:54.410 I did all of that just to show you that this isn't a new 00:03:54.410 --> 00:03:57.190 concept: this is just the conservation of energy. 00:03:57.190 --> 00:04:00.120 The only new thing I did is I divided-- we have this notion 00:04:00.120 --> 00:04:03.200 of the cross-sectional area, and we have this notion of 00:04:03.200 --> 00:04:06.190 pressure-- so where does that help us? 00:04:06.190 --> 00:04:10.320 This actually tells us-- and you can do this example in 00:04:10.320 --> 00:04:13.660 multiple situations, but I like to think of if we didn't 00:04:13.660 --> 00:04:16.170 have gravity first, because gravity tends to confuse 00:04:16.170 --> 00:04:19.070 things, but we'll introduce gravity in a video or two-- is 00:04:19.070 --> 00:04:25.490 that when you have any external pressure onto a 00:04:25.490 --> 00:04:29.490 liquid, onto an incompressible fluid, that pressure is 00:04:29.490 --> 00:04:32.960 distributed evenly throughout the fluid. 00:04:32.960 --> 00:04:36.770 That's what we essentially just proved just using the law 00:04:36.770 --> 00:04:40.030 of conservation of energy, and everything we know about work. 00:04:40.030 --> 00:04:43.370 What I just said is called Pascal's principle: if any 00:04:43.370 --> 00:04:46.390 external pressure is applied to a fluid, that pressure is 00:04:46.390 --> 00:04:49.660 distributed throughout the fluid equally. 00:04:49.660 --> 00:04:51.640 Another way to think about it-- we proved it with this 00:04:51.640 --> 00:04:59.710 little drawing here-- is, let's say that I have a tube, 00:04:59.710 --> 00:05:01.520 and at the end of the tube is a balloon. 00:05:01.520 --> 00:05:04.720 Let's say I'm doing this on the Space Shuttle. 00:05:04.720 --> 00:05:09.320 It's saying that if I increase-- say I have some 00:05:09.320 --> 00:05:10.570 piston here. 00:05:14.350 --> 00:05:17.560 This is stable, and I have water 00:05:17.560 --> 00:05:18.810 throughout this whole thing. 00:05:22.480 --> 00:05:25.950 Let me see if I can use that field function again-- oh no, 00:05:25.950 --> 00:05:29.760 there must have been a hole in my drawing. 00:05:29.760 --> 00:05:31.120 Let me just draw the water. 00:05:31.120 --> 00:05:36.970 I have water throughout this whole thing, and all Pascal's 00:05:36.970 --> 00:05:39.410 principle is telling us that if I were to apply some 00:05:39.410 --> 00:05:51.360 pressure here, that that net pressure, that extra pressure 00:05:51.360 --> 00:05:55.040 I'm applying, is going to compress this little bit. 00:05:55.040 --> 00:05:56.770 That extra compression is going to be distributed 00:05:56.770 --> 00:05:58.130 through the whole balloon. 00:05:58.130 --> 00:06:00.330 Let's say that this right here is rigid-- it's some kind of 00:06:00.330 --> 00:06:01.450 middle structure. 00:06:01.450 --> 00:06:06.150 The rest of the balloon is going to expand uniformly, so 00:06:06.150 --> 00:06:08.830 that increased pressure I'm doing is going through the 00:06:08.830 --> 00:06:09.100 whole thing. 00:06:09.100 --> 00:06:12.980 It's not like the balloon will get longer, or that the 00:06:12.980 --> 00:06:16.010 pressure is just translated down here, or that just up 00:06:16.010 --> 00:06:17.615 here the balloon's going to get wider and it's just going 00:06:17.615 --> 00:06:19.170 to stay the same length there. 00:06:19.170 --> 00:06:22.800 Hopefully, that gives you a little bit of intuition. 00:06:22.800 --> 00:06:25.050 Going back to what I had drawn before, that's actually 00:06:25.050 --> 00:06:28.430 interesting, because that's actually another simple or 00:06:28.430 --> 00:06:32.220 maybe not so simple machine that we've constructed. 00:06:32.220 --> 00:06:36.190 I almost defined it as a simple machine when I 00:06:36.190 --> 00:06:37.100 initially drew it. 00:06:37.100 --> 00:06:40.520 Let's draw that weird thing again, where it looks like 00:06:40.520 --> 00:06:44.440 this, where I have water in it. 00:06:54.390 --> 00:06:56.678 Let's make sure I fill it, so that when I do the fill, it 00:06:56.678 --> 00:06:59.720 will completely fill, and doesn't fill other things. 00:07:02.450 --> 00:07:05.820 This is cool, because this is now another simple machine. 00:07:05.820 --> 00:07:15.320 We know that the pressure in is equal to the pressure out. 00:07:21.330 --> 00:07:26.990 And pressure is force divided by area, so the force in, 00:07:26.990 --> 00:07:32.080 divided by the area in, is equal to the force out divided 00:07:32.080 --> 00:07:34.200 by the area out. 00:07:37.350 --> 00:07:40.200 Let me give you an example: let's say that I were to apply 00:07:40.200 --> 00:07:49.450 with a pressure in equal to 10 pascals. 00:07:49.450 --> 00:07:51.590 That's a new word, and it's named after Pascal's 00:07:51.590 --> 00:07:55.090 principle, for Blaise Pascal. 00:07:55.090 --> 00:07:56.070 What is a pascal? 00:07:56.070 --> 00:08:02.310 That is just equal to 10 newtons per meter squared. 00:08:02.310 --> 00:08:06.460 That's all a pascal is-- it's a newton per meter squared, 00:08:06.460 --> 00:08:08.770 it's a very natural unit. 00:08:08.770 --> 00:08:12.930 Let's say my pressure in is 10 pascals, and let's say that my 00:08:12.930 --> 00:08:20.930 input area is 2 square meters. 00:08:20.930 --> 00:08:22.560 If I looked the surface of the water there it would be 2 00:08:22.560 --> 00:08:31.500 square meters, and let's say that my output area is equal 00:08:31.500 --> 00:08:38.640 to 4 meters squared. 00:08:41.780 --> 00:08:45.550 What I'm saying is that I can push on a piston here, and 00:08:45.550 --> 00:08:50.470 that the water's going to push up with some piston here. 00:08:50.470 --> 00:08:53.220 First of all, I told you what my input pressure is-- what's 00:08:53.220 --> 00:08:55.840 my input force? 00:08:55.840 --> 00:09:00.900 Input pressure is equal to input force divided by input 00:09:00.900 --> 00:09:06.260 area, so 10 pascals is equal to my input force divided by 00:09:06.260 --> 00:09:09.470 my area, so I multiply both sides by 2. 00:09:09.470 --> 00:09:13.640 I get input force is equal to 20 newtons. 00:09:13.640 --> 00:09:15.650 My question to you is what is the output force? 00:09:15.650 --> 00:09:17.810 How much force is the system going to push 00:09:17.810 --> 00:09:20.180 upwards at this end? 00:09:20.180 --> 00:09:24.880 We know that must if my input pressure was 10 pascals, my 00:09:24.880 --> 00:09:28.270 output pressure would also be 10 pascals. 00:09:28.270 --> 00:09:34.390 So I also have 10 pascals is equal to my out force over my 00:09:34.390 --> 00:09:37.760 out cross-sectional area. 00:09:37.760 --> 00:09:40.880 So I'll have a piston here, and it goes up like that. 00:09:40.880 --> 00:09:46.710 That's 4 meters, so I do 4 times 10, and so I get 40 00:09:46.710 --> 00:09:49.450 newtons is equal to my output force. 00:09:49.450 --> 00:09:50.740 So what just happened here? 00:09:50.740 --> 00:09:55.920 I inputted-- so my input force is equal to 20 newtons, and my 00:09:55.920 --> 00:10:00.550 output force is equal to 40 newtons, so I just doubled my 00:10:00.550 --> 00:10:03.840 force, or essentially I had a mechanical advantage of 2. 00:10:03.840 --> 00:10:07.530 This is an example of a simple machine, and 00:10:07.530 --> 00:10:09.040 it's a hydraulic machine. 00:10:09.040 --> 00:10:10.310 Anyway, I've just run out of time. 00:10:10.310 --> 00:10:11.560 I'll see you in the next video.

Pressure and Pascal's principle (part 1)

https://www.youtube.com/watch?v=Pn5YEMwQb4Y

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WEBVTT Kind: captions Language: en 00:00:00.700 --> 00:00:04.100 Let's learn a little bit about fluids. 00:00:04.100 --> 00:00:07.940 You probably have some notion of what a fluid is, but let's 00:00:07.940 --> 00:00:10.110 talk about it in the physics sense, or maybe even the 00:00:10.110 --> 00:00:12.280 chemistry sense, depending on in what context you're 00:00:12.280 --> 00:00:13.470 watching this video. 00:00:13.470 --> 00:00:14.680 So a fluid is anything that takes the 00:00:14.680 --> 00:00:16.030 shape of its container. 00:00:16.030 --> 00:00:28.550 For example, if I had a glass sphere, and let's say that I 00:00:28.550 --> 00:00:31.110 completely filled this glass sphere with water. 00:00:31.110 --> 00:00:32.640 I was going to say that we're in a zero gravity environment, 00:00:32.640 --> 00:00:34.130 but you really don't even need that. 00:00:34.130 --> 00:00:39.300 Let's say that every cubic centimeter or cubic meter of 00:00:39.300 --> 00:00:41.010 this glass sphere is filled with water. 00:00:44.380 --> 00:00:46.810 Let's say that it's not a glass, but a rubber sphere. 00:00:46.810 --> 00:00:49.700 If I were to change the shape of the sphere, but not really 00:00:49.700 --> 00:00:53.520 change the volume-- if I were to change the shape of the 00:00:53.520 --> 00:00:56.730 sphere where it looks like this now-- the water would 00:00:56.730 --> 00:01:01.220 just change its shape with the container. 00:01:01.220 --> 00:01:03.620 The water would just change in the shape of the container, 00:01:03.620 --> 00:01:07.410 and in this case, I have green water. 00:01:07.410 --> 00:01:11.580 The same is also true if that was oxygen, or if that was 00:01:11.580 --> 00:01:13.110 just some gas. 00:01:13.110 --> 00:01:16.450 It would fill the container, and in this situation, it 00:01:16.450 --> 00:01:20.130 would also fill the newly shaped container. 00:01:20.130 --> 00:01:26.070 A fluid, in general, takes the shape of its container. 00:01:31.420 --> 00:01:34.510 And I just gave you two examples of fluids-- you have 00:01:34.510 --> 00:01:41.370 liquids, and you have gases. 00:01:41.370 --> 00:01:43.590 Those are two types of fluid: both of those things take the 00:01:43.590 --> 00:01:45.180 shape of the container. 00:01:45.180 --> 00:01:48.180 What's the difference between a liquid and a gas, then? 00:01:48.180 --> 00:01:55.760 A gas is compressible, which means that I could actually 00:01:55.760 --> 00:02:00.110 decrease the volume of this container and the gas will 00:02:00.110 --> 00:02:02.500 just become denser within the container. 00:02:02.500 --> 00:02:05.750 You can think of it as if I blew air into a balloon-- you 00:02:05.750 --> 00:02:07.240 could squeeze that balloon a little bit. 00:02:07.240 --> 00:02:09.800 There's air in there, and at some point the pressure might 00:02:09.800 --> 00:02:11.200 get high enough to pop the balloon, but 00:02:11.200 --> 00:02:12.400 you can squeeze it. 00:02:12.400 --> 00:02:13.780 A liquid is incompressible. 00:02:21.080 --> 00:02:23.020 How do I know that a liquid is incompressible? 00:02:23.020 --> 00:02:25.750 Imagine the same balloon filled with water-- completely 00:02:25.750 --> 00:02:26.660 filled with water. 00:02:26.660 --> 00:02:30.760 If you squeezed on that balloon from every side-- let 00:02:30.760 --> 00:02:33.550 me pick a different color-- I have this balloon, and it was 00:02:33.550 --> 00:02:34.660 filled with water. 00:02:34.660 --> 00:02:37.370 If you squeezed on this balloon from every side, you 00:02:37.370 --> 00:02:39.900 would not be able to change the volume of this balloon. 00:02:39.900 --> 00:02:42.360 No matter what you do, you would not be able to change 00:02:42.360 --> 00:02:44.870 the volume of this balloon, no matter how much force or 00:02:44.870 --> 00:02:48.190 pressure you put from any side on it, while if this was 00:02:48.190 --> 00:02:53.650 filled with gas-- and magenta, blue in for gas-- you actually 00:02:53.650 --> 00:02:56.120 could decrease the volume by just increasing the pressure 00:02:56.120 --> 00:02:59.520 on all sides of the balloon. 00:02:59.520 --> 00:03:00.780 You can actually squeeze it, and make the 00:03:00.780 --> 00:03:02.060 entire volume smaller. 00:03:02.060 --> 00:03:03.930 That's the difference between a liquid and a gas-- gas is 00:03:03.930 --> 00:03:06.870 compressible, liquid isn't, and we'll learn later that you 00:03:06.870 --> 00:03:09.710 can turn a liquid into a gas, gas into a liquid, and turn 00:03:09.710 --> 00:03:11.850 liquids into solids, but we'll learn all about that later. 00:03:11.850 --> 00:03:15.620 This is a pretty good working definition of that. 00:03:15.620 --> 00:03:17.920 Let's use that, and now we're going to actually just focus 00:03:17.920 --> 00:03:20.740 on the liquids to see if we could learn a little bit about 00:03:20.740 --> 00:03:25.410 liquid motion, or maybe even fluid motion in general. 00:03:25.410 --> 00:03:34.470 Let me draw something else-- let's say I had a situation 00:03:34.470 --> 00:03:40.160 where I have this weird shaped object which tends to show up 00:03:40.160 --> 00:03:43.050 in a lot of physics books, which I'll draw in yellow. 00:03:43.050 --> 00:03:45.700 This weird shaped container where it's relatively narrow 00:03:45.700 --> 00:03:51.250 there, and then it goes and U-turns into 00:03:51.250 --> 00:03:54.010 a much larger opening. 00:03:58.480 --> 00:04:04.780 Let's say that the area of this opening is A1, and the 00:04:04.780 --> 00:04:09.110 area of this opening is A2-- this one is bigger. 00:04:09.110 --> 00:04:15.550 Now let's fill this thing with some liquid, which will be 00:04:15.550 --> 00:04:18.930 blue-- so that's my liquid. 00:04:23.890 --> 00:04:26.740 Let me see if they have this tool-- there 00:04:26.740 --> 00:04:27.500 you go, look at that. 00:04:27.500 --> 00:04:28.880 I filled it with liquid so quickly. 00:04:32.430 --> 00:04:35.190 This was liquid-- it's not just a fluid, and so what's 00:04:35.190 --> 00:04:36.250 the important thing about liquid? 00:04:36.250 --> 00:04:38.950 It's incompressible. 00:04:38.950 --> 00:04:44.610 Let's take what we know about force-- actually about work-- 00:04:44.610 --> 00:04:47.990 and see if we can come up with any rules about force and 00:04:47.990 --> 00:04:49.250 pressure with liquids. 00:04:49.250 --> 00:04:50.680 So what do we know about work? 00:04:50.680 --> 00:04:54.490 Work is force times distance, or you can also view it as the 00:04:54.490 --> 00:04:57.910 energy put into the system-- I'll write it down here. 00:04:57.910 --> 00:05:03.480 Work is equal to force times distance. 00:05:03.480 --> 00:05:08.790 We learned in mechanical advantage that the work in-- 00:05:08.790 --> 00:05:13.470 I'll do it with that I-- is equal to work out. 00:05:13.470 --> 00:05:15.500 The force times the distance that you've put into a system 00:05:15.500 --> 00:05:16.940 is equal to the force times the distance 00:05:16.940 --> 00:05:17.490 you put out of it. 00:05:17.490 --> 00:05:19.820 And you might want to review the work chapters on that. 00:05:19.820 --> 00:05:21.700 That's just the little law of conservation of energy, 00:05:21.700 --> 00:05:24.030 because work in is just the energy that you're putting 00:05:24.030 --> 00:05:26.160 into a system-- it's measured in joules-- and the work out 00:05:26.160 --> 00:05:28.010 is the energy that comes out of the system. 00:05:28.010 --> 00:05:30.910 And that's just saying that no energy is destroyed or 00:05:30.910 --> 00:05:34.270 created, it just turns into different forms. Let's just 00:05:34.270 --> 00:05:36.940 use this definition: the force times distance in is equal to 00:05:36.940 --> 00:05:38.260 force times distance out. 00:05:52.570 --> 00:05:56.080 Let's say that I pressed with some force 00:05:56.080 --> 00:05:57.510 on this entire surface. 00:05:57.510 --> 00:06:02.360 Let's say I had a piston-- let me see if I can draw a piston, 00:06:02.360 --> 00:06:04.320 and what's a good color for a piston-- so let's add a 00:06:04.320 --> 00:06:05.690 magenta piston right here. 00:06:08.230 --> 00:06:14.350 I push down on this magenta piston, and so I pushed down 00:06:14.350 --> 00:06:20.380 on this with a force of F1. 00:06:20.380 --> 00:06:25.790 Let's say I push it a distance of D1-- 00:06:25.790 --> 00:06:26.970 that's its initial position. 00:06:26.970 --> 00:06:30.460 Its final position-- let's see what color, and the hardest 00:06:30.460 --> 00:06:32.980 part of these videos is picking the color-- after I 00:06:32.980 --> 00:06:36.500 pushed, the piston goes this far. 00:06:36.500 --> 00:06:41.120 This is the distance that I pushed it-- this is D1. 00:06:41.120 --> 00:06:46.130 The water is here and I push the water down D1 meters. 00:06:46.130 --> 00:06:50.890 In this situation, my work in is F1 times D1. 00:06:50.890 --> 00:06:55.540 Let me ask you a question: how much water did I displace? 00:06:55.540 --> 00:06:57.650 How much total water did I displace? 00:06:57.650 --> 00:06:59.370 Well, it's this volume? 00:06:59.370 --> 00:07:02.370 I took this entire volume and pushed it down, so what's the 00:07:02.370 --> 00:07:05.530 volume right there that I displaced? 00:07:05.530 --> 00:07:09.360 The volume there is going to be-- the initial volume that 00:07:09.360 --> 00:07:14.510 I'm displacing, or the volume displaced, has 00:07:14.510 --> 00:07:16.640 to equal this distance. 00:07:16.640 --> 00:07:21.250 This is a cylinder of liquid, so this distance times the 00:07:21.250 --> 00:07:24.040 area of the container at that point. 00:07:24.040 --> 00:07:25.565 I'm assuming that it's constant at that point, and 00:07:25.565 --> 00:07:32.620 then it changes after that, so it equals area 1 times 00:07:32.620 --> 00:07:36.680 distance 1. 00:07:36.680 --> 00:07:41.510 We also know that that liquid has to go someplace, because 00:07:41.510 --> 00:07:42.990 what do we know about a liquid? 00:07:42.990 --> 00:07:47.510 We can't compress it, you can't change its total volume, 00:07:47.510 --> 00:07:50.830 so all of that volume is going to have to go someplace else. 00:07:50.830 --> 00:07:53.060 This is where the liquid was, and the liquid is going to 00:07:53.060 --> 00:07:57.400 rise some level-- let's say that it gets to this level, 00:07:57.400 --> 00:08:00.640 and this is its new level. 00:08:00.640 --> 00:08:05.260 It's going to change some distance here, it's going to 00:08:05.260 --> 00:08:07.930 change some distance there, and how do we know what 00:08:07.930 --> 00:08:09.400 distance that's going to be? 00:08:09.400 --> 00:08:12.960 The volume that it changes here has to go someplace. 00:08:12.960 --> 00:08:14.910 You can say, that's going to push on that, that's all going 00:08:14.910 --> 00:08:17.230 to push, and that liquid has to go someplace. 00:08:17.230 --> 00:08:19.470 Essentially it's going to end up-- it might not be the exact 00:08:19.470 --> 00:08:21.630 same molecules, but that might displace some liquid here, 00:08:21.630 --> 00:08:23.530 that's going to displace some liquid here and here and here 00:08:23.530 --> 00:08:25.700 and here and all the way until the liquid up here gets 00:08:25.700 --> 00:08:27.880 displaced and gets pushed upward. 00:08:27.880 --> 00:08:30.550 The volume that you're pushing down here is the same volume 00:08:30.550 --> 00:08:32.470 that goes up right here. 00:08:32.470 --> 00:08:37.830 So what's the volume-- what's the change in volume, or how 00:08:37.830 --> 00:08:40.610 much volume did you push up here? 00:08:40.610 --> 00:08:44.580 This volume here is going to be the distance 2 times this 00:08:44.580 --> 00:08:49.990 larger area, so we could say volume 2 is going to be equal 00:08:49.990 --> 00:08:55.080 to the distance 2 times this larger area. 00:08:55.080 --> 00:08:58.170 We know that this liquid is incompressible, so this volume 00:08:58.170 --> 00:09:01.750 has to be the same as this volume. 00:09:01.750 --> 00:09:05.470 We know that these two quantities are equal to each 00:09:05.470 --> 00:09:12.410 other, so area 1 times distance 1 is going to be 00:09:12.410 --> 00:09:16.120 equal to this area times this distance. 00:09:21.600 --> 00:09:22.580 Let's see what we can do. 00:09:22.580 --> 00:09:25.540 We know this, that the force in times the distance in is 00:09:25.540 --> 00:09:28.550 equal to the force out times the distance out. 00:09:28.550 --> 00:09:31.010 Let's take this equation-- I'm going to switch back to green 00:09:31.010 --> 00:09:33.990 just so we don't lose track of things-- 00:09:33.990 --> 00:09:37.510 and divide both sides. 00:09:37.510 --> 00:09:39.230 Let's rewrite it-- so let's say I 00:09:39.230 --> 00:09:41.730 rewrote each input force. 00:09:41.730 --> 00:09:43.530 Actually, I'm about to run out of time, so I'll continue this 00:09:43.530 --> 00:09:44.400 into the next video. 00:09:44.400 --> 00:09:45.980 See you soon.

Harmonic motion part 3 (no calculus)

https://www.youtube.com/watch?v=oqBHBO8cqLI

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https://www.youtube.com/api/timedtext?v=oqBHBO8cqLI&ei=FGWUZZ7zM9qpp-oPxcONmAg&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249220&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E9110ED0C35CFCCBB291D45BAE1220718FD385E9.837A673412C7A36553CC6D8A1A09303B66D7471D&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.100 --> 00:00:01.810 Welcome back. 00:00:01.810 --> 00:00:03.860 And if you were covering your eyes because you didn't want 00:00:03.860 --> 00:00:06.250 to see calculus, I think you can open your eyes again. 00:00:06.250 --> 00:00:08.830 There shouldn't be any significant displays of 00:00:08.830 --> 00:00:10.330 calculus in this video. 00:00:10.330 --> 00:00:12.550 But just to review what we went over, we just said, OK if 00:00:12.550 --> 00:00:14.760 we have a spring-- and I drew it vertically this time-- but 00:00:14.760 --> 00:00:16.750 pretend like there's no gravity, or maybe pretend like 00:00:16.750 --> 00:00:19.440 we're viewing-- we're looking at the top of a table, because 00:00:19.440 --> 00:00:21.080 we don't want to look at the effect of 00:00:21.080 --> 00:00:21.990 a spring and gravity. 00:00:21.990 --> 00:00:23.450 We just want to look at a spring by itself. 00:00:23.450 --> 00:00:25.760 So this could be in deep space, or something else. 00:00:25.760 --> 00:00:26.780 But we're not thinking about gravity. 00:00:26.780 --> 00:00:28.670 But I drew it vertically just so that we can get more 00:00:28.670 --> 00:00:30.080 intuition for this curve. 00:00:30.080 --> 00:00:33.520 Well, we started off saying is if I have a spring and 0-- x 00:00:33.520 --> 00:00:35.740 equals 0 is kind of the natural resting point of the 00:00:35.740 --> 00:00:38.430 spring, if I just let this mass-- if I didn't pull on the 00:00:38.430 --> 00:00:39.380 spring at all. 00:00:39.380 --> 00:00:41.340 But I have a mass attached to the spring, and if I were to 00:00:41.340 --> 00:00:46.060 stretch the spring to point A, we said, well what happens? 00:00:46.060 --> 00:00:49.990 Well, it starts with very little velocity, but there's a 00:00:49.990 --> 00:00:52.100 restorative force, that's going to be pulling it back 00:00:52.100 --> 00:00:53.430 towards this position. 00:00:53.430 --> 00:00:56.170 So that force will accelerate the mass, accelerate the mass, 00:00:56.170 --> 00:00:59.620 accelerate the mass, until it gets right here. 00:00:59.620 --> 00:01:03.040 And then it'll have a lot of velocity here, but then it'll 00:01:03.040 --> 00:01:04.360 start decelerating. 00:01:04.360 --> 00:01:06.680 And then it'll decelerate, decelerate, decelerate. 00:01:06.680 --> 00:01:08.560 Its velocity will stop, and it'll come back up. 00:01:08.560 --> 00:01:10.625 And if we drew this as a function of time, 00:01:10.625 --> 00:01:12.060 this is what happens. 00:01:12.060 --> 00:01:15.010 It starts moving very slowly, accelerates. 00:01:15.010 --> 00:01:18.300 At this point, at x equals 0, it has its maximum speed. 00:01:18.300 --> 00:01:21.010 So the rate of change of velocity-- or the rate of 00:01:21.010 --> 00:01:25.230 change of position is fastest. And we can see the slope is 00:01:25.230 --> 00:01:26.770 very fast right here. 00:01:26.770 --> 00:01:30.240 And then, we start slowing down again, slowing down, 00:01:30.240 --> 00:01:31.990 until we get back to the spot of A. 00:01:31.990 --> 00:01:35.500 And then we keep going up and down, up and down, like that. 00:01:35.500 --> 00:01:40.280 And we showed that actually, the equation for the mass's 00:01:40.280 --> 00:01:45.320 position as a function of time is x of t-- and we used a 00:01:45.320 --> 00:01:47.620 little bit of differential equations to prove it. 00:01:47.620 --> 00:01:50.310 But this equation-- not that I recommend that you memorize 00:01:50.310 --> 00:01:51.950 anything-- but this is a pretty 00:01:51.950 --> 00:01:53.640 useful equation to memorize. 00:01:53.640 --> 00:01:58.390 Because you can use it to pretty much figure out 00:01:58.390 --> 00:02:05.210 anything-- about the position, or of the mass at any given 00:02:05.210 --> 00:02:09.490 time, or the frequency of this oscillatory motion, or 00:02:09.490 --> 00:02:10.120 anything else. 00:02:10.120 --> 00:02:12.005 Even the velocity, if you know a little bit of calculus, you 00:02:12.005 --> 00:02:13.720 can figure out the velocity at anytime, of the object. 00:02:13.720 --> 00:02:16.030 And that's pretty neat. 00:02:16.030 --> 00:02:18.740 So what can we do now? 00:02:18.740 --> 00:02:20.690 Well, let's try to figure out the period of 00:02:20.690 --> 00:02:26.220 this oscillating system. 00:02:26.220 --> 00:02:28.450 And just so you know-- I know I put the label harmonic 00:02:28.450 --> 00:02:30.980 motion on all of these-- this is simple harmonic motion. 00:02:30.980 --> 00:02:34.470 Simple harmonic motion is something that can be 00:02:34.470 --> 00:02:36.690 described by a trigonometric function like this. 00:02:36.690 --> 00:02:39.720 And it just oscillates back and forth, back and forth. 00:02:39.720 --> 00:02:41.850 And so, what we're doing is harmonic motion. 00:02:41.850 --> 00:02:44.100 And now, let's figure out what this period is. 00:02:44.100 --> 00:02:47.670 Remember we said that after T seconds, it gets back to its 00:02:47.670 --> 00:02:50.210 original position, and then after another T seconds, it 00:02:50.210 --> 00:02:51.860 gets back to its original position. 00:02:51.860 --> 00:02:53.540 Let's figure out with this T is. 00:02:53.540 --> 00:02:55.320 And that's essentially its period, right? 00:02:55.320 --> 00:02:57.830 What's the period of a function? 00:02:57.830 --> 00:03:00.650 It's how long it takes to get back to your starting point. 00:03:00.650 --> 00:03:06.560 Or how long it takes for the whole cycle to happen once. 00:03:06.560 --> 00:03:08.140 So what is this T? 00:03:08.140 --> 00:03:09.140 So let me ask you a question. 00:03:09.140 --> 00:03:11.320 What are all the points-- that if this is a 00:03:11.320 --> 00:03:12.950 cosine function, right? 00:03:12.950 --> 00:03:18.290 What are all of the points at which cosine is equal to 1? 00:03:18.290 --> 00:03:20.470 Or this function would be equal to A, right? 00:03:20.470 --> 00:03:22.860 Because whenever cosine is equal to 1, this whole 00:03:22.860 --> 00:03:24.415 function is equal to A. 00:03:24.415 --> 00:03:25.930 And it's these points. 00:03:25.930 --> 00:03:31.450 Well cosine is equal to 1 when-- so, theta-- let's say, 00:03:31.450 --> 00:03:36.430 when is cosine of theta equal to 1? 00:03:36.430 --> 00:03:38.890 So, at what angles is this true? 00:03:38.890 --> 00:03:42.150 Well it's true at theta is equal to 0, right? 00:03:42.150 --> 00:03:44.200 Cosine of 0 is 1. 00:03:44.200 --> 00:03:46.430 Cosine of 2 pi is also 1, right? 00:03:46.430 --> 00:03:48.400 We could just keep going around that unit circle. 00:03:48.400 --> 00:03:50.770 You should watch the unit circle video if this makes no 00:03:50.770 --> 00:03:51.260 sense to you. 00:03:51.260 --> 00:03:53.600 Or the graphing trig functions. 00:03:53.600 --> 00:03:55.850 It's also true at 4 pi. 00:03:55.850 --> 00:03:59.790 Really, any multiple of 2 pi, this is true. 00:03:59.790 --> 00:04:00.320 Right? 00:04:00.320 --> 00:04:04.310 Cosine of that angle is equal to 1. 00:04:04.310 --> 00:04:05.900 So the same thing is true. 00:04:05.900 --> 00:04:14.400 This function, x of t, is equal to A at what points? 00:04:14.400 --> 00:04:18.980 x of t is equal to A whenever this expression-- within the 00:04:18.980 --> 00:04:24.540 cosines-- whenever this expression is equal to 0, 2 00:04:24.540 --> 00:04:27.990 pi, 4 pi, et cetera. 00:04:27.990 --> 00:04:30.610 And this first time that it cycles, right, from 0 to 2 00:04:30.610 --> 00:04:36.650 pi-- from 0 to T, that'll be at 2 pi, right? 00:04:36.650 --> 00:04:41.730 So this whole expression will equal A, when k-- and that's 00:04:41.730 --> 00:04:42.970 these points, right? 00:04:42.970 --> 00:04:44.760 That's when this function is equal to A. 00:04:44.760 --> 00:04:46.930 It'll happen again over here someplace. 00:04:46.930 --> 00:04:50.260 When this little internal expression is equal to 2 pi, 00:04:50.260 --> 00:04:52.180 or really any multiple of 2 pi. 00:04:52.180 --> 00:04:56.010 So we could say, so x of t is equal to A when the square 00:04:56.010 --> 00:05:03.460 root of k over m times t, is equal to 2 pi. 00:05:03.460 --> 00:05:07.330 Or another way of thinking about it, is let's multiply 00:05:07.330 --> 00:05:10.510 both sides of this equation times the inverse of the 00:05:10.510 --> 00:05:12.450 square root of k over m. 00:05:12.450 --> 00:05:20.410 And you get, t is equal to 2 pi times the square root-- and 00:05:20.410 --> 00:05:21.840 it's going to be the inverse of this, right? 00:05:21.840 --> 00:05:25.550 Of m over k. 00:05:25.550 --> 00:05:28.440 And there we have the period of this function. 00:05:28.440 --> 00:05:31.460 This is going to be equal to 2 pi times the square 00:05:31.460 --> 00:05:33.860 root of m over k. 00:05:33.860 --> 00:05:41.475 So if someone tells you, well I have a spring that I'm going 00:05:41.475 --> 00:05:43.925 to pull from some-- I'm going to stretch it, or compress it 00:05:43.925 --> 00:05:46.080 a little bit, then I let go-- what is the period? 00:05:46.080 --> 00:05:49.180 How long does it take for the spring to go back to its 00:05:49.180 --> 00:05:49.990 original position? 00:05:49.990 --> 00:05:52.480 It'll keep doing that, as we have no friction, or no 00:05:52.480 --> 00:05:54.060 gravity, or any air resistance, or 00:05:54.060 --> 00:05:54.790 anything like that. 00:05:54.790 --> 00:05:56.700 Air resistance really is just a form of friction. 00:05:56.700 --> 00:05:58.940 You could immediately-- if you memorize this formula, 00:05:58.940 --> 00:06:01.360 although you should know where it comes from-- you could 00:06:01.360 --> 00:06:03.840 immediately say, well I know how long the period is. 00:06:03.840 --> 00:06:06.390 It's 2 pi times m over k. 00:06:06.390 --> 00:06:09.090 That's how long it's going to take the spring to get back-- 00:06:09.090 --> 00:06:11.560 to complete one cycle. 00:06:11.560 --> 00:06:13.520 And then what about the frequency? 00:06:13.520 --> 00:06:16.320 If you wanted to know cycles per second, well that's just 00:06:16.320 --> 00:06:19.200 the inverse of the period, right? 00:06:19.200 --> 00:06:22.190 So if I wanted to know the frequency, that equals 1 over 00:06:22.190 --> 00:06:23.820 the period, right? 00:06:23.820 --> 00:06:26.860 Period is given in seconds per cycle. 00:06:26.860 --> 00:06:33.340 So frequency is cycles per second, and this 00:06:33.340 --> 00:06:35.880 is seconds per cycle. 00:06:35.880 --> 00:06:38.970 So frequency is just going to be 1 over this. 00:06:38.970 --> 00:06:44.500 Which is 1 over 2 pi times the square root of k over m. 00:06:44.500 --> 00:06:46.180 That's the frequency. 00:06:46.180 --> 00:06:50.110 But I have always had trouble memorizing this, and this. 00:06:50.110 --> 00:06:50.620 You always [UNINTELLIGIBLE] 00:06:50.620 --> 00:06:52.240 k over m, and m over k, and all of that. 00:06:52.240 --> 00:06:54.650 All you have to really memorize is this. 00:06:54.650 --> 00:06:56.840 And even that, you might even have an intuition 00:06:56.840 --> 00:06:57.660 as to why it's true. 00:06:57.660 --> 00:06:59.240 You can even go to the differential equations if you 00:06:59.240 --> 00:07:00.680 want to reprove it to yourself. 00:07:00.680 --> 00:07:04.020 Because if you have this, you really can answer any question 00:07:04.020 --> 00:07:07.990 about the position of the mass, at any time. 00:07:07.990 --> 00:07:10.150 The velocity of the mass, at any time, just by taking the 00:07:10.150 --> 00:07:10.980 derivative. 00:07:10.980 --> 00:07:13.140 Or the period, or the frequency of the function. 00:07:13.140 --> 00:07:14.810 As long as you know how to take the period and frequency 00:07:14.810 --> 00:07:16.180 of trig functions. 00:07:16.180 --> 00:07:19.010 You can watch my videos, and watch my trig videos, to get a 00:07:19.010 --> 00:07:20.340 refresher on that. 00:07:20.340 --> 00:07:22.850 One thing that's pretty interesting about this, is 00:07:22.850 --> 00:07:26.930 notice that the frequency and the period, right? 00:07:26.930 --> 00:07:28.900 This is the period of the function, that's how long it 00:07:28.900 --> 00:07:30.470 takes do one cycle. 00:07:30.470 --> 00:07:33.090 This is how many cycles it does in one second-- both of 00:07:33.090 --> 00:07:35.010 them are independent of A. 00:07:35.010 --> 00:07:37.830 So it doesn't matter, I could stretch it only a little bit, 00:07:37.830 --> 00:07:40.750 like there, and it'll take the same amount of time to go 00:07:40.750 --> 00:07:43.560 back, and come back like that, as it would if I 00:07:43.560 --> 00:07:44.470 stretch it a lot. 00:07:44.470 --> 00:07:45.290 It would just do that. 00:07:45.290 --> 00:07:48.800 If I stretched it just a little bit, the function would 00:07:48.800 --> 00:07:51.450 look like this. 00:07:51.450 --> 00:07:53.340 Make sure I do this right. 00:07:53.340 --> 00:07:55.030 I'm not doing that right. 00:07:55.030 --> 00:07:56.280 Edit, undo. 00:07:58.680 --> 00:08:00.800 If I just do it a little bit, the amplitude is going to be 00:08:00.800 --> 00:08:03.180 less, but the function is going to essentially do the 00:08:03.180 --> 00:08:04.890 same thing. 00:08:04.890 --> 00:08:08.190 It's just going to do that. 00:08:08.190 --> 00:08:10.090 So it's going to take the same amount of time to complete the 00:08:10.090 --> 00:08:11.450 cycle, it'll just have a lower amplitude. 00:08:11.450 --> 00:08:14.890 So that's interesting to me, that how much I stretch it, 00:08:14.890 --> 00:08:18.000 it's not going to make it take longer or less time to 00:08:18.000 --> 00:08:19.850 complete one cycle. 00:08:19.850 --> 00:08:21.500 That's interesting. 00:08:21.500 --> 00:08:26.360 And so if I just told you, that I actually start having 00:08:26.360 --> 00:08:27.930 objects compressed, right? 00:08:27.930 --> 00:08:33.419 So in that case, let's say my A is minus 3. 00:08:33.419 --> 00:08:37.490 I have a spring constant of-- let's say k is, 00:08:37.490 --> 00:08:39.280 I don't know, 10. 00:08:39.280 --> 00:08:43.830 And I have a mass of 2 kilograms. Then I could 00:08:43.830 --> 00:08:47.100 immediately tell you what the equation of the position as a 00:08:47.100 --> 00:08:48.630 function of time at any point is. 00:08:48.630 --> 00:08:53.120 It's going to be x of t will equal-- I'm running out of 00:08:53.120 --> 00:08:57.310 space-- so x of t would equal-- this is just basic 00:08:57.310 --> 00:09:03.450 subsitution-- minus 3 cosine of 10 divided by 2, right? k 00:09:03.450 --> 00:09:04.970 over m, is 5. 00:09:04.970 --> 00:09:07.600 So square root of 5t. 00:09:07.600 --> 00:09:10.370 I know that's hard to read, but you get the point. 00:09:10.370 --> 00:09:11.860 I just substituted that. 00:09:11.860 --> 00:09:14.550 But the important thing to know is this-- this is, I 00:09:14.550 --> 00:09:17.360 think, the most important thing-- and then if given a 00:09:17.360 --> 00:09:19.690 trig function, you have trouble remembering how to 00:09:19.690 --> 00:09:21.690 figure out the period or frequency-- although I always 00:09:21.690 --> 00:09:26.120 just think about, when does this expression equal 1? 00:09:26.120 --> 00:09:29.952 And then you can figure out-- when does it equal 1, or when 00:09:29.952 --> 00:09:32.416 does it equal 0-- and you can figure out its period. 00:09:32.416 --> 00:09:33.360 If you don't have it, 00:09:33.360 --> 00:09:36.170 you can memorize this formula for period, and this formula 00:09:36.170 --> 00:09:38.830 for frequency, but I think that might be a waste of your 00:09:38.830 --> 00:09:39.830 brain space. 00:09:39.830 --> 00:09:43.090 Anyway, I'll see you in the next video.

Harmonic motion part 2 (calculus)

https://www.youtube.com/watch?v=xoUppFlif04

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https://www.youtube.com/api/timedtext?v=xoUppFlif04&ei=FGWUZYXzMoHNp-oP6ei7qAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249220&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=54E0E1BB4DE83FD1EEFDD4BAE2EEBC66F51847A3.2422FA0D2041B3BFEA308C19F52B9FA4F9C42BEC&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.780 --> 00:00:04.310 So where I left off in the last video, I'd just rewritten 00:00:04.310 --> 00:00:05.440 the spring equation. 00:00:05.440 --> 00:00:08.900 And I just wrote force is mass times acceleration. 00:00:08.900 --> 00:00:11.980 And I was in the process of saying, well if x is a 00:00:11.980 --> 00:00:14.000 function of t, what's acceleration? 00:00:14.000 --> 00:00:17.170 Well, velocity is this derivative of x with respect 00:00:17.170 --> 00:00:17.790 to time, right? 00:00:17.790 --> 00:00:20.060 Your change in position over change of time. 00:00:20.060 --> 00:00:23.350 And acceleration is the derivative of velocity, or the 00:00:23.350 --> 00:00:25.635 second derivative of position. 00:00:25.635 --> 00:00:29.350 So you take the derivative twice of x of t, right? 00:00:29.350 --> 00:00:35.930 So let's rewrite this equation in those terms. Let me erase 00:00:35.930 --> 00:00:37.570 all this--I actually want to keep all of this, just so we 00:00:37.570 --> 00:00:40.710 remember what we're talking about this whole time. 00:00:40.710 --> 00:00:43.800 Let me see if I can erase it cleanly. 00:00:43.800 --> 00:00:45.050 That's pretty good. 00:00:48.260 --> 00:00:50.270 Let me erase all of this. 00:00:56.600 --> 00:00:57.850 All of this. 00:00:57.850 --> 00:00:59.100 I'll even erase this. 00:01:02.430 --> 00:01:05.280 That's pretty good, all right. 00:01:05.280 --> 00:01:08.350 Now back to work. 00:01:08.350 --> 00:01:11.070 So, we know that-- or hopefully we know-- that 00:01:11.070 --> 00:01:13.390 acceleration is the second derivative of x as 00:01:13.390 --> 00:01:14.060 a function of t. 00:01:14.060 --> 00:01:18.910 So we can rewrite this as mass times the second 00:01:18.910 --> 00:01:21.290 derivative of x. 00:01:21.290 --> 00:01:23.870 So I'll write that as-- well, I think the easiest notation 00:01:23.870 --> 00:01:26.940 would just be x prime prime. 00:01:26.940 --> 00:01:29.640 That's just the second derivative of x as 00:01:29.640 --> 00:01:31.180 a function of t. 00:01:31.180 --> 00:01:33.530 I'll write the function notation, just so you remember 00:01:33.530 --> 00:01:36.190 this is a function of time. 00:01:36.190 --> 00:01:43.520 Is equal to minus k times x of t. 00:01:43.520 --> 00:01:46.080 And what you see here, what I've just written, this is 00:01:46.080 --> 00:01:50.270 actually a differential equation. 00:01:50.270 --> 00:01:51.650 And so what is a differential equation? 00:01:51.650 --> 00:01:55.030 Well, it's an equation where, in one expression, or in one 00:01:55.030 --> 00:01:57.560 equation, on both sides of this, you not only have a 00:01:57.560 --> 00:02:01.100 function, but you have derivatives of that function. 00:02:01.100 --> 00:02:05.670 And the solution to a differential equation isn't 00:02:05.670 --> 00:02:06.880 just a number, right? 00:02:06.880 --> 00:02:10.250 A solution to equations that we've done in the past are 00:02:10.250 --> 00:02:14.420 numbers, essentially, or a set of numbers, or maybe a line. 00:02:14.420 --> 00:02:17.660 But the solution to differential equations is 00:02:17.660 --> 00:02:20.250 actually going to be a function, or a class of 00:02:20.250 --> 00:02:22.470 functions, or a set of functions. 00:02:22.470 --> 00:02:25.700 So it'll take a little time to get your head around it, but 00:02:25.700 --> 00:02:28.100 this is as good an example as ever to be exposed to it. 00:02:28.100 --> 00:02:31.390 And we're not going to solve this differential equation 00:02:31.390 --> 00:02:32.130 analytically. 00:02:32.130 --> 00:02:35.550 We're going to use our intuition behind what we did 00:02:35.550 --> 00:02:37.710 earlier in the previous video. 00:02:37.710 --> 00:02:41.040 We're going to use that to guess at what a solution to 00:02:41.040 --> 00:02:42.710 this differential equation is. 00:02:42.710 --> 00:02:46.070 And then, if it works out, then we'll have a little bit 00:02:46.070 --> 00:02:46.640 more intuition. 00:02:46.640 --> 00:02:49.140 And then we'll actually know what the position is, at any 00:02:49.140 --> 00:02:52.870 given time, of this mass attached to the spring. 00:02:52.870 --> 00:02:53.700 So this is exciting. 00:02:53.700 --> 00:02:56.140 This is a differential equation. 00:02:56.140 --> 00:02:58.580 When we drew the position-- our intuition for the position 00:02:58.580 --> 00:03:01.070 over time-- our intuition tells us that it's a cosine 00:03:01.070 --> 00:03:02.790 function, with amplitude A. 00:03:02.790 --> 00:03:08.130 So we said it's A cosine omega t, where this is the angular 00:03:08.130 --> 00:03:11.400 velocity of-- well, I don't want to go into that just yet, 00:03:11.400 --> 00:03:13.080 we'll get a little bit more intuition in a second. 00:03:13.080 --> 00:03:17.490 And now, what we can do is, let's test this expression-- 00:03:17.490 --> 00:03:23.980 this function-- to see if it satisfies this equation. 00:03:23.980 --> 00:03:25.230 Right? 00:03:28.260 --> 00:03:39.880 If we say that x of t is equal to A cosine of wt, what is the 00:03:39.880 --> 00:03:42.530 derivative of this? x prime of t. 00:03:42.530 --> 00:03:45.450 And you could review the derivative 00:03:45.450 --> 00:03:47.560 videos to remember this. 00:03:47.560 --> 00:03:50.100 Well, it's the derivative of the inside, so it'll be that 00:03:50.100 --> 00:03:53.680 omega, times the outside scalar. 00:03:53.680 --> 00:03:56.650 A omega. 00:03:56.650 --> 00:03:58.915 And then the derivative-- I'm just doing the chain rule-- 00:03:58.915 --> 00:04:01.700 the derivative of cosine of t is minus sine of whatever's in 00:04:01.700 --> 00:04:03.070 the inside. 00:04:03.070 --> 00:04:04.780 I'll put the minus outside. 00:04:04.780 --> 00:04:11.040 So it's minus sine of wt. 00:04:11.040 --> 00:04:15.510 And then, if we want the second derivative-- so that's 00:04:15.510 --> 00:04:17.120 x prime prime of t. 00:04:20.899 --> 00:04:22.490 Let me do this in a different color, just so it doesn't get 00:04:22.490 --> 00:04:23.430 monotonous. 00:04:23.430 --> 00:04:25.540 That's the derivative of this, right? 00:04:25.540 --> 00:04:28.270 So what's the derivative of-- these are just 00:04:28.270 --> 00:04:29.260 scalar values, right? 00:04:29.260 --> 00:04:30.570 These are just constants. 00:04:30.570 --> 00:04:32.870 So the derivative of the inside is an omega. 00:04:32.870 --> 00:04:35.270 I multiply the omega times the scalar constant. 00:04:35.270 --> 00:04:43.190 I get minus A omega squared. 00:04:43.190 --> 00:04:45.350 And then the derivative of sine is just cosine. 00:04:45.350 --> 00:04:46.910 But the minus is still there, because I had the minus to 00:04:46.910 --> 00:04:47.950 begin with. 00:04:47.950 --> 00:04:54.422 Minus cosine of omega t. 00:04:54.422 --> 00:04:56.560 Now let's see if this is true. 00:04:56.560 --> 00:05:07.840 So if this is true, I should be able to say that m times 00:05:07.840 --> 00:05:10.740 the second derivative of x of t, which is in this case is 00:05:10.740 --> 00:05:21.750 this, times minus Aw squared cosine wt. 00:05:21.750 --> 00:05:31.430 That should be equal to minus k times my original function-- 00:05:31.430 --> 00:05:32.380 times x of t. 00:05:32.380 --> 00:05:34.020 And x of t is a cosine wt. 00:05:37.210 --> 00:05:39.530 I'm running out of space. 00:05:39.530 --> 00:05:41.590 Hopefully you understand what I'm saying. 00:05:41.590 --> 00:05:44.870 I just substituted x prime prime, the second derivative, 00:05:44.870 --> 00:05:51.310 into this, and I just substituted x of t, which I 00:05:51.310 --> 00:05:53.800 guess that's that, in here. 00:05:53.800 --> 00:05:55.070 And now I got this. 00:05:55.070 --> 00:05:56.570 And let me see if I can rewrite. 00:05:56.570 --> 00:05:58.670 Maybe I can get rid of the spring up here. 00:05:58.670 --> 00:05:59.510 I'm trying to look for space. 00:05:59.510 --> 00:06:00.930 I don't want to get rid of this, because I think this 00:06:00.930 --> 00:06:04.200 gives us some intuition of what we're doing. 00:06:04.200 --> 00:06:06.250 One of those days that I wish I had a larger blackboard. 00:06:10.680 --> 00:06:13.160 Erase the spring. 00:06:13.160 --> 00:06:16.490 Hopefully you can remember that image in your mind. 00:06:16.490 --> 00:06:19.190 And actually, I can erase that. 00:06:19.190 --> 00:06:21.945 I can erase that. 00:06:21.945 --> 00:06:24.600 I can erase all of this, just so I have some space, without 00:06:24.600 --> 00:06:27.150 getting rid of that nice curve I took the time to draw in the 00:06:27.150 --> 00:06:28.650 last video. 00:06:28.650 --> 00:06:29.750 Almost there. 00:06:29.750 --> 00:06:31.860 OK. 00:06:31.860 --> 00:06:33.110 Back to work. 00:06:35.470 --> 00:06:37.420 Make sure my pen feels right, OK. 00:06:37.420 --> 00:06:42.960 So all I did is I took-- we said that by the spring 00:06:42.960 --> 00:06:46.900 constant, if you rewrite force as mass times acceleration, 00:06:46.900 --> 00:06:47.580 you get this. 00:06:47.580 --> 00:06:49.340 Which is essentially a differential equation, I just 00:06:49.340 --> 00:06:52.220 rewrote acceleration as the second derivative. 00:06:52.220 --> 00:06:56.450 Then I took a guess, that this is x of t, just based on our 00:06:56.450 --> 00:06:58.500 intuition of the drawing. 00:06:58.500 --> 00:06:59.680 I took a guess. 00:06:59.680 --> 00:07:01.430 And then I took the second derivative of it. 00:07:01.430 --> 00:07:01.490 Right? 00:07:01.490 --> 00:07:04.160 This is the first derivative, this is the second derivative. 00:07:04.160 --> 00:07:06.300 And then I substituted the second derivative here, and I 00:07:06.300 --> 00:07:07.450 substituted the function here. 00:07:07.450 --> 00:07:08.770 And this is what I got. 00:07:08.770 --> 00:07:12.500 And so let me see if I can simplify that a little bit. 00:07:12.500 --> 00:07:28.550 So if I rewrite there, I get minus mAw squared cosine of wt 00:07:28.550 --> 00:07:37.720 is equal to minus kA cosine of wt. 00:07:37.720 --> 00:07:38.900 Well it looks good so far. 00:07:38.900 --> 00:07:41.685 Let's see, we can get rid of the minus signs on both sides. 00:07:46.450 --> 00:07:47.680 Get rid of the A's on both sides. 00:07:47.680 --> 00:07:47.750 Right? 00:07:47.750 --> 00:07:50.335 We can divide both sides by A. 00:07:50.335 --> 00:07:54.090 Let me do this in black, just so it really erases it. 00:07:54.090 --> 00:07:57.780 So if we get rid of A on both sides, we're left with that. 00:08:01.210 --> 00:08:04.690 And then-- so let's see, we have mw squared cosine of 00:08:04.690 --> 00:08:08.620 omega t is equal to k cosine of omega t. 00:08:08.620 --> 00:08:12.660 So this equation holds true if what is true? 00:08:12.660 --> 00:08:21.120 This equation holds true if mw squared-- or omega squared, I 00:08:21.120 --> 00:08:21.520 think that's omega. 00:08:21.520 --> 00:08:24.796 I always forget my-- is equal to k. 00:08:24.796 --> 00:08:29.110 Or another way of saying it, if omega squared is 00:08:29.110 --> 00:08:32.909 equal to k over m. 00:08:32.909 --> 00:08:40.309 Or, omega is equal to the square root of k over m. 00:08:40.309 --> 00:08:41.220 So there we have it. 00:08:41.220 --> 00:08:44.080 We have figured out what x of t has to be. 00:08:44.080 --> 00:08:47.650 We said that this differential equation is true, if this is x 00:08:47.650 --> 00:08:49.840 of t, and omega is equal to this. 00:08:49.840 --> 00:08:55.520 So now we've figured out the actual function that describes 00:08:55.520 --> 00:08:58.350 the position of that spring as a function of time. 00:08:58.350 --> 00:09:04.886 x of t is going to be equal to-- we were right about the 00:09:04.886 --> 00:09:07.590 A, and that's just intuition, right, because the amplitude 00:09:07.590 --> 00:09:13.820 of this cosine function is A-- A cosine-- and instead of 00:09:13.820 --> 00:09:18.880 writing w, we can now write the square root of k over m. 00:09:18.880 --> 00:09:26.060 The square root of k over m t. 00:09:26.060 --> 00:09:27.490 That to me is amazing. 00:09:27.490 --> 00:09:34.140 We have now, using not too sophisticated calculus, solved 00:09:34.140 --> 00:09:34.980 a differential equation. 00:09:34.980 --> 00:09:38.860 And now can-- if you tell me at 5.8 seconds, where is x, I 00:09:38.860 --> 00:09:39.920 can tell you. 00:09:39.920 --> 00:09:42.180 And I just realized that I am now running out of time, so I 00:09:42.180 --> 00:09:44.210 will see you in the next video.

Introduction to harmonic motion

https://www.youtube.com/watch?v=Nk2q-_jkJVs

vtt

https://www.youtube.com/api/timedtext?v=Nk2q-_jkJVs&ei=FGWUZeyqO_6vp-oPr6Cx2AE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249221&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=905E9E1F3901B424DB11861948A91D6A6E738C9E.BCB2C5A95EA81082D12DABD511F141C860221782&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.140 --> 00:00:04.080 Let's see if we can use what we know about springs now to 00:00:04.080 --> 00:00:05.640 get a little intuition about how the 00:00:05.640 --> 00:00:06.810 spring moves over time. 00:00:06.810 --> 00:00:07.600 And hopefully we'll learn a little bit 00:00:07.600 --> 00:00:08.660 about harmonic motion. 00:00:08.660 --> 00:00:11.010 We'll actually even step into the world of differential 00:00:11.010 --> 00:00:11.930 equations a little bit. 00:00:11.930 --> 00:00:14.200 And don't get daunted when we get there. 00:00:14.200 --> 00:00:15.950 Or just close your eyes when it happens. 00:00:15.950 --> 00:00:18.340 Anyway, so I've drawn a spring, like I've done in the 00:00:18.340 --> 00:00:19.410 last couple of videos. 00:00:19.410 --> 00:00:23.150 And 0, this point in the x-axis, that's where the 00:00:23.150 --> 00:00:25.500 spring's natural resting state is. 00:00:25.500 --> 00:00:28.890 And in this example I have a mass, mass m, 00:00:28.890 --> 00:00:30.220 attached to the spring. 00:00:30.220 --> 00:00:31.480 And I've stretched the string. 00:00:31.480 --> 00:00:32.680 I've essentially pulled it. 00:00:32.680 --> 00:00:35.480 So the mass is now sitting at point A. 00:00:35.480 --> 00:00:36.710 So what's going to happen to this? 00:00:36.710 --> 00:00:40.330 Well, as we know, the force, the restorative force of the 00:00:40.330 --> 00:00:45.230 spring, is equal to minus some 00:00:45.230 --> 00:00:47.360 constant, times the x position. 00:00:47.360 --> 00:00:48.860 The x position starting at A. 00:00:48.860 --> 00:00:50.910 So initially the spring is going to pull 00:00:50.910 --> 00:00:52.870 back this way, right? 00:00:52.870 --> 00:00:54.750 The spring is going to pull back this way. 00:00:54.750 --> 00:00:57.280 It's going to get faster and faster and faster and faster. 00:00:57.280 --> 00:00:58.710 And we learned that at this point, it has a lot of 00:00:58.710 --> 00:00:59.870 potential energy. 00:00:59.870 --> 00:01:02.110 At this point, when it kind of gets back to its resting 00:01:02.110 --> 00:01:06.940 state, it'll have a lot of velocity and a lot of kinetic 00:01:06.940 --> 00:01:08.680 energy, but very little potential energy. 00:01:08.680 --> 00:01:10.780 But then its momentum is going to keep it going, and it's 00:01:10.780 --> 00:01:14.950 going to compress the spring all the way, until all of that 00:01:14.950 --> 00:01:16.930 kinetic energy is turned back into potential energy. 00:01:16.930 --> 00:01:19.370 Then the process will start over again. 00:01:19.370 --> 00:01:22.550 So let's see if we can just get an intuition for what x 00:01:22.550 --> 00:01:24.160 will look like as a function of time. 00:01:24.160 --> 00:01:29.980 So our goal is to figure out x of t, x as a function of time. 00:01:29.980 --> 00:01:31.900 That's going to be our goal on this video and 00:01:31.900 --> 00:01:33.610 probably the next few. 00:01:33.610 --> 00:01:37.770 So let's just get an intuition for what's happening here. 00:01:37.770 --> 00:01:41.340 So let me try to graph x as a function of time. 00:01:41.340 --> 00:01:46.030 So time is the independent variable. 00:01:46.030 --> 00:01:49.440 And I'll start at time is equal to 0. 00:01:49.440 --> 00:01:52.060 So this is the time axis. 00:01:52.060 --> 00:01:53.100 Let me draw the x-axis. 00:01:53.100 --> 00:01:55.420 This might be a little unusual for you, for me to draw the 00:01:55.420 --> 00:01:57.860 x-axis in the vertical, but that's because x is the 00:01:57.860 --> 00:02:01.530 dependent variable in this situation. 00:02:01.530 --> 00:02:05.880 So that's the x-axis, very unusually. 00:02:05.880 --> 00:02:08.940 Or we could say x of t, just so you know x is a function of 00:02:08.940 --> 00:02:12.100 time, x of t. 00:02:12.100 --> 00:02:15.500 And this state, that I've drawn here, this is at time 00:02:15.500 --> 00:02:16.490 equals 0, right? 00:02:16.490 --> 00:02:17.400 So this is at 0. 00:02:17.400 --> 00:02:19.340 Let me switch colors. 00:02:19.340 --> 00:02:24.020 So at time equals 0, what is the x position of the mass? 00:02:24.020 --> 00:02:26.300 Well the x position is A, right? 00:02:26.300 --> 00:02:30.890 So if I draw this, this is A. 00:02:30.890 --> 00:02:32.280 Actually, let me draw a line there. 00:02:32.280 --> 00:02:34.700 That might come in useful. 00:02:34.700 --> 00:02:37.560 This is A. 00:02:37.560 --> 00:02:40.030 And then this is going to be-- let me try to make it 00:02:40.030 --> 00:02:44.420 relatively-- that is negative A. 00:02:44.420 --> 00:02:45.670 That's minus A. 00:02:49.070 --> 00:02:52.200 So at time t equals 0, where is it? 00:02:52.200 --> 00:02:52.920 Well it's at A. 00:02:52.920 --> 00:02:57.530 So this is where the graph is, right? 00:02:57.530 --> 00:02:59.880 Actually, let's do something interesting. 00:02:59.880 --> 00:03:01.740 Let's define the period. 00:03:01.740 --> 00:03:03.880 So the period I'll do with a capital T. 00:03:03.880 --> 00:03:07.950 Let's say the period is how long it takes for this mass to 00:03:07.950 --> 00:03:09.140 go from this position. 00:03:09.140 --> 00:03:11.020 It's going to accelerate, accelerate, accelerate, 00:03:11.020 --> 00:03:12.180 accelerate. 00:03:12.180 --> 00:03:14.800 Be going really fast at this point, all kinetic energy. 00:03:14.800 --> 00:03:17.080 And then start slowing down, slowing down, slowing down, 00:03:17.080 --> 00:03:17.800 slowing down. 00:03:17.800 --> 00:03:20.390 And then do that whole process all the way back. 00:03:20.390 --> 00:03:22.780 Let's say T is the amount of time it takes to do that whole 00:03:22.780 --> 00:03:24.560 process, right? 00:03:24.560 --> 00:03:31.850 So at time 0 today, and then we also know that at time T-- 00:03:31.850 --> 00:03:38.500 this is time T-- it'll also be at A, right? 00:03:38.500 --> 00:03:40.660 I'm just trying to graph some points that I know of this 00:03:40.660 --> 00:03:43.050 function and just see if I can get some intuition of what 00:03:43.050 --> 00:03:46.590 this function might be analytically. 00:03:46.590 --> 00:03:52.060 So if it takes T seconds to go there and back, it takes T 00:03:52.060 --> 00:03:54.090 over 2 seconds to get here, right? 00:03:54.090 --> 00:03:56.470 The same amount of time it takes to get here was also the 00:03:56.470 --> 00:03:58.770 same amount of time it takes to get back. 00:03:58.770 --> 00:04:05.880 So at T over 2 what's going to be the x position? 00:04:05.880 --> 00:04:08.780 Well at T over 2, the block is going to be here. 00:04:08.780 --> 00:04:10.690 It will have compressed all the way over here. 00:04:10.690 --> 00:04:12.550 So at T over 2, it'll have been here. 00:04:15.440 --> 00:04:18.680 And then at the points in between, it will be at x 00:04:18.680 --> 00:04:20.880 equals 0, right? 00:04:20.880 --> 00:04:23.310 It'll be there and there. 00:04:23.310 --> 00:04:24.600 Hopefully that makes sense. 00:04:24.600 --> 00:04:26.520 So now we know these points. 00:04:26.520 --> 00:04:28.900 But let's think about what the actual function looks like. 00:04:28.900 --> 00:04:30.880 Will it just be a straight line down, then a straight 00:04:30.880 --> 00:04:33.230 line up, and then the straight line down, and then a 00:04:33.230 --> 00:04:34.690 straight line up. 00:04:34.690 --> 00:04:37.220 That would imply-- think about it-- if you have a straight 00:04:37.220 --> 00:04:39.920 line down that whole time, that means that you would have 00:04:39.920 --> 00:04:43.910 a constant rate of change of your x value. 00:04:43.910 --> 00:04:45.550 Or another way of thinking about that is that you would 00:04:45.550 --> 00:04:48.270 have a constant velocity, right? 00:04:48.270 --> 00:04:51.200 Well do we have a constant velocity this entire time? 00:04:51.200 --> 00:04:51.880 Well, no. 00:04:51.880 --> 00:04:55.200 We know that at this point right here you have a very 00:04:55.200 --> 00:04:57.650 high velocity, right? 00:04:57.650 --> 00:04:58.850 You have a very high velocity. 00:04:58.850 --> 00:05:00.700 We know at this point you have a very low velocity. 00:05:00.700 --> 00:05:03.410 So you're accelerating this entire time. 00:05:03.410 --> 00:05:05.150 And you actually, the more you think about it, you're 00:05:05.150 --> 00:05:09.480 actually accelerating at a decreasing rate. 00:05:09.480 --> 00:05:11.610 But you're accelerating the entire time. 00:05:11.610 --> 00:05:15.120 And then you're accelerating and then you're decelerating 00:05:15.120 --> 00:05:16.180 this entire time. 00:05:16.180 --> 00:05:19.420 So your actual rate of change of x is not constant, so you 00:05:19.420 --> 00:05:21.740 wouldn't have a zigzag pattern, right? 00:05:21.740 --> 00:05:24.850 And it'll keep going here and then you'll have a point here. 00:05:24.850 --> 00:05:25.840 So what's happening? 00:05:25.840 --> 00:05:27.980 When you start off, you're going very slow. 00:05:27.980 --> 00:05:29.860 Your change of x is very slow. 00:05:29.860 --> 00:05:32.380 And then you start accelerating. 00:05:32.380 --> 00:05:36.290 And then, once you get to this point, right here, you start 00:05:36.290 --> 00:05:37.540 decelerating. 00:05:39.410 --> 00:05:44.050 Until at this point, your velocity is exactly 0. 00:05:44.050 --> 00:05:46.820 So your rate of change, or your slope, is going to be 0. 00:05:46.820 --> 00:05:49.890 And then you're going to start accelerating back. 00:05:49.890 --> 00:05:51.610 Your velocity is going to get faster, faster, faster. 00:05:51.610 --> 00:05:53.870 It's going to be really fast at this point. 00:05:53.870 --> 00:05:57.850 And then you'll start decelerating at that point. 00:05:57.850 --> 00:05:59.750 So at this point, what does this point correspond to? 00:05:59.750 --> 00:06:00.820 You're back at A. 00:06:00.820 --> 00:06:04.350 So at this point your velocity is now 0 again. 00:06:04.350 --> 00:06:06.350 So the rate of change of x is 0. 00:06:06.350 --> 00:06:08.650 And now you're going to start accelerating. 00:06:08.650 --> 00:06:11.290 Your slope increases, increases, increases. 00:06:11.290 --> 00:06:14.500 This is the point of highest kinetic energy right here. 00:06:14.500 --> 00:06:17.220 Then your velocity starts slowing down. 00:06:17.220 --> 00:06:20.420 And notice here, your slope at these points is 0. 00:06:20.420 --> 00:06:21.640 So that means you have no kinetic 00:06:21.640 --> 00:06:22.710 energy at those points. 00:06:22.710 --> 00:06:25.330 And it just keeps on going. 00:06:25.330 --> 00:06:27.590 On and on and on and on and on. 00:06:27.590 --> 00:06:28.970 So what does this look like? 00:06:28.970 --> 00:06:31.050 Well, I haven't proven it to you, but out of all the 00:06:31.050 --> 00:06:34.610 functions that I have in my repertoire, this looks an 00:06:34.610 --> 00:06:36.700 awful lot like a trigonometric function. 00:06:36.700 --> 00:06:38.860 And if I had to pick one, I would pick cosine. 00:06:38.860 --> 00:06:40.100 Well why? 00:06:40.100 --> 00:06:44.210 Because when cosine is 0-- I'll write it down here-- 00:06:44.210 --> 00:06:47.360 cosine of 0 is equal to 1, right? 00:06:47.360 --> 00:06:50.610 So when t equals 0, this function is equal to A. 00:06:50.610 --> 00:06:59.880 So this function probably looks something like A cosine 00:06:59.880 --> 00:07:05.730 of-- and I'll just use the variable omega t-- it probably 00:07:05.730 --> 00:07:08.650 looks something like that, this function. 00:07:08.650 --> 00:07:10.570 And we'll learn in a second that it looks 00:07:10.570 --> 00:07:11.110 exactly like that. 00:07:11.110 --> 00:07:12.460 But I want to prove it to you, so don't just 00:07:12.460 --> 00:07:13.690 take my word for it. 00:07:13.690 --> 00:07:17.490 So let's just figure out how we can figure out what w is. 00:07:17.490 --> 00:07:21.100 And it's probably a function of the mass of this object and 00:07:21.100 --> 00:07:22.910 also probably a function of the spring 00:07:22.910 --> 00:07:24.400 constant, but I'm not sure. 00:07:24.400 --> 00:07:26.620 So let's see what we can figure out. 00:07:26.620 --> 00:07:31.000 Well now I'm about to embark into a little bit of calculus. 00:07:31.000 --> 00:07:32.380 Actually, a decent bit of calculus. 00:07:32.380 --> 00:07:34.470 And we'll actually even touch on differential equations. 00:07:34.470 --> 00:07:36.790 This might be the first differential equation you see 00:07:36.790 --> 00:07:39.620 in your life, so it's a momentous occasion. 00:07:39.620 --> 00:07:41.120 But let's just move forward. 00:07:41.120 --> 00:07:42.670 Close your eyes if you don't want to be confused, or go 00:07:42.670 --> 00:07:46.290 watch the calculus videos at least so you know what a 00:07:46.290 --> 00:07:47.510 derivative is. 00:07:47.510 --> 00:07:52.190 So let's write this seemingly simple equation, or let's 00:07:52.190 --> 00:07:54.570 rewrite it in ways that we know. 00:07:54.570 --> 00:07:57.620 So what's the definition of force? 00:07:57.620 --> 00:08:00.030 Force is mass times acceleration, right? 00:08:00.030 --> 00:08:05.820 So we can rewrite Hooke's law as-- let me switch colors-- 00:08:05.820 --> 00:08:11.020 mass times acceleration is equal to minus the spring 00:08:11.020 --> 00:08:15.600 constant, times the position, right? 00:08:15.600 --> 00:08:17.840 And I'll actually write the position as a function of t, 00:08:17.840 --> 00:08:19.010 just so you remember. 00:08:19.010 --> 00:08:22.230 We're so used to x being the independent variable, that if 00:08:22.230 --> 00:08:24.470 I didn't write that function of t, it might get confusing. 00:08:24.470 --> 00:08:26.770 You're like, oh I thought x is the independent variable. 00:08:26.770 --> 00:08:28.350 No. 00:08:28.350 --> 00:08:31.150 Because in this function that we want to figure out, we want 00:08:31.150 --> 00:08:33.270 to know what happens as a function of time? 00:08:33.270 --> 00:08:35.280 So actually this is also maybe a good review 00:08:35.280 --> 00:08:38.080 of parametric equations. 00:08:38.080 --> 00:08:39.630 This is where we get into calculus. 00:08:39.630 --> 00:08:40.880 What is acceleration? 00:08:44.670 --> 00:08:51.650 If I call my position x, my position is equal to x as a 00:08:51.650 --> 00:08:52.980 function of t, right? 00:08:52.980 --> 00:08:56.500 I put in some time, and it tells me what my x value is. 00:08:56.500 --> 00:08:57.510 That's my position. 00:08:57.510 --> 00:08:58.890 What's my velocity? 00:08:58.890 --> 00:09:02.300 Well my velocity is the derivative of this, right? 00:09:02.300 --> 00:09:06.320 My velocity, at any given point, is going to be the 00:09:06.320 --> 00:09:07.890 derivative of this function. 00:09:07.890 --> 00:09:11.130 The rate of change of this function with respect to t. 00:09:11.130 --> 00:09:13.340 So I would take the rate of change with 00:09:13.340 --> 00:09:17.080 respect to t, x of t. 00:09:17.080 --> 00:09:23.030 And I could write that as dx, dt. 00:09:23.030 --> 00:09:24.490 And then what's acceleration? 00:09:24.490 --> 00:09:26.410 Well acceleration is just the rate of change 00:09:26.410 --> 00:09:28.340 of velocity, right? 00:09:28.340 --> 00:09:30.560 So it would be taking the derivative of this. 00:09:30.560 --> 00:09:32.940 Or another way of doing it, it's like taking the second 00:09:32.940 --> 00:09:36.460 derivative of the position function, right? 00:09:36.460 --> 00:09:41.850 So in this situation, acceleration is equal to, we 00:09:41.850 --> 00:09:44.930 could write it as-- I'm just showing you all different 00:09:44.930 --> 00:09:49.650 notations-- x prime prime of t, second derivative of x with 00:09:49.650 --> 00:09:50.480 respect to t. 00:09:50.480 --> 00:09:53.930 Or-- these are just notational-- d squared x over 00:09:53.930 --> 00:09:55.910 dt squared. 00:09:55.910 --> 00:09:57.000 So that's the second derivative. 00:09:57.000 --> 00:09:58.280 Oh it looks like I'm running out of time. 00:09:58.280 --> 00:09:59.480 So I'll see you in the next video. 00:09:59.480 --> 00:10:01.510 Remember what I just wrote. just wrote

Spring potential energy example (mistake in math)

https://www.youtube.com/watch?v=P3QV9ktuYlQ

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:01.460 Welcome back. 00:00:01.460 --> 00:00:04.490 So let's do a potential energy problem with 00:00:04.490 --> 00:00:05.510 a compressed spring. 00:00:05.510 --> 00:00:08.060 So let's make this an interesting problem. 00:00:08.060 --> 00:00:10.220 Let's say I have a loop-d-loop. 00:00:10.220 --> 00:00:12.120 A loop-d-loop made out of ice. 00:00:12.120 --> 00:00:15.080 And I made it out of ice so that we don't have friction. 00:00:15.080 --> 00:00:16.330 Let me draw my loop-d-loop. 00:00:19.900 --> 00:00:22.850 There's the loop, there's the d-loop. 00:00:22.850 --> 00:00:24.020 All right. 00:00:24.020 --> 00:00:29.475 And let's say this loop-d-loop has a radius of 1 meter. 00:00:29.475 --> 00:00:34.220 Let's say this is-- this right here-- is 1 meter. 00:00:34.220 --> 00:00:36.700 So of course the loop-d-loop is 2 meters high. 00:00:40.050 --> 00:00:42.380 And let's say I have a spring here-- it's 00:00:42.380 --> 00:00:43.760 a compressed spring. 00:00:43.760 --> 00:00:45.200 Let's say this is the wall. 00:00:45.200 --> 00:00:46.942 This is my spring, it's compressed, so it's 00:00:46.942 --> 00:00:49.000 all tight like that. 00:00:49.000 --> 00:00:53.050 And let's say its spring constant, k, is, 00:00:53.050 --> 00:00:56.150 I don't know, 10. 00:00:56.150 --> 00:00:59.650 Attached to that compressed spring-- so I have a block of 00:00:59.650 --> 00:01:03.860 ice, because I need ice on ice, so I have no friction. 00:01:03.860 --> 00:01:07.740 This is my block of ice, shining. 00:01:07.740 --> 00:01:16.930 And let's say the block of ice is, I don't know, 4 kilograms. 00:01:16.930 --> 00:01:19.860 And we also know that we are on Earth, and that's 00:01:19.860 --> 00:01:21.330 important, because this problem might have been 00:01:21.330 --> 00:01:23.660 different if we were on another planet. 00:01:23.660 --> 00:01:28.520 And my question to you is how much do we have to compress 00:01:28.520 --> 00:01:31.380 the spring-- so, let's say that the spring's natural 00:01:31.380 --> 00:01:36.000 state was here, right, if we didn't push on it. 00:01:36.000 --> 00:01:37.230 And now it's here. 00:01:37.230 --> 00:01:38.740 So what is this distance? 00:01:38.740 --> 00:01:42.800 How much do I have to compress this spring, in order for when 00:01:42.800 --> 00:01:47.615 I let go of the spring, the block goes with enough speed 00:01:47.615 --> 00:01:50.920 and enough energy, that it's able to complete the 00:01:50.920 --> 00:01:56.320 loop-d-loop, and reach safely to the other end? 00:01:56.320 --> 00:01:58.640 So, how do we do this problem? 00:01:58.640 --> 00:02:02.440 Well, in order-- any loop-d-loop problem, the hard 00:02:02.440 --> 00:02:04.870 part is completing the high point of the 00:02:04.870 --> 00:02:07.240 loop-d-loop, right? 00:02:07.240 --> 00:02:09.490 The hard part is making sure you have enough velocity at 00:02:09.490 --> 00:02:12.050 this point, so that you don't fall down. 00:02:12.050 --> 00:02:15.430 Your velocity has to offset the downward acceleraton, in 00:02:15.430 --> 00:02:17.530 which case-- and here, is going to be the centripetal 00:02:17.530 --> 00:02:19.320 acceleration, right? 00:02:19.320 --> 00:02:20.740 So that's one thing to think about. 00:02:20.740 --> 00:02:23.180 And you might say, wow this is complicated, I have a spring 00:02:23.180 --> 00:02:25.150 here, it's going to accelerate the block. 00:02:25.150 --> 00:02:26.720 And then the block's going to get here, and then it's going 00:02:26.720 --> 00:02:28.720 to decelerate, decelerate. 00:02:28.720 --> 00:02:30.720 This is probably where it's going to be at its slowest, 00:02:30.720 --> 00:02:32.610 then it's going to accelerate back here. 00:02:32.610 --> 00:02:34.430 It's a super complicated problem. 00:02:34.430 --> 00:02:36.400 And in physics, whenever you have a super complicated 00:02:36.400 --> 00:02:38.980 problem, it's probably because you are approaching it in a 00:02:38.980 --> 00:02:40.810 super complicated way, but there might be a 00:02:40.810 --> 00:02:41.610 simple way to do it. 00:02:41.610 --> 00:02:44.980 And that's using energy-- potential and kinetic energy. 00:02:44.980 --> 00:02:47.280 And what we learned when we learned about potential and 00:02:47.280 --> 00:02:50.190 kinetic energy, is that the total energy in the system 00:02:50.190 --> 00:02:51.520 doesn't change. 00:02:51.520 --> 00:02:53.370 It just gets converted from one form to another. 00:02:53.370 --> 00:02:55.820 So it goes from potential energy to kinetic 00:02:55.820 --> 00:02:58.680 energy, or to heat. 00:02:58.680 --> 00:02:59.890 And we assume that there's no heat, 00:02:59.890 --> 00:03:00.780 because there's no friction. 00:03:00.780 --> 00:03:02.940 So let's do this problem. 00:03:02.940 --> 00:03:05.970 So what we want to know is, how much do I have to compress 00:03:05.970 --> 00:03:06.760 this spring? 00:03:06.760 --> 00:03:09.580 So what I'm essentially saying is, how much potential energy 00:03:09.580 --> 00:03:13.680 do I have to start off with-- with this compressed spring-- 00:03:13.680 --> 00:03:15.900 in order to make it up here? 00:03:15.900 --> 00:03:17.310 So what's the potential energy? 00:03:17.310 --> 00:03:19.675 Let's say I compress the spring x meters. 00:03:22.340 --> 00:03:24.880 And in the last video, how much potential energy 00:03:24.880 --> 00:03:26.410 would I then have? 00:03:26.410 --> 00:03:28.720 Well, we learned that the potential energy of a 00:03:28.720 --> 00:03:32.040 compressed spring-- and I'll call this the initial 00:03:32.040 --> 00:03:37.110 potential energy-- the initial potential energy, with an i-- 00:03:37.110 --> 00:03:42.720 is equal to 1/2 kx squared. 00:03:42.720 --> 00:03:44.180 And we know what k is. 00:03:44.180 --> 00:03:47.140 I told you that the spring constant for the spring is 10. 00:03:47.140 --> 00:03:52.990 So my initial potential energy is going to be 1/2 times 10, 00:03:52.990 --> 00:03:54.240 times x squared. 00:03:58.010 --> 00:04:00.340 So what are all of the energy components here? 00:04:00.340 --> 00:04:02.520 Well, obviously, at this point, the block's going to 00:04:02.520 --> 00:04:05.160 have to be moving, in order to not fall down. 00:04:05.160 --> 00:04:07.990 So it's going to have some velocity, v. 00:04:07.990 --> 00:04:10.770 It's going tangential to the loop-d-loop. 00:04:10.770 --> 00:04:14.020 And it also is going to have some potential energy still. 00:04:14.020 --> 00:04:15.850 And where is that potential energy coming from? 00:04:15.850 --> 00:04:18.790 Well, it's going to come because it's up in the air. 00:04:18.790 --> 00:04:22.089 It's above the surface of the loop-d-loop. 00:04:22.089 --> 00:04:24.780 So it's going to have some gravitational potential 00:04:24.780 --> 00:04:26.450 energy, right? 00:04:26.450 --> 00:04:31.370 So at this point, we're going to have some kinetic energy. 00:04:31.370 --> 00:04:34.460 We'll call that-- well, I'll just call that kinetic energy 00:04:34.460 --> 00:04:36.690 final-- because this is while we care about alpha, maybe 00:04:36.690 --> 00:04:38.410 here it might be the kinetic energy final, but I'll just 00:04:38.410 --> 00:04:40.240 define this as kinetic energy final. 00:04:40.240 --> 00:04:45.580 And then plus the potential energy final. 00:04:45.580 --> 00:04:48.480 And that of course, has to add up to 10x squared. 00:04:48.480 --> 00:04:51.510 And this, of course, now, this was kind of called the spring 00:04:51.510 --> 00:04:52.850 potential energy, and now this is 00:04:52.850 --> 00:04:55.080 gravitational potential energy. 00:04:55.080 --> 00:04:57.780 So what's the energy at this point? 00:04:57.780 --> 00:04:59.660 Well, what's kinetic energy? 00:04:59.660 --> 00:05:06.590 Kinetic energy final is going to have to be 1/2 times the 00:05:06.590 --> 00:05:11.200 mass times the velocity squared, right? 00:05:11.200 --> 00:05:13.690 And then what's the potential energy at this point? 00:05:13.690 --> 00:05:16.660 It's gravitational potential energy, so it's the mass times 00:05:16.660 --> 00:05:19.380 gravity times this height. 00:05:19.380 --> 00:05:21.150 Right? 00:05:21.150 --> 00:05:22.070 So I'll write that here. 00:05:22.070 --> 00:05:27.250 Potential energy final is going to be mass times gravity 00:05:27.250 --> 00:05:29.940 times the height, which also stands for Mass General 00:05:29.940 --> 00:05:33.020 Hospital, anyway. 00:05:33.020 --> 00:05:35.750 You can tell my wife's a doctor, so my 00:05:35.750 --> 00:05:38.130 brain just-- anyway. 00:05:38.130 --> 00:05:41.360 So let's figure out the kinetic energy at this point. 00:05:41.360 --> 00:05:44.320 So what does the velocity have to be? 00:05:44.320 --> 00:05:46.430 Well, we have to figure out what the centripetal 00:05:46.430 --> 00:05:50.580 acceleration is, and then, given that, we can figure out 00:05:50.580 --> 00:05:51.120 the velocity. 00:05:51.120 --> 00:05:52.915 Because we know that the centripetal acceleration-- and 00:05:52.915 --> 00:05:55.730 I'll change colors for variety-- centripetal 00:05:55.730 --> 00:06:00.830 acceleration has to be the velocity squared, over the 00:06:00.830 --> 00:06:03.900 radius, right? 00:06:03.900 --> 00:06:06.780 Or we could say-- and what is the centripetal acceleration 00:06:06.780 --> 00:06:07.490 at this point? 00:06:07.490 --> 00:06:09.450 Well it's just the acceleration of gravity, 9.8 00:06:09.450 --> 00:06:11.410 meters per second squared. 00:06:11.410 --> 00:06:14.750 So 9.8 meters per second squared is equal to v 00:06:14.750 --> 00:06:16.470 squared over r. 00:06:16.470 --> 00:06:18.900 And what's the radius of this loop-d-loop? 00:06:18.900 --> 00:06:20.420 Well it's 1. 00:06:20.420 --> 00:06:21.940 So v squared over r is just going to 00:06:21.940 --> 00:06:23.420 be equal to v squared. 00:06:23.420 --> 00:06:26.110 So v squared equals 9.8-- we could take the square root, or 00:06:26.110 --> 00:06:27.740 we could just substitute the 9.8 straight into this 00:06:27.740 --> 00:06:29.420 equation, right? 00:06:29.420 --> 00:06:36.930 So the kinetic energy final is going to be equal to 1/2 times 00:06:36.930 --> 00:06:45.050 the mass times 4 times v squared times 9.8. 00:06:45.050 --> 00:06:50.770 And that equals-- let's just use g for 9.8, because I think 00:06:50.770 --> 00:06:53.110 that might keep it interesting. 00:06:53.110 --> 00:06:54.490 So this is just g, right? 00:06:54.490 --> 00:06:56.340 So it's 2 times g. 00:06:56.340 --> 00:07:03.610 So the kinetic energy final is equal to 2g-- and g is 00:07:03.610 --> 00:07:06.680 normally kilogram meters per second squared, but now it's 00:07:06.680 --> 00:07:07.600 energy, right? 00:07:07.600 --> 00:07:09.360 So it's going to be in joules. 00:07:09.360 --> 00:07:11.640 But it's 2g, right? 00:07:11.640 --> 00:07:13.260 And what is the potential energy at this point? 00:07:13.260 --> 00:07:18.470 Well, it's the mass, which is 4, times g times the height, 00:07:18.470 --> 00:07:19.490 which is 2. 00:07:19.490 --> 00:07:22.290 So it's equal to 8g. 00:07:22.290 --> 00:07:22.800 Right. 00:07:22.800 --> 00:07:24.770 So what's the total energy at this point? 00:07:24.770 --> 00:07:29.080 The kinetic energy is 2g, the potential energy is 8g, so the 00:07:29.080 --> 00:07:32.730 total energy at this point is 10g. 00:07:32.730 --> 00:07:36.580 10g total energy. 00:07:36.580 --> 00:07:38.950 So if the total energy at this point is 10g, and we didn't 00:07:38.950 --> 00:07:42.000 lose any energy to friction and heat, and all of that. 00:07:42.000 --> 00:07:44.800 So then the total energy at this point has also 00:07:44.800 --> 00:07:46.240 got to equal 10g. 00:07:46.240 --> 00:07:49.530 And at this point we have no kinetic energy, because this 00:07:49.530 --> 00:07:51.400 block hasn't started moving yet. 00:07:51.400 --> 00:07:53.210 So all the energy is a potential energy. 00:07:53.210 --> 00:07:56.280 So this also has to equal 10g. 00:07:56.280 --> 00:07:58.620 And this g, I keep saying, is just 9.8. 00:07:58.620 --> 00:08:00.750 I just wanted to do that just so you see that it's a 00:08:00.750 --> 00:08:04.140 multiple of 9.8, just for you to think about. 00:08:04.140 --> 00:08:04.900 So what do we have here? 00:08:04.900 --> 00:08:05.110 [? I'll do ?] 00:08:05.110 --> 00:08:07.060 these numbers worked out well. 00:08:07.060 --> 00:08:09.410 So let's divide both sides by 10. 00:08:09.410 --> 00:08:13.730 You get x squared is equal to g, which is 9.8. 00:08:13.730 --> 00:08:16.560 So the x is going to be equal to the square root of g, which 00:08:16.560 --> 00:08:19.420 is going to be equal to what? 00:08:19.420 --> 00:08:24.350 Let's see-- if I take 9.8, take the square root of it, 00:08:24.350 --> 00:08:26.360 it's like 3.13. 00:08:26.360 --> 00:08:30.170 So x is 3.13. 00:08:30.170 --> 00:08:34.049 So we just did a fairly-- what seemed to be a difficult 00:08:34.049 --> 00:08:35.179 problem, but it wasn't so bad. 00:08:35.179 --> 00:08:37.500 We just said that, well the energy in the beginning has to 00:08:37.500 --> 00:08:40.340 be the energy at any point in this, assuming that none of 00:08:40.340 --> 00:08:42.340 the energy is lost to heat. 00:08:42.340 --> 00:08:45.990 And so we just figured out that if we compress this 00:08:45.990 --> 00:08:48.540 spring, with the spring constant of 10. 00:08:48.540 --> 00:08:54.920 If we compress it 3.3 meters-- 3.13 meters-- we will have 00:08:54.920 --> 00:08:57.670 created enough potential energy-- and in this case, the 00:08:57.670 --> 00:09:01.730 potential energy is 10 times 9.8, so roughly 98 joules. 00:09:01.730 --> 00:09:06.400 98 joules of potential energy to carry this object all the 00:09:06.400 --> 00:09:09.150 way with enough velocity at the top of the loop-d-loop to 00:09:09.150 --> 00:09:11.330 complete it, and then come back down safely. 00:09:11.330 --> 00:09:13.280 And so if we wanted to think about it, what's the kinetic 00:09:13.280 --> 00:09:14.140 energy at this point? 00:09:14.140 --> 00:09:16.730 Well we figured out it was 2 times g, so 00:09:16.730 --> 00:09:23.120 it's like 19.6 joules. 00:09:23.120 --> 00:09:24.020 Right. 00:09:24.020 --> 00:09:30.590 And then at this point, it is 98 joules. 00:09:30.590 --> 00:09:30.950 Right? 00:09:30.950 --> 00:09:32.000 Did I do that right? 00:09:32.000 --> 00:09:35.160 Well, anyway I'm running out of time, so I hope I did do 00:09:35.160 --> 00:09:36.150 that last part right. 00:09:36.150 --> 00:09:38.130 But I'll see you in the next video.

Potential energy stored in a spring

https://www.youtube.com/watch?v=eVl5zs6Lqy0

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https://www.youtube.com/api/timedtext?v=eVl5zs6Lqy0&ei=FGWUZZeMO5uLp-oPsLqruA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249221&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=CE6B47E942669C0E7720E571B83CCED5E6C3BAA7.25A848128482EE4A33540EA7D23A26035E1A898F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.800 --> 00:00:01.770 Welcome back. 00:00:01.770 --> 00:00:03.750 So we have this green spring here, and let's see, 00:00:03.750 --> 00:00:04.910 there's a wall here. 00:00:04.910 --> 00:00:06.080 This connected to the wall. 00:00:06.080 --> 00:00:09.690 And let's say that this is where the spring is naturally. 00:00:09.690 --> 00:00:12.560 So if I were not to push on the spring, it would stretch 00:00:12.560 --> 00:00:14.040 all the way out here. 00:00:14.040 --> 00:00:16.410 But in this situation, I pushed on the spring, so it 00:00:16.410 --> 00:00:19.560 has a displacement of x to the left. 00:00:19.560 --> 00:00:21.450 And we'll just worry about magnitude, so we won't worry 00:00:21.450 --> 00:00:23.770 too much about direction. 00:00:23.770 --> 00:00:25.680 So what I want to do is think a little bit-- well, first I 00:00:25.680 --> 00:00:29.710 want to graph how much force I've applied at different 00:00:29.710 --> 00:00:31.920 points as I compress this spring. 00:00:31.920 --> 00:00:35.140 And then I want to use that graph to maybe figure out how 00:00:35.140 --> 00:00:38.270 much work we did in compressing the spring. 00:00:38.270 --> 00:00:45.300 So let's look at-- I know I'm compressing to the left. 00:00:45.300 --> 00:00:48.990 Maybe I should compress to the right, so that you can-- well, 00:00:48.990 --> 00:00:51.050 we're just worrying about the magnitude of the x-axis. 00:00:51.050 --> 00:00:53.960 Let's draw a little graph here. 00:00:53.960 --> 00:00:59.050 That's my y-axis, x-axis. 00:01:03.960 --> 00:01:09.350 So this axis is how much I've compressed it, x, and then 00:01:09.350 --> 00:01:14.170 this axis, the y-axis, is how much force I have to apply. 00:01:14.170 --> 00:01:18.850 So when the spring was initially all the way out 00:01:18.850 --> 00:01:21.180 here, to compress it a little bit, how much force 00:01:21.180 --> 00:01:22.670 do I have to apply? 00:01:22.670 --> 00:01:25.950 Well, this was its natural state, right? 00:01:25.950 --> 00:01:29.190 And we know from-- well, Hooke's Law told us that the 00:01:29.190 --> 00:01:34.570 restorative force-- I'll write a little r down here-- is 00:01:34.570 --> 00:01:39.400 equal to negative K, where K is the spring constant, times 00:01:39.400 --> 00:01:42.010 the displacement, right? 00:01:42.010 --> 00:01:44.580 That's the restorative force, so that's the force that the 00:01:44.580 --> 00:01:47.470 spring applies to whoever's pushing on it. 00:01:47.470 --> 00:01:50.260 The force to compress it is just the same thing, but it's 00:01:50.260 --> 00:01:51.980 going in the same direction as the x. 00:01:51.980 --> 00:01:55.310 If I'm moving the spring, if I'm compressing the spring to 00:01:55.310 --> 00:01:58.310 the left, then the force I'm applying is also to the left. 00:01:58.310 --> 00:02:01.030 So I'll call that the force of compression. 00:02:01.030 --> 00:02:03.290 The force of compression is going to be 00:02:03.290 --> 00:02:05.270 equal to K times x. 00:02:05.270 --> 00:02:07.440 And when the spring is compressed and not 00:02:07.440 --> 00:02:09.680 accelerating in either direction, the force of 00:02:09.680 --> 00:02:11.320 compression is going to be equal to 00:02:11.320 --> 00:02:12.730 the restorative force. 00:02:12.730 --> 00:02:16.120 So what I want to do here is plot the force of compression 00:02:16.120 --> 00:02:17.450 with respect to x. 00:02:17.450 --> 00:02:20.320 And I should have drawn it the other way, but I think you 00:02:20.320 --> 00:02:22.810 understand that x is increasing to the left in my 00:02:22.810 --> 00:02:23.760 example, right? 00:02:23.760 --> 00:02:30.270 This is where x is equal to 0 right here. 00:02:30.270 --> 00:02:33.220 And say, this might be x is equal to 10 because we've 00:02:33.220 --> 00:02:36.030 compressed it by 10 meters. 00:02:36.030 --> 00:02:38.530 So let's see how much force we've applied. 00:02:38.530 --> 00:02:43.120 So when x is 0, which is right here, how much force do we 00:02:43.120 --> 00:02:45.300 need to apply to compress the spring? 00:02:45.300 --> 00:02:48.600 Well, if we give zero force, the spring won't move, but if 00:02:48.600 --> 00:02:52.640 we just give a little, little bit of force, if we just give 00:02:52.640 --> 00:02:55.370 infinitesimal, super-small amount of force, we'll 00:02:55.370 --> 00:02:58.980 compress the spring just a little bit, right? 00:02:58.980 --> 00:03:01.530 Because at that point, the force of compression is going 00:03:01.530 --> 00:03:03.340 to be pretty much zero. 00:03:03.340 --> 00:03:06.830 So when the spring is barely compressed, we're going to 00:03:06.830 --> 00:03:12.380 apply a little, little bit of force, so almost at zero. 00:03:12.380 --> 00:03:15.330 To displace the spring zero, we apply zero force. 00:03:15.330 --> 00:03:17.630 To displace the spring a little bit, we have to apply a 00:03:17.630 --> 00:03:19.290 little bit more force. 00:03:19.290 --> 00:03:22.830 To displace soon. the spring 1 meter, so if this is say, 1 00:03:22.830 --> 00:03:28.870 meter, how much force will we have to 00:03:28.870 --> 00:03:31.380 apply to keep it there? 00:03:31.380 --> 00:03:36.690 So let's say if this is 1 meter, the force of 00:03:36.690 --> 00:03:38.630 compression is going to be K times 1, so it's 00:03:38.630 --> 00:03:39.880 just going to be K. 00:03:42.770 --> 00:03:46.790 And realize, you didn't apply zero and then apply K force. 00:03:46.790 --> 00:03:49.650 You keep applying a little bit more force. 00:03:49.650 --> 00:03:52.410 Every time you compress the spring a little bit, it takes 00:03:52.410 --> 00:03:55.670 a little bit more force to compress it a little bit more. 00:03:55.670 --> 00:03:58.950 So to compress it 1 meters, you need to apply K. 00:03:58.950 --> 00:04:01.770 And to get it there, you have to keep increasing the amount 00:04:01.770 --> 00:04:02.735 of force you apply. 00:04:02.735 --> 00:04:10.330 At 2 meters, you would've been up to 2K, et cetera. 00:04:10.330 --> 00:04:11.900 I think you see a line is forming. 00:04:11.900 --> 00:04:14.650 Let me draw that line. 00:04:14.650 --> 00:04:17.480 The line looks something like that. 00:04:17.480 --> 00:04:20.920 And so this is how much force you need to apply as a 00:04:20.920 --> 00:04:24.210 function of the displacement of the spring from its natural 00:04:24.210 --> 00:04:26.190 rest state, right? 00:04:26.190 --> 00:04:28.530 And here I have positive x going to the right, but in 00:04:28.530 --> 00:04:30.790 this case, positive x is to the left. 00:04:30.790 --> 00:04:32.470 I'm just measuring its actual displacement. 00:04:32.470 --> 00:04:36.460 I'm not worried too much about direction right now. 00:04:36.460 --> 00:04:38.320 So I just want you to think a little bit about what's 00:04:38.320 --> 00:04:39.350 happening here. 00:04:39.350 --> 00:04:42.330 You just have to slowly keep on-- you could apply a very 00:04:42.330 --> 00:04:43.920 large force initially. 00:04:43.920 --> 00:04:46.820 If you apply a very large force initially, the spring 00:04:46.820 --> 00:04:48.620 will actually accelerate much faster, because you're 00:04:48.620 --> 00:04:52.150 applying a much larger force than its restorative force, 00:04:52.150 --> 00:04:54.210 and so it might accelerate and then it'll spring back, and 00:04:54.210 --> 00:04:55.910 actually, we'll do a little example of that. 00:04:55.910 --> 00:04:59.260 But really, just to displace the spring a certain distance, 00:04:59.260 --> 00:05:01.730 you have to just gradually increase the force, just so 00:05:01.730 --> 00:05:04.530 that you offset the restorative force. 00:05:04.530 --> 00:05:06.780 Hopefully, that makes sense, and you understand that the 00:05:06.780 --> 00:05:09.280 force just increases proportionally as a function 00:05:09.280 --> 00:05:11.250 of the distance, and that's just because 00:05:11.250 --> 00:05:12.700 this is a linear equation. 00:05:12.700 --> 00:05:14.430 And what's the slope of this? 00:05:14.430 --> 00:05:17.250 Well, slope is rise over run, right? 00:05:17.250 --> 00:05:23.350 So if I run 1, this is 1, what's my rise? 00:05:23.350 --> 00:05:24.340 It's K. 00:05:24.340 --> 00:05:28.590 So the slope of this graph is K. 00:05:28.590 --> 00:05:31.400 So using this graph, let's figure out how much work we 00:05:31.400 --> 00:05:35.400 need to do to compress this spring. 00:05:35.400 --> 00:05:39.650 I don't know, let's say this is x0. 00:05:39.650 --> 00:05:41.115 So x is where it's the general variable. 00:05:41.115 --> 00:05:42.840 X0 is a particular value for x. 00:05:42.840 --> 00:05:44.390 That could be 10 or whatever. 00:05:44.390 --> 00:05:46.360 Let's see how much work we need. 00:05:46.360 --> 00:05:48.380 So what's the definition of work? 00:05:48.380 --> 00:05:52.750 Work is equal to the force in the direction of your 00:05:52.750 --> 00:05:57.770 displacement times the displacement, right? 00:05:57.770 --> 00:05:59.550 So let's see how much we've displaced. 00:05:59.550 --> 00:06:05.370 So when we go from zero to here, we've 00:06:05.370 --> 00:06:06.810 displaced this much. 00:06:06.810 --> 00:06:08.950 And what was the force of the displacement? 00:06:08.950 --> 00:06:11.890 Well, the force was gradually increasing the entire time, so 00:06:11.890 --> 00:06:16.330 the force is going to be be roughly about that big. 00:06:16.330 --> 00:06:17.920 I'm approximating. 00:06:17.920 --> 00:06:19.960 And I'll show you that you actually have to approximate. 00:06:19.960 --> 00:06:22.480 So the force is kind of that square right there. 00:06:25.980 --> 00:06:31.560 And then to displace the next little distance-- that's not 00:06:31.560 --> 00:06:34.830 bright enough-- my force is going to increase a little 00:06:34.830 --> 00:06:35.770 bit, right? 00:06:35.770 --> 00:06:38.120 So this is the force, this is the distance. 00:06:38.120 --> 00:06:41.000 So if you you see, the work I'm doing is actually going to 00:06:41.000 --> 00:06:42.880 be the area under the curve, each of 00:06:42.880 --> 00:06:43.890 these rectangles, right? 00:06:43.890 --> 00:06:46.080 Because the height of the rectangle is the force I'm 00:06:46.080 --> 00:06:50.340 applying and the width is the distance, right? 00:06:50.340 --> 00:06:52.960 So the work is just going to be the sum of all of these 00:06:52.960 --> 00:06:53.740 rectangles. 00:06:53.740 --> 00:06:56.450 And the rectangles I drew are just kind of approximations, 00:06:56.450 --> 00:06:57.590 because they don't get right under the line. 00:06:57.590 --> 00:06:59.590 You have to keep making the rectangle smaller, smaller, 00:06:59.590 --> 00:07:02.660 smaller, and smaller, and just sum up more and more and more 00:07:02.660 --> 00:07:04.360 rectangles, right? 00:07:04.360 --> 00:07:07.880 And actually I'm touching on integral calculus right now. 00:07:07.880 --> 00:07:09.130 But if you don't know integral calculus, 00:07:09.130 --> 00:07:09.950 don't worry about it. 00:07:09.950 --> 00:07:12.310 But the bottom line is the work we're doing-- hopefully I 00:07:12.310 --> 00:07:16.290 showed you-- is just going to be the area under this line. 00:07:16.290 --> 00:07:20.910 So the work I'm doing to displace the spring x meters 00:07:20.910 --> 00:07:26.020 is the area from here to here. 00:07:26.020 --> 00:07:27.430 And what's that area? 00:07:27.430 --> 00:07:30.590 Well, this is a triangle, so we just need to know the base, 00:07:30.590 --> 00:07:32.810 the height, and multiply it times 1/2, right? 00:07:32.810 --> 00:07:34.530 That's just the area of a triangle. 00:07:34.530 --> 00:07:36.050 So what's the base? 00:07:36.050 --> 00:07:39.860 So this is just x0. 00:07:39.860 --> 00:07:41.520 What's the height? 00:07:41.520 --> 00:07:44.360 Well, we know the slope is K, so this height is going to be 00:07:44.360 --> 00:07:47.750 x0 times K. 00:07:47.750 --> 00:07:51.370 So this point right here is the point x0, and 00:07:51.370 --> 00:07:54.610 then x0 times K. 00:07:54.610 --> 00:07:57.840 And so what's the area under the curve, which is the total 00:07:57.840 --> 00:08:02.090 work I did to compress the spring x0 meters? 00:08:02.090 --> 00:08:09.500 Well, it's the base, x0, times the height, x0, times K. 00:08:09.500 --> 00:08:12.280 And then, of course, multiply by 1/2, because we're dealing 00:08:12.280 --> 00:08:14.420 with a triangle, right? 00:08:14.420 --> 00:08:19.230 So that equals 1/2K x0 squared. 00:08:19.230 --> 00:08:22.050 And for those of you who know calculus, that, of course, is 00:08:22.050 --> 00:08:25.910 the same thing as the integral of Kx dx. 00:08:25.910 --> 00:08:27.080 And that should make sense. 00:08:27.080 --> 00:08:28.180 Each of these are little dx's. 00:08:28.180 --> 00:08:29.490 But I don't want to go too much into calculus now. 00:08:29.490 --> 00:08:31.340 It'll confuse people. 00:08:31.340 --> 00:08:33.950 So that's the total work necessary to compress the 00:08:33.950 --> 00:08:36.950 spring by distance of x0. 00:08:36.950 --> 00:08:38.659 Or if we set a distance of x, you can just get 00:08:38.659 --> 00:08:40.340 rid of this 0 here. 00:08:40.340 --> 00:08:41.770 And why is that useful? 00:08:41.770 --> 00:08:44.700 Because the work necessary to compress the spring that much 00:08:44.700 --> 00:08:47.800 is also how much potential energy there is 00:08:47.800 --> 00:08:49.340 stored in the spring. 00:08:49.340 --> 00:08:57.750 So if I told you that I had a spring and its spring constant 00:08:57.750 --> 00:09:03.890 is 10, and I compressed it 5 meters, so x is equal to 5 00:09:03.890 --> 00:09:08.800 meters, at the time that it's compressed, how much potential 00:09:08.800 --> 00:09:10.780 energy is in that spring? 00:09:10.780 --> 00:09:15.940 We can just say the potential energy is equal to 1/2K times 00:09:15.940 --> 00:09:18.790 x squared equals 1/2. 00:09:18.790 --> 00:09:25.050 K is 10 times 25, and that equals 125. 00:09:25.050 --> 00:09:26.920 And, of course, work and potential energy 00:09:26.920 --> 00:09:29.980 are measured in joules. 00:09:29.980 --> 00:09:33.110 So this is really what you just have to memorize. 00:09:33.110 --> 00:09:34.240 Or hopefully you don't memorize it. 00:09:34.240 --> 00:09:36.700 Hopefully, you understand where I got it, and that's why 00:09:36.700 --> 00:09:38.030 I spent 10 minutes doing it. 00:09:38.030 --> 00:09:39.940 But this is how much work is necessary to compress the 00:09:39.940 --> 00:09:43.030 spring to that point and how much potential energy is 00:09:43.030 --> 00:09:45.790 stored once it is compressed to that point, or actually 00:09:45.790 --> 00:09:47.490 stretched that much. 00:09:47.490 --> 00:09:48.520 We've been compressing, but you can 00:09:48.520 --> 00:09:50.690 also stretch the spring. 00:09:50.690 --> 00:09:53.730 If you know that, then we can start doing some problems with 00:09:53.730 --> 00:09:56.060 potential energy in springs, which I will 00:09:56.060 --> 00:09:57.110 do in the next video. 00:09:57.110 --> 00:09:58.360 See

Intro to springs and Hooke's law

https://www.youtube.com/watch?v=ZzwuHS9ldbY

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https://www.youtube.com/api/timedtext?v=ZzwuHS9ldbY&ei=FWWUZdW0JM6sp-oP_YmruAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249221&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=23E385313903231FF4DDD5D8A0021368FBF93735.1728C13156B36B45CD7CDF1EEAC9572F615E2D9F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.790 --> 00:00:02.876 Let's learn a little bit about springs. 00:00:02.876 --> 00:00:05.100 So let's say I have a spring. 00:00:05.100 --> 00:00:08.870 Let me draw the ground so that we know what's going on with 00:00:08.870 --> 00:00:09.490 the spring. 00:00:09.490 --> 00:00:13.280 So let me see, this is the floor. 00:00:13.280 --> 00:00:16.830 That's the floor, and I have a spring. 00:00:16.830 --> 00:00:18.680 It's along the floor. 00:00:18.680 --> 00:00:21.000 I'll use a thicker one, just to show it's a spring. 00:00:21.000 --> 00:00:24.060 Let's say the spring looks something like this. 00:00:24.060 --> 00:00:26.170 Whoops, I'm still using the line tool. 00:00:26.170 --> 00:00:27.430 So the spring looks like this. 00:00:27.430 --> 00:00:32.299 This is my spring, my amazingly drawn spring. 00:00:32.299 --> 00:00:36.170 Let's say at this end it's attached to a wall. 00:00:36.170 --> 00:00:38.980 That's a wall. 00:00:38.980 --> 00:00:41.470 And so this is a spring when I don't have any force acting on 00:00:41.470 --> 00:00:44.880 it, this is just the natural state of the spring. 00:00:44.880 --> 00:00:50.050 And we could call this, where it just naturally rests, this 00:00:50.050 --> 00:00:51.340 tip of the spring. 00:00:51.340 --> 00:00:57.640 And let's say that when I were to apply a force of 5 Newtons 00:00:57.640 --> 00:01:00.410 into the spring, it looks something like this. 00:01:00.410 --> 00:01:03.210 Redraw everything. 00:01:03.210 --> 00:01:11.355 So when I apply a force of 5 Newtons-- I'll draw the wall 00:01:11.355 --> 00:01:12.700 in magenta now. 00:01:16.180 --> 00:01:19.300 When I apply a force of 5 Newtons, the 00:01:19.300 --> 00:01:20.550 spring looks like this. 00:01:25.510 --> 00:01:26.880 It compresses, right? 00:01:26.880 --> 00:01:28.060 We're all familiar with this. 00:01:28.060 --> 00:01:30.320 We sit on a bed every day or a sofa. 00:01:30.320 --> 00:01:31.620 So let's say it compresses to here. 00:01:37.220 --> 00:01:40.960 If this was the normal resting-- so this is where the 00:01:40.960 --> 00:01:44.000 spring was when I applied no force, but when I applied 5 00:01:44.000 --> 00:01:53.420 Newtons in that direction, let's say that this distance 00:01:53.420 --> 00:02:01.290 right here is 10 meters. 00:02:01.290 --> 00:02:03.940 And so a typical question that you'll see, and we'll explain 00:02:03.940 --> 00:02:08.759 how to do it, is a spring compresses or elongates when 00:02:08.759 --> 00:02:11.950 you apply a certain force by some distance. 00:02:11.950 --> 00:02:14.010 How much will it compress when you apply a different force? 00:02:14.010 --> 00:02:16.615 So my question is how much will it compress when I apply 00:02:16.615 --> 00:02:19.260 a 10-Newton force? 00:02:19.260 --> 00:02:22.480 So your intuition that it'll compress more is correct, but 00:02:22.480 --> 00:02:27.220 is it linear to how much I compress it? 00:02:27.220 --> 00:02:30.150 Is it a square of how much I compress it? 00:02:30.150 --> 00:02:32.175 How does it relate? 00:02:32.175 --> 00:02:34.480 I think you probably could guess. 00:02:34.480 --> 00:02:36.880 It's actually worth an experiment. 00:02:36.880 --> 00:02:39.280 Or you could just keep watching the video. 00:02:39.280 --> 00:02:42.020 So let's say I apply a 10-Newton force. 00:02:42.020 --> 00:02:43.180 What will the spring look like? 00:02:43.180 --> 00:02:44.740 Well, it'll be more compressed. 00:02:49.900 --> 00:02:55.780 Drop my force to 10 Newtons. 00:02:55.780 --> 00:02:58.060 And if this was the natural place where the spring would 00:02:58.060 --> 00:03:00.580 rest, what is this distance? 00:03:00.580 --> 00:03:02.280 Well, it turns out that it is linear. 00:03:02.280 --> 00:03:03.590 What do I mean by linear? 00:03:03.590 --> 00:03:07.860 Well, it means that the more the force-- it's equally 00:03:07.860 --> 00:03:10.750 proportional to how much the spring will compress. 00:03:10.750 --> 00:03:12.060 And it actually works the other way. 00:03:12.060 --> 00:03:15.040 If you applied 5 Newtons in this direction, to the right, 00:03:15.040 --> 00:03:18.670 you would have gone 10 meters in this direction. 00:03:18.670 --> 00:03:20.940 So it goes whether you're elongating the spring or 00:03:20.940 --> 00:03:24.450 compressing the spring within some reasonable tolerance. 00:03:24.450 --> 00:03:26.620 We've all had this experience. 00:03:26.620 --> 00:03:29.300 If you compress something too much or you stretch it too 00:03:29.300 --> 00:03:32.070 much, it doesn't really go back to where it was before. 00:03:32.070 --> 00:03:34.430 But within some reasonable tolerance, it's proportional. 00:03:34.430 --> 00:03:35.830 So what does that mean? 00:03:35.830 --> 00:03:43.990 That means that the restoring force of the spring is minus 00:03:43.990 --> 00:03:47.840 some number, times the displacement of the spring. 00:03:47.840 --> 00:03:49.040 So what does this mean? 00:03:49.040 --> 00:03:52.110 So in this example right here, what was the displacement of 00:03:52.110 --> 00:03:53.140 the spring? 00:03:53.140 --> 00:03:56.528 Well, if we take positive x to the right and negative x to 00:03:56.528 --> 00:04:00.980 the left, the displacement of the spring was what? 00:04:00.980 --> 00:04:05.010 The displacement, in this example right here, x is equal 00:04:05.010 --> 00:04:06.090 to minus 10, right? 00:04:06.090 --> 00:04:08.180 Because I went 10 to the left. 00:04:08.180 --> 00:04:13.520 And so it says that the restorative force is going to 00:04:13.520 --> 00:04:21.310 be equal to minus K times how much it's 00:04:21.310 --> 00:04:23.260 distorted times minus 10. 00:04:23.260 --> 00:04:27.470 So the minuses cancel out, so it equals 10K. 00:04:27.470 --> 00:04:30.420 What's the restorative force in this example? 00:04:30.420 --> 00:04:32.210 Well, you might say, it's 5 Newtons, just because that's 00:04:32.210 --> 00:04:37.550 the only force I've drawn here, and you would be to some 00:04:37.550 --> 00:04:38.160 degree correct. 00:04:38.160 --> 00:04:40.260 And actually, since we're doing positive and negative, 00:04:40.260 --> 00:04:43.570 and this 5 Newton is to the left, so to the negative 00:04:43.570 --> 00:04:45.770 x-direction, actually, I should call this minus 5 00:04:45.770 --> 00:04:48.090 Newtons and I should call this minus 10 Newtons, because 00:04:48.090 --> 00:04:50.340 obviously, these are vectors and we're going to the left. 00:04:50.340 --> 00:04:53.630 I picked the convention that to the left means negative. 00:04:53.630 --> 00:04:54.980 So what's the restorative force? 00:04:54.980 --> 00:04:57.820 Well, in this example-- and we assume that K is a positive 00:04:57.820 --> 00:05:00.500 number for our purposes. 00:05:00.500 --> 00:05:02.370 In this example, the restorative force is a 00:05:02.370 --> 00:05:03.660 positive number. 00:05:03.660 --> 00:05:05.310 So what is the restorative force? 00:05:05.310 --> 00:05:08.970 So that's actually the force, the counteracting force, of 00:05:08.970 --> 00:05:10.030 the spring. 00:05:10.030 --> 00:05:12.680 That's what this formula gives us. 00:05:12.680 --> 00:05:15.220 So if this spring is stationary when I apply this 00:05:15.220 --> 00:05:18.030 5-Newton force, that means that there must be another 00:05:18.030 --> 00:05:19.830 equal and opposite force that's 00:05:19.830 --> 00:05:22.160 positive 5 Newtons, right? 00:05:22.160 --> 00:05:24.460 If there weren't, the spring would keep compressing. 00:05:24.460 --> 00:05:27.240 And if the force was more than 5 Newtons, the spring would go 00:05:27.240 --> 00:05:28.620 back this way. 00:05:28.620 --> 00:05:32.010 So the fact that I know that when I apply a 5-Newton force 00:05:32.010 --> 00:05:34.920 to the left, or a negative 5-Newton force, the spring is 00:05:34.920 --> 00:05:37.130 no longer moving, it means that there must be-- or no 00:05:37.130 --> 00:05:39.830 longer accelerating, actually, it means that there must be an 00:05:39.830 --> 00:05:42.330 equal and opposite force to the right, and that's the 00:05:42.330 --> 00:05:43.650 restorative force. 00:05:43.650 --> 00:05:47.370 Another way to think about it is if I were to let-- well, I 00:05:47.370 --> 00:05:48.510 won't go in there now. 00:05:48.510 --> 00:05:50.990 So in this case, the restorative force is 5 00:05:50.990 --> 00:05:52.950 Newtons, so we can solve for K. 00:05:52.950 --> 00:05:56.940 We could say 5 is equal to 10K. 00:05:56.940 --> 00:05:58.110 Divide both sides by 10. 00:05:58.110 --> 00:05:59.380 You get K is equal to 1/2. 00:06:03.850 --> 00:06:06.610 So now we can use that information to figure out what 00:06:06.610 --> 00:06:10.750 is the displacement when I apply a 00:06:10.750 --> 00:06:12.050 negative 10-Newton force? 00:06:12.050 --> 00:06:15.410 When I push the spring in with 10 Newtons in 00:06:15.410 --> 00:06:16.790 the leftward direction? 00:06:16.790 --> 00:06:19.050 So first of all, what's the restorative force here? 00:06:19.050 --> 00:06:22.790 Well, if the spring is no longer accelerating in either 00:06:22.790 --> 00:06:25.710 direction, or the tip of the spring is no longer 00:06:25.710 --> 00:06:28.120 accelerating in either direction, we know that the 00:06:28.120 --> 00:06:30.540 restorative force must be counterbalancing this force 00:06:30.540 --> 00:06:32.290 that I'm compressing with, right? 00:06:32.290 --> 00:06:35.810 The force that the spring wants to expand back with is 00:06:35.810 --> 00:06:39.030 10 Newtons, positive 10 Newtons, right? 00:06:39.030 --> 00:06:42.550 And we know the spring constant, this K for this 00:06:42.550 --> 00:06:46.780 spring, for this material, whatever it might be, is 1/2. 00:06:46.780 --> 00:06:53.320 So we know the restorative force is equal to 1/2 times 00:06:53.320 --> 00:06:55.280 the distance, right? 00:06:55.280 --> 00:06:58.940 And the formula is minus K, right? 00:06:58.940 --> 00:07:00.670 And then, what is the restorative 00:07:00.670 --> 00:07:02.340 force in this example? 00:07:02.340 --> 00:07:06.190 Well I said it's 10 Newtons, so we know that 10 Newtons is 00:07:06.190 --> 00:07:07.440 equal to minus 1/2x. 00:07:09.950 --> 00:07:11.750 And so what is x? 00:07:11.750 --> 00:07:14.030 Well, multiply both sides by minus 1/2, and 00:07:14.030 --> 00:07:15.920 you get minus 20. 00:07:15.920 --> 00:07:18.455 I'm sorry, multiply both sides by minus 2, you get minus 20 00:07:18.455 --> 00:07:19.705 is equal to x. 00:07:22.900 --> 00:07:25.700 So x goes to the left 20 units. 00:07:25.700 --> 00:07:28.390 So that's all that it's telling us. 00:07:28.390 --> 00:07:32.710 And this law is called Hooke's Law, and it's named after-- 00:07:32.710 --> 00:07:35.810 I'll read it-- a physicist in the 17th century, a British 00:07:35.810 --> 00:07:39.170 physicist. And he figured out that the amount of force 00:07:39.170 --> 00:07:44.460 necessary to keep a spring compressed is proportional to 00:07:44.460 --> 00:07:46.420 how much you've compressed it. 00:07:46.420 --> 00:07:48.570 And that's all that this formula says. 00:07:48.570 --> 00:07:51.150 And that negative number, remember, this formula gives 00:07:51.150 --> 00:07:52.710 us the restorative force. 00:07:52.710 --> 00:07:55.910 So it says that the force is always in the opposite 00:07:55.910 --> 00:07:57.860 direction of how much you displace it. 00:07:57.860 --> 00:08:01.470 So, for example, if you were to displace this spring in 00:08:01.470 --> 00:08:04.590 this direction, if you were to apply a force and x were a 00:08:04.590 --> 00:08:08.620 positive and you were to go in that direction, the force-- no 00:08:08.620 --> 00:08:09.150 wait, sorry. 00:08:09.150 --> 00:08:11.320 This is where the spring rests. 00:08:11.320 --> 00:08:14.260 If you were to apply some force and take the spring out 00:08:14.260 --> 00:08:18.610 to here, this negative number tells us that the spring will 00:08:18.610 --> 00:08:21.610 essentially try to pull back with the restorative force in 00:08:21.610 --> 00:08:24.190 the other direction. 00:08:24.190 --> 00:08:26.750 Let's do one more problem and I think this 00:08:26.750 --> 00:08:29.570 will be clear to you. 00:08:29.570 --> 00:08:33.460 So let's say I have a spring, and all of these problems kind 00:08:33.460 --> 00:08:35.090 of go along. 00:08:35.090 --> 00:08:40.280 So let's say when I apply a force of 2 Newtons, so this is 00:08:40.280 --> 00:08:44.880 what I apply when I apply a force of 2 Newtons. 00:08:44.880 --> 00:08:46.180 Well, let's say it this way. 00:08:46.180 --> 00:08:49.740 Let's say when I stretch the spring. 00:08:49.740 --> 00:08:55.650 Let's say this is the spring, and when I apply a force of 2 00:08:55.650 --> 00:09:05.940 Newtons to the right, the spring gets stretched 1 meter. 00:09:05.940 --> 00:09:09.280 So first of all, let's figure out what K is. 00:09:09.280 --> 00:09:14.030 So if the spring is stretched by 1 meter, out here, its 00:09:14.030 --> 00:09:18.650 restorative force will be 2 Newtons back this way, right? 00:09:18.650 --> 00:09:22.620 So its restorative force, this 2 Newtons, will equal minus K 00:09:22.620 --> 00:09:24.530 times how much I displaced it. 00:09:24.530 --> 00:09:28.050 Well I, displaced it by 1 meter, so then we multiply 00:09:28.050 --> 00:09:32.380 both sides by negative 1, and we get K is equal to minus 2. 00:09:32.380 --> 00:09:37.350 So then we can use Hooke's Law to note the equation for 00:09:37.350 --> 00:09:40.450 this-- to figure out the restorative force for this 00:09:40.450 --> 00:09:44.110 particular spring, and it would be minus 2x. 00:09:44.110 --> 00:09:46.530 And then I said, well, how much force would I have to 00:09:46.530 --> 00:09:49.730 apply to distort the spring by 2 meters? 00:09:49.730 --> 00:09:52.420 Well, it's 2 times 2, it would be 4. 00:09:52.420 --> 00:09:57.330 4 Newtons to displace it by 2 meters, and, of course, the 00:09:57.330 --> 00:09:59.580 restorative force will then be in the opposite direction, and 00:09:59.580 --> 00:10:01.620 that's where we get the negative number. 00:10:01.620 --> 00:10:02.800 Anyway, I've run out of time. 00:10:02.800 --> 00:10:04.860 I'll see you in the next video.

Introduction to Newton's law of gravitation

https://www.youtube.com/watch?v=391txUI76gM

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WEBVTT Kind: captions Language: en 00:00:00.660 --> 00:00:02.590 We're now going to learn a little bit about gravity. 00:00:02.590 --> 00:00:05.590 And just so you know, gravity is something that, especially 00:00:05.590 --> 00:00:08.620 in introductory physics or even reasonably advanced 00:00:08.620 --> 00:00:11.600 physics, we can learn how to calculate it, we can learn how 00:00:11.600 --> 00:00:14.650 to realize what are the important variables in it, but 00:00:14.650 --> 00:00:17.050 it's something that's really not well understood. 00:00:17.050 --> 00:00:20.940 Even once you learn general relativity, if you do get 00:00:20.940 --> 00:00:23.880 there, I have to say, you can kind of say, oh, well, it's 00:00:23.880 --> 00:00:26.120 the warping of space time and all of this, but it's hard to 00:00:26.120 --> 00:00:31.190 get an intuition of why two objects, just because they 00:00:31.190 --> 00:00:33.690 have this thing called mass, they are 00:00:33.690 --> 00:00:34.640 attracted to each other. 00:00:34.640 --> 00:00:38.170 It's really, at least to me, a little bit mystical. 00:00:38.170 --> 00:00:42.240 But with that said, let's learn to deal with gravity. 00:00:42.240 --> 00:00:45.140 And we'll do that learning Newton's Law of Gravity, and 00:00:45.140 --> 00:00:48.620 this works for most purposes. 00:00:48.620 --> 00:00:50.640 So Newton's Law of Gravity says that the force between 00:00:50.640 --> 00:00:55.140 two masses, and that's the gravitational force, is equal 00:00:55.140 --> 00:00:58.990 to the gravitational constant G times the mass of the first 00:00:58.990 --> 00:01:03.500 object times the mass of the second object divided by the 00:01:03.500 --> 00:01:06.890 distance between the two objects squared. 00:01:06.890 --> 00:01:07.860 So that's simple enough. 00:01:07.860 --> 00:01:10.320 So let's play around with this, and see if we can get 00:01:10.320 --> 00:01:13.490 some results that look reasonably familiar to us. 00:01:13.490 --> 00:01:17.120 So let's use this formula to figure out what the 00:01:17.120 --> 00:01:21.070 acceleration, the gravitational acceleration, is 00:01:21.070 --> 00:01:23.120 at the surface of the Earth. 00:01:23.120 --> 00:01:26.130 So let's draw the Earth, just so we know what 00:01:26.130 --> 00:01:27.380 we're talking about. 00:01:29.830 --> 00:01:31.630 So that's my Earth. 00:01:31.630 --> 00:01:34.060 And let's say we want to figure out the gravitational 00:01:34.060 --> 00:01:36.260 acceleration on Sal. 00:01:36.260 --> 00:01:37.510 That's me. 00:01:41.000 --> 00:01:45.770 And so how do we apply this equation to figure out how 00:01:45.770 --> 00:01:49.490 much I'm accelerating down towards the center of Earth or 00:01:49.490 --> 00:01:51.950 the Earth's center of mass? 00:01:51.950 --> 00:01:55.890 The force is equal to-- so what's this big G thing? 00:01:55.890 --> 00:02:00.010 The G is the universal gravitational constant. 00:02:00.010 --> 00:02:03.330 Although, as far as I know, and I'm not an expert on this, 00:02:03.330 --> 00:02:08.620 I actually think its measurement can change. 00:02:08.620 --> 00:02:11.390 It's not truly, truly a constant, or I guess when on 00:02:11.390 --> 00:02:12.840 different scales, it can be a little bit different. 00:02:12.840 --> 00:02:16.700 But for our purposes, it is a constant, and the constant in 00:02:16.700 --> 00:02:21.520 most physics classes, is this: 6.67 times 10 to the negative 00:02:21.520 --> 00:02:24.546 11th meters cubed per kilogram seconds squared. 00:02:24.546 --> 00:02:27.470 I know these units are crazy, but all you have to realize is 00:02:27.470 --> 00:02:30.005 these are just the units needed, that when you multiply 00:02:30.005 --> 00:02:33.040 it times a mass and a mass divided by a distance squared, 00:02:33.040 --> 00:02:35.930 you get Newtons, or kilogram meters per second squared. 00:02:35.930 --> 00:02:38.360 So we won't worry so much about the units right now. 00:02:38.360 --> 00:02:40.610 Just realize that you're going to have to work with meters in 00:02:40.610 --> 00:02:42.420 kilograms seconds. 00:02:42.420 --> 00:02:44.620 So let's just write that number down. 00:02:44.620 --> 00:02:47.440 I'll change colors to keep it interesting. 00:02:47.440 --> 00:02:55.080 6.67 times 10 to the negative 11th, and we want to know the 00:02:55.080 --> 00:03:00.190 acceleration on Sal, so m1 is the mass of Sal. 00:03:00.190 --> 00:03:02.490 And I don't feel like revealing my mass in this 00:03:02.490 --> 00:03:05.240 video, so I'll just leave it as a variable. 00:03:05.240 --> 00:03:06.430 And then what's the mass 2? 00:03:06.430 --> 00:03:07.950 It's the mass of Earth. 00:03:07.950 --> 00:03:08.820 And I wrote that here. 00:03:08.820 --> 00:03:10.250 I looked it up on Wikipedia. 00:03:10.250 --> 00:03:13.410 This is the mass of Earth. 00:03:13.410 --> 00:03:18.890 So I multiply it times the mass of Earth, times 5.97 00:03:18.890 --> 00:03:21.670 times 10 to the 24th kilograms-- weighs a little 00:03:21.670 --> 00:03:25.700 bit, not weighs, is a little bit more massive than Sal-- 00:03:25.700 --> 00:03:27.900 divided by the distance squared. 00:03:27.900 --> 00:03:29.420 Now, you might say, well, what's the distance between 00:03:29.420 --> 00:03:30.990 someone standing on the Earth and the Earth? 00:03:30.990 --> 00:03:33.080 Well, it's zero because they're touching the Earth. 00:03:33.080 --> 00:03:35.130 But it's important to realize that the distance between the 00:03:35.130 --> 00:03:38.570 two objects, especially when we're talking about the 00:03:38.570 --> 00:03:40.910 universal law of gravitation, is the distance between their 00:03:40.910 --> 00:03:42.310 center of masses. 00:03:42.310 --> 00:03:44.290 For all general purposes, my center of mass, maybe it's 00:03:44.290 --> 00:03:46.610 like three feet above the ground, because 00:03:46.610 --> 00:03:48.500 I'm not that tall. 00:03:48.500 --> 00:03:50.510 It's probably a little bit lower than that, actually. 00:03:50.510 --> 00:03:52.760 Anyway, my center of mass might be three feet above the 00:03:52.760 --> 00:03:54.380 ground, and where's Earth's center of mass? 00:03:54.380 --> 00:03:56.330 Well, it's at the center of Earth, so we have to know the 00:03:56.330 --> 00:03:59.800 radius of Earth, right? 00:03:59.800 --> 00:04:03.120 So the radius of Earth is-- I also looked it up on 00:04:03.120 --> 00:04:07.290 Wikipedia-- 6,371 kilometers. 00:04:07.290 --> 00:04:08.490 How many meters is that? 00:04:08.490 --> 00:04:11.100 It's 6 million meters, right? 00:04:11.100 --> 00:04:13.470 And then, you know, the extra meter to get to my center of 00:04:13.470 --> 00:04:16.079 mass, we can ignore for now, because it would be .001, so 00:04:16.079 --> 00:04:16.970 we'll ignore that for now. 00:04:16.970 --> 00:04:18.120 So it's 6-- and soon. 00:04:18.120 --> 00:04:20.290 I'll write it in scientific notation since everything else 00:04:20.290 --> 00:04:25.230 is in scientific notation-- 6.371 times 10 to the sixth 00:04:25.230 --> 00:04:25.860 meters, right? 00:04:25.860 --> 00:04:29.610 6,000 kilometers is 6 million meters. 00:04:29.610 --> 00:04:30.660 So let's write that down. 00:04:30.660 --> 00:04:37.580 So the distance is going to be 6.37 times 10 00:04:37.580 --> 00:04:40.470 to the sixth meters. 00:04:40.470 --> 00:04:41.430 We have to square that. 00:04:41.430 --> 00:04:44.560 Remember, it's distance squared. 00:04:44.560 --> 00:04:47.830 So let's see if we can simplify this a little bit. 00:04:47.830 --> 00:04:52.670 Let's just multiply those top numbers first. Force is equal 00:04:52.670 --> 00:04:53.910 to-- let's bring the variable out. 00:04:53.910 --> 00:04:58.550 Mass of Sal times-- let's do this top part. 00:04:58.550 --> 00:05:17.470 So we have 6.67 times 5.97 is equal to 39.82. 00:05:17.470 --> 00:05:19.670 And I just multiplied this times this, so now I have to 00:05:19.670 --> 00:05:21.160 multiply the 10's. 00:05:21.160 --> 00:05:23.590 So 10 to the negative 11th times 10 to the negative 24th. 00:05:23.590 --> 00:05:25.065 We can just add the exponents. 00:05:25.065 --> 00:05:26.850 They have the same base. 00:05:26.850 --> 00:05:28.360 So what's 24 minus 11? 00:05:28.360 --> 00:05:31.030 It's 10 to the 13th, right? 00:05:31.030 --> 00:05:32.610 And then what does the denominator look like? 00:05:32.610 --> 00:05:35.510 It's going to be the 6.37 squared times 10 00:05:35.510 --> 00:05:37.270 to the sixth squared. 00:05:37.270 --> 00:05:40.010 So it's going to be-- whatever this is is going to be like 37 00:05:40.010 --> 00:05:44.900 or something-- times-- what's 10 to the sixth squared? 00:05:44.900 --> 00:05:47.650 It's 10 to the 12th, right? 00:05:47.650 --> 00:05:49.120 10 to the 12th. 00:05:49.120 --> 00:05:52.586 So let's figure out what 6.37 squared is. 00:05:52.586 --> 00:05:56.434 This little calculator I have doesn't have squared, so I 00:05:56.434 --> 00:06:09.650 have to-- so it's 40.58. 00:06:09.650 --> 00:06:12.740 And so simplifying it, the force is equal to the mass of 00:06:12.740 --> 00:06:31.990 Sal times-- let's divide, 39.82 divided by 40.58 is 00:06:31.990 --> 00:06:38.290 equal to 9.81. 00:06:38.290 --> 00:06:40.130 That's just this divided by this. 00:06:40.130 --> 00:06:44.770 And then 10 to the 13th divided by 10 to the 12th. 00:06:44.770 --> 00:06:46.200 Actually no, this isn't 9.81. 00:06:46.200 --> 00:06:48.150 Sorry, it's 0.981. 00:06:48.150 --> 00:06:51.700 0.981, and then 10 to the 13th divided by 10 to the 12th is 00:06:51.700 --> 00:06:52.560 just 10, right? 00:06:52.560 --> 00:06:57.160 10 to the first, times 10, so what's 0.981 times 10? 00:06:57.160 --> 00:07:03.710 Well, the force is equal to 9.81 times the mass of Sal. 00:07:03.710 --> 00:07:04.550 And where does this get us? 00:07:04.550 --> 00:07:06.780 How can we figure out the acceleration right now? 00:07:06.780 --> 00:07:11.220 Well, force is just mass times acceleration, right? 00:07:11.220 --> 00:07:14.800 So that's also going to just be equal to the acceleration 00:07:14.800 --> 00:07:19.790 of gravity-- that's supposed to be a small g there-- times 00:07:19.790 --> 00:07:22.310 the mass of Sal, right? 00:07:22.310 --> 00:07:25.265 So we know the gravitational force is 9.81 times the mass 00:07:25.265 --> 00:07:27.570 of Sal, and we also know that that's the same thing as the 00:07:27.570 --> 00:07:29.760 acceleration of gravity times the mass of Sal. 00:07:29.760 --> 00:07:31.840 We can divide both sides by the mass of Sal, and we have 00:07:31.840 --> 00:07:33.390 the acceleration of gravity. 00:07:33.390 --> 00:07:35.820 And if we had used the units the whole way, you would have 00:07:35.820 --> 00:07:38.470 seen that it is kilograms meters per second squared. 00:07:38.470 --> 00:07:41.880 And we have just shown that, at least based on the numbers 00:07:41.880 --> 00:07:46.090 that they've given in Wikipedia, the acceleration of 00:07:46.090 --> 00:07:49.170 gravity on the surface of the Earth is almost exactly what 00:07:49.170 --> 00:07:51.450 we've been using in all the projectile motion problems. 00:07:51.450 --> 00:07:55.290 It's 9.8 meters per second squared. 00:07:55.290 --> 00:07:57.570 That's exciting. 00:07:57.570 --> 00:07:59.830 So let's do another quick problem with gravity, because 00:07:59.830 --> 00:08:02.570 I've got two minutes. 00:08:02.570 --> 00:08:06.390 Let's say there's another planet called the 00:08:06.390 --> 00:08:08.380 planet Small Earth. 00:08:08.380 --> 00:08:15.760 And let's say the radius of Small Earth is equal to 1/2 00:08:15.760 --> 00:08:19.380 the radius of Earth and the mass of Small Earth is equal 00:08:19.380 --> 00:08:22.280 to 1/2 the mass of Earth. 00:08:22.280 --> 00:08:27.750 So what's the pull of gravity on any object, say same 00:08:27.750 --> 00:08:29.170 object, on this? 00:08:29.170 --> 00:08:31.880 How much smaller would it be on this planet? 00:08:31.880 --> 00:08:33.952 Well, actually let me save that to the next video, 00:08:33.952 --> 00:08:34.789 because I hate being rushed. 00:08:34.789 --> 00:08:36.500 So I'll see you

Gravitation (part 2)

https://www.youtube.com/watch?v=8i0j3j16yFk

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https://www.youtube.com/api/timedtext?v=8i0j3j16yFk&ei=YmeUZfKvGqOCmLAP_b6MwAU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D451ECB502099D2563D9BC48A944C48D3B37A73C.61A26026C61BA264414180FF2A6C68BA445658C8&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.900 --> 00:00:01.550 Welcome back. 00:00:01.550 --> 00:00:05.620 So I was trying to rush and finish a problem in the last 00:00:05.620 --> 00:00:08.189 two minutes of the video, and I realize that's just bad 00:00:08.189 --> 00:00:09.980 teaching, because I end up rushing. 00:00:09.980 --> 00:00:11.810 So this is the problem we were going to work on, and you'll 00:00:11.810 --> 00:00:12.780 see a lot of these. 00:00:12.780 --> 00:00:17.890 They just want you to become familiar with the variables in 00:00:17.890 --> 00:00:19.350 Newton's law of gravitation. 00:00:19.350 --> 00:00:23.010 So I said that there's two planets, one is Earth. 00:00:23.010 --> 00:00:27.100 Now I have time to draw things, so that's Earth. 00:00:27.100 --> 00:00:28.830 And then there's Small Earth. 00:00:28.830 --> 00:00:32.470 And Small Earth-- well, maybe I'll just call it the small 00:00:32.470 --> 00:00:34.370 planet, so we don't get confused. 00:00:34.370 --> 00:00:37.600 It's green, showing that there's probably 00:00:37.600 --> 00:00:40.300 life on that planet. 00:00:40.300 --> 00:00:42.550 Let's say it has 1/2 the radius, and 1/2 the mass. 00:00:45.300 --> 00:00:46.625 So if you think about it, it's probably a 00:00:46.625 --> 00:00:48.000 lot denser than Earth. 00:00:48.000 --> 00:00:49.550 That's a good problem to think about. 00:00:49.550 --> 00:00:51.150 How much denser is it, right? 00:00:51.150 --> 00:00:55.520 Because if you have 1/2 the radius, your volume is much 00:00:55.520 --> 00:00:56.560 less than 1/2. 00:00:56.560 --> 00:00:58.173 I don't want to go into that now, but that's something for 00:00:58.173 --> 00:00:58.950 you to think about. 00:00:58.950 --> 00:01:01.900 But my question is what fraction, if I'm standing on 00:01:01.900 --> 00:01:06.260 the surface of this-- so the same person, so Sal, if I'm on 00:01:06.260 --> 00:01:13.900 Earth, what fraction is the pull when I'm on this small 00:01:13.900 --> 00:01:15.390 green planet? 00:01:15.390 --> 00:01:18.170 So what is the pull on me on Earth? 00:01:18.170 --> 00:01:23.620 Well, it's just going to be-- my weight on Earth, the force 00:01:23.620 --> 00:01:28.510 on Earth, is going to be equal to the gravitational constant 00:01:28.510 --> 00:01:35.160 times my mass, mass of me. 00:01:35.160 --> 00:01:41.930 So m sub m times the mass of Earth divided by what? 00:01:41.930 --> 00:01:44.010 We learned in the last video, divided by the distance 00:01:44.010 --> 00:01:45.730 between me and the center of the mass of Earth. 00:01:45.730 --> 00:01:49.740 Really, my center of mass and the center of mass of Earth. 00:01:49.740 --> 00:01:52.820 But this is between the surface of the Earth, and I'd 00:01:52.820 --> 00:01:56.030 like to think that I'm not short, but it's negligible 00:01:56.030 --> 00:01:58.380 between my center of mass and the surface, so we'll just 00:01:58.380 --> 00:02:00.690 consider the radius of the Earth. 00:02:00.690 --> 00:02:05.290 So we divide it by the radius of the Earth squared. 00:02:08.770 --> 00:02:11.820 Using these same variables, what's going to be the force 00:02:11.820 --> 00:02:13.900 on this other planet? 00:02:13.900 --> 00:02:16.490 So the force on the other planet, this green planet-- 00:02:16.490 --> 00:02:20.930 I'll do it in green-- and we're calling it the small 00:02:20.930 --> 00:02:23.710 planet, it equals what? 00:02:23.710 --> 00:02:26.810 It equals the gravitational constant again. 00:02:26.810 --> 00:02:29.240 And my mass doesn't change when I go from one planet to 00:02:29.240 --> 00:02:31.850 another, right? 00:02:31.850 --> 00:02:33.620 Its mass now is what? 00:02:33.620 --> 00:02:37.710 We would write it m sub s here, right? 00:02:37.710 --> 00:02:39.300 This is the small planet. 00:02:39.300 --> 00:02:41.850 And we wrote right here that it's 1/2 the mass of Earth, so 00:02:41.850 --> 00:02:42.930 I'll just write that. 00:02:42.930 --> 00:02:44.700 So it's 1/2 the mass of Earth. 00:02:48.190 --> 00:02:49.900 And what's its radius? 00:02:49.900 --> 00:02:51.750 What's the radius now? 00:02:51.750 --> 00:02:54.100 I could just write the radius of the small planet squared, 00:02:54.100 --> 00:02:54.990 but I'll say, well, we know. 00:02:54.990 --> 00:02:57.920 It's 1/2 the radius of Earth, so let's put that in there. 00:02:57.920 --> 00:03:00.290 So 1/2 radius of Earth. 00:03:00.290 --> 00:03:02.420 We have to square it. 00:03:02.420 --> 00:03:04.270 Let's see what this simplifies to. 00:03:04.270 --> 00:03:12.760 This equals-- so we can take this 1/2 here-- 1/2G mass of 00:03:12.760 --> 00:03:18.140 me times mass of Earth over-- what's 1/2 squared? 00:03:18.140 --> 00:03:19.260 It's 1/4. 00:03:19.260 --> 00:03:26.580 Over 1/4 radius of Earth squared. 00:03:26.580 --> 00:03:30.330 And what's 1/2 divided by 1/4? 00:03:30.330 --> 00:03:32.420 1/4 goes into 1/2 two times, right? 00:03:32.420 --> 00:03:33.880 Or another way you can think about it is if you have a 00:03:33.880 --> 00:03:35.500 fraction in the denominator, when you put it in the 00:03:35.500 --> 00:03:38.020 numerator, you flip it and it becomes 4. 00:03:38.020 --> 00:03:39.060 So 4 times 1/2 is 2. 00:03:39.060 --> 00:03:41.760 Either way, it's just math. 00:03:41.760 --> 00:03:45.640 So the force on the small planet is going to be equal to 00:03:45.640 --> 00:03:53.030 1/2 divided by 1/4 is 2 times G, mass of me, times mass of 00:03:53.030 --> 00:03:56.930 Earth, divided by the radius of Earth squared. 00:03:56.930 --> 00:04:01.650 And if we look up here, this is the same 00:04:01.650 --> 00:04:05.445 thing as this, right? 00:04:05.445 --> 00:04:07.130 It's identical. 00:04:07.130 --> 00:04:10.820 So we know that the force that applied to me when I'm on the 00:04:10.820 --> 00:04:15.660 surface of the small planet is actually two times the force 00:04:15.660 --> 00:04:19.180 applied on Earth, when I go to Earth. 00:04:19.180 --> 00:04:20.610 And that's something interesting to think about, 00:04:20.610 --> 00:04:24.830 because you might have said initially, wow, you know, the 00:04:24.830 --> 00:04:27.040 mass of the object matters a lot in gravity. 00:04:27.040 --> 00:04:29.270 The more massive the object, the more it's 00:04:29.270 --> 00:04:31.330 going to pull on me. 00:04:31.330 --> 00:04:33.640 But what we see here is that actually, no. 00:04:33.640 --> 00:04:36.180 When I'm on the surface of this smaller planet, it's 00:04:36.180 --> 00:04:37.830 pulling even harder on me. 00:04:37.830 --> 00:04:38.830 And why is that? 00:04:38.830 --> 00:04:41.990 Well, because I'm actually closer to its center of mass. 00:04:41.990 --> 00:04:45.260 And as we talked about earlier in this video, this object is 00:04:45.260 --> 00:04:46.650 probably a lot denser. 00:04:46.650 --> 00:04:50.350 You could say it's only 1/2 the mass, but it's much less 00:04:50.350 --> 00:04:51.780 than 1/2 of the volume, right? 00:04:51.780 --> 00:04:55.090 Because the volume is the cube of the radius and all of that. 00:04:55.090 --> 00:04:56.920 I don't want to confuse you, but this is just something to 00:04:56.920 --> 00:04:57.610 think about. 00:04:57.610 --> 00:04:59.480 So not only does the mass matter, but the 00:04:59.480 --> 00:05:01.970 radius matters a lot. 00:05:01.970 --> 00:05:03.980 And the radius is actually the square, so it actually 00:05:03.980 --> 00:05:06.350 matters even more. 00:05:06.350 --> 00:05:09.800 So that's something that's pretty 00:05:09.800 --> 00:05:10.430 interesting to think about. 00:05:10.430 --> 00:05:14.080 And these are actually very common problems when they just 00:05:14.080 --> 00:05:16.680 want to tell you, oh, you go to a planet that is two times 00:05:16.680 --> 00:05:21.140 the mass of another planet, et cetera, et cetera, what is the 00:05:21.140 --> 00:05:23.090 difference in force between the two? 00:05:23.090 --> 00:05:25.570 And one thing I want you to realize, actually, before I 00:05:25.570 --> 00:05:29.050 finish this video since I do have some extra time, when we 00:05:29.050 --> 00:05:30.890 think about gravity, especially with planets and 00:05:30.890 --> 00:05:33.500 all of that, you always feel like, oh, it's 00:05:33.500 --> 00:05:36.400 Earth pulling on me. 00:05:36.400 --> 00:05:41.750 Let's say that this is the Earth, and the Earth is huge, 00:05:41.750 --> 00:05:45.730 and this is a tiny spaceship right here. 00:05:45.730 --> 00:05:48.310 It's traveling. 00:05:48.310 --> 00:05:50.190 You always think that Earth is pulling on 00:05:50.190 --> 00:05:51.150 the spaceship, right? 00:05:51.150 --> 00:05:53.350 The gravitational force of Earth. 00:05:53.350 --> 00:05:57.070 But it actually turns out, when we looked at the formula, 00:05:57.070 --> 00:05:57.980 the formula is symmetric. 00:05:57.980 --> 00:05:59.800 It's not really saying one is pulling on the other. 00:05:59.800 --> 00:06:01.570 They're actually saying this is the force 00:06:01.570 --> 00:06:03.420 between the two objects. 00:06:03.420 --> 00:06:04.960 They're attracted to each other. 00:06:04.960 --> 00:06:13.580 So if the Earth is pulling on me with the force of 500 00:06:13.580 --> 00:06:16.820 Newtons, it actually turns out that I am pulling on the Earth 00:06:16.820 --> 00:06:18.920 with an equal and opposite force of 5 Newtons. 00:06:18.920 --> 00:06:20.420 We're pulling towards each other. 00:06:20.420 --> 00:06:23.140 It just feels like the Earth is, at least from my point of 00:06:23.140 --> 00:06:25.220 view, that the Earth is pulling to me. 00:06:25.220 --> 00:06:29.410 And we're actually both being pulled towards the combined 00:06:29.410 --> 00:06:30.180 center of mass. 00:06:30.180 --> 00:06:33.050 So in this situation, let's say the Earth is pulling on 00:06:33.050 --> 00:06:37.220 the spaceship with the force of-- I don't know. 00:06:37.220 --> 00:06:40.210 I'm making up numbers now, but let's say 00:06:40.210 --> 00:06:43.970 it's 1 million Newtons. 00:06:43.970 --> 00:06:46.320 It actually turns out that the spaceship will be pulling on 00:06:46.320 --> 00:06:51.950 the Earth with the same force of 1 million Newtons. 00:06:51.950 --> 00:06:55.610 And they're both going to be moved to the combined system's 00:06:55.610 --> 00:06:56.900 center of mass. 00:06:56.900 --> 00:07:00.320 And the combined system's center of mass since the Earth 00:07:00.320 --> 00:07:02.910 is so much more massive is going to be very close to 00:07:02.910 --> 00:07:03.790 Earth's center of mass. 00:07:03.790 --> 00:07:05.770 It's probably going to be very close to 00:07:05.770 --> 00:07:06.450 Earth's center of mass. 00:07:06.450 --> 00:07:07.900 It's going to be like right there, right? 00:07:07.900 --> 00:07:12.070 So in this situation, Earth won't be doing a lot of 00:07:12.070 --> 00:07:15.930 moving, but it will be pulled in the direction of the 00:07:15.930 --> 00:07:18.340 spaceship, and the spaceship will try to go to Earth's 00:07:18.340 --> 00:07:20.680 center of mass, but at some point, probably the 00:07:20.680 --> 00:07:24.750 atmosphere, or the rock that it runs into, it won't be able 00:07:24.750 --> 00:07:27.640 to go much further and it might crash 00:07:27.640 --> 00:07:28.610 right around there. 00:07:28.610 --> 00:07:31.420 Anyway, I wanted just to give you the sense that it's not 00:07:31.420 --> 00:07:33.240 necessarily one object just pulling on the other. 00:07:33.240 --> 00:07:35.230 They're pulling towards each other to their 00:07:35.230 --> 00:07:36.850 combined center of masses. 00:07:36.850 --> 00:07:40.650 It would make a lot more sense if they had just two people 00:07:40.650 --> 00:07:42.210 floating in space, they actually would have some 00:07:42.210 --> 00:07:43.900 gravity towards each other. 00:07:43.900 --> 00:07:47.110 It's almost a little romantic. 00:07:47.110 --> 00:07:48.970 They would float to each other. 00:07:48.970 --> 00:07:52.910 And actually, you could figure it out. 00:07:52.910 --> 00:07:55.030 I don't have the time to do it, but you could use this 00:07:55.030 --> 00:07:58.010 formula and use the constant, and you could figure out, 00:07:58.010 --> 00:08:00.350 well, what is the gravitational attraction 00:08:00.350 --> 00:08:01.560 between two people? 00:08:01.560 --> 00:08:04.080 And what you'll see is that between two people floating in 00:08:04.080 --> 00:08:06.850 space, there are other forms of attraction that are 00:08:06.850 --> 00:08:09.150 probably stronger than their 00:08:09.150 --> 00:08:11.380 gravitational attraction, anyway. 00:08:11.380 --> 00:08:13.520 I'll let you ponder that and I will see 00:08:13.520 --> 00:08:15.410 you in the next video.

Calculus Proof that a=v^2/r

https://www.youtube.com/watch?v=YRBRarbMCyE

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https://www.youtube.com/api/timedtext?v=YRBRarbMCyE&ei=YmeUZfbRG8iMp-oP_rCFmAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=04F699E4D4B1DB193DBF2C3CFF6E69261ED9CA02.76AD16D43C95D4920E4A305EF95439544FD75921&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.760 --> 00:00:01.480 Welcome back. 00:00:01.480 --> 00:00:06.050 Well, I'm now going to prove to you that the magnitude of 00:00:06.050 --> 00:00:09.630 centripetal acceleration is equal to the magnitude of the 00:00:09.630 --> 00:00:13.150 velocity when you're going around the circle divided-- 00:00:13.150 --> 00:00:15.590 velocity squared divided by the radius. 00:00:15.590 --> 00:00:17.590 So let's start with the drawing, just so that we know 00:00:17.590 --> 00:00:22.200 what we're doing, just as much for me as it is for you. 00:00:22.200 --> 00:00:24.260 So that's the circle, and you can guess that's 00:00:24.260 --> 00:00:25.840 going to be our path. 00:00:25.840 --> 00:00:30.280 And let's call p our position vector. 00:00:30.280 --> 00:00:34.820 And this is the center of the circle right here. 00:00:34.820 --> 00:00:38.390 Let me do the position vector in magenta. 00:00:38.390 --> 00:00:42.070 So let's say this is p, my position. 00:00:42.070 --> 00:00:43.320 Let me draw that bolder. 00:00:50.320 --> 00:00:51.780 So that's the vector p. 00:00:56.840 --> 00:01:00.920 And let me define a few other things over here. 00:01:00.920 --> 00:01:04.580 So let's say that the angle that it's forming with the 00:01:04.580 --> 00:01:06.430 positive x-axis-- let's say this is the 00:01:06.430 --> 00:01:12.580 positive x-axis-- is theta. 00:01:12.580 --> 00:01:13.580 That's theta right there. 00:01:13.580 --> 00:01:15.910 Let's say the radius of the circle is r. 00:01:15.910 --> 00:01:18.580 So we have an object. 00:01:18.580 --> 00:01:19.630 It's right here. 00:01:19.630 --> 00:01:22.920 This is its position defined by this position vector, and 00:01:22.920 --> 00:01:23.750 it's spinning around. 00:01:23.750 --> 00:01:26.130 So its position vector is going-- at some point, the 00:01:26.130 --> 00:01:27.820 arrow's going to be pointed there and then there. 00:01:27.820 --> 00:01:29.740 It's just going to be going around and around the circle. 00:01:29.740 --> 00:01:33.520 But this is its position vector at some moment in time. 00:01:33.520 --> 00:01:39.480 So what is that position vector in our bracket notation 00:01:39.480 --> 00:01:40.410 of vectors? 00:01:40.410 --> 00:01:42.460 We have to figure out its x and y-components. 00:01:42.460 --> 00:01:50.320 Its x-component is right here, or you could almost say its 00:01:50.320 --> 00:01:53.600 i-component, if we were doing engineering notation. 00:01:53.600 --> 00:01:58.140 That's its x-component and that's its y-component, right? 00:01:58.140 --> 00:02:00.940 So in our bracket notation or whatever, I always forget what 00:02:00.940 --> 00:02:05.700 I call things, p is this. 00:02:05.700 --> 00:02:11.240 p, which is our position vector, what's its 00:02:11.240 --> 00:02:12.150 x-component? 00:02:12.150 --> 00:02:15.550 It's the radius times the cosine of theta. 00:02:15.550 --> 00:02:18.050 This should be second nature to you at this point. 00:02:18.050 --> 00:02:20.880 Radius times cosine of theta. 00:02:20.880 --> 00:02:21.980 What's its y-component? 00:02:21.980 --> 00:02:22.750 It's this. 00:02:22.750 --> 00:02:23.480 It's just this, right? 00:02:23.480 --> 00:02:25.020 That's its y-component. 00:02:25.020 --> 00:02:27.670 Radius times the sine of theta. 00:02:31.240 --> 00:02:32.030 Fair enough. 00:02:32.030 --> 00:02:33.330 Hopefully, that makes sense to you so far. 00:02:33.330 --> 00:02:37.670 I just defined its position vector and I drew it out here 00:02:37.670 --> 00:02:41.680 visually, and then I also wrote it analytically in its x 00:02:41.680 --> 00:02:45.230 and y-components, as a sum of its x y-components. 00:02:45.230 --> 00:02:46.250 Well, that's good and everything. 00:02:46.250 --> 00:02:47.420 So let's see if we can figure out what its 00:02:47.420 --> 00:02:49.330 velocity vector is. 00:02:49.330 --> 00:02:50.600 Well, what is velocity? 00:02:50.600 --> 00:02:52.890 Velocity is actually just a change in position. 00:02:52.890 --> 00:02:54.480 Actually, now we are actually dealing with 00:02:54.480 --> 00:02:55.980 velocity, not speed. 00:02:55.980 --> 00:02:57.650 We will actually get a vector. 00:02:57.650 --> 00:02:59.640 So what is the velocity vector? 00:02:59.640 --> 00:03:01.520 So the velocity vector is going to be 00:03:01.520 --> 00:03:02.415 at any given point. 00:03:02.415 --> 00:03:04.540 I'll do it in a different color. 00:03:04.540 --> 00:03:06.550 The velocity vector is going to be tangent to the circle. 00:03:06.550 --> 00:03:08.530 It's going to look something like that. 00:03:08.530 --> 00:03:10.770 That's going to be the velocity vector. 00:03:10.770 --> 00:03:15.240 So the velocity vector is equal to the change in the 00:03:15.240 --> 00:03:17.750 position over time. 00:03:17.750 --> 00:03:22.120 So let's take the derivative of the position vector with 00:03:22.120 --> 00:03:24.410 respect to time. 00:03:24.410 --> 00:03:25.500 And how do we take a derivative? 00:03:25.500 --> 00:03:27.220 Well, we could just take the derivative of the x and 00:03:27.220 --> 00:03:29.880 y-components separately, and I'll show you how we do it in 00:03:29.880 --> 00:03:30.520 this notation. 00:03:30.520 --> 00:03:31.540 And if you think about it, it should make a 00:03:31.540 --> 00:03:33.550 little bit of sense. 00:03:33.550 --> 00:03:35.870 So the radius is constant. 00:03:35.870 --> 00:03:38.040 As we go around the circle, the radius isn't changing. 00:03:38.040 --> 00:03:39.980 Another thing to keep in mind, we're going around. 00:03:39.980 --> 00:03:43.010 We're spinning around the circle at a constant rate. 00:03:43.010 --> 00:03:44.410 So my speed isn't changing. 00:03:44.410 --> 00:03:46.680 My velocity is obviously changing, because the 00:03:46.680 --> 00:03:49.920 direction is changing, but the actual rate at which I'm 00:03:49.920 --> 00:03:51.560 spinning, or the angular velocity, is 00:03:51.560 --> 00:03:52.460 going to be a constant. 00:03:52.460 --> 00:03:54.780 That's something to keep in mind for what we get 00:03:54.780 --> 00:03:55.820 to in the next step. 00:03:55.820 --> 00:03:58.560 But anyway, let's take the derivative. 00:03:58.560 --> 00:04:04.260 So let's take the derivative of the x term first. Well, r 00:04:04.260 --> 00:04:06.050 is just a constant so it doesn't change. 00:04:06.050 --> 00:04:08.290 We can just take the r out. 00:04:08.290 --> 00:04:11.110 And then what's the derivative of cosine theta 00:04:11.110 --> 00:04:13.100 with respect to time? 00:04:13.100 --> 00:04:14.180 Not with respect to theta. 00:04:14.180 --> 00:04:15.840 Remember, we're taking the derivative 00:04:15.840 --> 00:04:17.209 with respect to time. 00:04:17.209 --> 00:04:18.930 So we do the chain rule. 00:04:18.930 --> 00:04:27.860 It's the derivative of theta with respect to time times the 00:04:27.860 --> 00:04:32.110 derivative of this term with respect to theta. 00:04:32.110 --> 00:04:33.840 And what's the derivative of this? 00:04:33.840 --> 00:04:37.910 What's minus sine of theta? 00:04:37.910 --> 00:04:39.560 Let me put the minus out here. 00:04:39.560 --> 00:04:41.670 Minus sine of theta. 00:04:41.670 --> 00:04:43.360 I just didn't want to put the minus right in front of the 00:04:43.360 --> 00:04:45.320 sine, because you would think it's minus sine theta. 00:04:45.320 --> 00:04:48.560 So it's minus r times the rate of change of the angle with 00:04:48.560 --> 00:04:52.620 respect to time times sine of theta, the rate at which this 00:04:52.620 --> 00:04:54.730 term is changing with respect to theta. 00:04:54.730 --> 00:04:57.600 Let's do the same thing on the y side. 00:04:57.600 --> 00:04:59.100 r is a constant. 00:04:59.100 --> 00:05:02.330 Chain rule: the rate at which theta is changing 00:05:02.330 --> 00:05:03.580 with respect to time. 00:05:06.265 --> 00:05:06.460 you. 00:05:06.460 --> 00:05:07.970 And then what's the derivative of sine of theta 00:05:07.970 --> 00:05:08.720 with respect to theta? 00:05:08.720 --> 00:05:11.250 Well, that's cosine of theta. 00:05:11.250 --> 00:05:14.570 This was just the chain rule that I did. 00:05:14.570 --> 00:05:17.230 Let's see if we can simplify that a little bit. 00:05:17.230 --> 00:05:20.690 So d theta, dt in both of these, that's the same thing 00:05:20.690 --> 00:05:22.000 as angular velocity. 00:05:22.000 --> 00:05:24.500 And watch the video on angular velocity if that doesn't make 00:05:24.500 --> 00:05:25.580 sense to you. 00:05:25.580 --> 00:05:30.170 But we can simplify this as just w, angular velocity, and 00:05:30.170 --> 00:05:32.060 that's going to be a constant. 00:05:32.060 --> 00:05:33.770 And we have an r there. 00:05:33.770 --> 00:05:37.120 Let's take a wr out of both sides. 00:05:37.120 --> 00:05:45.450 So we have the velocity vector is equal to-- the velocity 00:05:45.450 --> 00:05:51.410 vector is equal to wr-- actually, let's take minus wr 00:05:51.410 --> 00:05:57.480 out, so minus wr, And then this term is sine of theta. 00:06:00.345 --> 00:06:03.350 And we're taking a minus wr, so the wr goes away, and then 00:06:03.350 --> 00:06:05.050 we introduce a minus sign here, so it's 00:06:05.050 --> 00:06:06.300 minus cosine of theta. 00:06:11.110 --> 00:06:11.470 Good enough. 00:06:11.470 --> 00:06:13.290 And the reason why we were able to take-- and this w, 00:06:13.290 --> 00:06:14.730 remember, is going to be a constant. 00:06:14.730 --> 00:06:17.070 It's not changing with respect to time. 00:06:17.070 --> 00:06:20.630 The angle is changing with respect to time, but not the 00:06:20.630 --> 00:06:21.660 rate of change of the angle. 00:06:21.660 --> 00:06:23.630 It's spinning at a constant rate. 00:06:23.630 --> 00:06:25.160 So what's the acceleration vector going to be? 00:06:25.160 --> 00:06:27.430 The acceleration vector, and I'll switch colors again to 00:06:27.430 --> 00:06:28.920 keep it interesting. 00:06:28.920 --> 00:06:32.200 The acceleration vector is just the derivative of the 00:06:32.200 --> 00:06:33.845 velocity vector with respect to time. 00:06:39.430 --> 00:06:44.066 And that equals-- this is this constant term, so let's just 00:06:44.066 --> 00:06:44.940 leave it on the outside. 00:06:44.940 --> 00:06:47.770 It's minus wr. 00:06:47.770 --> 00:06:50.040 And chain rule again. 00:06:50.040 --> 00:06:52.228 If we're taking the derivative with respect to time, first we 00:06:52.228 --> 00:06:53.950 have to take the derivative of theta with respect to time, 00:06:53.950 --> 00:06:54.950 and we don't know what that is. 00:06:54.950 --> 00:06:56.240 That's just going to be w. 00:06:56.240 --> 00:07:03.620 d theta dt and then times this expression, the derivative of 00:07:03.620 --> 00:07:05.835 this expression with respect to theta. 00:07:05.835 --> 00:07:10.160 The sine of theta derivative is just cosine of theta. 00:07:10.160 --> 00:07:15.220 And then on the y-side, what's the derivative of theta with 00:07:15.220 --> 00:07:16.810 respect to time? 00:07:16.810 --> 00:07:20.560 It's just going to be omega. 00:07:20.560 --> 00:07:22.680 And what's the derivative of minus cosine theta with 00:07:22.680 --> 00:07:23.520 respect to theta? 00:07:23.520 --> 00:07:24.770 Well, that's sine theta. 00:07:27.610 --> 00:07:32.270 And once again, this is w and this is w. 00:07:32.270 --> 00:07:34.320 We could take the w out of the equation. 00:07:34.320 --> 00:07:38.090 We get the acceleration vector is equal to-- take the w out 00:07:38.090 --> 00:07:44.480 of the x and y-components-- is equal to minus w squared r 00:07:44.480 --> 00:07:51.470 times cosine theta sine theta. 00:07:51.470 --> 00:07:54.680 Or we could take this r and multiply it times both of the 00:07:54.680 --> 00:07:57.760 x and y-components, and we have the acceleration vector 00:07:57.760 --> 00:08:05.920 is equal to minus w squared r cosine theta r sine theta. 00:08:05.920 --> 00:08:08.690 Now does this thing here look familiar? 00:08:08.690 --> 00:08:09.210 Well, sure. 00:08:09.210 --> 00:08:12.330 That was our original position vector. 00:08:12.330 --> 00:08:15.420 So we could say that the acceleration vector is equal 00:08:15.420 --> 00:08:20.450 to minus our angular velocity squared times 00:08:20.450 --> 00:08:23.390 the position vector. 00:08:23.390 --> 00:08:26.390 And that makes actually a lot of sense, because w squared is 00:08:26.390 --> 00:08:27.630 going to be a positive term. 00:08:27.630 --> 00:08:32.510 And what it's saying is that the direction of this vector 00:08:32.510 --> 00:08:34.083 is going to be the negative of the direction of 00:08:34.083 --> 00:08:34.580 the position vector. 00:08:34.580 --> 00:08:37.054 So if our position vector is going outward, our 00:08:37.054 --> 00:08:40.559 acceleration vector, which I'll draw in green, is going 00:08:40.559 --> 00:08:41.880 to be going inward. 00:08:41.880 --> 00:08:43.789 The acceleration vector is going to be inward. 00:08:43.789 --> 00:08:46.460 So it is what we wanted to see. 00:08:46.460 --> 00:08:49.160 Let's see if we can express this as a function of the 00:08:49.160 --> 00:08:50.420 magnitudes. 00:08:50.420 --> 00:08:53.830 So we'll also say that the magnitude of the acceleration 00:08:53.830 --> 00:08:57.545 vector, and that's just a without an arrow on top; I 00:08:57.545 --> 00:09:02.120 could put brackets around it, is equal to the negative 00:09:02.120 --> 00:09:05.050 angular velocity squared times the magnitude of 00:09:05.050 --> 00:09:07.860 the position vector. 00:09:07.860 --> 00:09:09.890 Well, what's the magnitude of this position vector? 00:09:09.890 --> 00:09:11.320 How long is it? 00:09:11.320 --> 00:09:14.470 Well, its magnitude is r by definition on the 00:09:14.470 --> 00:09:15.640 beginning of the thing. 00:09:15.640 --> 00:09:17.450 So this is just r. 00:09:17.450 --> 00:09:21.390 So acceleration is equal to the negative angular velocity 00:09:21.390 --> 00:09:23.210 squared times r. 00:09:23.210 --> 00:09:25.460 And what's angular velocity? 00:09:25.460 --> 00:09:28.360 Well, we learned in that video that angular velocity-- I'll 00:09:28.360 --> 00:09:30.970 do it right here-- is equal to v/r. 00:09:30.970 --> 00:09:33.080 If we just talk about the magnitudes, not the vectors. 00:09:33.080 --> 00:09:34.580 Remember, if we're not drawing an arrow on top these 00:09:34.580 --> 00:09:36.950 variables, they're just scalar quantities. 00:09:36.950 --> 00:09:43.550 So the acceleration is equal to-- actually, we can get rid 00:09:43.550 --> 00:09:45.770 of the negative sign because we're not worried about 00:09:45.770 --> 00:09:46.940 direction right now. 00:09:46.940 --> 00:09:48.190 Well, we can keep it there. 00:09:48.190 --> 00:09:49.170 It doesn't matter. 00:09:49.170 --> 00:09:56.470 So we get v squared, because the magnitude of this, 00:09:56.470 --> 00:09:58.850 absolute value is essentially magnitude. 00:09:58.850 --> 00:10:02.780 So you get v squared over r squared times r. 00:10:02.780 --> 00:10:05.840 So r times r squared, you get the acceleration is v 00:10:05.840 --> 00:10:07.780 squared over r. 00:10:07.780 --> 00:10:09.400 And that's what we set out to prove. 00:10:09.400 --> 00:10:10.470 And I'm out of time now. 00:10:10.470 --> 00:10:11.610 So I'll see you in the next video. 00:10:11.610 --> 00:10:13.180 Hopefully, I didn't confuse

Conservation of angular momemtum

https://www.youtube.com/watch?v=s_R8d3isJDA

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.770 --> 00:00:07.140 We learned in the videos on torque that torque equals-- 00:00:07.140 --> 00:00:10.940 let me actually draw a picture, so you remember what 00:00:10.940 --> 00:00:12.170 we're talking about with torque. 00:00:12.170 --> 00:00:18.580 So let's say-- so that's the arm, let's say right here is 00:00:18.580 --> 00:00:21.120 what its pivot is. 00:00:21.120 --> 00:00:22.510 That's its pivot. 00:00:22.510 --> 00:00:30.420 And let's say that I'm putting a force right there, and it's 00:00:30.420 --> 00:00:31.216 perpendicular to this arm. 00:00:31.216 --> 00:00:35.540 And let's say the length of this arm is r. 00:00:35.540 --> 00:00:45.220 And this force is f. 00:00:45.220 --> 00:00:48.000 So we know that the torque is this force times 00:00:48.000 --> 00:00:49.730 the distance, right? 00:00:55.510 --> 00:00:59.030 And then we also know that-- what is force? 00:00:59.030 --> 00:01:01.390 Well, that is equal to mass times 00:01:01.390 --> 00:01:03.980 acceleration times distance. 00:01:03.980 --> 00:01:05.730 So torque is equal to mass times 00:01:05.730 --> 00:01:08.090 acceleration times distance. 00:01:08.090 --> 00:01:10.900 And then so what is acceleration? 00:01:10.900 --> 00:01:15.540 Well that's equal to mass times change in velocity over 00:01:15.540 --> 00:01:22.240 change in time, times distance, right? 00:01:22.240 --> 00:01:24.020 So we learned all that from our torque chapter, and you 00:01:24.020 --> 00:01:26.420 might want to review it, just to get an intuition of what 00:01:26.420 --> 00:01:27.180 torque is good for. 00:01:27.180 --> 00:01:29.850 But in general, if something isn't spinning, you apply 00:01:29.850 --> 00:01:32.080 torque, and you'll get it spinning. 00:01:32.080 --> 00:01:34.560 Or if something is spinning already, if you apply torque 00:01:34.560 --> 00:01:36.560 in the direction that it's spinning, it'll spin faster. 00:01:36.560 --> 00:01:39.280 Or if you go in the opposite direction, it'll spin slower. 00:01:39.280 --> 00:01:42.170 And what I'm showing you here, is that what happens if you 00:01:42.170 --> 00:01:43.690 apply no torque? 00:01:43.690 --> 00:01:47.310 Well if you apply no torque, then we know that this 00:01:47.310 --> 00:01:51.070 quantity is 0. 00:01:51.070 --> 00:01:54.270 Or another way to think about it-- actually why did I write 00:01:54.270 --> 00:01:57.530 d, it could be d, but I shouldn't have called this r, 00:01:57.530 --> 00:01:59.590 it should be d. 00:01:59.590 --> 00:02:03.950 So if this is 0-- if we are applying no torque, right-- 00:02:03.950 --> 00:02:04.920 what do we know? 00:02:04.920 --> 00:02:06.850 We know that the change in velocity over change in time, 00:02:06.850 --> 00:02:09.660 times this distance, won't change-- that 00:02:09.660 --> 00:02:11.760 this quantity is 0. 00:02:11.760 --> 00:02:19.020 So we know that the velocity times the distance is going to 00:02:19.020 --> 00:02:20.270 be a constant. 00:02:24.230 --> 00:02:27.660 And that comes from what I just talked about. 00:02:27.660 --> 00:02:31.070 It falls out of Newton's laws, but it applies to spinning. 00:02:31.070 --> 00:02:33.690 An object that's not spinning will tend to not stay 00:02:33.690 --> 00:02:35.480 spinning, and an object that is spinning will 00:02:35.480 --> 00:02:37.190 tend to stay spinning. 00:02:37.190 --> 00:02:39.470 So in this case, if this object at this point right 00:02:39.470 --> 00:02:42.090 here-- so we're in a case where there's no torque, so 00:02:42.090 --> 00:02:44.770 this force is 0-- there's no force applying here. 00:02:44.770 --> 00:02:48.190 And whatever this object's velocity was-- its tangential 00:02:48.190 --> 00:02:52.490 velocity-- it's going to stay at that velocity, right? 00:02:52.490 --> 00:02:54.300 It's just going to keep spinning at that velocity. 00:02:54.300 --> 00:02:56.940 If I apply more torque it'll go even faster, if I apply 00:02:56.940 --> 00:03:00.600 less torque, it'll slow down a little bit. 00:03:00.600 --> 00:03:02.510 But we know that this velocity times 00:03:02.510 --> 00:03:03.440 the distance is constant. 00:03:03.440 --> 00:03:05.850 And actually, I don't know why I took this m out-- we know 00:03:05.850 --> 00:03:08.060 that the mass times the velocity times the distance, 00:03:08.060 --> 00:03:09.320 is constant. 00:03:09.320 --> 00:03:10.520 Right? 00:03:10.520 --> 00:03:11.350 So what does that tell us? 00:03:11.350 --> 00:03:18.400 Well, we learned in the angular velocity video-- my 00:03:18.400 --> 00:03:20.530 mind's a little slow today-- that what? 00:03:20.530 --> 00:03:27.220 Angular velocity is equal to velocity divided by the 00:03:27.220 --> 00:03:30.330 radius, and this case, the radius is this distance, so we 00:03:30.330 --> 00:03:32.730 could also write it as velocity over distance. 00:03:32.730 --> 00:03:34.510 And when I talk about radius, it's just the radius of the 00:03:34.510 --> 00:03:37.760 circle that you're spinning around in, right? 00:03:37.760 --> 00:03:39.620 This could be the circle, up here. 00:03:39.620 --> 00:03:41.920 So this d, that's the same thing as the radius. 00:03:41.920 --> 00:03:44.280 I'm just switching letters to confuse you. 00:03:44.280 --> 00:03:48.210 But let's see if we can write something, if we can change 00:03:48.210 --> 00:03:50.670 this expression to include angular velocity. 00:03:50.670 --> 00:03:52.800 You'll see where I'm going in a second. 00:03:52.800 --> 00:03:55.070 So let's solve for v. 00:03:55.070 --> 00:03:58.560 So let's multiply both sides of this, times d. 00:03:58.560 --> 00:04:01.450 So you get dw is equal to v. 00:04:01.450 --> 00:04:04.890 Right, I just took this d, put it on this side. 00:04:04.890 --> 00:04:11.740 So let's write that here-- m times dw-- right, I just 00:04:11.740 --> 00:04:15.560 replaced the v-- times d, is equal to a constant. 00:04:15.560 --> 00:04:20.660 Assuming that we have no net torque on the system. 00:04:20.660 --> 00:04:22.130 And so what does that get us? 00:04:22.130 --> 00:04:26.310 Well that gets us m times the angular velocity times d 00:04:26.310 --> 00:04:28.490 squared is equal to a constant. 00:04:31.340 --> 00:04:32.170 So what does this tell us? 00:04:32.170 --> 00:04:35.400 This tells us the mass of an object spinning-- let me 00:04:35.400 --> 00:04:42.050 rewrite this, because I think-- so what we know is 00:04:42.050 --> 00:04:47.300 that the mass of an object spinning, times how fast it is 00:04:47.300 --> 00:04:52.020 spinning, times the distance to the center of its 00:04:52.020 --> 00:04:54.110 rotation-- and actually I'm going to change that d to an 00:04:54.110 --> 00:04:57.370 r, I don't know why I even used d to begin with-- times 00:04:57.370 --> 00:04:57.720 that squared. 00:04:57.720 --> 00:05:03.950 That's going to be equal to a constant, assuming no net 00:05:03.950 --> 00:05:05.720 force-- no net torque. 00:05:05.720 --> 00:05:08.970 Another way of looking at that, if we just wanted to go 00:05:08.970 --> 00:05:11.420 to angular velocity from the get-go, we could have said 00:05:11.420 --> 00:05:17.870 torque is equal to mass times change in velocity, times 00:05:17.870 --> 00:05:21.840 change in time, times the radius to the center of where 00:05:21.840 --> 00:05:23.400 you're rotating around. 00:05:23.400 --> 00:05:29.810 And change in velocity is just the same thing as mass times 00:05:29.810 --> 00:05:34.920 change in angular velocity, times r. 00:05:34.920 --> 00:05:36.460 Change in time, and then there's another 00:05:36.460 --> 00:05:38.480 r, that's this r. 00:05:38.480 --> 00:05:40.996 Because velocity is angular velocity times r, and we're 00:05:40.996 --> 00:05:43.560 assuming r doesn't change, so any change in velocity is just 00:05:43.560 --> 00:05:46.260 going to be a change in the angular velocity. 00:05:46.260 --> 00:05:48.880 And then we would get the same thing that we just got here. 00:05:48.880 --> 00:05:51.040 If we have no net torque, we're going to have no change 00:05:51.040 --> 00:05:52.840 in angular velocity, so angular 00:05:52.840 --> 00:05:54.100 velocity will be a constant. 00:05:54.100 --> 00:05:57.690 So you'll get mass times angular velocity, times-- and 00:05:57.690 --> 00:06:00.780 then you have this r, and this r-- times r squared, is going 00:06:00.780 --> 00:06:02.790 to be equal to a constant. 00:06:02.790 --> 00:06:04.430 Where am I going with all of this? 00:06:04.430 --> 00:06:06.930 Well let's think about something. 00:06:06.930 --> 00:06:12.870 Let's say that I have some object traveling in a circle. 00:06:12.870 --> 00:06:15.080 I do everything for a reason, and you'll see my 00:06:15.080 --> 00:06:18.290 reason right now. 00:06:18.290 --> 00:06:34.120 Let's say that-- so let's say this is some type of 00:06:34.120 --> 00:06:38.740 retractable pole, and I'm an ice skater. 00:06:38.740 --> 00:06:42.230 And I'm not using the ice skater's body right now, 00:06:42.230 --> 00:06:43.690 because it'll become complicated. 00:06:43.690 --> 00:06:47.070 Let's say this is some type of robot arm, and that's its 00:06:47.070 --> 00:06:48.050 joint right there. 00:06:48.050 --> 00:06:49.440 And it's holding a mass out here. 00:06:53.030 --> 00:06:55.730 This is neat because after this concept, you'll 00:06:55.730 --> 00:06:58.660 understand what goes on in figure skating, and then the 00:06:58.660 --> 00:06:59.930 Olympics-- oh no, the Winter Olympics are 00:06:59.930 --> 00:07:00.980 over, aren't they? 00:07:00.980 --> 00:07:04.830 Anyway, let's say that this object is spinning 00:07:04.830 --> 00:07:06.080 around at some rate. 00:07:08.750 --> 00:07:17.320 It's spinning around at 10 radians a second. 00:07:17.320 --> 00:07:19.330 That's its angular velocity. 00:07:19.330 --> 00:07:22.570 And let's say, right now its rotational 00:07:22.570 --> 00:07:29.560 distance is 10 feet. 00:07:29.560 --> 00:07:38.665 So this is spinning on an ice skating rink, because you 00:07:38.665 --> 00:07:40.280 don't want friction and all of that. 00:07:40.280 --> 00:07:43.920 So what's its current angular momentum? 00:07:43.920 --> 00:07:47.690 So that's what this term right here is, angular momentum. 00:07:50.400 --> 00:07:52.790 So what's its current angular momentum? 00:07:52.790 --> 00:08:00.880 Well its mass times 10, times-- 10 is its-- actually, 00:08:00.880 --> 00:08:02.920 let's make this radius a different number, let's call 00:08:02.920 --> 00:08:05.810 it 8 feet, just you know what I'm doing. 00:08:05.810 --> 00:08:10.310 So its angular velocity is 10, and then its radius 00:08:10.310 --> 00:08:12.560 is 8, so times 64. 00:08:12.560 --> 00:08:15.250 So it equals 640 times mass. 00:08:15.250 --> 00:08:16.535 This is its angular momentum. 00:08:19.790 --> 00:08:22.130 Now what happens if this arm, for whatever reason, 00:08:22.130 --> 00:08:26.900 shortens-- and maybe it does something like this, the arm 00:08:26.900 --> 00:08:27.950 kind of bends. 00:08:27.950 --> 00:08:29.760 And then the mass comes here, it comes in 00:08:29.760 --> 00:08:32.340 closer to the center. 00:08:32.340 --> 00:08:33.900 I'll write that in a different color. 00:08:33.900 --> 00:08:36.280 The mass comes closer to the center, so now 00:08:36.280 --> 00:08:37.659 the radius is 4. 00:08:37.659 --> 00:08:40.870 But I've had no net torque on the system, all I've done is 00:08:40.870 --> 00:08:45.450 change how far it is from the center of rotation. 00:08:45.450 --> 00:08:47.660 How much faster is it going to spin now? 00:08:47.660 --> 00:08:48.760 Let's think about it. 00:08:48.760 --> 00:08:51.310 Its angular momentum won't change, this is a constant, 00:08:51.310 --> 00:08:53.680 which is its angular momentum. 00:08:53.680 --> 00:08:54.940 That won't change. 00:08:54.940 --> 00:09:01.810 So we now know that the mass times the angular-- the new 00:09:01.810 --> 00:09:05.470 angular momentum, we'll write that angular momentum 1-- 00:09:05.470 --> 00:09:08.890 times the new distance squared-- times 16-- is also 00:09:08.890 --> 00:09:11.780 going to be equal to 640m. 00:09:11.780 --> 00:09:13.860 The angular momentum doesn't change. 00:09:13.860 --> 00:09:17.000 Let's cross out m from both sides, and then divide both 00:09:17.000 --> 00:09:18.330 sides by 16. 00:09:18.330 --> 00:09:23.790 We now have that w1 is equal to 16 goes into 640. 00:09:23.790 --> 00:09:24.390 So what happened? 00:09:24.390 --> 00:09:27.540 Originally, I was going around at 10 radians per second, when 00:09:27.540 --> 00:09:33.400 I halved the radius-- when I got half as close to the 00:09:33.400 --> 00:09:36.200 center of my rotation, I'm actually spinning around 4 00:09:36.200 --> 00:09:37.540 times as much. 00:09:37.540 --> 00:09:40.620 And that's because this term is a quadratic term. 00:09:40.620 --> 00:09:44.860 And you probably have observed this behavior before, when you 00:09:44.860 --> 00:09:48.140 see the ice skater skating around, and they're spinning 00:09:48.140 --> 00:09:50.950 with their arms wide open, and then they pull their arms in, 00:09:50.950 --> 00:09:53.130 and they go a lot, lot, lot, lot faster. 00:09:53.130 --> 00:09:56.700 And that's because their angular velocity, or the rate 00:09:56.700 --> 00:10:01.980 at which they're spinning, is proportional to the square of 00:10:01.980 --> 00:10:05.680 the radius around their axis of rotation. 00:10:05.680 --> 00:10:07.770 Anyway, I hope I didn't confuse you, and I'll do some 00:10:07.770 --> 00:10:09.620 more problems with this in the future, but I've 00:10:09.620 --> 00:10:10.390 just run out of time. 00:10:10.390 --> 00:10:11.840 See you soon.

Introduction to angular velocity

https://www.youtube.com/watch?v=X4UTe1fZUzI

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https://www.youtube.com/api/timedtext?v=X4UTe1fZUzI&ei=YmeUZcrBHJO5vdIPpOWKiA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=4FB6F7FBF76B8A0D391FF6E1996206266E4E4CF8.06E762F4F7CE31A91E9CA6F5603F2854E10FF0FA&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.740 --> 00:00:04.300 Well, we've done a lot of work with how fast something moves, 00:00:04.300 --> 00:00:07.250 let's see if we can work with how fast something spins. 00:00:07.250 --> 00:00:09.820 Let's see what we can do. 00:00:09.820 --> 00:00:11.970 Since we're going to be working with things spinning, 00:00:11.970 --> 00:00:14.080 let me draw a circle. 00:00:14.080 --> 00:00:17.370 Since things that spin go in circles. 00:00:21.300 --> 00:00:23.810 And let me just draw the positive x-axis because it'll 00:00:23.810 --> 00:00:26.340 come in handy in a second. 00:00:26.340 --> 00:00:29.120 That's the positive x-axis. 00:00:29.120 --> 00:00:32.610 And let's say that I have an object, and the circle is the 00:00:32.610 --> 00:00:33.860 object's path. 00:00:33.860 --> 00:00:36.705 So let's say this is the object. 00:00:36.705 --> 00:00:38.780 And it's going around in a circle in a 00:00:38.780 --> 00:00:39.755 counterclockwise direction. 00:00:39.755 --> 00:00:41.030 Not squiggly counterclockwise, it's just 00:00:41.030 --> 00:00:43.620 going around this way. 00:00:43.620 --> 00:00:46.740 Let's say I wanted to figure out, or I wanted to quantify 00:00:46.740 --> 00:00:50.300 how much, or how fast this thing is spinning. 00:00:50.300 --> 00:00:53.300 So one thing that you're probably familiar with is 00:00:53.300 --> 00:00:56.230 revolutions per second, or rotations per second. 00:00:56.230 --> 00:00:59.130 So let's write that down, let's just say for the sake of 00:00:59.130 --> 00:01:03.110 argument that this was moving at, I don't know, 1 revolution 00:01:03.110 --> 00:01:03.630 per second. 00:01:03.630 --> 00:01:06.380 So after 1 second it goes back, then another second. 00:01:06.380 --> 00:01:07.810 So that's how fast it's spinning, 1 00:01:07.810 --> 00:01:09.570 revolution per second. 00:01:09.570 --> 00:01:12.800 1 revolution-- I'll just put rev-- per second. 00:01:15.420 --> 00:01:17.210 So let's see if we can quantify that in angles, and 00:01:17.210 --> 00:01:19.130 we'll do it in radians, but you could always convert it 00:01:19.130 --> 00:01:21.440 back to degrees, if you want. 00:01:24.630 --> 00:01:27.660 I don't know if you can see that line. 00:01:27.660 --> 00:01:31.350 Let's just say that theta is the angle between the radius 00:01:31.350 --> 00:01:33.180 from the center to that object, and 00:01:33.180 --> 00:01:35.840 the positive x-axis. 00:01:35.840 --> 00:01:38.980 So if this object is travelling at 1 revolution per 00:01:38.980 --> 00:01:43.090 second, how many radians per second is it traveling? 00:01:43.090 --> 00:01:46.120 Well, how many radians are there in a revolution? 00:01:46.120 --> 00:01:49.740 Well there's 2 pi radians in a revolution, right? 00:01:49.740 --> 00:01:52.980 1 go-around in a circle is 2 pi radians. 00:01:52.980 --> 00:02:01.950 So we could say, so this equals 1 rev per second, times 00:02:01.950 --> 00:02:07.680 2 pi radians per rev, right? 00:02:07.680 --> 00:02:10.060 And then the revolutions will cancel out. 00:02:10.060 --> 00:02:12.230 And you have 1 times 2 pi, so you have 2 00:02:12.230 --> 00:02:14.620 pi radians per second. 00:02:14.620 --> 00:02:25.354 So this equals 2 pi radians per second. 00:02:25.354 --> 00:02:30.250 So that's interesting, we now know exactly after 5 seconds 00:02:30.250 --> 00:02:31.810 how many radians it has gone. 00:02:31.810 --> 00:02:34.050 Or after half a second, how many radians has it gone. 00:02:34.050 --> 00:02:35.650 But that might be vaguely useful. 00:02:35.650 --> 00:02:38.770 Let's see if we can somehow convert from this notion of 00:02:38.770 --> 00:02:44.530 how fast something is spinning to its actual speed. 00:02:44.530 --> 00:02:46.950 I was tempted to say velocity, but its velocity is always 00:02:46.950 --> 00:02:49.190 changing, because the direction is always changing. 00:02:49.190 --> 00:02:51.600 But the magnitude of the velocity is staying the same, 00:02:51.600 --> 00:02:52.740 so its speed is staying the same. 00:02:52.740 --> 00:02:55.680 But we'll say v for speed, because that's what they tend 00:02:55.680 --> 00:02:59.200 to do in most formulas that you'll see. 00:02:59.200 --> 00:03:02.710 So let's think about it this way, in 1 revolution-- so 00:03:02.710 --> 00:03:06.760 there's a couple ways you can think about this, but as we go 00:03:06.760 --> 00:03:11.010 1 revolution, how far has this object traveled? 00:03:11.010 --> 00:03:15.250 Well, it's traveled the circumference of this circle. 00:03:15.250 --> 00:03:16.770 And in order to know the circumference, we have to know 00:03:16.770 --> 00:03:17.680 the radius of the circle. 00:03:17.680 --> 00:03:21.540 So let's say that the radius is r-- let's say it's in 00:03:21.540 --> 00:03:22.870 meters, r meters. 00:03:25.700 --> 00:03:34.510 So how many meters will I travel in 1 second, then? 00:03:34.510 --> 00:03:36.870 Well, you could do the same thing up here. 00:03:36.870 --> 00:03:46.360 1 revolution per second, times 2 pi r, where r is the 00:03:46.360 --> 00:03:50.810 radius-- whoops, 2 pi r, you can ignore that line-- meters 00:03:50.810 --> 00:03:52.950 per revolution, that's just the circumference of the 00:03:52.950 --> 00:03:54.820 thing, of the circle. 00:03:54.820 --> 00:03:59.450 And that equals-- the revolutions cancel out-- 2 pi 00:03:59.450 --> 00:04:04.610 r meters per second. 00:04:04.610 --> 00:04:08.290 So it's interesting, given the radius and how many 00:04:08.290 --> 00:04:11.690 revolutions per second, we can now figure out its velocity. 00:04:11.690 --> 00:04:17.490 So this right here is how fast it's spinning, and this is the 00:04:17.490 --> 00:04:20.290 object's actual speed, right? 00:04:20.290 --> 00:04:23.030 And this term of how fast something's spinning, that's 00:04:23.030 --> 00:04:24.620 called angular velocity. 00:04:24.620 --> 00:04:26.620 And of course you know that the term for how fast 00:04:26.620 --> 00:04:30.240 something is actually moving is velocity. 00:04:30.240 --> 00:04:36.240 And just so you know, the term for angular velocity is this 00:04:36.240 --> 00:04:40.310 curvy w, I think that's lower case omega, 00:04:40.310 --> 00:04:43.050 that's angular velocity. 00:04:43.050 --> 00:04:49.070 So in this case, angular velocity is equal to 2 pi 00:04:49.070 --> 00:04:52.200 radians per second. 00:04:52.200 --> 00:04:54.060 And what's the velocity equal to, or at least the magnitude 00:04:54.060 --> 00:04:56.590 of the velocity-- I know the direction's always changing. 00:04:56.590 --> 00:05:02.270 Well, we know that the velocity is equal to 2 pi r 00:05:02.270 --> 00:05:03.920 meters per second. 00:05:03.920 --> 00:05:05.860 So if we just ignore the units for a second, where do you see 00:05:05.860 --> 00:05:07.590 the difference between the angular 00:05:07.590 --> 00:05:09.620 velocity and the velocity? 00:05:09.620 --> 00:05:12.850 The angular velocity in this case is 2 pi, and the 00:05:12.850 --> 00:05:14.680 velocity is 2 pi r. 00:05:14.680 --> 00:05:17.630 So in general, if you just multiply the angular velocity 00:05:17.630 --> 00:05:19.530 times r, you get the velocity. 00:05:19.530 --> 00:05:23.650 So angular velocity times the radius is equal to velocity. 00:05:23.650 --> 00:05:26.275 Or you can divide both sides of that by r, and you get the 00:05:26.275 --> 00:05:28.920 angular velocity is equal to the velocity 00:05:28.920 --> 00:05:30.490 divided by the radius. 00:05:30.490 --> 00:05:34.250 And this is a formula that you should know by heart, although 00:05:34.250 --> 00:05:36.820 it's good to know where it came from. 00:05:36.820 --> 00:05:39.670 I guess I did it this way to maybe give you an intuition, 00:05:39.670 --> 00:05:42.580 because I always have to work with numbers. 00:05:42.580 --> 00:05:44.650 Especially when I'm new to a concept-- so that's why I said 00:05:44.650 --> 00:05:46.390 1 revolution per second, instead of just putting 00:05:46.390 --> 00:05:50.170 everything as a variable-- but another way to think about it 00:05:50.170 --> 00:05:52.840 is, what is the definition of a radian? 00:05:52.840 --> 00:06:03.370 By definition, a radian-- if this angle is x radians, it's 00:06:03.370 --> 00:06:09.860 an angle, and it also tells us that the arc that is kind of 00:06:09.860 --> 00:06:14.590 projected by this angle, is equal to x radiuses. 00:06:17.100 --> 00:06:21.700 So if each radius is 2 meters, it would be x times 2 meters. 00:06:21.700 --> 00:06:23.700 So if this is x radians, then this is going to 00:06:23.700 --> 00:06:26.130 be x times r meters. 00:06:26.130 --> 00:06:34.860 And that actually comes from the definition of the radian. 00:06:34.860 --> 00:06:37.410 And that might be more intuitive to you, than the 00:06:37.410 --> 00:06:39.230 original explanation, or less, so hopefully one 00:06:39.230 --> 00:06:40.250 of those two works. 00:06:40.250 --> 00:06:43.730 But as you can see, if this angle is x, and this distance 00:06:43.730 --> 00:06:52.050 is x times r, and if omega is change in that angle, over 00:06:52.050 --> 00:06:53.300 change in time. 00:06:57.450 --> 00:07:02.890 Then we know this is true too, that velocity is just change 00:07:02.890 --> 00:07:04.160 in this, over change in time, right. 00:07:04.160 --> 00:07:09.680 Velocity is change in-- the radius doesn't change-- change 00:07:09.680 --> 00:07:13.280 in x times r, divided by change in time. 00:07:13.280 --> 00:07:15.760 And we know once again that this is omega. 00:07:15.760 --> 00:07:19.680 So another way we just showed again, that omega times the 00:07:19.680 --> 00:07:21.015 radius is equal to the velocity. 00:07:21.015 --> 00:07:23.860 Or the angular velocity times the radius is 00:07:23.860 --> 00:07:25.240 equal to the velocity. 00:07:25.240 --> 00:07:27.520 And this is a useful thing to learn, we'll see it in a 00:07:27.520 --> 00:07:31.020 couple of things, when I do the proof for centripetal 00:07:31.020 --> 00:07:33.600 acceleration in calculus, I'm going to use this fact. 00:07:33.600 --> 00:07:36.510 And when I-- and actually I'm probably going to record that 00:07:36.510 --> 00:07:39.020 video now-- I'm actually going to show you the law of 00:07:39.020 --> 00:07:41.370 conservation of angular momentum, which is very 00:07:41.370 --> 00:07:44.260 similar to the law of conservation of momentum, but 00:07:44.260 --> 00:07:45.250 it deals with things spinning. 00:07:45.250 --> 00:07:47.820 And this notion of angular velocity is 00:07:47.820 --> 00:07:49.090 going to come in useful. 00:07:49.090 --> 00:07:51.440 So this is the important takeaway, that w 00:07:51.440 --> 00:07:52.670 equals v over r. 00:07:52.670 --> 00:07:55.070 And hopefully my video has not confused you, and has shown 00:07:55.070 --> 00:07:59.350 you that w, the rate at which the angle is changing, is 00:07:59.350 --> 00:08:01.700 equal to the velocity of the object, or the magnitude of 00:08:01.700 --> 00:08:05.330 the velocity, divided by the radius of the 00:08:05.330 --> 00:08:07.280 circle that it's spinning. 00:08:07.280 --> 00:08:09.420 I'll see you in the next video.

Visual Proof: a= v^2/r

https://www.youtube.com/watch?v=TNX-Z6XR3gA

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https://www.youtube.com/api/timedtext?v=TNX-Z6XR3gA&ei=YmeUZcK_G93oxN8Pr9ePgAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=536FC14C8945CC62C2F9BBD9A42568868C6B7102.ECB61EEE4D63B46E604B8C9EAC5321710D61714C&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:01.550 Welcome back. 00:00:01.550 --> 00:00:03.670 In the last couple of videos, actually the very first video 00:00:03.670 --> 00:00:06.080 on centripetal acceleration, I told you that the necessary 00:00:06.080 --> 00:00:08.580 centripetal acceleration, and we're just talking about the 00:00:08.580 --> 00:00:12.630 magnitude, and actually the c tells you the direction, it's 00:00:12.630 --> 00:00:16.430 centripetal so its inward acceleration, it equals the 00:00:16.430 --> 00:00:18.610 velocity squared over the radius. 00:00:18.610 --> 00:00:21.170 I told you that the real rigorous proof has to be done 00:00:21.170 --> 00:00:21.620 with calculus. 00:00:21.620 --> 00:00:24.130 But I looked on Wikipedia and there's actually a pretty neat 00:00:24.130 --> 00:00:28.230 proof, although when you read Wikipedia, it's not so obvious 00:00:28.230 --> 00:00:29.150 on what they're saying. 00:00:29.150 --> 00:00:30.810 So I thought I would do a video on it, because this is 00:00:30.810 --> 00:00:33.990 cool and you don't need calculus to understand it. 00:00:33.990 --> 00:00:35.430 So let's just do something. 00:00:35.430 --> 00:00:40.285 Let's just plot the distance vector, the velocity vectors 00:00:40.285 --> 00:00:41.890 and the acceleration vectors as something 00:00:41.890 --> 00:00:42.920 goes around a circle. 00:00:42.920 --> 00:00:45.770 Let me draw two circles. 00:00:45.770 --> 00:00:50.340 So let's say this is my first circle. 00:00:50.340 --> 00:00:55.190 Let me draw another circle, another color just for fun. 00:00:55.190 --> 00:00:57.170 OK. 00:00:57.170 --> 00:00:58.420 This is my other circle. 00:01:02.210 --> 00:01:05.770 This is the center of the circle. 00:01:05.770 --> 00:01:08.410 And this is the center of the circle. 00:01:08.410 --> 00:01:12.340 And so what's the position vector in any point in time? 00:01:12.340 --> 00:01:15.450 Well, the position vector, you could just 00:01:15.450 --> 00:01:17.190 draw it as a radius. 00:01:19.950 --> 00:01:22.940 So the position vector at any given time looks 00:01:22.940 --> 00:01:23.710 something like this. 00:01:23.710 --> 00:01:25.600 So let's say initially this is the position vector. 00:01:29.420 --> 00:01:31.200 That's the initial position vector. 00:01:31.200 --> 00:01:35.680 Its magnitude is the radius of the circle. 00:01:35.680 --> 00:01:39.360 And this direction is right here in the positive 00:01:39.360 --> 00:01:40.480 x-direction. 00:01:40.480 --> 00:01:43.380 And at that point, what is the velocity vector? 00:01:43.380 --> 00:01:44.870 Well, the velocity-- let's assume we're going 00:01:44.870 --> 00:01:45.880 counterclockwise. 00:01:45.880 --> 00:01:47.740 I don't know why I assumed that. 00:01:47.740 --> 00:01:49.720 It could go the other way. 00:01:49.720 --> 00:01:52.346 Let's say that this is the velocity vector at that point. 00:01:52.346 --> 00:01:56.830 The velocity vector is going to look something like that. 00:01:59.810 --> 00:02:02.580 That is the velocity vector. 00:02:02.580 --> 00:02:04.440 That's v. 00:02:04.440 --> 00:02:07.510 It's going tangent to the circle. 00:02:07.510 --> 00:02:09.940 Let's plot the velocity vector as a function of 00:02:09.940 --> 00:02:10.919 time on this circle. 00:02:10.919 --> 00:02:13.870 So if that's the velocity vector of that time, I can 00:02:13.870 --> 00:02:15.100 draw the velocity vector here. 00:02:15.100 --> 00:02:15.880 This the the same vector. 00:02:15.880 --> 00:02:18.570 Remember, I'm just saying, at a particular time, what does 00:02:18.570 --> 00:02:20.710 the distance vector look like and what does the velocity 00:02:20.710 --> 00:02:21.400 vector look like? 00:02:21.400 --> 00:02:25.660 So at that time, the velocity vector looks like this. 00:02:25.660 --> 00:02:27.570 I'm trying to make it the same size to show you 00:02:27.570 --> 00:02:28.190 it's the same vector. 00:02:28.190 --> 00:02:29.650 And I'm doing it in the same color to show you it's the 00:02:29.650 --> 00:02:32.560 same vector. 00:02:32.560 --> 00:02:35.300 This is the exact same vector. 00:02:35.300 --> 00:02:36.360 This is the actual circle. 00:02:36.360 --> 00:02:39.500 Like if you were to draw it, this is the path of the dot or 00:02:39.500 --> 00:02:40.990 whatever is moving around the circle. 00:02:40.990 --> 00:02:44.510 And this, you can just view it as I'm plotting the velocity 00:02:44.510 --> 00:02:47.790 over time, the velocity vector over time. 00:02:47.790 --> 00:02:52.470 So let's say a few seconds later, or a few moments later, 00:02:52.470 --> 00:02:55.460 what does the radius vector look like? 00:02:55.460 --> 00:02:57.030 Well, then the radius vector looks like this. 00:03:04.216 --> 00:03:07.620 I'm trying to write as neatly as possible. 00:03:07.620 --> 00:03:10.150 And what does the velocity vector look like? 00:03:10.150 --> 00:03:12.700 Go back to the purple. 00:03:12.700 --> 00:03:15.850 The velocity vector once again is tangent to the circle. 00:03:15.850 --> 00:03:19.160 And it'll look something like that. 00:03:19.160 --> 00:03:20.160 It's going to have the same magnitude, 00:03:20.160 --> 00:03:21.410 just different direction. 00:03:23.700 --> 00:03:25.870 Actually, let me do it in a different color, just to show 00:03:25.870 --> 00:03:26.590 you what I'm doing. 00:03:26.590 --> 00:03:31.250 So let me do it in brown. 00:03:31.250 --> 00:03:34.020 So the velocity vector is going to look 00:03:34.020 --> 00:03:36.190 something like this. 00:03:36.190 --> 00:03:40.040 It's brown, supposed to be a brown. 00:03:40.040 --> 00:03:40.770 That's the velocity vector. 00:03:40.770 --> 00:03:44.260 So after a few seconds, or a few moments, where is the 00:03:44.260 --> 00:03:46.270 velocity vector here? 00:03:46.270 --> 00:03:47.300 Well, it's going to look like this. 00:03:47.300 --> 00:03:48.850 Remember, this vector I'm just plotting 00:03:48.850 --> 00:03:51.840 here after a few seconds. 00:03:51.840 --> 00:03:54.320 Let me use that line tool. 00:03:54.320 --> 00:03:58.630 So it's the same magnitude and now the velocity is just at a 00:03:58.630 --> 00:03:59.290 different angle. 00:03:59.290 --> 00:04:01.615 It should be the exact same angle as what I just drew in 00:04:01.615 --> 00:04:02.865 the other circle. 00:04:07.790 --> 00:04:11.660 So that's this velocity vector. 00:04:11.660 --> 00:04:13.460 So when we start here, the velocity is going to change at 00:04:13.460 --> 00:04:14.370 the circle. 00:04:14.370 --> 00:04:17.019 After a few moments, the object's rotating around the 00:04:17.019 --> 00:04:19.390 circle, so now the velocity is the same magnitude. 00:04:19.390 --> 00:04:21.380 It's just switched directions. 00:04:21.380 --> 00:04:22.250 And so what's happening here? 00:04:22.250 --> 00:04:25.370 When we plot the velocity vector over time, it has the 00:04:25.370 --> 00:04:29.750 same magnitude, so it'll draw out a circle, but its 00:04:29.750 --> 00:04:32.680 direction changes. 00:04:32.680 --> 00:04:36.110 Just as in this case, what takes us from this point to 00:04:36.110 --> 00:04:36.910 this point? 00:04:36.910 --> 00:04:38.960 This was the velocity vector and the velocity vector's 00:04:38.960 --> 00:04:39.690 always changing. 00:04:39.690 --> 00:04:42.300 But, in general, this is the change in position. 00:04:42.300 --> 00:04:43.680 And what causes the change in position? 00:04:43.680 --> 00:04:45.940 Well, the velocity, or at least the speed, because the 00:04:45.940 --> 00:04:47.720 vector's always changing. 00:04:47.720 --> 00:04:50.390 So in this case, what's changing the velocity? 00:04:50.390 --> 00:04:54.270 Well, just like velocity changes radius, acceleration 00:04:54.270 --> 00:04:56.670 changes velocity. 00:04:56.670 --> 00:05:00.910 Let me draw an acceleration vector, and acceleration is 00:05:00.910 --> 00:05:04.690 going to be in this direction. 00:05:04.690 --> 00:05:06.930 Because if the velocity is changing from here to here, 00:05:06.930 --> 00:05:10.410 the acceleration is going along the direction of the 00:05:10.410 --> 00:05:12.950 change in velocity. 00:05:12.950 --> 00:05:14.380 So the acceleration vector might look 00:05:14.380 --> 00:05:15.010 something like that. 00:05:15.010 --> 00:05:17.200 It's going to be tangent to this velocity path. 00:05:21.700 --> 00:05:23.940 And that's interesting, too, because if this is the 00:05:23.940 --> 00:05:27.000 acceleration vector when the velocity is here, so when the 00:05:27.000 --> 00:05:29.100 velocity is here, the acceleration vector's going 00:05:29.100 --> 00:05:32.010 directly to the left, so that means that acceleration 00:05:32.010 --> 00:05:34.890 vector's going directly to left, and so this is also the 00:05:34.890 --> 00:05:38.120 acceleration vector, which coincides with what we learned 00:05:38.120 --> 00:05:39.610 about centripetal acceleration. 00:05:39.610 --> 00:05:41.610 The acceleration has to be going inwards. 00:05:41.610 --> 00:05:43.300 And we see that when we actually plot 00:05:43.300 --> 00:05:44.720 the velocity vector. 00:05:44.720 --> 00:05:49.430 And if we plot acceleration vector here, once again it's 00:05:49.430 --> 00:05:51.700 going to be going tangent to this plot of 00:05:51.700 --> 00:05:52.920 the velocity vector. 00:05:52.920 --> 00:05:53.880 Let me do it in a different color. 00:05:53.880 --> 00:05:54.602 I've already used that color. 00:05:54.602 --> 00:05:56.280 I'll do it in yellow. 00:05:56.280 --> 00:05:57.870 So then the acceleration vector is going to look 00:05:57.870 --> 00:05:58.460 something like this. 00:05:58.460 --> 00:05:59.530 Remember, acceleration is nothing 00:05:59.530 --> 00:06:00.930 but change in velocity. 00:06:00.930 --> 00:06:03.915 So at this point, when this is the velocity vector, the 00:06:03.915 --> 00:06:06.130 acceleration vector will just be this. 00:06:06.130 --> 00:06:10.540 Once again, it's just the opposite direction of the 00:06:10.540 --> 00:06:11.500 position vector. 00:06:11.500 --> 00:06:12.720 So why am I doing all this? 00:06:12.720 --> 00:06:14.240 Well, I'm doing all of this to set up the 00:06:14.240 --> 00:06:16.700 analogy to show you. 00:06:16.700 --> 00:06:22.210 As this object completes one entire path, what's happening 00:06:22.210 --> 00:06:22.960 on this circle? 00:06:22.960 --> 00:06:25.890 Well, this circle, the velocity is completing one 00:06:25.890 --> 00:06:27.480 entire path, right? 00:06:27.480 --> 00:06:31.220 However long it takes to go around this circle is the same 00:06:31.220 --> 00:06:35.840 amount of time it takes to go from this velocity back to 00:06:35.840 --> 00:06:36.630 this velocity. 00:06:36.630 --> 00:06:38.950 The magnitude is the same the whole time, but the direction 00:06:38.950 --> 00:06:39.990 is changing. 00:06:39.990 --> 00:06:43.010 So if this takes 10 seconds to go around the circle in real 00:06:43.010 --> 00:06:46.910 position, then it takes 10 seconds for the acceleration 00:06:46.910 --> 00:06:50.210 to change the direction of this velocity enough that it 00:06:50.210 --> 00:06:53.910 goes back to the original velocity direction. 00:06:53.910 --> 00:06:55.950 So why am I doing all of that? 00:06:55.950 --> 00:07:01.110 Well, how long does it take for the object to do one 00:07:01.110 --> 00:07:03.660 rotation around this path? 00:07:03.660 --> 00:07:06.650 Well, it's just the distance divided by the velocity. 00:07:06.650 --> 00:07:10.740 So the time to do one revolution around this path is 00:07:10.740 --> 00:07:12.190 the distance-- well, that's just the 00:07:12.190 --> 00:07:13.420 circumference of the circle. 00:07:13.420 --> 00:07:21.920 Well, that's 2 pi r divided by your speed, which is v. 00:07:21.920 --> 00:07:26.190 And how much time does it take to go around this path? 00:07:26.190 --> 00:07:31.080 Well, we know it's going be the same time, and now we're 00:07:31.080 --> 00:07:33.400 going to say the time's going to be the change 00:07:33.400 --> 00:07:34.770 in velocity, right? 00:07:34.770 --> 00:07:37.435 In this circle, this is the change in distance, but it's 00:07:37.435 --> 00:07:40.170 the change in velocity. 00:07:40.170 --> 00:07:42.530 So here we had this much change in velocity. 00:07:42.530 --> 00:07:44.450 I know this might be a little non-intuitive. 00:07:44.450 --> 00:07:46.330 Then we have a little bit more change in velocity, a little 00:07:46.330 --> 00:07:48.260 bit more change in velocity, a little bit 00:07:48.260 --> 00:07:49.810 more change in velocity. 00:07:49.810 --> 00:07:53.790 So the total change in velocity is just going to be 00:07:53.790 --> 00:07:56.740 the circumference of this circle. 00:07:56.740 --> 00:07:58.720 And what's the circumference of this circle? 00:07:58.720 --> 00:08:01.870 The radius of the circle is the magnitude of the velocity. 00:08:01.870 --> 00:08:06.140 So it's 2 pi times the velocity of the circle. 00:08:06.140 --> 00:08:10.570 And so, what is the amount of time it takes a do one 00:08:10.570 --> 00:08:11.170 revolution? 00:08:11.170 --> 00:08:13.280 Well, it's the total change in velocity. 00:08:13.280 --> 00:08:16.010 And that's 2 pi times the magnitude of the velocity 00:08:16.010 --> 00:08:19.110 divided by acceleration. 00:08:19.110 --> 00:08:21.490 And if this doesn't make complete sense, just remember, 00:08:21.490 --> 00:08:24.980 acceleration is change in velocity over change in time. 00:08:29.290 --> 00:08:32.520 I'm glossing over it a little bit, because you might say, 00:08:32.520 --> 00:08:34.299 oh, well, the net change in velocity is zero. 00:08:34.299 --> 00:08:38.059 But we're actually more concerned about what's the 00:08:38.059 --> 00:08:40.100 total change in the velocity? 00:08:40.100 --> 00:08:43.559 So it changes a lot, then it goes back to its original, so 00:08:43.559 --> 00:08:45.420 that the net is still zero. 00:08:45.420 --> 00:08:48.200 But this should hopefully give you a little intuition. 00:08:48.200 --> 00:08:49.250 So think about what I said. 00:08:49.250 --> 00:08:52.760 But the bottom line is we know that this expression and this 00:08:52.760 --> 00:08:56.080 expression have to be equal, because in the same amount of 00:08:56.080 --> 00:09:00.050 time it takes the object to do one complete revolution on 00:09:00.050 --> 00:09:03.730 this circle, its velocity, direction, has also done one 00:09:03.730 --> 00:09:04.560 complete revolution. 00:09:04.560 --> 00:09:07.070 So we can set these two equal to each other. 00:09:07.070 --> 00:09:16.440 So we could say 2 pi r over v is equal to 2 pi v over a. 00:09:16.440 --> 00:09:17.750 Let me switch colors. 00:09:17.750 --> 00:09:21.490 We can cross the 2 pi out on both sides. 00:09:21.490 --> 00:09:25.530 And then we could multiply both sides times v. 00:09:25.530 --> 00:09:28.970 You get r is equal to v squared over a. 00:09:28.970 --> 00:09:32.440 Multiply both sides times a, I'm doing this little bit 00:09:32.440 --> 00:09:36.120 circuitously, and you get ar is equal to v squared. 00:09:36.120 --> 00:09:38.940 Divide both sides by r, and you get a is equal to v 00:09:38.940 --> 00:09:40.850 squared over r. 00:09:40.850 --> 00:09:43.010 So hopefully, that gives you a little bit of intuition about 00:09:43.010 --> 00:09:44.650 why this works. 00:09:44.650 --> 00:09:46.240 And I got this from Wikipedia, so I want to 00:09:46.240 --> 00:09:47.210 give them proper credit. 00:09:47.210 --> 00:09:50.550 And hopefully, my explanation helps clarify what the people 00:09:50.550 --> 00:09:52.940 on Wikipedia are talking about a little bit. 00:09:52.940 --> 00:09:55.880 Anyway, I'll also do the calculus proof, because that's 00:09:55.880 --> 00:09:57.720 more just straight up math. 00:09:57.720 --> 00:10:00.070 And I'll see you in the next video.

Centripetal Acceleration (part 3)

https://www.youtube.com/watch?v=p3AGlD6g8X8

vtt

https://www.youtube.com/api/timedtext?v=p3AGlD6g8X8&ei=YmeUZZGsHPirp-oPq6600AM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=CFE695B858785F9210307F13A452CBE896805118.1C7727B6A6A79C7FF39E74CFE6883105802EBA4F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:01.460 Welcome back. 00:00:01.460 --> 00:00:04.250 I thought I'd do a couple of centripetal acceleration 00:00:04.250 --> 00:00:06.250 problems. Some of the more common ones that you 00:00:06.250 --> 00:00:07.300 maybe see in school. 00:00:07.300 --> 00:00:10.015 So let's say I have a, well it could be a hot wheel, or it 00:00:10.015 --> 00:00:11.640 could be a car, a real car. 00:00:11.640 --> 00:00:13.195 And say it's on a track. 00:00:13.195 --> 00:00:15.520 So we draw the track. 00:00:15.520 --> 00:00:16.550 What's a good color for a track. 00:00:16.550 --> 00:00:17.390 I don't know, yellow. 00:00:17.390 --> 00:00:18.480 It's a yellow track. 00:00:18.480 --> 00:00:20.770 So this is going to be a side view. 00:00:20.770 --> 00:00:23.160 And we all remember this from our Hot Wheels days. 00:00:23.160 --> 00:00:26.840 And the track does a loop-d-loop, like that. 00:00:26.840 --> 00:00:28.840 And then I have a car, and let's say it's going at a 00:00:28.840 --> 00:00:29.700 constant velocity. 00:00:29.700 --> 00:00:31.980 We're not going to worry about the car's, or at least a 00:00:31.980 --> 00:00:34.270 constant speed, right, cause the direction of velocity 00:00:34.270 --> 00:00:35.080 might change. 00:00:35.080 --> 00:00:36.410 Just not the magnitude. 00:00:36.410 --> 00:00:38.460 So this is the car. 00:00:38.460 --> 00:00:41.120 And my question to you is, how fast does it have to be going? 00:00:41.120 --> 00:00:44.580 What's its speed have to be for it to not fall when it 00:00:44.580 --> 00:00:46.800 gets to this point in the loop-d-loop? 00:00:46.800 --> 00:00:47.950 So it's going to go like this. 00:00:47.950 --> 00:00:50.960 This is going to be the path of the car. 00:00:50.960 --> 00:00:53.800 How fast does it need to go? 00:00:53.800 --> 00:00:56.180 So let's just think about it. 00:00:56.180 --> 00:00:58.340 Of course, I have to give a little bit of information. 00:00:58.340 --> 00:01:07.590 Let's say the radius of this loop-d-loop, I don't know, 00:01:07.590 --> 00:01:08.790 let's say it's 20 feet. 00:01:08.790 --> 00:01:09.930 I'm making this up on the fly. 00:01:09.930 --> 00:01:11.760 I hope the numbers work out well. 00:01:11.760 --> 00:01:16.020 So what we have to figure out is essentially, what is the 00:01:16.020 --> 00:01:18.260 centripetal acceleration going to be on this car? 00:01:18.260 --> 00:01:19.980 And if we know the centripetal acceleration and we know the 00:01:19.980 --> 00:01:22.370 radius, we can figure out the velocity. 00:01:22.370 --> 00:01:24.710 So let's just think about what happens to the car as it goes 00:01:24.710 --> 00:01:26.400 up the loop-d-loop. 00:01:26.400 --> 00:01:28.950 At this point, let's say when the car's right here-- so this 00:01:28.950 --> 00:01:33.490 is the car-- what is making the car's velocity change? 00:01:33.490 --> 00:01:36.170 Cause at this point the car's velocity is like this. 00:01:36.170 --> 00:01:38.820 And then at this point, the car's velocity is like this. 00:01:38.820 --> 00:01:40.750 At this point, the car's velocity is like this. 00:01:40.750 --> 00:01:44.720 It's always going to be tangent to the loop-d-loop. 00:01:44.720 --> 00:01:46.950 Well, this point, down here, is actually going to be the 00:01:46.950 --> 00:01:49.580 normal force of the loop-d-loop right? 00:01:49.580 --> 00:01:53.650 As the car is kind of driving into the slope, the slope is 00:01:53.650 --> 00:01:56.590 putting upward pressure on the car that's making it go in the 00:01:56.590 --> 00:01:58.280 circular path. 00:01:58.280 --> 00:02:01.460 But as we go beyond this point, we see something 00:02:01.460 --> 00:02:02.820 interesting happening. 00:02:02.820 --> 00:02:08.639 There's going to be the normal force of the loop-d-loop 00:02:08.639 --> 00:02:09.949 itself, of the surface. 00:02:09.949 --> 00:02:12.030 And then we're also going to start having gravity pulling 00:02:12.030 --> 00:02:13.050 down on the car. 00:02:13.050 --> 00:02:15.590 And some portion of the gravity will kind of pull to 00:02:15.590 --> 00:02:16.350 the center. 00:02:16.350 --> 00:02:18.440 And it's a little complicated at this point. 00:02:18.440 --> 00:02:23.400 But at this point-- let me draw the car, let me draw the 00:02:23.400 --> 00:02:29.540 upside down car-- what are the forces? 00:02:29.540 --> 00:02:35.630 Well, if we want the car to just not fall, right, the only 00:02:35.630 --> 00:02:37.480 force is going to be the force of gravity, or the 00:02:37.480 --> 00:02:39.380 acceleration of gravity. 00:02:39.380 --> 00:02:42.110 If the car is going fast enough to offset the 00:02:42.110 --> 00:02:46.060 acceleration of gravity, it'll stay on the loop-d-loop. 00:02:46.060 --> 00:02:49.310 So let's figure out how fast it has to be going to offset 00:02:49.310 --> 00:02:51.060 the acceleration of gravity. 00:02:51.060 --> 00:02:53.620 Well, we know from the previous video, that the 00:02:53.620 --> 00:02:58.020 centripetal acceleration is equal to the magnitude-- I'm 00:02:58.020 --> 00:02:59.800 going to put an absolute value around that so you know it's 00:02:59.800 --> 00:03:01.180 not V for velocity vector, it's the 00:03:01.180 --> 00:03:02.850 magnitude of the velocity. 00:03:02.850 --> 00:03:04.870 I know I keep changing conventions, but that's good 00:03:04.870 --> 00:03:07.720 for you cause you'll see different things-- divided by 00:03:07.720 --> 00:03:08.200 the radius. 00:03:08.200 --> 00:03:09.260 Velocity squared over radius. 00:03:09.260 --> 00:03:10.250 Remember, this isn't the vector. 00:03:10.250 --> 00:03:12.140 This is just the magnitude. 00:03:12.140 --> 00:03:14.710 And once again this is also just the magnitude of the 00:03:14.710 --> 00:03:15.680 acceleration vector. 00:03:15.680 --> 00:03:16.780 So we know what the acceleration 00:03:16.780 --> 00:03:18.030 vector is at this point. 00:03:20.730 --> 00:03:23.340 Well, we're just worried about the magnitude. 00:03:23.340 --> 00:03:25.670 We'll say it's 32 feet per second squared. 00:03:25.670 --> 00:03:28.890 That's the acceleration of gravity on the surface of the 00:03:28.890 --> 00:03:29.910 earth, at sea level. 00:03:29.910 --> 00:03:32.300 30 feet per second squared. 00:03:32.300 --> 00:03:35.860 And that equals the velocity squared-- I'll get rid of the 00:03:35.860 --> 00:03:40.410 absolute value sign, or the magnitude sign-- divided by 00:03:40.410 --> 00:03:43.270 the radius, divided by 20. 00:03:43.270 --> 00:03:45.380 And so multiply both sides by 20. 00:03:45.380 --> 00:03:50.645 I get 640 is equal to velocity squared. 00:03:53.240 --> 00:04:04.170 And so 640 square root is 25.3. 00:04:04.170 --> 00:04:11.366 So the car has to be going 25.3 feet per second. 00:04:11.366 --> 00:04:12.780 So now I'll ask you a question. 00:04:12.780 --> 00:04:15.500 What happens if it's going at 20 feet per second? 00:04:15.500 --> 00:04:17.640 Well, if it's going 20 feet per second, some place around 00:04:17.640 --> 00:04:21.470 here, let me draw the path of the slower car going 20 feet 00:04:21.470 --> 00:04:22.240 per second. 00:04:22.240 --> 00:04:24.830 If the car's going 20 feet per second, it'll probably make it 00:04:24.830 --> 00:04:26.870 pretty far, and then some place around here, it's just 00:04:26.870 --> 00:04:29.870 going to start falling and then it'll fall down. 00:04:29.870 --> 00:04:31.490 That's a car going 20 feet per second. 00:04:34.210 --> 00:04:35.390 I'll ask you another question. 00:04:35.390 --> 00:04:38.590 What's going to happen if the car goes faster? 00:04:38.590 --> 00:04:40.950 Let's say what happens if the velocity is-- and I'll do it 00:04:40.950 --> 00:04:46.990 in a different color-- I don't know, 50 feet per second? 00:04:46.990 --> 00:04:48.750 So this is a super fast car. 00:04:52.660 --> 00:04:54.800 So if this was just a simple orbit 00:04:54.800 --> 00:04:56.080 problem, what would happen? 00:04:56.080 --> 00:04:59.550 We have this centripetal acceleration from gravity, but 00:04:59.550 --> 00:05:02.600 that alone isn't enough to offset its velocity. 00:05:02.600 --> 00:05:05.120 So if we have no other forces other than gravity, the car 00:05:05.120 --> 00:05:09.520 would kind of exit its orbit, you could kind of say right? 00:05:09.520 --> 00:05:11.620 It would actually go out of its path. 00:05:11.620 --> 00:05:13.190 It might do something like this. 00:05:13.190 --> 00:05:15.160 Whoops, its path might look something like this. 00:05:15.160 --> 00:05:17.390 I know I draw these very messy things, but it might actually 00:05:17.390 --> 00:05:19.170 fly out, right? 00:05:19.170 --> 00:05:21.700 So what's keeping it from flying out? 00:05:21.700 --> 00:05:25.430 Well, the actual surface of the loop-d-loop right? 00:05:25.430 --> 00:05:29.150 So we have something very interesting going on here. 00:05:29.150 --> 00:05:30.300 Well, let's figure it out. 00:05:30.300 --> 00:05:32.580 What is the centripetal acceleration have to be? 00:05:32.580 --> 00:05:34.630 Well, if the car's going 50 feet per second, in order to 00:05:34.630 --> 00:05:38.840 keep it going in a circle, the acceleration has to be 50 00:05:38.840 --> 00:05:43.860 squared over the radius, which is 20. 00:05:43.860 --> 00:05:50.630 50 times 50, so that's 25, two 0's over 20. 00:05:50.630 --> 00:05:52.020 The 0's cancel. 00:05:52.020 --> 00:05:53.960 So the acceleration is going have to be 00:05:53.960 --> 00:05:59.490 125 feet per second. 00:05:59.490 --> 00:06:02.090 The inward acceleration, the centripetal acceleration has 00:06:02.090 --> 00:06:04.980 to be 125 feet per second. 00:06:04.980 --> 00:06:06.340 So what's going on? 00:06:06.340 --> 00:06:07.930 Gravity is only going to provide 32 feet 00:06:07.930 --> 00:06:09.220 per second of that. 00:06:09.220 --> 00:06:12.330 So the rest of it is actually going to be the normal force 00:06:12.330 --> 00:06:15.740 of the surface of the loop-d-loop or of the surface 00:06:15.740 --> 00:06:17.690 of the road, cause the car is going to push. 00:06:17.690 --> 00:06:22.020 The car is going to want to kind of exit its orbit. 00:06:22.020 --> 00:06:23.570 It's going to do something like that. 00:06:23.570 --> 00:06:26.610 And what's keeping it from doing that is the road. 00:06:26.610 --> 00:06:28.630 The road is keeping the car contained. 00:06:28.630 --> 00:06:34.530 And so it's essentially putting enough extra normal 00:06:34.530 --> 00:06:39.050 force onto the car and so that force will kind of be applied 00:06:39.050 --> 00:06:41.850 to the tires to offset that. 00:06:41.850 --> 00:06:44.580 And there you have it. 00:06:44.580 --> 00:06:47.680 So think of it this way. 00:06:47.680 --> 00:06:54.380 If you were tied to the loop-d-loop on the inside, 00:06:54.380 --> 00:06:55.700 right here. 00:06:55.700 --> 00:06:57.930 If this is some kind of western movie where the 00:06:57.930 --> 00:07:01.130 heroine is about to die, and there's a car coming that's 00:07:01.130 --> 00:07:02.780 going to roll her over. 00:07:02.780 --> 00:07:08.950 If that car is going at exactly 25.3 feet per second, 00:07:08.950 --> 00:07:10.130 it'll probably just kind of bump 00:07:10.130 --> 00:07:11.140 over the heroine actually. 00:07:11.140 --> 00:07:12.780 The heroine probably will cause the car to fall and she 00:07:12.780 --> 00:07:14.500 probably won't die. 00:07:14.500 --> 00:07:18.120 But if that car's going at 50 feet per second, then there's 00:07:18.120 --> 00:07:23.750 enough normal force so the road is pushing on the car, 00:07:23.750 --> 00:07:25.720 and of course is an equal and opposite force, so the car is 00:07:25.720 --> 00:07:26.600 pushing down. 00:07:26.600 --> 00:07:28.410 And that amount of the car pushing down 00:07:28.410 --> 00:07:31.030 would squish the heroine. 00:07:31.030 --> 00:07:34.240 So I don't know if that was disturbing or useful or 00:07:34.240 --> 00:07:36.060 confusing, but this is just another way 00:07:36.060 --> 00:07:37.000 to think about things. 00:07:37.000 --> 00:07:38.550 Anyway, I'll see you in the next video.

Centripetal Acceleration (part 2)

https://www.youtube.com/watch?v=UmiotSKSRvw

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https://www.youtube.com/api/timedtext?v=UmiotSKSRvw&ei=ZGeUZfO3DO2Ip-oP0veBsAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249812&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=29D98A576A4EC7929A714472A7A411FF1BFD8315.29A6CF6745C97F16CC5BDF3DFFFAC3551FC559C3&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:01.450 Welcome back. 00:00:01.450 --> 00:00:04.582 So where I left off, I was hopefully trying to give you 00:00:04.582 --> 00:00:07.650 an intuition of this inward acceleration. 00:00:07.650 --> 00:00:09.820 In this case, I have a rock spinning around, and I'm 00:00:09.820 --> 00:00:11.020 holding it on a string. 00:00:11.020 --> 00:00:15.100 And I want to think about what the inward acceleration is 00:00:15.100 --> 00:00:16.440 dependent on. 00:00:16.440 --> 00:00:19.570 And so we talked a little bit about its velocity in these 00:00:19.570 --> 00:00:21.250 multiple red arrows. 00:00:21.250 --> 00:00:22.740 These are potential velocity vectors. 00:00:22.740 --> 00:00:25.760 So the point I was trying to make is, if the velocity 00:00:25.760 --> 00:00:30.070 vector is bigger, to change its direction requires even 00:00:30.070 --> 00:00:31.550 more acceleration. 00:00:31.550 --> 00:00:34.230 Because the change in the x direction is bigger, and then 00:00:34.230 --> 00:00:35.780 the change in the y direction is bigger. 00:00:35.780 --> 00:00:39.580 Because for example, right here, it's y-- I'll draw it in 00:00:39.580 --> 00:00:41.540 a different color-- it's y velocity vector. 00:00:41.540 --> 00:00:44.350 And I'll do it for the big velocity. 00:00:44.350 --> 00:00:46.100 Its y velocity vector was here. 00:00:46.100 --> 00:00:49.720 And then now its y velocity vector is almost essentially 00:00:49.720 --> 00:00:52.150 the whole vector. 00:00:52.150 --> 00:00:54.490 So when the velocity increases, and let's say this 00:00:54.490 --> 00:00:55.940 is the case of the big arrow. 00:00:55.940 --> 00:00:59.480 If the velocity was less, like the small arrow, its velocity 00:00:59.480 --> 00:01:05.890 of vectors was here, and it would have to go to here. 00:01:05.890 --> 00:01:08.480 So I don't know if it's completely obvious, but the 00:01:08.480 --> 00:01:11.220 change in velocity, for example in the y direction 00:01:11.220 --> 00:01:14.770 over this course of time, has to be more. 00:01:14.770 --> 00:01:17.140 And likewise, the change in x direction over this, and this 00:01:17.140 --> 00:01:20.500 might be more obvious, because here this is the x velocity 00:01:20.500 --> 00:01:22.810 vector if the velocity was less. 00:01:22.810 --> 00:01:25.840 And this is the x velocity vector if it was more. 00:01:25.840 --> 00:01:27.020 And that's going to 0. 00:01:27.020 --> 00:01:29.060 Between when the dot goes from here to here. 00:01:29.060 --> 00:01:31.920 Here it has no x component. 00:01:31.920 --> 00:01:34.672 So clearly, I would have had to decelerate in the x 00:01:34.672 --> 00:01:39.420 direction more, if we have a larger velocity vector. 00:01:39.420 --> 00:01:41.315 I'm either giving you an intuition, or I'm confusing 00:01:41.315 --> 00:01:43.450 you, but I'll keep going. 00:01:43.450 --> 00:01:48.220 So velocity clearly-- the greater the velocity, the more 00:01:48.220 --> 00:01:51.290 I'm going to have to pull in on this one. 00:01:51.290 --> 00:01:53.800 The more the inward acceleration has to be to keep 00:01:53.800 --> 00:01:55.730 this thing going in the circle. 00:01:55.730 --> 00:01:57.840 And not only does the acceleration have to be bigger 00:01:57.840 --> 00:02:00.295 just to get that velocity vector down, but you've got to 00:02:00.295 --> 00:02:05.490 realize when, between this point and this point, the 00:02:05.490 --> 00:02:08.360 faster I go, the less time it's going to take me to get 00:02:08.360 --> 00:02:10.340 from here to here. 00:02:10.340 --> 00:02:13.980 So not only am I going to have to change the velocity more, I 00:02:13.980 --> 00:02:16.670 have to do that in less time. 00:02:16.670 --> 00:02:20.310 So the velocity is affecting this acceleration that I need 00:02:20.310 --> 00:02:21.390 to pull in two ways. 00:02:21.390 --> 00:02:24.650 So the higher the velocity, I have to change more of the 00:02:24.650 --> 00:02:27.290 velocity, and I have to do it in less time. 00:02:27.290 --> 00:02:30.170 And it actually turns out that the acceleration is 00:02:30.170 --> 00:02:34.410 proportional to the velocity squared. 00:02:34.410 --> 00:02:35.680 And then there's another term, and I won't 00:02:35.680 --> 00:02:36.360 prove it in this video. 00:02:36.360 --> 00:02:38.240 I'm going to prove it in another video. 00:02:38.240 --> 00:02:40.360 And that might be out of your scope just now, because it 00:02:40.360 --> 00:02:42.120 requires some calculus. 00:02:42.120 --> 00:02:43.420 But I want to give you an intuition. 00:02:43.420 --> 00:02:45.670 So we know that the acceleration is dependent on 00:02:45.670 --> 00:02:46.880 the velocity squared. 00:02:46.880 --> 00:02:49.800 And I want to make that point of why it's not just velocity, 00:02:49.800 --> 00:02:55.390 because there's two ways that the velocity is affecting how 00:02:55.390 --> 00:02:56.980 much of an acceleration I need. 00:02:56.980 --> 00:02:58.880 One, is just the magnitude of the velocity. 00:02:58.880 --> 00:03:02.170 The more that is, the more I have to accelerate to change 00:03:02.170 --> 00:03:03.070 this direction. 00:03:03.070 --> 00:03:06.800 And the second is, between this point and this point, I 00:03:06.800 --> 00:03:09.190 have less time to change that velocity. 00:03:09.190 --> 00:03:11.470 So the velocity affects it in two ways. 00:03:11.470 --> 00:03:14.370 And that's where you get an intuition for the v squared. 00:03:14.370 --> 00:03:17.780 The other thing that matters is actually the radius. 00:03:17.780 --> 00:03:20.960 So let's say this is the radius. 00:03:20.960 --> 00:03:22.990 And the acceleration is inversely 00:03:22.990 --> 00:03:24.850 proportional to the radius. 00:03:24.850 --> 00:03:26.180 And why is that? 00:03:26.180 --> 00:03:28.700 Well, it's kind of like the second, I guess we can call 00:03:28.700 --> 00:03:32.940 it, velocity argument I gave. The bigger the radius. 00:03:32.940 --> 00:03:35.260 So let's say that, let's look at this rate and let's look at 00:03:35.260 --> 00:03:37.480 another radius that is bigger. 00:03:37.480 --> 00:03:39.350 Let's say that this is another radius. 00:03:39.350 --> 00:03:41.040 Let's say if the string was longer and 00:03:41.040 --> 00:03:42.290 this was another case. 00:03:44.860 --> 00:03:46.830 Let's says this was the case. 00:03:46.830 --> 00:03:48.830 But the velocity was the same. 00:03:48.830 --> 00:03:53.110 So this is the case with a larger-- so let's say it's the 00:03:53.110 --> 00:03:54.930 same velocity vector. 00:03:54.930 --> 00:03:56.890 So we have the same thing at play, to go from 00:03:56.890 --> 00:03:58.600 there to say, there. 00:04:01.190 --> 00:04:05.320 We have to change the velocity the same magnitude. 00:04:05.320 --> 00:04:10.250 As we had to change the velocity from here to here. 00:04:13.050 --> 00:04:14.920 But one thing changes. 00:04:14.920 --> 00:04:18.099 This distance is longer. 00:04:18.099 --> 00:04:20.430 So if this velocity is the same as this velocity, but 00:04:20.430 --> 00:04:23.830 this distance is longer, you have more time 00:04:23.830 --> 00:04:25.550 to change the velocity. 00:04:25.550 --> 00:04:29.140 So the larger the radius, the more time you have to change 00:04:29.140 --> 00:04:30.240 the velocity. 00:04:30.240 --> 00:04:32.780 And so the less acceleration you need. 00:04:32.780 --> 00:04:35.590 Hopefully that makes sense. 00:04:35.590 --> 00:04:37.170 Let me repeat it. 00:04:37.170 --> 00:04:40.280 Because I think it's nice to have the intuition. 00:04:40.280 --> 00:04:43.730 Between this point and this point, unless you have a 00:04:43.730 --> 00:04:48.040 smaller radius, if I want to change the acceleration vector 00:04:48.040 --> 00:04:55.380 from that to that, I have however long it takes to go 00:04:55.380 --> 00:04:56.030 this distance. 00:04:56.030 --> 00:05:00.660 I have that long to change its velocity vector. 00:05:00.660 --> 00:05:02.990 But if the radius was a little longer. 00:05:02.990 --> 00:05:05.840 And I had the same velocity, so here the velocity is that. 00:05:05.840 --> 00:05:08.680 And then here, the velocity is that. 00:05:08.680 --> 00:05:12.180 I have a little bit more time to change its velocity. 00:05:12.180 --> 00:05:15.470 And remember, acceleration is just change in velocity over 00:05:15.470 --> 00:05:16.890 change in time. 00:05:16.890 --> 00:05:21.600 And so if I have the same change in velocity, but I have 00:05:21.600 --> 00:05:23.250 more time to do it with. 00:05:23.250 --> 00:05:25.170 This is a bigger number. 00:05:25.170 --> 00:05:28.370 The required acceleration is less. 00:05:28.370 --> 00:05:30.330 So maybe I've given you an intuition. 00:05:30.330 --> 00:05:31.630 And I will prove this for sure. 00:05:31.630 --> 00:05:38.150 Because this is kind of a very touchy, very soft way, and not 00:05:38.150 --> 00:05:42.000 very rigorous way of proving that the required acceleration 00:05:42.000 --> 00:05:44.490 to make something going in a circle is v squared over r. 00:05:44.490 --> 00:05:47.270 But accept that as a bit of faith, and I'll prove it. 00:05:47.270 --> 00:05:48.960 Especially once you learn your calculus, and you can watch 00:05:48.960 --> 00:05:49.810 that video. 00:05:49.810 --> 00:05:53.890 And we could do a couple of problems. And this is probably 00:05:53.890 --> 00:05:54.930 what you care about anyway. 00:05:54.930 --> 00:05:58.100 But I think it's nice to have that intuition. 00:05:58.100 --> 00:06:05.880 So once again, let's say we're in deep space. 00:06:08.700 --> 00:06:11.280 Let's do a little orbit problem. 00:06:11.280 --> 00:06:12.290 Let say that this is Earth. 00:06:12.290 --> 00:06:15.420 I will do Earth in blue. 00:06:15.420 --> 00:06:18.480 Let's say that is Earth. 00:06:18.480 --> 00:06:23.610 And I have a satellite that is going around Earth. 00:06:23.610 --> 00:06:29.040 And I'm not going to calculate its-- this is a satellite 00:06:29.040 --> 00:06:31.370 that's going around Earth. 00:06:31.370 --> 00:06:33.730 And I'm going to do a whole other video on actually 00:06:33.730 --> 00:06:37.300 figuring out the gravitational force depending on how far 00:06:37.300 --> 00:06:38.530 away you are from an object. 00:06:38.530 --> 00:06:43.010 But let's just assume that its gravitational force is similar 00:06:43.010 --> 00:06:44.800 to what it is on the surface of Earth. 00:06:44.800 --> 00:06:47.540 And that is a big assumption, or let's assume this is a 00:06:47.540 --> 00:06:50.170 different planet and I'm going to give you the gravitational 00:06:50.170 --> 00:06:51.300 acceleration. 00:06:51.300 --> 00:06:56.590 Let's say that the planet, and maybe this isn't Earth, is 00:06:56.590 --> 00:06:57.520 pulling inwards. 00:06:57.520 --> 00:06:59.340 It always pulls towards its center. 00:06:59.340 --> 00:07:02.220 And I'll do that in a different color. 00:07:02.220 --> 00:07:07.700 That it's pulling inwards at 30 feet per second. 00:07:07.700 --> 00:07:13.200 My question is, what does the velocity of the satellite have 00:07:13.200 --> 00:07:16.890 to be for the object to stay in orbit? 00:07:16.890 --> 00:07:19.670 Well, we could just apply the formula and then maybe we can 00:07:19.670 --> 00:07:21.380 have a little intuition. 00:07:21.380 --> 00:07:24.040 So acceleration is equal-- oh, I'm sorry. 00:07:24.040 --> 00:07:38.810 Let's say that the object is 6,000 feet from the center. 00:07:38.810 --> 00:07:41.970 And when we learn gravity, we'll learn actually that it's 00:07:41.970 --> 00:07:44.670 the distance from the radius that you're going around is 00:07:44.670 --> 00:07:45.640 actually relative to the center. 00:07:45.640 --> 00:07:47.260 Well, actually it's always going to be, 00:07:47.260 --> 00:07:49.900 because this is a circle. 00:07:49.900 --> 00:07:50.560 So, the radius. 00:07:50.560 --> 00:07:51.800 This is the center of that circle. 00:07:51.800 --> 00:07:52.760 You don't have to worry about that. 00:07:52.760 --> 00:07:54.330 It has nothing to do with gravity. 00:07:54.330 --> 00:07:57.990 OK, so the acceleration is v squared over r. 00:07:57.990 --> 00:08:02.590 Acceleration, we know is 30 feet per second. 00:08:02.590 --> 00:08:05.200 We want to figure out the velocity, and then the radius, 00:08:05.200 --> 00:08:09.330 I said, is 6,000 feet. 00:08:09.330 --> 00:08:10.820 So let's ignore the units for now. 00:08:10.820 --> 00:08:13.530 This could make things complex. 00:08:13.530 --> 00:08:16.260 Multiply both sides of this equation times 6,000, so 3 00:08:16.260 --> 00:08:18.080 times 6 is 18. 00:08:18.080 --> 00:08:18.980 You'll have four 0's. 00:08:18.980 --> 00:08:21.690 2, 3, 4. 00:08:21.690 --> 00:08:24.200 So v squared is equal to 18,000. 00:08:24.200 --> 00:08:26.120 Everything's in feet per second. 00:08:26.120 --> 00:08:27.110 And so what's v? 00:08:27.110 --> 00:08:30.922 So v is equal to the square root of 180,000. 00:08:30.922 --> 00:08:34.159 Let me get the calculator out, because I haven't memorized my 00:08:34.159 --> 00:08:37.470 perfect squares that high. 00:08:37.470 --> 00:08:39.100 1, 2, 3. 00:08:39.100 --> 00:08:42.770 Take the square root, so 424. 00:08:42.770 --> 00:08:48.530 So the required velocity is 424 feet per second. 00:08:48.530 --> 00:08:51.660 So if the satellite is at that velocity, it will go in 00:08:51.660 --> 00:08:54.970 perfect orbit around this planet. 00:08:54.970 --> 00:08:56.860 Now what happens if this object's 00:08:56.860 --> 00:09:00.560 velocity is a little slower. 00:09:00.560 --> 00:09:02.280 What happens-- let me do it in a different color so you know 00:09:02.280 --> 00:09:03.680 what I'm talking about. 00:09:03.680 --> 00:09:05.790 What happens if its velocity is a little 00:09:05.790 --> 00:09:06.400 bit less than that? 00:09:06.400 --> 00:09:11.820 Let's say its is 300 feet per second. 00:09:11.820 --> 00:09:14.720 Well, then it's not going to be able to travel far enough 00:09:14.720 --> 00:09:20.180 tangent to the circle to-- well, it'll 00:09:20.180 --> 00:09:21.630 essentially just do this. 00:09:21.630 --> 00:09:24.480 It'll always be getting a little bit closer. 00:09:24.480 --> 00:09:28.520 And it'll spiral in and it'll hit the planet. 00:09:28.520 --> 00:09:32.510 And another way to think about what the required velocity is, 00:09:32.510 --> 00:09:36.520 you have to go fast enough so that you're always falling in. 00:09:36.520 --> 00:09:39.180 But every time you fall in, you want to go in this case, 00:09:39.180 --> 00:09:42.440 the right enough so that you're an equal distant-- 00:09:42.440 --> 00:09:44.540 you're still the same distance away from the center. 00:09:44.540 --> 00:09:46.630 So that your distance from the center never changes if you're 00:09:46.630 --> 00:09:47.950 going fast enough. 00:09:47.950 --> 00:09:50.950 And then what happens if its velocity is too fast? 00:09:50.950 --> 00:09:53.530 What happens in the case where its velocity is 00:09:53.530 --> 00:09:56.800 600 feet per second? 00:09:56.800 --> 00:10:01.010 Well, in that case, the object will have enough velocity to 00:10:01.010 --> 00:10:04.020 actually escape the orbit of the planet. 00:10:04.020 --> 00:10:06.020 But anyway, that's enough about that. 00:10:06.020 --> 00:10:08.390 I'm actually out of time, so I'll see 00:10:08.390 --> 00:10:09.910 you in the next video.

Introduction to centripetal acceleration (part 1)

https://www.youtube.com/watch?v=GBGGh2Ie4d0

vtt

https://www.youtube.com/api/timedtext?v=GBGGh2Ie4d0&ei=YmeUZf-DHNLWxN8P4OWRiAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=63F76482490ABB6CDC03E89D5D630AA513B4654E.33A13C4F44A9CCD3BA2B539AE16EDE16B151CAE3&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.760 --> 00:00:03.960 Let's learn a little bit about centripetal acceleration. 00:00:03.960 --> 00:00:05.690 So let's say I have a dot. 00:00:05.690 --> 00:00:06.550 It could be anything. 00:00:06.550 --> 00:00:08.160 And we're in deep space, so we're not going to think about 00:00:08.160 --> 00:00:10.080 gravity and all these things yet. 00:00:10.080 --> 00:00:13.540 Let's say I have some object that's floating through space 00:00:13.540 --> 00:00:14.370 with some velocity. 00:00:14.370 --> 00:00:15.210 Let me draw the object. 00:00:15.210 --> 00:00:17.800 It's this dot. 00:00:17.800 --> 00:00:19.250 And it has some velocity. 00:00:19.250 --> 00:00:23.660 And I will draw its velocity vector. 00:00:23.660 --> 00:00:27.200 Let's say that this is-- I'll draw this really big. 00:00:27.200 --> 00:00:28.020 That's not big. 00:00:28.020 --> 00:00:29.270 Let me make a thicker line. 00:00:31.510 --> 00:00:32.140 There you go. 00:00:32.140 --> 00:00:34.000 Thicker line. 00:00:34.000 --> 00:00:35.950 If you can see that. 00:00:35.950 --> 00:00:38.550 So let's say that's the dot's velocity vector. 00:00:41.560 --> 00:00:43.130 And I have a question for you. 00:00:43.130 --> 00:00:45.190 So this is what the dot's doing at exactly this moment. 00:00:45.190 --> 00:00:48.170 Let's say a little bit later, I want the dot to be doing 00:00:48.170 --> 00:00:49.700 something like this. 00:00:49.700 --> 00:00:55.010 I want its velocity vector to look like-- i want its 00:00:55.010 --> 00:00:58.240 velocity factor to look like this. 00:00:58.240 --> 00:00:59.490 A little bit later. 00:01:01.900 --> 00:01:05.630 And its color changes to yellow. 00:01:05.630 --> 00:01:06.450 That's out of the scope. 00:01:06.450 --> 00:01:08.270 Color changing is out of the scope of this lecture. 00:01:08.270 --> 00:01:11.620 But let's say a little bit later I want its velocity 00:01:11.620 --> 00:01:16.880 vector to look something like that. 00:01:16.880 --> 00:01:18.720 So I was a little careful, because I want it to have the 00:01:18.720 --> 00:01:21.950 same magnitude, but just a little different direction. 00:01:21.950 --> 00:01:23.980 I want it to tilt into the right a little bit. 00:01:23.980 --> 00:01:27.650 Let me draw that. 00:01:27.650 --> 00:01:30.780 And my question is, what had to be the acceleration acting, 00:01:30.780 --> 00:01:33.630 when this was its velocity vector, in order to get its 00:01:33.630 --> 00:01:37.070 velocity vector to look like this a little bit later? 00:01:37.070 --> 00:01:38.840 Well, right here, its velocity vector is 00:01:38.840 --> 00:01:41.840 completely in the x direction. 00:01:41.840 --> 00:01:43.510 And there's no y component. 00:01:43.510 --> 00:01:45.140 Let's break down this velocity vector 00:01:45.140 --> 00:01:47.240 into the x and y component. 00:01:47.240 --> 00:01:50.818 So its x component looks something like this. 00:01:50.818 --> 00:01:54.725 And its y component looks something like this. 00:01:57.530 --> 00:02:01.140 So in order to change its velocity, something would have 00:02:01.140 --> 00:02:04.710 to-- Because clearly here its y component was 0. 00:02:04.710 --> 00:02:06.260 There's no y component. 00:02:06.260 --> 00:02:07.570 And now there is a y component. 00:02:07.570 --> 00:02:10.310 So the acceleration that you would have to apply to this 00:02:10.310 --> 00:02:13.720 dot would have to have some y component. 00:02:13.720 --> 00:02:17.090 The acceleration would have to have some y component. 00:02:17.090 --> 00:02:18.700 Let's say that. 00:02:18.700 --> 00:02:20.660 This is acceleration, that's why it's not the same size. 00:02:20.660 --> 00:02:21.593 Actually let me draw acceleration 00:02:21.593 --> 00:02:23.370 in a different color. 00:02:23.370 --> 00:02:25.160 And I'll try to do this in a bunch of different ways, so 00:02:25.160 --> 00:02:28.036 that you get an intuition for it. 00:02:28.036 --> 00:02:30.690 I'm doing it in a slightly different blue. 00:02:30.690 --> 00:02:33.430 So you have to accelerate in the y direction a bit, and we 00:02:33.430 --> 00:02:36.210 don't know how long, but long enough to get its velocity in 00:02:36.210 --> 00:02:38.360 the y direction this big. 00:02:38.360 --> 00:02:39.370 And then what would you have to 00:02:39.370 --> 00:02:41.460 accelerate in the x direction? 00:02:41.460 --> 00:02:44.760 Well, your x component shrinks a little bit. 00:02:44.760 --> 00:02:47.170 This was its x component, and now since you're tilted down, 00:02:47.170 --> 00:02:49.510 you get more y and less x. 00:02:49.510 --> 00:02:53.770 So what direction does acceleration have to go in to 00:02:53.770 --> 00:02:59.280 shrink the x component? 00:02:59.280 --> 00:03:00.690 You have to go opposite. 00:03:00.690 --> 00:03:04.020 So your acceleration is going to go that way a little bit. 00:03:04.020 --> 00:03:06.415 And so if you were to add these two vectors, you would 00:03:06.415 --> 00:03:08.620 have to accelerate something-- the acceleration vector would 00:03:08.620 --> 00:03:09.870 look something like this. 00:03:17.350 --> 00:03:20.740 So I'm-- And this might be a little confusing to you, but 00:03:20.740 --> 00:03:25.385 all I'm showing is that, I want you to think about-- Say 00:03:25.385 --> 00:03:28.060 an object is traveling through space at a constant velocity, 00:03:28.060 --> 00:03:30.540 what does the acceleration vector have to look like in 00:03:30.540 --> 00:03:33.310 order for that object to curve a little bit? 00:03:33.310 --> 00:03:36.670 So for example, if this was the-- let me draw, let's say 00:03:36.670 --> 00:03:39.400 this object was here. 00:03:39.400 --> 00:03:42.530 And if we had no force on it, it would just keep moving in 00:03:42.530 --> 00:03:43.930 this direction. 00:03:43.930 --> 00:03:46.220 Its path would look like this. 00:03:46.220 --> 00:03:47.260 I'll draw the path in brown. 00:03:47.260 --> 00:03:48.210 The path would look like this. 00:03:48.210 --> 00:03:49.330 It would be here, and then it would be here 00:03:49.330 --> 00:03:50.400 a few seconds later. 00:03:50.400 --> 00:03:52.220 It would just keep going in the direction that it's at. 00:03:52.220 --> 00:03:54.430 And we know that from Newton's law, an object in motion tends 00:03:54.430 --> 00:03:55.730 to stay in motion. 00:03:55.730 --> 00:03:59.120 And the only way you can have a change in velocity is if you 00:03:59.120 --> 00:04:01.840 have some net force and some net 00:04:01.840 --> 00:04:04.160 acceleration on the object. 00:04:04.160 --> 00:04:06.410 So in order for the object to curve, in order for the 00:04:06.410 --> 00:04:10.080 object's path to look something like this-- so 00:04:10.080 --> 00:04:11.010 here's the other path. 00:04:11.010 --> 00:04:15.600 Let's say the object's path is like this. 00:04:15.600 --> 00:04:17.480 And let's say it keeps curving. 00:04:17.480 --> 00:04:18.620 The object's path is like that. 00:04:18.620 --> 00:04:19.940 It keeps curving. 00:04:19.940 --> 00:04:21.070 Goes there, to there. 00:04:21.070 --> 00:04:22.670 It's going in a curved motion. 00:04:22.670 --> 00:04:24.720 What has to keep happening? 00:04:24.720 --> 00:04:27.240 Well, in order for the object's velocity to go from 00:04:27.240 --> 00:04:31.330 this direction, in the x direction, in order for its 00:04:31.330 --> 00:04:33.750 velocity to go from this, to go from this. 00:04:33.750 --> 00:04:36.130 You have to accelerate a little bit inwards. 00:04:36.130 --> 00:04:36.990 We drew that right here. 00:04:36.990 --> 00:04:39.450 You have to have an inward acceleration. 00:04:39.450 --> 00:04:41.800 And now its velocity-- and now if you did nothing else, and 00:04:41.800 --> 00:04:44.730 the object will just keep going in this direction. 00:04:44.730 --> 00:04:47.560 So in order to make it curve little more, you've got to 00:04:47.560 --> 00:04:48.580 pull in a little bit again. 00:04:48.580 --> 00:04:51.770 So you have to have-- and then if you don't pull in again, 00:04:51.770 --> 00:04:53.620 it'll just keep going in this direction. 00:04:53.620 --> 00:04:55.560 So you got to keep pulling inwards on it. 00:04:58.530 --> 00:05:01.250 And I think you're starting to get a sense of what I'm 00:05:01.250 --> 00:05:04.930 saying, so if you want the object to go in a circle, for 00:05:04.930 --> 00:05:09.210 example, there has to be constant inward force always 00:05:09.210 --> 00:05:10.560 pulling on the object. 00:05:10.560 --> 00:05:15.190 Because if there isn't that constant inward-- so this is 00:05:15.190 --> 00:05:17.270 this off white color, this is when there's a 00:05:17.270 --> 00:05:18.310 constant inward force. 00:05:18.310 --> 00:05:20.030 The object's going to travel in a circular path. 00:05:23.910 --> 00:05:26.600 Well, what if that inward force doesn't exist? 00:05:26.600 --> 00:05:28.710 Let's say that inward force is there, and then all of a 00:05:28.710 --> 00:05:31.460 sudden when the object's here, the inward force disappears. 00:05:31.460 --> 00:05:34.100 Then the object's just going to go in a straight line. 00:05:34.100 --> 00:05:37.280 Tangent to the circular path. 00:05:37.280 --> 00:05:37.830 And that makes more sense. 00:05:37.830 --> 00:05:38.980 And I'll go into it in a little bit 00:05:38.980 --> 00:05:39.720 more detail in a second. 00:05:39.720 --> 00:05:42.170 But if you are spinning an object around in 00:05:42.170 --> 00:05:43.490 a circle on a string. 00:05:43.490 --> 00:05:46.490 And the string is providing the inward force, and as soon 00:05:46.490 --> 00:05:49.730 as you let go of the string, the object goes off in a 00:05:49.730 --> 00:05:50.500 straight line. 00:05:50.500 --> 00:05:53.320 And so that that's what we're talking about. 00:05:53.320 --> 00:05:56.215 So what does this inward acceleration has-- actually, 00:05:56.215 --> 00:05:58.170 before I go into actually calculating what that inward 00:05:58.170 --> 00:06:00.940 acceleration has to be, let's think about what happens when 00:06:00.940 --> 00:06:02.860 that inwards acceleration isn't enough 00:06:02.860 --> 00:06:04.110 or if it's too much? 00:06:09.650 --> 00:06:12.290 So I just said, if I have an-- let's say this is like the 00:06:12.290 --> 00:06:13.510 center of our rotation. 00:06:13.510 --> 00:06:16.160 Let's say this is where the inward force is coming from. 00:06:16.160 --> 00:06:17.140 Right there. 00:06:17.140 --> 00:06:18.770 And let's say this is my object. 00:06:18.770 --> 00:06:21.070 Let say it's a spaceship. 00:06:21.070 --> 00:06:22.320 That's my spaceship. 00:06:25.110 --> 00:06:27.440 Fire coming out of it. 00:06:27.440 --> 00:06:29.410 And actually the fire would only go on an impulse. 00:06:29.410 --> 00:06:31.300 Because once you have a little fire coming out in space, 00:06:31.300 --> 00:06:33.480 there's no wind resistance, so it'll keep just going. 00:06:33.480 --> 00:06:35.850 You don't have to keep-- so the science fiction movies, 00:06:35.850 --> 00:06:38.310 where the fire just keeps going, that 00:06:38.310 --> 00:06:39.010 doesn't make sense. 00:06:39.010 --> 00:06:41.400 You would just have an impulse of fire that would accelerate 00:06:41.400 --> 00:06:43.690 you, and then you wouldn't have to keep fueling the fire 00:06:43.690 --> 00:06:44.820 in the back of the spaceship. 00:06:44.820 --> 00:06:47.455 But anyway, maybe you realize it already, or that's out of 00:06:47.455 --> 00:06:50.180 the scope of this lecture, but let's say that the object's 00:06:50.180 --> 00:06:52.580 velocity is like that. 00:06:52.580 --> 00:06:54.420 And I don't know, let's say this is a planet, or it's some 00:06:54.420 --> 00:06:57.830 kind of weird forcefield, or something. 00:06:57.830 --> 00:07:02.520 If this forcefield is a weak forcefield, if this object is 00:07:02.520 --> 00:07:06.100 moving really fast, and this is a weak forcefield that's 00:07:06.100 --> 00:07:08.540 always pulling in towards it, the object's path is going to 00:07:08.540 --> 00:07:09.790 look something like this. 00:07:14.250 --> 00:07:15.490 It's not going to be a circle. 00:07:15.490 --> 00:07:16.750 It's going to kind of spiral outwards. 00:07:20.530 --> 00:07:22.620 So that's kind of a weak inward force. 00:07:22.620 --> 00:07:25.693 If it's a really strong inward force, the object's path is 00:07:25.693 --> 00:07:26.650 going to look like this. 00:07:26.650 --> 00:07:29.250 And I'll do that in yellow. 00:07:29.250 --> 00:07:31.290 It's actually going to fall in. 00:07:31.290 --> 00:07:34.310 It's going to spiral into the object. 00:07:34.310 --> 00:07:37.540 And if it's just right, given how far the object is, and 00:07:37.540 --> 00:07:40.100 I'll give you a formula for what just right implies. 00:07:40.100 --> 00:07:43.680 But if the inward force is just right, the object's going 00:07:43.680 --> 00:07:46.960 to essentially orbit around the subject. 00:07:46.960 --> 00:07:48.380 It's going to go in a perfect circle. 00:07:48.380 --> 00:07:53.620 Its path will be like that. 00:07:53.620 --> 00:07:56.460 So let's think about what that inward force has to be. 00:07:56.460 --> 00:07:58.730 And we're not going to-- well, this could have been gravity, 00:07:58.730 --> 00:08:01.870 this inward force, but I won't work with gravity yet, because 00:08:01.870 --> 00:08:04.300 gravity, actually the force changes depending on how close 00:08:04.300 --> 00:08:05.190 you get to it. 00:08:05.190 --> 00:08:07.020 So I won't deal with gravity yet. 00:08:07.020 --> 00:08:10.970 But it could've been anything, it's just a tractor beam. 00:08:10.970 --> 00:08:15.360 So let's try to get an intuition of how strong that 00:08:15.360 --> 00:08:20.280 inward force is, and what variables it's dependent on. 00:08:20.280 --> 00:08:26.100 So let's say I have a let's say this is my hand, and I 00:08:26.100 --> 00:08:27.350 have a string. 00:08:30.920 --> 00:08:32.950 And on that string, I have a rock. 00:08:39.049 --> 00:08:41.980 And my question is, if I want this rock just spinning around 00:08:41.980 --> 00:08:44.400 in a circle, and it'll spin around in a perfect circle. 00:08:44.400 --> 00:08:46.720 We've all done this before. 00:08:46.720 --> 00:08:51.180 If I want the rock to spin around in that circle, what 00:08:51.180 --> 00:08:54.330 has to be the force that I pull on this string? 00:08:54.330 --> 00:08:58.320 Or essentially, how much does the inward acceleration have 00:08:58.320 --> 00:09:00.280 to be on that rock? 00:09:00.280 --> 00:09:04.870 Let's get an intuition for maybe what has to happen. 00:09:04.870 --> 00:09:08.430 So think about it, if the rock is moving, let's say that 00:09:08.430 --> 00:09:10.750 right here, at this point, its velocity looks 00:09:10.750 --> 00:09:12.700 something like that. 00:09:12.700 --> 00:09:15.310 At this point, it's velocity looks something like this. 00:09:15.310 --> 00:09:18.480 And then when the rock is here, its velocity looks 00:09:18.480 --> 00:09:19.730 something like this. 00:09:21.950 --> 00:09:24.650 So to move from this velocity to this velocity, if the 00:09:24.650 --> 00:09:27.990 magnitude is really big, I'm going to have to apply more 00:09:27.990 --> 00:09:29.990 acceleration to change direction. 00:09:29.990 --> 00:09:32.170 Hopefully that makes a little sense. 00:09:32.170 --> 00:09:34.130 To go from this velocity vector to 00:09:34.130 --> 00:09:35.730 this velocity vector. 00:09:35.730 --> 00:09:38.390 If this vector was even bigger, if to say the vector 00:09:38.390 --> 00:09:41.330 looked like this, it had more magnitude. 00:09:41.330 --> 00:09:43.910 I would have to pull inward with even more force and 00:09:43.910 --> 00:09:47.000 accelerate inward at an even higher rate. 00:09:47.000 --> 00:09:50.290 So definitely, the higher the velocity the more I'm going to 00:09:50.290 --> 00:09:51.410 have to pull in. 00:09:51.410 --> 00:09:54.400 And also the higher the velocity, the shorter the time 00:09:54.400 --> 00:09:55.580 between this and this. 00:09:55.580 --> 00:09:57.620 This object is moving around in a circle. 00:09:57.620 --> 00:09:59.540 So there's kind of two components that are 00:09:59.540 --> 00:10:02.180 affecting-- velocity affects the acceleration 00:10:02.180 --> 00:10:03.840 I need in two ways. 00:10:03.840 --> 00:10:05.620 Well, actually I'm running out of time, so I'll continue this 00:10:05.620 --> 00:10:07.220 in the next video.

Projectile motion with ordered set notation

https://www.youtube.com/watch?v=jl_gQ-eL3xo

vtt

https://www.youtube.com/api/timedtext?v=jl_gQ-eL3xo&ei=ZGeUZdGiGI-gp-oPjbS7kA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249812&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=ADAD551018494912823AB7BC22B5673F1590F58C.7F936F778C9C5FE9ACD71E21BD3B7804794A39F3&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.790 --> 00:00:01.530 Welcome back. 00:00:01.530 --> 00:00:03.920 I now want to introduce you really just to a different 00:00:03.920 --> 00:00:06.330 notation for writing vectors, and then we'll do that same 00:00:06.330 --> 00:00:08.540 problem or a slight variation on that problem 00:00:08.540 --> 00:00:09.680 using the new notation. 00:00:09.680 --> 00:00:11.760 This is just to expose you to things, so that you don't get 00:00:11.760 --> 00:00:14.830 confused if your teacher uses a different notation than what 00:00:14.830 --> 00:00:15.770 I've been doing. 00:00:15.770 --> 00:00:18.090 So when we did the unit vectors, we learned that we 00:00:18.090 --> 00:00:21.550 can express a vector as a component of its x- and 00:00:21.550 --> 00:00:22.130 y-components. 00:00:22.130 --> 00:00:30.270 So let's say I had a vector-- let me just pick a random 00:00:30.270 --> 00:00:31.090 vector just to show you. 00:00:31.090 --> 00:00:37.780 So say I had vector a and that equals 2 times the unit vector 00:00:37.780 --> 00:00:41.930 i plus 3 times the unit vector j. 00:00:41.930 --> 00:00:43.430 That's the unit vector notation, and I actually 00:00:43.430 --> 00:00:45.090 looked it up on Wikipedia, and they actually called it the 00:00:45.090 --> 00:00:46.060 engineering notation. 00:00:46.060 --> 00:00:49.170 That's probably why I used it because I am an engineer, or I 00:00:49.170 --> 00:00:52.830 was an engineer before managing money. 00:00:52.830 --> 00:00:55.350 But another way to write this, and I call this the bracket 00:00:55.350 --> 00:00:59.770 notation, or the ordered pair notation, is you could also 00:00:59.770 --> 00:01:01.020 write it like this. 00:01:03.140 --> 00:01:05.050 We have this one bracket. 00:01:05.050 --> 00:01:09.180 That's the x-component, that's the y-component. 00:01:09.180 --> 00:01:11.130 It almost looks like a coordinate pair, but since 00:01:11.130 --> 00:01:12.990 they have the brackets, you know it's a vector. 00:01:12.990 --> 00:01:15.290 But you would draw it the exact same way. 00:01:15.290 --> 00:01:17.570 So given that, let's do that same problem 00:01:17.570 --> 00:01:19.150 that we had just done. 00:01:19.150 --> 00:01:20.660 Hopefully, this make sense to you. 00:01:20.660 --> 00:01:22.230 It's just a different way of writing it. 00:01:22.230 --> 00:01:24.170 Instead of an i and a j, you just write these brackets. 00:01:24.170 --> 00:01:26.611 Instead of a plus, you write a comma. 00:01:26.611 --> 00:01:29.120 Let me clear this. 00:01:29.120 --> 00:01:30.230 I'm going to do a slight variation. 00:01:30.230 --> 00:01:32.140 This was actually the second part of that problem. 00:01:32.140 --> 00:01:33.560 My cousin gave these problems to me. 00:01:33.560 --> 00:01:36.890 They're pretty good, so I figure I'd stick with them. 00:01:36.890 --> 00:01:42.000 So in the old problem, let me draw my coordinate axes again. 00:01:44.885 --> 00:01:47.370 That's the y-axis. 00:01:47.370 --> 00:01:48.915 That's the x-axis. 00:01:53.670 --> 00:01:56.740 So in the old problem, I started off with a ball that 00:01:56.740 --> 00:01:57.970 was 4 feet off the ground. 00:01:57.970 --> 00:02:00.310 So let's say that's 4. 00:02:00.310 --> 00:02:07.760 And I hit it at 120 feet per second at a 30-degree angle. 00:02:07.760 --> 00:02:10.710 So that's a 30-degree angle like that. 00:02:13.332 --> 00:02:17.170 Its' a 30-degree angle to the horizontal. 00:02:17.170 --> 00:02:23.580 And there's a fence 350 feet away that's 30 feet high. 00:02:23.580 --> 00:02:24.660 It's roughly around there. 00:02:24.660 --> 00:02:26.020 That's 30. 00:02:26.020 --> 00:02:27.845 And what we need to do is figure out whether the ball 00:02:27.845 --> 00:02:28.770 can clear the fence. 00:02:28.770 --> 00:02:31.190 We figured out the last time when we used the unit vector 00:02:31.190 --> 00:02:33.430 notation that it doesn't clear the fence. 00:02:33.430 --> 00:02:36.180 But in this problem, or the second part of this problem, 00:02:36.180 --> 00:02:38.300 they said that there's a 5 meter per second 00:02:38.300 --> 00:02:40.340 wind gust to the right. 00:02:40.340 --> 00:02:45.260 So there's a wind gust of 5 meters per second right when I 00:02:45.260 --> 00:02:46.190 hit the ball. 00:02:46.190 --> 00:02:48.305 And you could go into the complications of how much does 00:02:48.305 --> 00:02:49.620 that accelerate the ball? 00:02:49.620 --> 00:02:51.410 Or what's the air resistance of the ball? 00:02:51.410 --> 00:02:53.410 I think for the simplicity of the problem, they're just 00:02:53.410 --> 00:02:56.670 saying that the x-component of the ball's velocity right 00:02:56.670 --> 00:03:00.370 after you hit it increases by 5 meters per second. 00:03:00.370 --> 00:03:01.540 I think that's their point. 00:03:01.540 --> 00:03:03.690 So let's go back and do the problem the exact same way 00:03:03.690 --> 00:03:05.240 that we did it the last time, but we'll 00:03:05.240 --> 00:03:07.090 use a different notation. 00:03:07.090 --> 00:03:09.750 So we can write that equation that I had written before, 00:03:09.750 --> 00:03:16.110 that the position at any given time as a function of t is 00:03:16.110 --> 00:03:19.870 equal to the initial position-- that's an i right 00:03:19.870 --> 00:03:24.480 there-- plus the initial velocity. 00:03:24.480 --> 00:03:26.300 These are all vectors. 00:03:26.300 --> 00:03:36.040 Initial velocity times t plus the acceleration vector over 00:03:36.040 --> 00:03:38.820 2t squared. 00:03:38.820 --> 00:03:40.330 So what's the initial position? 00:03:40.330 --> 00:03:43.140 And now we're going to use some of our new notation. 00:03:43.140 --> 00:03:50.690 The initial position when I hit the ball, its x-component 00:03:50.690 --> 00:03:52.630 is 0, right? 00:03:52.630 --> 00:03:55.130 It's almost like its coordinate, and they're not 00:03:55.130 --> 00:03:57.010 that different of a notation. 00:03:57.010 --> 00:04:00.920 And then the y-position is 4. 00:04:00.920 --> 00:04:02.510 Easy enough. 00:04:02.510 --> 00:04:05.710 What's its initial velocity? 00:04:05.710 --> 00:04:07.470 Let me do it. 00:04:07.470 --> 00:04:10.810 So we can split it up into the x- and the y-components. 00:04:10.810 --> 00:04:18.420 The y-component is 120 sine of 30 degrees and then the x 00:04:18.420 --> 00:04:24.240 component is 120 cosine of 30 degrees. 00:04:24.240 --> 00:04:26.410 That's just the x-component after I hit it. 00:04:26.410 --> 00:04:27.860 But then they say there's this wind gust so it's 00:04:27.860 --> 00:04:29.220 going to be plus 5. 00:04:29.220 --> 00:04:31.020 I think that's their point when they say that there's 00:04:31.020 --> 00:04:33.130 this wind gust. They say that right when you hit it, for 00:04:33.130 --> 00:04:35.150 some reason in the x-direction, it accelerates a 00:04:35.150 --> 00:04:38.750 little bit by 5 meters per second. 00:04:38.750 --> 00:04:42.140 So the velocity vector. 00:04:42.140 --> 00:04:44.140 This notation actually is better, because it takes less 00:04:44.140 --> 00:04:46.170 space up, and you don't have all these i's and j's and 00:04:46.170 --> 00:04:47.880 pluses confusing everything. 00:04:47.880 --> 00:04:49.690 So the initial velocity vector, what's its 00:04:49.690 --> 00:04:50.690 x-component? 00:04:50.690 --> 00:04:52.590 It's 120 cosine of 30. 00:04:52.590 --> 00:04:58.310 Cosine of 30 is square root of 3/2 times 120 is 60 square 00:04:58.310 --> 00:05:00.370 roots of 3, and then you add 5 to it. 00:05:00.370 --> 00:05:00.990 So what is that? 00:05:00.990 --> 00:05:03.650 Let me just solve it right now. 00:05:03.650 --> 00:05:15.130 So 3 times the square root of 3 times 60 plus 5. 00:05:15.130 --> 00:05:16.730 So let's just round up and make it easier. 00:05:16.730 --> 00:05:18.315 It's 109 meters per second. 00:05:18.315 --> 00:05:21.940 108.9, so let's just say 109. 00:05:21.940 --> 00:05:26.010 So the x-component of the velocity is 109. 00:05:26.010 --> 00:05:29.030 And the y-component was just 120 times the sine of 30. 00:05:29.030 --> 00:05:32.660 Well, sine of 30 is 1/2, so this is 60. 00:05:32.660 --> 00:05:36.050 Oh, sorry, this should be brackets, although some people 00:05:36.050 --> 00:05:38.060 actually write the parentheses there so it looks just like 00:05:38.060 --> 00:05:39.930 coordinates, but I like to keep it with these brackets so 00:05:39.930 --> 00:05:41.410 that you don't think that these are coordinates since 00:05:41.410 --> 00:05:43.750 you know these are vectors. 00:05:43.750 --> 00:05:45.790 And a position vector is really the same thing as a 00:05:45.790 --> 00:05:46.600 position coordinate. 00:05:46.600 --> 00:05:49.270 But a velocity vector is obviously not a coordinate. 00:05:49.270 --> 00:05:51.640 What's the acceleration vector? 00:05:51.640 --> 00:05:53.505 Well, the acceleration vector, as we said, goes straight-- 00:05:53.505 --> 00:05:54.540 that's not straight down. 00:05:54.540 --> 00:05:59.520 This is straight down at minus 32 feet per second squared. 00:05:59.520 --> 00:06:01.880 That's the acceleration of gravity on Earth. 00:06:01.880 --> 00:06:06.785 So the acceleration vector is equal to -- it has no 00:06:06.785 --> 00:06:12.390 x-component and its y-component is minus 32. 00:06:12.390 --> 00:06:14.560 So now let's put these back in that original equation. 00:06:14.560 --> 00:06:17.650 So our position vector, and I'll switch colors to keep 00:06:17.650 --> 00:06:19.680 things from getting monotonous. 00:06:19.680 --> 00:06:23.320 Our position vector-- these are little arrows or one-sided 00:06:23.320 --> 00:06:33.040 arrows-- equals my initial position, and that's 0, 4 plus 00:06:33.040 --> 00:06:43.660 my initial velocity vector, 109, 60 times t, and I'm 00:06:43.660 --> 00:06:49.640 running out of space, plus at squared over two, so t squared 00:06:49.640 --> 00:06:59.740 over 2 times my acceleration vector, 0 minus 32. 00:06:59.740 --> 00:07:02.350 This is actually a little cleaner way of writing it, but 00:07:02.350 --> 00:07:03.800 this is exactly what we did when we did 00:07:03.800 --> 00:07:04.600 it with unit vectors. 00:07:04.600 --> 00:07:07.030 Instead of writing i's and j's, we're just writing the 00:07:07.030 --> 00:07:08.670 numbers in brackets here. 00:07:08.670 --> 00:07:10.510 So let's see if we can simplify this. 00:07:10.510 --> 00:07:14.090 So let me write it in a different color, so that you 00:07:14.090 --> 00:07:15.440 know I'm doing. 00:07:15.440 --> 00:07:26.060 OK, so our position vector t is equal to 0, 4 plus-- and 00:07:26.060 --> 00:07:28.680 now we can distribute this t, multiply it times both of 00:07:28.680 --> 00:07:40.990 these-- plus 109t, 60t plus-- and we can distribute this t 00:07:40.990 --> 00:07:42.030 squared over 2. 00:07:42.030 --> 00:07:43.970 Well, that times 0 is 0. 00:07:43.970 --> 00:07:51.210 And then that times minus 32 is minus 16t squared. 00:07:51.210 --> 00:07:53.540 Now we can add the vectors. 00:07:53.540 --> 00:07:57.330 So the position at any t. 00:07:57.330 --> 00:07:59.950 So let's add all the x-components of the vectors. 00:07:59.950 --> 00:08:06.720 0, 109t, 0, so we just get 109t. 00:08:06.720 --> 00:08:09.110 And then what's the y-components? 00:08:09.110 --> 00:08:23.320 4 plus 60t minus 16t squared. 00:08:23.320 --> 00:08:24.710 And there we go. 00:08:24.710 --> 00:08:27.730 We've defined the position vector at a 00:08:27.730 --> 00:08:28.880 function of any time. 00:08:28.880 --> 00:08:29.770 So let's solve the problem. 00:08:29.770 --> 00:08:32.260 Now that they have this wind gust and our x velocity's 00:08:32.260 --> 00:08:34.380 going a little faster, let's see if we can clear the fence. 00:08:34.380 --> 00:08:37.610 So how long does it take to get to 350 feet in the 00:08:37.610 --> 00:08:38.450 x-direction? 00:08:38.450 --> 00:08:42.159 Well, this number right here has to equal 350. 00:08:42.159 --> 00:08:47.480 So we have 109t has to be equal to 350. 00:08:47.480 --> 00:08:49.820 And so what's 350 divided by 109? 00:08:49.820 --> 00:08:59.040 350 divided by 109 is equal to 3.2 seconds. 00:08:59.040 --> 00:09:03.070 t is equal to 3.2 seconds. 00:09:03.070 --> 00:09:06.120 And so what's the height at 3.2 seconds? 00:09:06.120 --> 00:09:07.370 So let's square that. 00:09:09.740 --> 00:09:21.235 3.2 times 3.2 equals times 16 equals 164. 00:09:21.235 --> 00:09:25.000 So this equals 164. 00:09:25.000 --> 00:09:27.546 And then what's 60 times 3.2? 00:09:27.546 --> 00:09:33.680 60 times 3.2 is equal to 192. 00:09:33.680 --> 00:09:35.090 So what do we get? 00:09:35.090 --> 00:09:50.990 We get 192 plus 4 minus 164 is equal to 32. 00:09:50.990 --> 00:10:00.350 So our position vector at time 3.2 seconds is equal to 350 00:10:00.350 --> 00:10:08.150 feet in the x-direction and 32 feet in the y-direction, and 00:10:08.150 --> 00:10:10.970 that will clear that 30-foot fence. 00:10:10.970 --> 00:10:13.860 Our ball's going to be two feet above the fence. 00:10:13.860 --> 00:10:15.390 Hope I didn't confuse you too much. 00:10:15.390 --> 00:10:16.980 See you soon.

Unit vector notation (part 2)

https://www.youtube.com/watch?v=595Tiga1gIg

vtt

https://www.youtube.com/api/timedtext?v=595Tiga1gIg&ei=YmeUZZaQHJycp-oP9Y-vqAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2F89FE5909ADF3BE9AB875E8FD06BCC61280BE31.4EB3B7BBF3A74FDDB932221EA28DF8826EFD55BD&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:01.790 Welcome back. 00:00:01.790 --> 00:00:04.700 In the last video, I at the end of the video, like I 00:00:04.700 --> 00:00:06.700 always do in the attempt to confuse you, I told you that 00:00:06.700 --> 00:00:09.060 if I had two vectors-- And let me just make up some new ones, 00:00:09.060 --> 00:00:12.520 so I can draw them visually in a second or two. 00:00:12.520 --> 00:00:14.920 Let's call the first vector a. 00:00:14.920 --> 00:00:15.910 Let me do a different color. 00:00:15.910 --> 00:00:19.490 This toothpaste color is getting monotonous. 00:00:19.490 --> 00:00:21.880 Let me do something that looks relaxing. 00:00:21.880 --> 00:00:25.910 Let's call a first vector a and, I don't know, let's make 00:00:25.910 --> 00:00:31.500 it interesting, let me say it's minus 3 times the unit 00:00:31.500 --> 00:00:37.520 vector i plus 2 times the unit vector j. 00:00:37.520 --> 00:00:40.570 And then I have vector b. 00:00:40.570 --> 00:00:48.120 And that is equal to, 2i, so two times the unit vector i. 00:00:48.120 --> 00:00:54.100 Plus, 4 times the unit vector j. 00:00:54.100 --> 00:00:56.550 In the last video I said, well, the whole reason why 00:00:56.550 --> 00:00:59.680 this unit vector notation is even -- Well, one of the 00:00:59.680 --> 00:01:01.860 reasons, we'll see that there many reasons why it's useful. 00:01:01.860 --> 00:01:04.390 One of the really cool things about it is, before when we 00:01:04.390 --> 00:01:07.170 added vectors, we would put them head to tails, and then 00:01:07.170 --> 00:01:08.960 draw it visually, and then we had this new vector. 00:01:08.960 --> 00:01:10.980 And we really had no way of expressing it 00:01:10.980 --> 00:01:12.130 without drawing it. 00:01:12.130 --> 00:01:15.850 But when we write things as multiples of the unit vectors. 00:01:15.850 --> 00:01:17.020 We don't have to draw it. 00:01:17.020 --> 00:01:19.290 And it's actually very easy to add vectors. 00:01:19.290 --> 00:01:19.890 And how do we do it? 00:01:19.890 --> 00:01:23.050 We just add the x components, and we add the y components. 00:01:23.050 --> 00:01:28.100 So we said that these two vectors, a plus b, these 00:01:28.100 --> 00:01:30.320 little weird arrows on top, that's just saying that those 00:01:30.320 --> 00:01:31.790 are vectors. 00:01:31.790 --> 00:01:33.040 That's equals. 00:01:38.290 --> 00:01:43.640 So it's minus 3, plus 2i, and I'm going to arbitrarily 00:01:43.640 --> 00:01:46.150 switch colors, because it's getting monotonous. 00:01:46.150 --> 00:01:49.450 Plus 2 plus 4j. 00:01:49.450 --> 00:01:51.680 We just added the x components, or the 00:01:51.680 --> 00:01:52.800 multiples of i. 00:01:52.800 --> 00:01:55.440 And we added the y components, or just the multiples of j. 00:01:55.440 --> 00:01:58.180 Because i was the unit vector in the x direction, and j was 00:01:58.180 --> 00:02:00.050 the unit vector in the y direction. 00:02:00.050 --> 00:02:02.560 And we get, what's minus 3 plus 2? 00:02:02.560 --> 00:02:03.600 That's minus 1. 00:02:03.600 --> 00:02:05.190 We get minus 1i. 00:02:05.190 --> 00:02:06.510 That could just be minus i. 00:02:06.510 --> 00:02:09.720 But I'll write the 1 because we're just getting warmed up 00:02:09.720 --> 00:02:10.419 with unit vectors. 00:02:10.419 --> 00:02:15.410 So minus 1i plus 6j. 00:02:15.410 --> 00:02:18.990 And when I did that, you might say, well, Sal, I can't just 00:02:18.990 --> 00:02:19.790 take your word for it. 00:02:19.790 --> 00:02:23.910 Because you seem not someone who 00:02:23.910 --> 00:02:27.090 should be believed blindly. 00:02:27.090 --> 00:02:31.200 So I think that's a valid opinion to have. So I will 00:02:31.200 --> 00:02:34.330 show you that this works, by adding the vectors visually. 00:02:34.330 --> 00:02:35.410 So let's draw it. 00:02:35.410 --> 00:02:37.640 And I think this will give you a little better sense of unit 00:02:37.640 --> 00:02:38.910 vectors generally. 00:02:38.910 --> 00:02:40.635 Let me draw the axes. 00:02:44.140 --> 00:02:47.820 So that's my y-axis. 00:02:47.820 --> 00:02:49.070 Let me draw my x-axis. 00:02:52.700 --> 00:02:55.400 I have to make sure have enough space to draw the unit 00:02:55.400 --> 00:02:57.890 vectors that we've drawn, or to draw the 00:02:57.890 --> 00:02:59.140 vectors that we've drawn. 00:03:01.690 --> 00:03:03.990 Just to show that the axes go on forever, I have to draw 00:03:03.990 --> 00:03:05.270 that arrow. 00:03:05.270 --> 00:03:13.433 All right, so let's say this is 1, 2, 3. 00:03:13.433 --> 00:03:21.140 This is 1, 2, 3, 4. 00:03:21.140 --> 00:03:30.360 And I draw 1, 2, 3, 4, 5, 6. 00:03:30.360 --> 00:03:33.110 I think we should be able to now add them. 00:03:33.110 --> 00:03:35.020 I didn't have to waste all this space down here. 00:03:35.020 --> 00:03:39.200 So let's just first draw the vectors, minus 3i plus 2j. 00:03:39.200 --> 00:03:46.070 So minus 3i, just this right here, is going to be a vector 00:03:46.070 --> 00:03:47.700 that looks something like this. 00:03:47.700 --> 00:03:50.600 So it's just minus 3 times the x vector, so 00:03:50.600 --> 00:03:52.320 it'll go to the left. 00:03:52.320 --> 00:03:55.780 Because i is 1 in the positive direction. 00:03:55.780 --> 00:03:58.180 If we put a negative there, it flips it over. 00:03:58.180 --> 00:03:59.690 Let me use a different color. 00:03:59.690 --> 00:04:05.940 So this is minus 3i, and then plus 2j. 00:04:05.940 --> 00:04:07.560 So plus 2j looks like this. 00:04:11.590 --> 00:04:14.300 If we were to add those two vectors visually, we can put 00:04:14.300 --> 00:04:15.170 them head to tails. 00:04:15.170 --> 00:04:17.750 And the way we can do that, we can either shift this vector 00:04:17.750 --> 00:04:19.730 up like this, and draw it up here. 00:04:19.730 --> 00:04:21.910 Or we could shift this vector, and put its tail 00:04:21.910 --> 00:04:22.810 its vector's head. 00:04:22.810 --> 00:04:25.010 But either way, let's shift this one up. 00:04:25.010 --> 00:04:27.950 So if we shifted up like that. 00:04:27.950 --> 00:04:30.160 Remember, we're just doing the head to tails, visual addition 00:04:30.160 --> 00:04:31.480 method of vectors. 00:04:31.480 --> 00:04:34.340 So I just put this tail to this head. 00:04:34.340 --> 00:04:35.260 And what do we get? 00:04:35.260 --> 00:04:37.400 So vector a will look like this, and I'm going to do it 00:04:37.400 --> 00:04:40.360 in the same color as vector a because I have a feeling that 00:04:40.360 --> 00:04:41.710 this diagram might get complicated. 00:04:44.900 --> 00:04:46.150 Well, I wanted to use the line tool. 00:04:50.180 --> 00:04:53.700 OK, so this is vector a. 00:04:53.700 --> 00:04:55.740 That's what vector a looks like. 00:04:55.740 --> 00:04:56.880 And so we worked backwards. 00:04:56.880 --> 00:04:59.260 I gave you the x component and the y component. 00:04:59.260 --> 00:05:01.310 And then I added them together by doing the head to tails 00:05:01.310 --> 00:05:04.990 method, and so this is what vector a would look like. 00:05:04.990 --> 00:05:07.440 And, instead of drawing it, a very easy representation is 00:05:07.440 --> 00:05:09.990 exactly what we did up here, a unit vector notation. 00:05:09.990 --> 00:05:11.800 And what's vector b look like? 00:05:11.800 --> 00:05:15.320 So it's 2i-- I'm going to do a completely different color. 00:05:15.320 --> 00:05:17.680 It's 2i, so it's this vector. 00:05:17.680 --> 00:05:19.500 2 times unit vector i. 00:05:19.500 --> 00:05:20.990 That's this. 00:05:20.990 --> 00:05:24.170 Plus 4j, 1, 2, 3, 4. 00:05:24.170 --> 00:05:26.870 So it looks like this. 00:05:26.870 --> 00:05:29.240 And let's take this one and shift it over to the left, so 00:05:29.240 --> 00:05:31.170 we can put its tail to the vector's head, so it would 00:05:31.170 --> 00:05:34.140 look like this. 00:05:34.140 --> 00:05:38.370 So vector b will look -- I'll do it in red. 00:05:38.370 --> 00:05:39.760 And I'll use a line tool. 00:05:39.760 --> 00:05:43.540 Vector b looks like this. 00:05:46.450 --> 00:05:48.500 I just put its components head to tails, and that's how 00:05:48.500 --> 00:05:50.170 I got vector b. 00:05:50.170 --> 00:05:53.480 And if I were to add them visually. 00:05:53.480 --> 00:05:55.640 I would do it the same way that I added its components. 00:05:55.640 --> 00:05:58.120 I would put the tail of one vector to the head of the 00:05:58.120 --> 00:06:00.100 other, and see if you get the resulting vector. 00:06:00.100 --> 00:06:00.930 So you could do it either way. 00:06:00.930 --> 00:06:02.270 Let's shift this a vector. 00:06:02.270 --> 00:06:04.760 Let's shift it in this direction. 00:06:04.760 --> 00:06:07.250 Remember, vectors, we're just giving the 00:06:07.250 --> 00:06:08.350 magnitude of direction. 00:06:08.350 --> 00:06:10.880 We're not necessarily giving a starting point. 00:06:10.880 --> 00:06:12.300 So you can shift them. 00:06:12.300 --> 00:06:15.600 You just can't change their orientation or their 00:06:15.600 --> 00:06:16.850 magnitudes. 00:06:16.850 --> 00:06:18.930 And that's actually how you add them, you shift them, and 00:06:18.930 --> 00:06:19.970 put them head to tails. 00:06:19.970 --> 00:06:21.690 That's when you add them visually. 00:06:21.690 --> 00:06:25.780 Let's put that a vector up here. 00:06:25.780 --> 00:06:30.015 So if we have the a vector, it looks something like this. 00:06:40.270 --> 00:06:41.700 And I want it to work out right. 00:06:41.700 --> 00:06:45.640 So the a vector looks something like that. 00:06:45.640 --> 00:06:49.280 And remember, all I did was I took the same vector, and I 00:06:49.280 --> 00:06:49.940 just shifted it. 00:06:49.940 --> 00:06:52.310 So that it can start at the head. 00:06:52.310 --> 00:06:55.160 So its tail can start at the head of the b vector. 00:06:55.160 --> 00:06:58.060 I just shifted the a vector, so this is still the a vector. 00:06:58.060 --> 00:06:59.030 By moving the vector around, you 00:06:59.030 --> 00:07:00.340 haven't changed the vector. 00:07:00.340 --> 00:07:02.980 I would only change the vector, if I scaled it, if I 00:07:02.980 --> 00:07:06.180 made it bigger or smaller, if I changed its orientation. 00:07:06.180 --> 00:07:09.600 And so visually, this is b, this is a, so if I add a to b, 00:07:09.600 --> 00:07:14.880 the resulting vector, going head to tails-- i'll do it in 00:07:14.880 --> 00:07:17.700 this green color --would look like this. 00:07:20.280 --> 00:07:23.710 It would look like that. 00:07:23.710 --> 00:07:25.940 So here we took all this trouble, and I had to draw 00:07:25.940 --> 00:07:27.630 these straight lines to visually 00:07:27.630 --> 00:07:28.710 add these two vectors. 00:07:28.710 --> 00:07:31.140 This green vector is a plus b. 00:07:31.140 --> 00:07:33.170 Let's see if this green vector is the same 00:07:33.170 --> 00:07:35.540 thing that we got here. 00:07:35.540 --> 00:07:39.070 Let's see if it's the same thing as this. 00:07:39.070 --> 00:07:44.660 So we got negative 1 times i, so negative 1 is here. 00:07:44.660 --> 00:07:47.440 And then we have 6j. 00:07:47.440 --> 00:07:48.530 Let me do it in another color. 00:07:48.530 --> 00:07:51.330 6j would look like this. 00:07:51.330 --> 00:07:52.520 6j looks like that. 00:07:52.520 --> 00:07:54.160 You put them heads to tails. 00:07:54.160 --> 00:07:58.020 And it would be something like this. 00:07:58.020 --> 00:07:59.320 And that is the green vector. 00:07:59.320 --> 00:08:01.840 And actually, just so you know, I know it didn't line up 00:08:01.840 --> 00:08:04.660 perfectly, and that's because I'm not drawing neatly, but 00:08:04.660 --> 00:08:08.420 these two points should actually be here if I were to 00:08:08.420 --> 00:08:09.980 have drawn this better. 00:08:09.980 --> 00:08:11.930 But I know this is very confusing, I 00:08:11.930 --> 00:08:12.840 had all these colors. 00:08:12.840 --> 00:08:15.620 But the whole point of it is, I wanted to show that you 00:08:15.620 --> 00:08:19.440 could visually draws vectors, and then shift them around, 00:08:19.440 --> 00:08:20.840 and then put them heads to tails. 00:08:20.840 --> 00:08:22.440 And then get the resulting vector. 00:08:22.440 --> 00:08:25.170 That's one way to add vectors, there's still no way to 00:08:25.170 --> 00:08:27.280 analytically represent it. 00:08:27.280 --> 00:08:30.510 Or you could just write any vector as its x and y 00:08:30.510 --> 00:08:33.340 components, and then the sum of the vectors is just going 00:08:33.340 --> 00:08:36.409 to be the sum of the x's and the sum of the y's. 00:08:36.409 --> 00:08:39.049 And that's a much cleaner, and a much easier, and much less 00:08:39.049 --> 00:08:44.840 prone to error, way of adding or subtracting two vectors. 00:08:44.840 --> 00:08:46.600 So hopefully that was convincing. 00:08:46.600 --> 00:08:51.660 That a plus b really is this vector. 00:08:51.660 --> 00:08:53.470 If it wasn't, I'm sorry. 00:08:53.470 --> 00:08:55.620 And I hope I didn't confuse you more. 00:08:55.620 --> 00:08:58.240 But now that we have this out of the way, and hopefully 00:08:58.240 --> 00:09:00.860 you're convinced that unit vector notation is useful. 00:09:00.860 --> 00:09:04.050 We can move on and maybe try to do some of our old 00:09:04.050 --> 00:09:06.370 projectile motion problems using this notation. 00:09:06.370 --> 00:09:09.140 And maybe it'll let us to do a little bit of 00:09:09.140 --> 00:09:10.590 extra stuff with it. 00:09:10.590 --> 00:09:12.100 See you soon.

Unit vector notation

https://www.youtube.com/watch?v=FaF3v-ezbSk

vtt

https://www.youtube.com/api/timedtext?v=FaF3v-ezbSk&ei=YmeUZa6VG9_ymLAPh_2FgAU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=679DB13AFB4B85E55F29FC639D0A12CB8F5C3049.8FD0B8F4538F6EE0DBAA2FFFB9B2A35A41E771D2&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.230 --> 00:00:03.140 Good afternoon. 00:00:03.140 --> 00:00:06.410 We've done a lot of work with vectors. 00:00:06.410 --> 00:00:10.050 In a lot of the problems, when we launch something into--- In 00:00:10.050 --> 00:00:11.920 the projectile motion problems, or when you were 00:00:11.920 --> 00:00:16.620 doing the incline plane problems. I always gave you a 00:00:16.620 --> 00:00:18.370 vector, like I would draw a vector like this. 00:00:18.370 --> 00:00:20.440 I would say something has a velocity of 00:00:20.440 --> 00:00:21.420 10 meters per second. 00:00:21.420 --> 00:00:22.900 It's at a 30 degree angle. 00:00:22.900 --> 00:00:25.060 And then I would break it up into the x and y components. 00:00:25.060 --> 00:00:31.550 So if I called this vector v, I would use a notation, v sub 00:00:31.550 --> 00:00:36.910 x, and the v sub x would have been this vector right here. 00:00:36.910 --> 00:00:39.200 v sub x would've been this vector down here. 00:00:39.200 --> 00:00:40.740 The x component of the vector. 00:00:43.280 --> 00:00:45.575 And then v sub y would have been the y component of the 00:00:45.575 --> 00:00:49.800 vector, and it would have been this vector. 00:00:49.800 --> 00:00:56.390 So this was v sub x, this was v sub y. 00:00:56.390 --> 00:00:58.610 And hopefully by now, it's second nature of how we would 00:00:58.610 --> 00:01:02.260 figure these things out. v sub x would be 10 times cosine of 00:01:02.260 --> 00:01:03.860 this angle. 00:01:03.860 --> 00:01:08.200 10 cosine of 30 degrees, which I think is square root of 3/2, 00:01:08.200 --> 00:01:09.890 but we're not worried about that right now. 00:01:09.890 --> 00:01:14.810 And v sub y would be 10 times the sine of that angle. 00:01:14.810 --> 00:01:17.740 This hopefully should be second nature to you. 00:01:17.740 --> 00:01:20.560 If it's not, you can just go through SOH-CAH-TOA and say, 00:01:20.560 --> 00:01:24.180 well, the sine of 30 degrees is the opposite of the 00:01:24.180 --> 00:01:24.890 hypotenuse. 00:01:24.890 --> 00:01:25.820 And you would get back to this. 00:01:25.820 --> 00:01:28.030 But we've reviewed all of that, and you should review 00:01:28.030 --> 00:01:31.160 the initial vector videos. 00:01:31.160 --> 00:01:34.590 But what I want you to do now, because this is useful for 00:01:34.590 --> 00:01:37.430 simple projectile motion problems-- But once we start 00:01:37.430 --> 00:01:40.700 dealing with more complicated vectors-- and maybe we're 00:01:40.700 --> 00:01:43.190 dealing with multi-dimensional of vectors, three-dimensional 00:01:43.190 --> 00:01:46.170 vectors, or we start doing linear algebra, where we do 00:01:46.170 --> 00:01:50.790 end dimensional factors --we need a coherent way, an 00:01:50.790 --> 00:01:52.450 analytical way, instead of having to always draw a 00:01:52.450 --> 00:01:55.660 picture of representing vectors. 00:01:55.660 --> 00:01:58.700 So what we do is, we use something I call, and I think 00:01:58.700 --> 00:02:00.630 everyone calls it, unit vector notation. 00:02:00.630 --> 00:02:01.860 So what does that mean? 00:02:01.860 --> 00:02:03.530 So we define these unit vectors. 00:02:03.530 --> 00:02:06.070 Let me draw some axes. 00:02:06.070 --> 00:02:07.720 And it's important to keep in mind, this might seem a little 00:02:07.720 --> 00:02:09.699 confusing at first, but this is no different than what 00:02:09.699 --> 00:02:12.350 we've been doing in our physics problem so far. 00:02:12.350 --> 00:02:20.980 Let me draw the axes right there. 00:02:20.980 --> 00:02:29.600 Let's say that this is 1, this is 0, this is 2. 00:02:29.600 --> 00:02:30.450 0, 1, 2. 00:02:30.450 --> 00:02:32.410 I don't know if must been writing an Arabic or 00:02:32.410 --> 00:02:33.820 something, going backwards. 00:02:33.820 --> 00:02:36.640 This is 0, 1, 2, that's not 20. 00:02:36.640 --> 00:02:42.240 And then let's say this is 1, this is 2, in the y direction. 00:02:42.240 --> 00:02:45.200 I'm going to define what I call the unit vectors in two 00:02:45.200 --> 00:02:46.020 dimensions. 00:02:46.020 --> 00:02:48.720 So I'm going to first define a vector. 00:02:48.720 --> 00:02:51.970 I'll call this vector i. 00:02:51.970 --> 00:02:53.220 And this is the vector. 00:02:58.175 --> 00:03:02.000 It just goes straight in the x direction, has no y component, 00:03:02.000 --> 00:03:04.140 and it has the magnitude of 1. 00:03:04.140 --> 00:03:06.450 And so this is i. 00:03:06.450 --> 00:03:10.470 We denote the unit vector by putting this little 00:03:10.470 --> 00:03:11.620 cap on top of it. 00:03:11.620 --> 00:03:12.540 There's multiple notations. 00:03:12.540 --> 00:03:15.480 Sometimes in the book, you'll see this i without the cap, 00:03:15.480 --> 00:03:16.410 and it's just boldface. 00:03:16.410 --> 00:03:17.370 There's some other notations. 00:03:17.370 --> 00:03:22.640 But if you see i, and not in the imaginary number sense, 00:03:22.640 --> 00:03:25.255 you should realize that that's the unit vector. 00:03:25.255 --> 00:03:28.960 It has magnitude 1 and it's completely in the x direction. 00:03:28.960 --> 00:03:32.200 And I'm going to define another vector, and that one 00:03:32.200 --> 00:03:33.710 is called j. 00:03:33.710 --> 00:03:37.750 And that is the same thing but in the y direction. 00:03:37.750 --> 00:03:40.200 That is the vector j. 00:03:40.200 --> 00:03:42.600 You put a little cap over it. 00:03:42.600 --> 00:03:44.290 So why did I do this? 00:03:44.290 --> 00:03:46.000 Well, if I'm dealing with two dimensions. 00:03:46.000 --> 00:03:48.325 And as later we'll see in three dimensions, so there 00:03:48.325 --> 00:03:50.090 will actually be a third dimension and we'll call that 00:03:50.090 --> 00:03:52.070 k, but don't worry about that right now. 00:03:52.070 --> 00:03:56.480 But if we're dealing in two dimensions, we can define any 00:03:56.480 --> 00:04:01.800 vector in terms of some sum of these two vectors. 00:04:01.800 --> 00:04:03.270 So how does that work? 00:04:03.270 --> 00:04:07.780 Well, this vector here, let's call it v. 00:04:07.780 --> 00:04:10.860 This vector, v, is the sum of its x 00:04:10.860 --> 00:04:12.200 component plus its y component. 00:04:12.200 --> 00:04:13.450 When you add vectors, you can put them head 00:04:13.450 --> 00:04:14.600 to tail like this. 00:04:14.600 --> 00:04:15.320 And that's the sum. 00:04:15.320 --> 00:04:18.589 So hopefully knowing what we already know, we knew that the 00:04:18.589 --> 00:04:21.079 vector, v, is equal to its x 00:04:21.079 --> 00:04:26.910 component plus its y component. 00:04:26.910 --> 00:04:28.270 When you add vectors, you essentially just put 00:04:28.270 --> 00:04:29.080 them head to tails. 00:04:29.080 --> 00:04:33.840 And then the resulting sum is where you end up. 00:04:33.840 --> 00:04:36.340 It would be if you added this vector, and then you put this 00:04:36.340 --> 00:04:37.170 tail to this head. 00:04:37.170 --> 00:04:37.910 And you end up there. 00:04:37.910 --> 00:04:38.480 So you end up there. 00:04:38.480 --> 00:04:40.300 So that's the vector. 00:04:40.300 --> 00:04:46.820 So can we define v sub x as some multiple of i, of this 00:04:46.820 --> 00:04:48.160 unit vector? 00:04:48.160 --> 00:04:49.070 Well, sure. 00:04:49.070 --> 00:04:53.610 v sub x completely goes in the x direction. 00:04:53.610 --> 00:04:56.590 But it doesn't have a magnitude of 1. 00:04:56.590 --> 00:05:01.150 It has a magnitude of 10 cosine 30 degrees. 00:05:01.150 --> 00:05:02.870 So its magnitude is ten. 00:05:02.870 --> 00:05:05.340 Let me draw the unit vector up here. 00:05:05.340 --> 00:05:07.640 This is the unit vector i. 00:05:07.640 --> 00:05:10.550 It's going to look something like this and this. 00:05:10.550 --> 00:05:13.120 So v sub x is in the exact same direction, and it's just 00:05:13.120 --> 00:05:14.820 a scaled version of this unit vector. 00:05:14.820 --> 00:05:19.120 And what multiple is it of that unit vector? 00:05:19.120 --> 00:05:21.320 Well, the unit vector has a magnitude of 1. 00:05:21.320 --> 00:05:23.720 This has a magnitude of 10 cosine of 30 degrees. 00:05:23.720 --> 00:05:27.130 I think that's like, 5 square roots of 3, or 00:05:27.130 --> 00:05:28.070 something like that. 00:05:28.070 --> 00:05:34.200 So we can write v sub x-- I keep switching colors to keep 00:05:34.200 --> 00:05:35.850 things interesting. 00:05:35.850 --> 00:05:42.680 We can write v sub x is equal to 10 cosine of 30 degrees 00:05:42.680 --> 00:05:45.610 times-- that's the degrees --times the unit vector i-- 00:05:45.610 --> 00:05:49.070 let me stay in that color, so you don't confused --times the 00:05:49.070 --> 00:05:50.420 unit vector i. 00:05:50.420 --> 00:05:52.010 Does that make sense? 00:05:52.010 --> 00:05:54.970 Well, the unit vector i goes in the exact same direction. 00:05:54.970 --> 00:05:57.980 But the x component of this vector is just a lot longer. 00:05:57.980 --> 00:06:01.630 It's 10 cosine 30 degrees long. 00:06:01.630 --> 00:06:05.080 And that's equal to-- cosine of 30 degrees is square root 00:06:05.080 --> 00:06:11.580 of 3/2 --so that's 5 square roots of 3 i. 00:06:11.580 --> 00:06:18.150 Similary, we can write the y component of this vector as 00:06:18.150 --> 00:06:19.400 some multiple of j. 00:06:23.110 --> 00:06:28.490 So we could say v sub y, the y component-- Well, what is sine 00:06:28.490 --> 00:06:29.250 of 30 degrees? 00:06:29.250 --> 00:06:31.400 Sine of 30 degrees is 1/2. 00:06:31.400 --> 00:06:35.470 1/2 times 10, so this is 5. 00:06:35.470 --> 00:06:39.740 So the y component goes completely in the y direction. 00:06:39.740 --> 00:06:42.880 So it's just going to be a multiple of this vector j, of 00:06:42.880 --> 00:06:44.410 the unit vector j. 00:06:44.410 --> 00:06:45.350 And what multiple is it? 00:06:45.350 --> 00:06:48.240 Well, it has length 5, while the unit vector 00:06:48.240 --> 00:06:49.550 has just length 1. 00:06:49.550 --> 00:06:54.590 So it's just 5 times the unit vector j. 00:06:54.590 --> 00:06:56.240 So how can we write vector v? 00:06:56.240 --> 00:06:59.030 Well, we know the vector v is the sum of its x component and 00:06:59.030 --> 00:07:00.930 its y component. 00:07:00.930 --> 00:07:03.840 And we also know, so this is a whole vector v. 00:07:03.840 --> 00:07:04.880 What's its x component? 00:07:04.880 --> 00:07:07.130 Its x component can be written as a multiple 00:07:07.130 --> 00:07:08.710 of the x unit vector. 00:07:08.710 --> 00:07:10.490 That's that right there. 00:07:10.490 --> 00:07:15.080 So you can write it as 5 square roots of 3 00:07:15.080 --> 00:07:19.330 i plus its y component. 00:07:19.330 --> 00:07:21.530 So what's its y component? 00:07:21.530 --> 00:07:24.880 Well, its y component is just a multiple of the y unit 00:07:24.880 --> 00:07:27.160 vector, which is called j, with the little 00:07:27.160 --> 00:07:28.680 funny hat on top. 00:07:28.680 --> 00:07:29.730 And that's just this. 00:07:29.730 --> 00:07:31.050 It's 5 times j. 00:07:34.970 --> 00:07:37.770 So what we've done now, by defining these unit vectors-- 00:07:37.770 --> 00:07:39.390 And I can switch this color just so you 00:07:39.390 --> 00:07:41.640 remember this is i. 00:07:41.640 --> 00:07:43.450 This unit vector is this. 00:07:43.450 --> 00:07:46.860 Using unit vectors in two dimensions, and we can 00:07:46.860 --> 00:07:50.140 eventually do them in multiple dimensions, we can 00:07:50.140 --> 00:07:55.130 analytically express any two dimensional vector. 00:07:55.130 --> 00:07:57.680 Instead of having to always draw it like we did before, 00:07:57.680 --> 00:08:00.710 and having to break out its components and 00:08:00.710 --> 00:08:01.810 always do it visually. 00:08:01.810 --> 00:08:05.660 We can stay in analytical mode and non graphical mode. 00:08:05.660 --> 00:08:09.930 And what makes this very useful is that if I can write 00:08:09.930 --> 00:08:13.480 a vector in this format, I can add them and subtract them 00:08:13.480 --> 00:08:18.830 without having to resort to visual means. 00:08:18.830 --> 00:08:20.080 And what do I mean by that? 00:08:23.180 --> 00:08:27.710 So if I had to find some vector a, is equal to, I don't 00:08:27.710 --> 00:08:33.870 know, 2i plus 3j. 00:08:33.870 --> 00:08:37.980 And I have some other vector b. 00:08:37.980 --> 00:08:39.980 This little arrow just means it's a vector. 00:08:39.980 --> 00:08:42.409 Sometimes you'll see it as a whole arrow. 00:08:42.409 --> 00:08:52.800 As, I don't know, 10i plus 2j. 00:08:52.800 --> 00:08:55.080 If I were to say what's the sum of these two 00:08:55.080 --> 00:08:57.650 vectors a plus b? 00:08:57.650 --> 00:09:00.460 Before we had this unit vector notation, we would have to 00:09:00.460 --> 00:09:02.320 draw them, and put them heads to tails. 00:09:02.320 --> 00:09:04.300 And you had to do it visually, and it would take 00:09:04.300 --> 00:09:04.870 you a lot of time. 00:09:04.870 --> 00:09:07.580 But once you have it broken up into the x and y components, 00:09:07.580 --> 00:09:10.170 you can just separately add the x and y components. 00:09:10.170 --> 00:09:18.330 So vector a plus vector b, that's just 2 plus 10 times i 00:09:18.330 --> 00:09:22.660 plus 3 plus 2 times j. 00:09:22.660 --> 00:09:27.820 And that's equal to 12i plus 5j. 00:09:27.820 --> 00:09:29.790 And something you might want to do, maybe I'll do it in the 00:09:29.790 --> 00:09:32.820 future video, is actually draw out these two vectors and add 00:09:32.820 --> 00:09:33.420 them visually. 00:09:33.420 --> 00:09:37.210 And you'll see that you get this exact answer. 00:09:37.210 --> 00:09:40.050 And as we go into further videos, or future videos, 00:09:40.050 --> 00:09:42.590 you'll see how this is super useful once we start doing 00:09:42.590 --> 00:09:45.090 more complicated physics problems, or once we start 00:09:45.090 --> 00:09:47.140 doing physics with calculus. 00:09:47.140 --> 00:09:50.840 Anyway, I'm about to run out of time on the ten minutes. 00:09:50.840 --> 00:09:52.090 So I'll see you in the next video.

Introduction to the yield curve

https://www.youtube.com/watch?v=b_cAxh44aNQ

vtt

https://www.youtube.com/api/timedtext?v=b_cAxh44aNQ&ei=YmeUZZXxHt2gp-oP7LCu2AY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=AB3134A2EEE1DD16A093A4538C75A4C3465CE24F.B183AB5FB33E34DA914D7733EEDD5A078E1DAC92&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.770 --> 00:00:01.750 Welcome back. 00:00:01.750 --> 00:00:04.780 Before we proceed further and get a little bit better 00:00:04.780 --> 00:00:08.000 understanding of why maybe some of these investors were 00:00:08.000 --> 00:00:10.330 so keen on investing in mortgage backed securities, 00:00:10.330 --> 00:00:13.330 essentially loaning this money to all these people who are 00:00:13.330 --> 00:00:16.090 buying these ever appreciating houses, I think we need to a 00:00:16.090 --> 00:00:18.110 few more tools in our tool belt. 00:00:18.110 --> 00:00:19.910 So I'm going to introduce you to the concept 00:00:19.910 --> 00:00:21.710 of the yield curve. 00:00:21.710 --> 00:00:22.890 You might have heard this before. 00:00:22.890 --> 00:00:25.970 You might have heard people on CNBC talk about it. 00:00:25.970 --> 00:00:29.460 And hopefully, after about the next five or ten minutes, you 00:00:29.460 --> 00:00:32.600 will know a lot about the yield curve. 00:00:32.600 --> 00:00:34.290 So when most people talk about the yield curve, they're 00:00:34.290 --> 00:00:35.950 talking about the treasury yield curve. 00:00:35.950 --> 00:00:37.930 And what does that mean? 00:00:37.930 --> 00:00:39.580 What is even a treasury? 00:00:39.580 --> 00:00:42.230 So these treasury securities, whether they're T-Bills, 00:00:42.230 --> 00:00:45.445 treasury bills, treasury notes, or treasury bonds. 00:00:57.010 --> 00:01:01.130 These are loans to the federal government. 00:01:01.130 --> 00:01:04.720 And these are considered risk-free. 00:01:04.720 --> 00:01:08.090 Because if you lend to the federal government and they're 00:01:08.090 --> 00:01:10.840 running short of cash, all they have to do is increase 00:01:10.840 --> 00:01:15.590 taxes on us the people and they can pay back your debt. 00:01:15.590 --> 00:01:19.400 So in dollar denominated terms, the treasury bills, 00:01:19.400 --> 00:01:22.060 notes, and bonds are about as safe as you can get in terms 00:01:22.060 --> 00:01:24.420 of lending your money to anyone. 00:01:24.420 --> 00:01:26.540 So when most people talk about the yield curve, they're 00:01:26.540 --> 00:01:28.630 talking about the risk-free yield curve. 00:01:28.630 --> 00:01:34.260 And they're talking about the curve for treasuries. 00:01:34.260 --> 00:01:36.770 So first, a little bit of definitions. 00:01:36.770 --> 00:01:40.470 What is the difference between treasury bills, treasury 00:01:40.470 --> 00:01:43.480 notes, and treasury bonds? 00:01:43.480 --> 00:01:45.790 They're pretty much all loans to the government. 00:01:45.790 --> 00:01:48.210 But they're loans for different amounts of time. 00:01:48.210 --> 00:01:55.580 So if I give a loan to the government for $1,000 for six 00:01:55.580 --> 00:01:57.500 months, that will be a treasury bill. 00:01:57.500 --> 00:02:00.710 So I will give the government $1,000, the government would 00:02:00.710 --> 00:02:02.180 give me a treasury bill. 00:02:02.180 --> 00:02:04.340 And that treasury bill from the government is essentially 00:02:04.340 --> 00:02:08.229 just an IOU saying that I'm going to give you your money 00:02:08.229 --> 00:02:12.740 back in six months with interest. Similarly, if it's 00:02:12.740 --> 00:02:15.010 three months, it's a three month treasury bill. 00:02:15.010 --> 00:02:19.590 Treasury notes are loans that are from one year to 10 years. 00:02:19.590 --> 00:02:24.110 So on this graph that we're going to make using the actual 00:02:24.110 --> 00:02:29.060 yield curve rates, from zero to one year-- and actually 00:02:29.060 --> 00:02:31.490 there's no zero year treasury bill. 00:02:31.490 --> 00:02:34.250 Actually, the shortest one is one month. 00:02:34.250 --> 00:02:36.830 This would be something like here on our graph. 00:02:36.830 --> 00:02:42.810 So from one month to one year, these are T-bills. 00:02:42.810 --> 00:02:46.090 And this is just definitional. 00:02:46.090 --> 00:02:49.560 Then from one year to 10 year, these are notes. 00:02:49.560 --> 00:02:55.670 Actually, I believe the one year itself is a note. 00:02:55.670 --> 00:02:57.440 Up to one year is a bill. 00:02:57.440 --> 00:02:58.520 Although, I might be wrong with that. 00:02:58.520 --> 00:02:59.490 Correct me if I'm wrong. 00:02:59.490 --> 00:03:00.700 That's just a definitional thing. 00:03:00.700 --> 00:03:03.950 From one to 10 year, these are called notes. 00:03:03.950 --> 00:03:06.850 And then when you go beyond 10 years, these are called 00:03:06.850 --> 00:03:09.470 treasury bonds. 00:03:09.470 --> 00:03:12.930 These are just definitional things to worry about. 00:03:12.930 --> 00:03:15.060 So with that out of the way, let's talk about what the 00:03:15.060 --> 00:03:17.840 yield curve is. 00:03:17.840 --> 00:03:19.570 I'll just give you a simple thought experiment. 00:03:19.570 --> 00:03:24.260 If I'm lending money to someone for a month versus 00:03:24.260 --> 00:03:28.020 lending money to that person for a year, in which situation 00:03:28.020 --> 00:03:31.080 am I probably taking on more risk? 00:03:31.080 --> 00:03:36.040 Well, sure, if I'm lending someone for a month, I know 00:03:36.040 --> 00:03:37.870 only so much can happen in that month. 00:03:37.870 --> 00:03:41.920 So I would expect to be paid less interest. Not just 00:03:41.920 --> 00:03:46.530 obviously in dollar terms, but even adjusted for time, I 00:03:46.530 --> 00:03:50.370 would expect less interest for that month. 00:03:50.370 --> 00:03:52.410 And this is actually an important point to make. 00:03:52.410 --> 00:03:55.880 When I say that I'm charging 6% interest for that month, 00:03:55.880 --> 00:03:57.730 that doesn't mean that after a month the person is going to 00:03:57.730 --> 00:03:59.650 pay me 6% on my money. 00:03:59.650 --> 00:04:03.040 It means that if I were to give that money to somebody 00:04:03.040 --> 00:04:04.820 for a month, and they were to pay it back. 00:04:04.820 --> 00:04:06.970 And then I were to give that money to, say, that same 00:04:06.970 --> 00:04:09.240 person, or another person, for a month, and I were to keep 00:04:09.240 --> 00:04:12.770 doing that for a year, then in aggregate I would get 6%. 00:04:12.770 --> 00:04:16.310 So that 6%, no matter what duration we talk about, 00:04:16.310 --> 00:04:19.550 whether one month, one year, five years, 15 years, when we 00:04:19.550 --> 00:04:22.190 talk about the interest rate, that's the rate that on 00:04:22.190 --> 00:04:24.160 average we would get for a year. 00:04:24.160 --> 00:04:26.440 It's the annualized interest rate. 00:04:26.440 --> 00:04:27.740 So going back to my question. 00:04:27.740 --> 00:04:30.970 If lend someone money, even the government, for a month, 00:04:30.970 --> 00:04:32.870 there's usually less risk in that. 00:04:32.870 --> 00:04:34.510 Because only so much could happen in a 00:04:34.510 --> 00:04:35.570 month versus in a year. 00:04:35.570 --> 00:04:38.190 In a year there might be more inflation, the dollar might 00:04:38.190 --> 00:04:42.120 collapse, I might be passing on better investments, I might 00:04:42.120 --> 00:04:44.880 need the cash in a year's time, while I have a lot of 00:04:44.880 --> 00:04:47.630 confidence that I don't need the cash in a month's time. 00:04:47.630 --> 00:04:51.690 So in general, you expect less interest when you loan money 00:04:51.690 --> 00:04:55.910 for a shorter period time than a longer period of time. 00:04:55.910 --> 00:04:57.645 And so let's draw the yield curve and see 00:04:57.645 --> 00:04:58.490 if this holds true. 00:04:58.490 --> 00:05:00.830 So I actually went to the treasury 00:05:00.830 --> 00:05:04.400 website, so that's treas.gov. 00:05:04.400 --> 00:05:05.490 And this is the yield curve. 00:05:05.490 --> 00:05:08.000 So they say on March 14, so this is 00:05:08.000 --> 00:05:09.860 the most recent number. 00:05:09.860 --> 00:05:10.830 And I'm going to plot this. 00:05:10.830 --> 00:05:15.520 They say, if you lend money to the government for one month, 00:05:15.520 --> 00:05:17.830 you'll get 1.2% on that money. 00:05:17.830 --> 00:05:20.740 And remember, if it's $1,000 it's not like I'm going to get 00:05:20.740 --> 00:05:23.830 1.2% on that $1,000 just after a month. 00:05:23.830 --> 00:05:26.320 If I kept doing it for a year, this is an annualized number, 00:05:26.320 --> 00:05:27.950 I'll get 1.2%. 00:05:27.950 --> 00:05:31.400 And so for three months, I get a little bit less. 00:05:31.400 --> 00:05:33.080 And then for six months I get more. 00:05:33.080 --> 00:05:35.710 And then it does seem that the overall trend is that I expect 00:05:35.710 --> 00:05:39.700 more and more money as I lend money to the government for 00:05:39.700 --> 00:05:41.530 larger and larger periods of time. 00:05:41.530 --> 00:05:44.490 And this is a little interesting anomaly that you 00:05:44.490 --> 00:05:46.610 get a little bit more interest for one 00:05:46.610 --> 00:05:47.500 month than three months. 00:05:47.500 --> 00:05:51.810 And we'll do a more advanced presentation later as to why 00:05:51.810 --> 00:05:56.170 you might get lower yields for longer duration investments. 00:05:56.170 --> 00:05:58.195 That's called an inverted yield curve. 00:05:58.195 --> 00:06:02.160 So let's just plot this and see what it looks like. 00:06:02.160 --> 00:06:03.530 So you saw where I got my data. 00:06:03.530 --> 00:06:06.000 So they say for one month I'd get 1.2%. 00:06:06.000 --> 00:06:07.030 So this is one month. 00:06:07.030 --> 00:06:09.800 It'd be right about here. 00:06:09.800 --> 00:06:12.370 Three months I get about the same thing. 00:06:12.370 --> 00:06:14.720 For six months I get 1.32%. 00:06:14.720 --> 00:06:16.910 Maybe that's like here. 00:06:16.910 --> 00:06:20.520 One year, I get one 1.37%. 00:06:20.520 --> 00:06:22.460 Maybe it's here. 00:06:22.460 --> 00:06:25.880 Five years, I get 2.37%. 00:06:25.880 --> 00:06:27.130 So that's maybe like here. 00:06:30.280 --> 00:06:31.760 And these aren't all of the durations. 00:06:31.760 --> 00:06:34.700 I'm just for simplicity not going to do all of them. 00:06:34.700 --> 00:06:37.030 For 10 years, 3.44%. 00:06:37.030 --> 00:06:40.850 So maybe that's here. 00:06:40.850 --> 00:06:44.410 For 20 years, I get 4.3%. 00:06:44.410 --> 00:06:45.870 Like that. 00:06:45.870 --> 00:06:50.540 And then for 30 years, I get 4.35%. 00:06:50.540 --> 00:06:53.400 So the current yield curve looks something like this. 00:07:03.480 --> 00:07:06.140 And so you now hopefully at least understand what the 00:07:06.140 --> 00:07:06.830 yield curve is. 00:07:06.830 --> 00:07:09.960 All it is, is using a simple graph. 00:07:09.960 --> 00:07:12.950 Someone can look at that graph and say, well, in general what 00:07:12.950 --> 00:07:17.340 type of rates am I getting for lending to the government? 00:07:17.340 --> 00:07:19.710 On a risk-free free basis, or as risk-free as anything we 00:07:19.710 --> 00:07:23.280 can expect, what type of rates am I getting when I lend to 00:07:23.280 --> 00:07:25.100 the government for different periods of time? 00:07:25.100 --> 00:07:26.730 And that's what the yield curve tells us. 00:07:26.730 --> 00:07:29.120 And in general, it's upwardly sloping. 00:07:29.120 --> 00:07:31.380 Because, as I said, when you lend money for a longer period 00:07:31.380 --> 00:07:34.000 of time, you're kind of taking on more risk. 00:07:34.000 --> 00:07:36.770 There's a lot more that you feel that could happen. 00:07:36.770 --> 00:07:38.020 You might need that cash. 00:07:40.630 --> 00:07:41.480 There might be inflation. 00:07:41.480 --> 00:07:42.560 The dollar might devalue. 00:07:42.560 --> 00:07:44.690 There's a lot of things that could happen. 00:07:44.690 --> 00:07:47.120 So the next question is, well, what 00:07:47.120 --> 00:07:49.500 determines this yield curve? 00:07:49.500 --> 00:07:55.890 Well, when the treasury, the government, needs to borrow 00:07:55.890 --> 00:07:59.370 money, what it does is say, hey everyone we need to borrow 00:07:59.370 --> 00:08:00.990 a billion dollars from you, because we 00:08:00.990 --> 00:08:02.510 can't control are spending. 00:08:02.510 --> 00:08:04.920 And they say we're going to borrow a billion dollars in 00:08:04.920 --> 00:08:05.960 one month notes. 00:08:05.960 --> 00:08:07.490 So this is one month notes. 00:08:07.490 --> 00:08:08.790 They're going to borrow a billion dollars. 00:08:08.790 --> 00:08:10.370 And they have an auction. 00:08:10.370 --> 00:08:14.340 And the world, investors from everywhere, they go in, they 00:08:14.340 --> 00:08:16.460 say, well, this is a safe place to put 00:08:16.460 --> 00:08:17.850 my cash for a month. 00:08:17.850 --> 00:08:21.490 And depending on the demand, that determines the rate. 00:08:21.490 --> 00:08:24.800 So if there are a lot of people who want to buy those 00:08:24.800 --> 00:08:28.940 one month treasuries, the rate might be a little bit lower. 00:08:28.940 --> 00:08:30.290 Does that make sense to you? 00:08:30.290 --> 00:08:31.170 Think about it. 00:08:31.170 --> 00:08:33.900 If a lot of people want to buy it, there's a lot of demand 00:08:33.900 --> 00:08:35.390 relative to the supply. 00:08:35.390 --> 00:08:38.840 So the government has to pay a lower interest rate on it. 00:08:38.840 --> 00:08:42.159 Similarly, if for whatever reason people don't want to 00:08:42.159 --> 00:08:44.630 keep their money in the dollar, they think the U.S. 00:08:44.630 --> 00:08:47.860 might default on their debt one day, and not that many 00:08:47.860 --> 00:08:51.480 people want to invest in the treasury, then that auction, 00:08:51.480 --> 00:08:53.510 the government is going to have to pay a higher interest 00:08:53.510 --> 00:08:56.340 rate to people for them to loan money to it. 00:08:56.340 --> 00:08:59.310 So maybe then the auction ends up up here. 00:08:59.310 --> 00:09:02.230 And similarly, the government does auctions for all of the 00:09:02.230 --> 00:09:03.380 different durations. 00:09:03.380 --> 00:09:05.150 And duration, I just mean the time period you're 00:09:05.150 --> 00:09:06.320 getting the loan for. 00:09:06.320 --> 00:09:08.370 So they do it for one month, three months, six months, one 00:09:08.370 --> 00:09:10.940 year, two year, three year, et cetera. 00:09:10.940 --> 00:09:17.930 Once the government has done that auction-- You give the 00:09:17.930 --> 00:09:19.010 money to the government, they give you an 00:09:19.010 --> 00:09:20.740 IOU called a T-bill. 00:09:20.740 --> 00:09:22.100 Then you could trade it with other people. 00:09:22.100 --> 00:09:24.930 And that's going to determine the rate in the short term. 00:09:24.930 --> 00:09:26.510 So the government does the auction. 00:09:26.510 --> 00:09:29.310 But then after the auction, and a lot of people had 00:09:29.310 --> 00:09:32.990 demand, but then a lot of people get freaked out. 00:09:32.990 --> 00:09:36.180 And the public markets, when you try to sell that treasury, 00:09:36.180 --> 00:09:37.480 will then expect. 00:09:37.480 --> 00:09:38.200 a higher yield. 00:09:38.200 --> 00:09:39.690 I know that might be a little complicated now. 00:09:39.690 --> 00:09:42.510 And I always start to jumble things when I run out of time. 00:09:42.510 --> 00:09:45.780 But hopefully at this point you have a sense of what the 00:09:45.780 --> 00:09:46.590 yield curve is. 00:09:46.590 --> 00:09:48.210 You have a sense of what treasury bills, treasury 00:09:48.210 --> 00:09:49.820 notes, and treasury bonds are. 00:09:49.820 --> 00:09:52.440 And you have some intuition on why the yield 00:09:52.440 --> 00:09:54.180 curve has this shape. 00:09:54.180 --> 00:09:56.110 See you in the next video.

Housing price conundrum (part 4)

https://www.youtube.com/watch?v=s6UYa2nwaDw

vtt

https://www.youtube.com/api/timedtext?v=s6UYa2nwaDw&ei=YmeUZYmbKubHp-oPxcGgwAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5C47968E1DF28382955C90529299E000E491702E.5C237CE2270CBD6B318FD5772E0D9BA0B8C60A95&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.670 --> 00:00:04.290 I'll now explain to you why, from 2000 to 2005, we had very 00:00:04.290 --> 00:00:06.390 low defaults on mortgages. 00:00:06.390 --> 00:00:14.490 Let's say that I buy a house for $1 million. 00:00:14.490 --> 00:00:15.610 I buy a $1 million house. 00:00:15.610 --> 00:00:18.280 So let's say the bank gives me $1 million. 00:00:18.280 --> 00:00:20.370 And then I'm willing to pay a percentage on it. 00:00:20.370 --> 00:00:23.700 So this is from the bank. 00:00:23.700 --> 00:00:25.130 This is me. 00:00:25.130 --> 00:00:27.190 And I use that to buy a house. 00:00:27.190 --> 00:00:28.360 I don't know if these diagrams help you. 00:00:28.360 --> 00:00:30.320 But you get the general idea. 00:00:30.320 --> 00:00:31.600 And the bank does that. 00:00:31.600 --> 00:00:36.330 And let's say, I don't know, a year later I lose my job. 00:00:36.330 --> 00:00:39.340 I just can't pay this mortgage anymore. 00:00:39.340 --> 00:00:41.420 So I have a couple of options. 00:00:41.420 --> 00:00:48.920 I can either sell the house and pay off the debt, or I 00:00:48.920 --> 00:00:50.780 guess I could just tell the bank, well I can't do 00:00:50.780 --> 00:00:52.570 anything, and I'm going to foreclose. 00:00:52.570 --> 00:00:53.760 And that would ruin my credit. 00:00:53.760 --> 00:00:54.440 It would hurt my credit. 00:00:54.440 --> 00:00:56.650 And I would lose all my down payment. 00:00:56.650 --> 00:00:59.470 So what are the circumstances that I can sell the house? 00:00:59.470 --> 00:01:01.650 Well, if I borrowed $1 million, as long as-- and 00:01:01.650 --> 00:01:02.940 let's say I didn't put any money down, just for 00:01:02.940 --> 00:01:04.190 simplicity. 00:01:04.190 --> 00:01:07.600 If I can sell the house for $1.1 million, well I 00:01:07.600 --> 00:01:08.640 would do it, right? 00:01:08.640 --> 00:01:10.910 Let me sell for $1.1 million. 00:01:10.910 --> 00:01:13.460 If I sell for $1.1 million, I pay the bank-- 00:01:13.460 --> 00:01:14.760 let me switch colors. 00:01:14.760 --> 00:01:19.900 I pay the bank $1 million, and I net $100,000. 00:01:19.900 --> 00:01:20.800 And everyone's happy. 00:01:20.800 --> 00:01:23.540 The bank got their money back, so they didn't lose any money 00:01:23.540 --> 00:01:24.690 on the transaction. 00:01:24.690 --> 00:01:26.360 I made $100,000. 00:01:26.360 --> 00:01:28.840 And so the whole reason why this worked out, even though 00:01:28.840 --> 00:01:30.970 maybe I was a credit risk, is because the 00:01:30.970 --> 00:01:32.970 housing prices went up. 00:01:32.970 --> 00:01:38.050 So when you have rising housing prices, the banks will 00:01:38.050 --> 00:01:40.100 not lose money lending you. 00:01:40.100 --> 00:01:43.270 Because if you can't pay, you just give back the house, the 00:01:43.270 --> 00:01:43.860 bank can sell it. 00:01:43.860 --> 00:01:45.170 Or, you won't even give back the house. 00:01:45.170 --> 00:01:47.190 You'll sell the house and you'll pay it off, even though 00:01:47.190 --> 00:01:48.800 you can't pay the mortgage anymore. 00:01:48.800 --> 00:01:51.200 The only situation where I would foreclose is if the 00:01:51.200 --> 00:01:54.070 market price of the house goes less than my loan. 00:01:54.070 --> 00:01:55.320 And that's actually the situation that 00:01:55.320 --> 00:01:56.400 we're facing now. 00:01:56.400 --> 00:01:58.340 So if, let's say that I can only sell 00:01:58.340 --> 00:02:00.400 this house for $900,000. 00:02:00.400 --> 00:02:01.430 Well, then I'm just going to give the 00:02:01.430 --> 00:02:04.000 keys back to the bank. 00:02:04.000 --> 00:02:05.840 That's actually called jingle mail, because you just mail 00:02:05.840 --> 00:02:07.030 the keys back. 00:02:07.030 --> 00:02:09.139 And then the bank sells the house for $900,000. 00:02:09.139 --> 00:02:10.610 And then they would take a loss. 00:02:10.610 --> 00:02:14.510 So when housing prices go down, that's the only 00:02:14.510 --> 00:02:16.810 situation where really you should have foreclosure. 00:02:16.810 --> 00:02:19.550 When housing prices soon. go up, the person who borrowed it 00:02:19.550 --> 00:02:22.700 is just going to sell the house and pay off the loan. 00:02:22.700 --> 00:02:24.070 And they are actually probably going to make some money. 00:02:24.070 --> 00:02:25.970 So there was every incentive to buy a house. 00:02:25.970 --> 00:02:29.390 So let's think about this whole dynamic over the last 00:02:29.390 --> 00:02:30.870 several videos that we've been building. 00:02:34.210 --> 00:02:42.300 So we said, from 2000 to 2004 housing prices went up. 00:02:45.180 --> 00:02:46.530 Let me do it like this. 00:02:46.530 --> 00:02:48.030 Let me change it a little bit. 00:02:52.950 --> 00:02:58.160 We can even say, from 2000 to 2006. 00:02:58.160 --> 00:03:00.300 So we know that housing prices went up. 00:03:06.660 --> 00:03:09.770 And why did why did housing prices go up? 00:03:09.770 --> 00:03:11.120 Well, we saw the data. 00:03:11.120 --> 00:03:13.240 It wasn't because people were earning more. 00:03:13.240 --> 00:03:16.010 It wasn't because the unemployment rate went down. 00:03:16.010 --> 00:03:17.940 It wasn't because the population increased. 00:03:17.940 --> 00:03:20.840 It wasn't because the supply of houses were limited. 00:03:20.840 --> 00:03:21.830 We disproved all that. 00:03:21.830 --> 00:03:24.830 We realize it was just because financing got easier. 00:03:24.830 --> 00:03:28.770 The standards for getting a loan went lower and lower. 00:03:28.770 --> 00:03:33.850 Financing got easier and easier. 00:03:33.850 --> 00:03:37.850 And because housing prices went up, what did that cause? 00:03:37.850 --> 00:03:40.150 We just said when housing prices go up, 00:03:40.150 --> 00:03:41.400 default rates go down. 00:03:46.760 --> 00:03:48.330 You could give a loan to someone 00:03:48.330 --> 00:03:49.890 who's a complete deadbeat. 00:03:49.890 --> 00:03:53.970 But as long as housing prices go up, if they lose their job, 00:03:53.970 --> 00:03:56.690 they can still sell that house and pay you back the loan. 00:03:56.690 --> 00:04:00.060 So housing prices going up makes sure there's no 00:04:00.060 --> 00:04:02.140 foreclosure, so defaults go down. 00:04:02.140 --> 00:04:04.950 So then the perceived risk goes down, of lending. 00:04:09.390 --> 00:04:17.360 Perceived lending risk goes down. 00:04:20.029 --> 00:04:23.040 So that makes more people willing to lend. 00:04:31.240 --> 00:04:34.450 And the corollary of more people willing to lend, is you 00:04:34.450 --> 00:04:36.410 that the actual standards go down. 00:04:38.910 --> 00:04:39.860 That's financing easier. 00:04:39.860 --> 00:04:40.600 We could actually write that. 00:04:40.600 --> 00:04:41.850 Standards go down. 00:04:44.990 --> 00:04:47.320 So you had this whole-- I guess you could argue whether 00:04:47.320 --> 00:04:50.230 this is a negative or a positive cycle. 00:04:50.230 --> 00:04:54.720 But you had this whole cycle occurring from the late '90s, 00:04:54.720 --> 00:04:57.770 but especially, it really got a lot of momentum at around 00:04:57.770 --> 00:05:00.090 2001, 2002, 2003. 00:05:00.090 --> 00:05:03.070 That financing got easier, despite the fact that people 00:05:03.070 --> 00:05:05.740 were earning less, population wasn't increasing that fast, 00:05:05.740 --> 00:05:06.920 that there were all of these new houses. 00:05:06.920 --> 00:05:09.170 And that caused housing prices to go up. 00:05:09.170 --> 00:05:11.850 Housing prices went up, then we had a lot fewer people 00:05:11.850 --> 00:05:13.300 defaulting on their loans. 00:05:13.300 --> 00:05:15.050 No one would default on their loans if they could sell it 00:05:15.050 --> 00:05:16.630 for more than the loan. 00:05:16.630 --> 00:05:19.420 Then a lot more people said, well these are super safe. 00:05:19.420 --> 00:05:23.910 And so the ratings agencies, Standard and Poor's and 00:05:23.910 --> 00:05:27.030 Moody's, were willing to give AAA ratings to more and more, 00:05:27.030 --> 00:05:29.030 what I would argue, are risky loans. 00:05:29.030 --> 00:05:31.070 So the perceived lending risk went down. 00:05:31.070 --> 00:05:32.920 Then more and more people liked this asset class. 00:05:32.920 --> 00:05:33.960 They said, wow, this is great. 00:05:33.960 --> 00:05:36.650 I can get a better return than I can get in a bank, or in 00:05:36.650 --> 00:05:39.080 Treasuries, or in a whole set of securities, even though 00:05:39.080 --> 00:05:41.750 these are very low-risk or perceived low-risk. 00:05:41.750 --> 00:05:43.990 So I want to funnel more and more money in here. 00:05:43.990 --> 00:05:46.890 And so the mortgage brokers and the investment banks said 00:05:46.890 --> 00:05:51.080 great, the only way we can get more volume to satisfy all 00:05:51.080 --> 00:05:53.700 these people who want to lend money-- the only way we can 00:05:53.700 --> 00:05:56.150 find more people to lend money to, is by 00:05:56.150 --> 00:05:57.460 lowering the standards. 00:05:57.460 --> 00:06:00.020 And this cycle went round and round and round. 00:06:00.020 --> 00:06:02.820 And it really started because this whole process of being 00:06:02.820 --> 00:06:05.970 able to take a bunch of people's mortgages together, 00:06:05.970 --> 00:06:09.630 package them up, and then turn them into securities and then 00:06:09.630 --> 00:06:12.270 sell them to a bunch of investors-- this was a 00:06:12.270 --> 00:06:15.630 quote-unquote innovation in the mid-'90s, or early '90s. 00:06:15.630 --> 00:06:16.810 I forgot exactly when. 00:06:16.810 --> 00:06:20.670 And it really started to take steam in the early part of 00:06:20.670 --> 00:06:21.860 this decade. 00:06:21.860 --> 00:06:26.000 So that's essentially why housing prices went up. 00:06:26.000 --> 00:06:29.200 And why kind of all of this silliness happened. 00:06:29.200 --> 00:06:32.270 And in the next video, I'll talk a little bit more about 00:06:32.270 --> 00:06:35.320 maybe who some of these investors were. 00:06:35.320 --> 00:06:39.300 And I'll tell you what a common hedge fund technique. 00:06:39.300 --> 00:06:40.980 And I think it's very important not to group all 00:06:40.980 --> 00:06:41.760 hedge funds together. 00:06:41.760 --> 00:06:43.000 There are some good ones. 00:06:43.000 --> 00:06:46.270 But what a common hedge fund technique was, to take 00:06:46.270 --> 00:06:49.110 advantage of this virtual cycle, to make the hedge fund 00:06:49.110 --> 00:06:50.980 founders very wealthy. 00:06:50.980 --> 00:06:52.450 I'll see

Housing price conundrum (part 3)

https://www.youtube.com/watch?v=aAfMps_VyOY

vtt

https://www.youtube.com/api/timedtext?v=aAfMps_VyOY&ei=YmeUZeKkKrvBmLAPq-aO0AU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2FD1B4B0AB93A1A305A9DFE1D92730F5661CD83B.09ECE39BDFE7CF39077A9CEA7F69800626BF89BA&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.720 --> 00:00:01.780 So just a bit of review. 00:00:01.780 --> 00:00:05.650 What happened from 2000 to 2004? 00:00:05.650 --> 00:00:08.029 Actually I should really say, from 2000 to 2006, because 00:00:08.029 --> 00:00:11.700 that really was when the housing bubble happened. 00:00:11.700 --> 00:00:13.270 Well financing got easier. 00:00:20.920 --> 00:00:23.640 Or essentially they lowered their standards. 00:00:23.640 --> 00:00:26.230 And it got progressively easier and easier and easier, 00:00:26.230 --> 00:00:27.400 every year we went. 00:00:27.400 --> 00:00:36.630 So then that allowed more people to bid on homes. 00:00:36.630 --> 00:00:39.110 So it increased the demand artificially in certain ways. 00:00:39.110 --> 00:00:41.670 Because we saw from that New York Times article that 00:00:41.670 --> 00:00:44.360 people's incomes weren't increasing, and the population 00:00:44.360 --> 00:00:46.160 wasn't increasing anywhere near as fast 00:00:46.160 --> 00:00:47.470 to soak up the supply. 00:00:47.470 --> 00:00:52.110 So all it did is allow people who were renting before, and 00:00:52.110 --> 00:00:53.900 who couldn't save the money for the down payment, now to 00:00:53.900 --> 00:00:54.490 participate. 00:00:54.490 --> 00:00:59.480 So now you had more people bidding for the same house, to 00:00:59.480 --> 00:01:01.550 bid up houses. 00:01:01.550 --> 00:01:03.530 But that led to the obvious question. 00:01:03.530 --> 00:01:08.050 Why did financing get easier and easier? 00:01:08.050 --> 00:01:11.980 So let's go back to the good old days, like the early '90s. 00:01:11.980 --> 00:01:13.370 Or actually, let's go even before that. 00:01:13.370 --> 00:01:16.990 Let's go to the classic, what happens to get a housing loan? 00:01:16.990 --> 00:01:20.260 Well traditionally, if I want to get a loan I 00:01:20.260 --> 00:01:21.535 would go to my bank. 00:01:26.500 --> 00:01:29.330 And that loan officer at the bank, he's going to be giving 00:01:29.330 --> 00:01:31.370 the bank's money for your house, right? 00:01:34.030 --> 00:01:35.210 He gives you money. 00:01:35.210 --> 00:01:38.200 And you're going to pay him interest. Right? 00:01:38.200 --> 00:01:39.800 This is me. 00:01:39.800 --> 00:01:43.390 And so that loan officer at the bank, he really cares that 00:01:43.390 --> 00:01:45.460 they're not going to lose money on the transaction. 00:01:45.460 --> 00:01:47.860 If he's going to give you $1 million, he wants to make sure 00:01:47.860 --> 00:01:50.820 that no matter what happens, if you lose your job, if you 00:01:50.820 --> 00:01:54.750 get arrested, if you skip town, that he's still going to 00:01:54.750 --> 00:01:56.260 be able to get his $1 million back. 00:01:56.260 --> 00:01:58.630 And if you go back to our equity and balance sheet 00:01:58.630 --> 00:02:01.180 presentations, that's why, back in the day, they made 00:02:01.180 --> 00:02:04.480 sure that you put 20%, 25% down payment on your house, 00:02:04.480 --> 00:02:06.650 that you had a good credit rating, that you had a good 00:02:06.650 --> 00:02:07.660 steady income. 00:02:07.660 --> 00:02:11.080 Because that banker, that loan officer, was 00:02:11.080 --> 00:02:11.890 going to be in trouble. 00:02:11.890 --> 00:02:14.860 And his bonus was based on how good the loans 00:02:14.860 --> 00:02:15.770 he gave held up. 00:02:15.770 --> 00:02:18.280 So that was the traditional model. 00:02:18.280 --> 00:02:21.190 What happened-- started to happen the mid-90s, especially 00:02:21.190 --> 00:02:24.760 in California, and then nationwide in about 2001, 00:02:24.760 --> 00:02:28.670 2002-- is you had what we call a securitization of the 00:02:28.670 --> 00:02:30.100 mortgage market. 00:02:30.100 --> 00:02:33.170 And this, in all fairness, this actually happened a while 00:02:33.170 --> 00:02:35.230 before, with things like Fannie Mae and Freddie Mac. 00:02:35.230 --> 00:02:37.830 And I'll do a completely separate video on those. 00:02:37.830 --> 00:02:40.000 But Fannie Mae and Freddie Mac essentially 00:02:40.000 --> 00:02:41.390 had the same standards. 00:02:41.390 --> 00:02:44.550 They had the standards of, we call them, conforming loans. 00:02:44.550 --> 00:02:46.740 I think the numbers -- you have to have 20% down. 00:02:46.740 --> 00:02:48.310 You have to have a certain credit score, 00:02:48.310 --> 00:02:49.550 certain steady income. 00:02:49.550 --> 00:02:53.190 So Fannie Mae and Freddie Mac were these entities that might 00:02:53.190 --> 00:02:56.420 buy the loan from your local banker. 00:02:56.420 --> 00:02:58.950 But their standards were just as high as the local banker's. 00:02:58.950 --> 00:03:03.110 And they were based -- I think they were actually, they had 00:03:03.110 --> 00:03:04.390 oversight by Congress. 00:03:04.390 --> 00:03:09.050 So they weren't giving away loans for free. 00:03:09.050 --> 00:03:11.116 But I'll do a whole other presentation on Fannie Mae and 00:03:11.116 --> 00:03:12.140 Freddie Mac. 00:03:12.140 --> 00:03:15.800 What you had happen in the late '90s, and especially in 00:03:15.800 --> 00:03:18.550 the early part of this decade, is you had a whole industry 00:03:18.550 --> 00:03:21.120 outside of the government-sponsored entities. 00:03:21.120 --> 00:03:23.550 The government-sponsored entities are Fannie Mae and 00:03:23.550 --> 00:03:24.510 Freddie Mac. 00:03:24.510 --> 00:03:28.280 And this is essentially -- instead of you going to your 00:03:28.280 --> 00:03:32.810 local bank for a loan -- this is me again -- I would go to 00:03:32.810 --> 00:03:35.700 my local mortgage broker. 00:03:35.700 --> 00:03:38.400 Countrywide is the most famous of them. 00:03:38.400 --> 00:03:39.440 I think they're CFC. 00:03:39.440 --> 00:03:40.840 That's their stock ticker. 00:03:40.840 --> 00:03:42.380 They're not bankrupt yet. 00:03:42.380 --> 00:03:44.630 So I would go to Countrywide. 00:03:44.630 --> 00:03:48.010 And essentially I would get $1 million from 00:03:48.010 --> 00:03:50.170 them, a home loan. 00:03:50.170 --> 00:03:51.470 I would get $1 million from them. 00:03:51.470 --> 00:03:54.520 And I'd agree to pay interest to Countrywide. 00:03:54.520 --> 00:03:57.890 But then Countrywide would do this like, a million times. 00:03:57.890 --> 00:03:59.600 So times a million, right? 00:03:59.600 --> 00:04:01.670 They'll give home loans to a million people, 00:04:01.670 --> 00:04:03.120 put them all together. 00:04:03.120 --> 00:04:05.320 And then they'll sell the loans. 00:04:05.320 --> 00:04:08.090 They'll sell the loans to like, let's say, Bear Stearns. 00:04:11.970 --> 00:04:14.060 So that's an investment bank. 00:04:14.060 --> 00:04:16.420 Let's call it Bear Stearns. 00:04:16.420 --> 00:04:17.890 Hope none of these people sue me. 00:04:17.890 --> 00:04:19.860 I guess they have bigger troubles now, then wondering 00:04:19.860 --> 00:04:21.579 about my YouTube videos. 00:04:21.579 --> 00:04:22.620 They sell it to Bear Stearns. 00:04:22.620 --> 00:04:26.070 And then Bear Stearns will package a bunch of these 00:04:26.070 --> 00:04:29.550 mortgages together, essentially IOU's from people. 00:04:29.550 --> 00:04:31.770 And then they would sell those to investors. 00:04:31.770 --> 00:04:33.980 Right? 00:04:33.980 --> 00:04:39.540 So essentially, instead of Countrywide being responsible 00:04:39.540 --> 00:04:44.580 for my loan, my payments now go to these investors. 00:04:44.580 --> 00:04:47.380 And you could watch the-- that says investors. 00:04:47.380 --> 00:04:49.190 I know my penmanship is horrible. 00:04:49.190 --> 00:04:51.030 But you should watch the videos on mortgage-backed 00:04:51.030 --> 00:04:53.530 securities and collateralized debt obligations, if you want 00:04:53.530 --> 00:04:55.800 to get a better understanding of exactly how 00:04:55.800 --> 00:04:57.080 the money flows go. 00:04:57.080 --> 00:04:59.810 But the bottom line is, because of this process, 00:04:59.810 --> 00:05:01.150 what's happening? 00:05:01.150 --> 00:05:04.860 Countrywide is just being a transactional. 00:05:04.860 --> 00:05:07.200 They're just doing the paperwork for my loan. 00:05:07.200 --> 00:05:08.920 They're temporarily holding the loan. 00:05:08.920 --> 00:05:11.780 And they're doing a little bit of due diligence. 00:05:11.780 --> 00:05:17.640 And in return for that, that Countrywide mortgage broker 00:05:17.640 --> 00:05:21.020 will just get a fixed fee for doing that transaction. 00:05:21.020 --> 00:05:25.000 Maybe they'll get like $5,000 for just doing the paperwork 00:05:25.000 --> 00:05:26.090 for my mortgage. 00:05:26.090 --> 00:05:26.570 Right? 00:05:26.570 --> 00:05:29.160 And then Bear Stearns will package a bunch of these 00:05:29.160 --> 00:05:31.710 mortgages up-- and now it's going to be in the billions-- 00:05:31.710 --> 00:05:35.320 and then repackage them and sell them to investors. 00:05:35.320 --> 00:05:37.700 In the process, Bear Stearns gets a cut. 00:05:37.700 --> 00:05:40.430 And Bear Stearns is doing this for millions of 00:05:40.430 --> 00:05:42.420 mortgages at a time. 00:05:42.420 --> 00:05:44.580 It's in the billions of dollars, and Bear 00:05:44.580 --> 00:05:45.600 Stearns gets a cut. 00:05:45.600 --> 00:05:48.570 So Bear Stearns essentially just gets a fee, like the 00:05:48.570 --> 00:05:49.180 mortgage broker. 00:05:49.180 --> 00:05:51.630 Of course it's a huge fee. 00:05:51.630 --> 00:05:53.290 And then the investors are going to 00:05:53.290 --> 00:05:54.766 get my interest payments. 00:05:54.766 --> 00:05:55.670 Right? 00:05:55.670 --> 00:05:58.220 And let's say if the interest rates, if I'm paying 7%, and 00:05:58.220 --> 00:06:00.710 the other million people are paying 7%, the investors are 00:06:00.710 --> 00:06:03.380 going to get 7% on their money. 00:06:03.380 --> 00:06:05.800 And that seems like a pretty reasonable proposition. 00:06:05.800 --> 00:06:08.960 And of course the investors would care that the money that 00:06:08.960 --> 00:06:10.950 they're essentially giving -- because they're giving money 00:06:10.950 --> 00:06:12.520 to the investment bankers who are giving money to 00:06:12.520 --> 00:06:13.250 Countrywide. 00:06:13.250 --> 00:06:16.100 And that's where my $1 million is essentially coming from. 00:06:16.100 --> 00:06:17.460 The only reason why the investors would give their 00:06:17.460 --> 00:06:21.490 money, is if they have a lot of confidence that these are 00:06:21.490 --> 00:06:23.580 really really good loans. 00:06:23.580 --> 00:06:25.150 Well the investors, they don't know who I am. 00:06:25.150 --> 00:06:27.745 They don't know what my job is, how likely I 00:06:27.745 --> 00:06:28.540 am to pay the loan. 00:06:28.540 --> 00:06:30.820 So the investors have to rely on someone to tell them that 00:06:30.820 --> 00:06:32.290 these are good loans. 00:06:32.290 --> 00:06:35.410 And that's where the rating agencies come in. 00:06:41.510 --> 00:06:45.170 And these are Standard & Poor's and Moody's. 00:06:45.170 --> 00:06:49.720 And they rate these assets, these mortgage-backed 00:06:49.720 --> 00:06:50.530 securities. 00:06:50.530 --> 00:06:53.100 And what they say is, well, they'll look at this big 00:06:53.100 --> 00:06:56.210 package of mortgages, these million mortgages that Bear 00:06:56.210 --> 00:06:57.300 Stearns has packaged together. 00:06:57.300 --> 00:06:59.430 And they'll look at the historical default rate. 00:06:59.430 --> 00:07:03.320 And they'll say, wow, these mortgages really haven't been 00:07:03.320 --> 00:07:04.050 defaulting. 00:07:04.050 --> 00:07:05.760 And you can think about why they haven't. 00:07:05.760 --> 00:07:07.590 Because housing prices been going up. 00:07:07.590 --> 00:07:10.060 So these mortgages really haven't been defaulting. 00:07:10.060 --> 00:07:11.770 There's a very high chance you're going to be able to get 00:07:11.770 --> 00:07:12.810 all your money back. 00:07:12.810 --> 00:07:14.870 So we're going to give these what they call, let's say they 00:07:14.870 --> 00:07:17.920 say AAA rating. 00:07:17.920 --> 00:07:20.730 So this investor, who knows, it could be the 00:07:20.730 --> 00:07:21.850 Central Bank of China. 00:07:21.850 --> 00:07:22.880 It could be a hedge fund. 00:07:22.880 --> 00:07:23.930 It could be a whole set of people. 00:07:23.930 --> 00:07:25.910 It might be the investment banks themselves. 00:07:25.910 --> 00:07:27.630 Sometimes they actually bought these just to 00:07:27.630 --> 00:07:29.090 make some extra money. 00:07:29.090 --> 00:07:31.270 These investors, they don't know who actually borrowed the 00:07:31.270 --> 00:07:34.330 money, or what kind of credit rating they had, or anything. 00:07:34.330 --> 00:07:36.170 But they just took a leap of faith. 00:07:36.170 --> 00:07:38.320 They said well, Standard & Poor's or 00:07:38.320 --> 00:07:39.650 Moody's did the work. 00:07:39.650 --> 00:07:43.320 They're telling me that this is AAA, which means the 00:07:43.320 --> 00:07:44.430 highest level of debt. 00:07:44.430 --> 00:07:45.490 Or you know, whatever they told them. 00:07:45.490 --> 00:07:47.140 Maybe it was A. 00:07:47.140 --> 00:07:49.860 I forget all the different qualities of debt. 00:07:49.860 --> 00:07:52.280 But they just took their word for it. 00:07:52.280 --> 00:07:55.525 And they got their 7% interest on their money-- whatever it 00:07:55.525 --> 00:07:57.000 was, 6% money. 00:07:57.000 --> 00:08:00.210 And that worked out pretty well. 00:08:00.210 --> 00:08:02.020 And so these guys, they liked the fact that they were 00:08:02.020 --> 00:08:02.790 getting the 7%. 00:08:02.790 --> 00:08:04.220 They said, this is a good asset class. 00:08:04.220 --> 00:08:06.150 So then they funneled even more money. 00:08:06.150 --> 00:08:10.680 So then there were even more investors that wanted do this. 00:08:10.680 --> 00:08:13.620 They're like, this is great, with very little risk I'm 00:08:13.620 --> 00:08:14.930 getting a pretty good return on my money. 00:08:14.930 --> 00:08:16.160 That's better than putting it in the bank. 00:08:16.160 --> 00:08:17.960 That's better than buying Treasury Bills, right? 00:08:17.960 --> 00:08:20.790 So then even more money flowed in. 00:08:20.790 --> 00:08:24.220 Well, more money wanted to invest in people's mortgages. 00:08:24.220 --> 00:08:27.130 But Countrywide would say, well, we're already giving 00:08:27.130 --> 00:08:29.300 mortgages to all the people who qualify. 00:08:29.300 --> 00:08:31.860 So in order to actually find more people who want mortgages 00:08:31.860 --> 00:08:33.480 from us, we'll just have to lower the 00:08:33.480 --> 00:08:34.770 standards a little bit. 00:08:34.770 --> 00:08:35.400 Right? 00:08:35.400 --> 00:08:37.080 And we can lower the standards, because we find 00:08:37.080 --> 00:08:39.130 even when we do lower the standards, no one's defaulting 00:08:39.130 --> 00:08:40.100 on their mortgages. 00:08:40.100 --> 00:08:42.350 And in the next video, I'll maybe give a little bit more 00:08:42.350 --> 00:08:42.610 [? color ?] 00:08:42.610 --> 00:08:43.250 why. 00:08:43.250 --> 00:08:46.190 So Countrywide will issue even more mortgages, and give them 00:08:46.190 --> 00:08:48.210 to these investors with even lower standards. 00:08:48.210 --> 00:08:50.050 Of course, the mortgage brokers at 00:08:50.050 --> 00:08:51.880 Countrywide, they love it. 00:08:51.880 --> 00:08:54.300 Because every time they do a transaction, they just get 00:08:54.300 --> 00:08:54.730 some money. 00:08:54.730 --> 00:08:56.800 And then they give the mortgage to the investment 00:08:56.800 --> 00:08:58.850 banker, which packages them up and then 00:08:58.850 --> 00:08:59.810 sells them to investors. 00:08:59.810 --> 00:09:01.490 So they get it off their hands. 00:09:01.490 --> 00:09:02.350 And they just got the fee. 00:09:02.350 --> 00:09:04.100 So they just collect the big cash. 00:09:04.100 --> 00:09:05.430 The investment banks love it, right? 00:09:05.430 --> 00:09:08.120 They just love doing the transactions, because they get 00:09:08.120 --> 00:09:10.380 more and more money every time they do the transactions. 00:09:10.380 --> 00:09:13.530 And for the moment, the investors seem pretty happy, 00:09:13.530 --> 00:09:17.960 because they keep giving money into this system, so to speak. 00:09:17.960 --> 00:09:19.840 Even though they might be reading the newspaper and 00:09:19.840 --> 00:09:21.450 seeing that the standards are going down. 00:09:21.450 --> 00:09:24.240 But they're consistently getting their return. 00:09:24.240 --> 00:09:27.575 And because the defaults were very low over this time period 00:09:27.575 --> 00:09:29.950 -- and I'll explain in the next video why the defaults 00:09:29.950 --> 00:09:33.520 were very low -- they felt that they were getting a good 00:09:33.520 --> 00:09:37.460 return, maybe 6% or 7%, on investments that had very, 00:09:37.460 --> 00:09:39.300 very low risk. 00:09:39.300 --> 00:09:44.680 So in the next video, I'll explain why the defaults were 00:09:44.680 --> 00:09:46.400 very low in that time period. 00:09:46.400 --> 00:09:48.030 See you soon.

Housing price conundrum (part 2)

https://www.youtube.com/watch?v=wYAhlTHIBT4

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https://www.youtube.com/api/timedtext?v=wYAhlTHIBT4&ei=YmeUZbr4Ks6sp-oP_YmruAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B690F91D1E05902650B82119098BA2C8AFB26DE2.C2F4A52B2454998EC2186EC69CF059B92FFADBBD&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.750 --> 00:00:03.880 Before I go a into an explanation of why housing 00:00:03.880 --> 00:00:08.340 prices skyrocketed from 2000 to 2006, I think it's a good 00:00:08.340 --> 00:00:11.250 idea to give a little history of what the housing market and 00:00:11.250 --> 00:00:14.130 the mortgage market used to be like before 00:00:14.130 --> 00:00:15.680 things got out of control. 00:00:15.680 --> 00:00:17.450 So let's go back to, say, I don't know. 00:00:17.450 --> 00:00:21.030 Let's go back to the late '70s-- 00:00:21.030 --> 00:00:22.840 maybe mid-'70s, actually. 00:00:22.840 --> 00:00:24.340 I remember my parents, they bought a house. 00:00:24.340 --> 00:00:25.330 We lived in New Orleans. 00:00:25.330 --> 00:00:27.390 And the house, if I remember correctly, it 00:00:27.390 --> 00:00:33.040 cost roughly $60,000. 00:00:33.040 --> 00:00:36.370 And back then, to buy a house -- and actually, for a while, 00:00:36.370 --> 00:00:38.390 until more recently -- in order to buy a house you had 00:00:38.390 --> 00:00:39.650 to put 25% down. 00:00:39.650 --> 00:00:42.430 So 25% of $60,000 is 1/4 of it. 00:00:42.430 --> 00:00:44.966 So you have to put $15,000 down. 00:00:44.966 --> 00:00:47.510 So you have to save up $15,000. 00:00:47.510 --> 00:00:50.660 And then you're going to get a mortgage on $45,000, right? 00:00:50.660 --> 00:00:52.300 $45,000, you're going to borrow. 00:00:52.300 --> 00:00:54.430 And I forgot the exact interest rates then, but I'm 00:00:54.430 --> 00:00:55.830 just going to throw out a number. 00:00:55.830 --> 00:00:58.090 This is really just for instructive purposes. 00:00:58.090 --> 00:00:59.920 Let's say interest rates back then, they were higher. 00:00:59.920 --> 00:01:02.190 They were like 9%, I think. 00:01:02.190 --> 00:01:04.260 So 9% on $45,000. 00:01:04.260 --> 00:01:05.900 How much interest am I going to pay? 00:01:05.900 --> 00:01:12.750 45,000 times 0.09. 00:01:12.750 --> 00:01:18.100 So I'm going to pay a little over $4,000 a year in 00:01:18.100 --> 00:01:30.240 interest. Or if I divide by 12, about $340 a month in 00:01:30.240 --> 00:01:31.320 interest. 00:01:31.320 --> 00:01:33.760 And I remember at the time, we actually moved out of our 00:01:33.760 --> 00:01:36.350 house and we rented it out, because we needed cash. 00:01:36.350 --> 00:01:39.280 And we rented out that exact same house -- and I this is in 00:01:39.280 --> 00:01:45.310 the late '70s or early '80s-- we rented out that exact same 00:01:45.310 --> 00:01:48.390 house for $900. 00:01:48.390 --> 00:01:54.680 The rent was $900. 00:01:54.680 --> 00:01:59.070 So this raises, I guess, a couple of questions. 00:01:59.070 --> 00:02:04.500 First of all, the big question is, why did those people who 00:02:04.500 --> 00:02:07.210 rented our house -- I mean, they paid $900 a month. 00:02:07.210 --> 00:02:10.289 They must have had a good income, for that time. 00:02:10.289 --> 00:02:12.970 Why were they willing to pay rent, when they could have 00:02:12.970 --> 00:02:15.380 bought a house, where the mortgage would have been -- 00:02:15.380 --> 00:02:16.840 interest plus a little principal -- it would have 00:02:16.840 --> 00:02:20.750 been no more than $400 a month? 00:02:20.750 --> 00:02:23.350 So why would you just throw away -- this is the classic 00:02:23.350 --> 00:02:26.640 rent-versus-buy argument -- why would you throw away $900, 00:02:26.640 --> 00:02:30.190 where you could actually build equity paying $400 a month for 00:02:30.190 --> 00:02:31.990 the exact same place? 00:02:31.990 --> 00:02:33.640 And you can think about that a little bit. 00:02:33.640 --> 00:02:35.510 But there's a bunch of reasons. 00:02:35.510 --> 00:02:38.550 What was necessary to buy a house then? 00:02:38.550 --> 00:02:43.180 Well, one, you needed a $15,000 down payment. 00:02:43.180 --> 00:02:46.100 Maybe these people had really good cash flow every month, 00:02:46.100 --> 00:02:50.210 but they just never had the circumstances, or maybe even 00:02:50.210 --> 00:02:53.740 the discipline, to save up $15,000. 00:02:53.740 --> 00:02:55.350 You also needed a really steady job. 00:02:55.350 --> 00:02:56.260 So you needed -- this is the down 00:02:56.260 --> 00:02:57.220 payment, this is one thing. 00:02:57.220 --> 00:02:58.470 You also needed a steady job. 00:03:01.330 --> 00:03:05.130 Maybe the people who were renting, they were working odd 00:03:05.130 --> 00:03:07.290 jobs, or they didn't have a steady income. 00:03:07.290 --> 00:03:07.850 Although I doubt it. 00:03:07.850 --> 00:03:09.750 I don't think we would have actually leased the house to 00:03:09.750 --> 00:03:10.910 them, had that been the case. 00:03:10.910 --> 00:03:12.160 They probably had that. 00:03:14.690 --> 00:03:16.510 And then the last thing you needed to get a mortgage, you 00:03:16.510 --> 00:03:17.800 needed good credit. 00:03:21.350 --> 00:03:22.840 And maybe these people didn't have that. 00:03:22.840 --> 00:03:25.500 Maybe they didn't pay some bills in the past. And they 00:03:25.500 --> 00:03:27.620 just couldn't find a bank that was willing to give them a 00:03:27.620 --> 00:03:30.990 loan, despite having a steady job and the $15,000 down. 00:03:30.990 --> 00:03:34.220 If you have to ask me, I think the biggest barrier for this 00:03:34.220 --> 00:03:36.080 family at that time was probably the 00:03:36.080 --> 00:03:37.830 $15,000 down payment. 00:03:37.830 --> 00:03:42.420 And frankly, they probably had trouble saving $15,000 because 00:03:42.420 --> 00:03:44.840 they were busy paying $900 in rent. 00:03:44.840 --> 00:03:48.400 So that was the circumstance throughout, actually, most of 00:03:48.400 --> 00:03:49.390 modern history. 00:03:49.390 --> 00:03:52.370 That you had this barrier towards buying a house. 00:03:52.370 --> 00:03:55.590 That it did make sense, that the conventional wisdom that 00:03:55.590 --> 00:03:58.650 it is better to buy than rent held. 00:03:58.650 --> 00:04:01.620 It's just, everyone knew that, but a lot of people just 00:04:01.620 --> 00:04:03.670 couldn't buy, even though they wanted to, because they didn't 00:04:03.670 --> 00:04:04.820 have the down payment. 00:04:04.820 --> 00:04:06.090 They didn't have the steady job. 00:04:06.090 --> 00:04:08.360 Or they didn't have the good credit. 00:04:08.360 --> 00:04:11.790 That was a circumstance then, and that lasted for some time. 00:04:11.790 --> 00:04:14.490 What happened in the early 2000s? 00:04:14.490 --> 00:04:16.480 And it actually happened in California in the mid-'90s. 00:04:16.480 --> 00:04:20.899 But it got more and more, I guess we could say, flagrant, 00:04:20.899 --> 00:04:23.380 as we went through the decade-- is that people 00:04:23.380 --> 00:04:24.800 started lowering these standards. 00:04:24.800 --> 00:04:27.120 And I'll do a whole other video on possibly why those 00:04:27.120 --> 00:04:28.330 standards were lowered. 00:04:28.330 --> 00:04:36.700 But let's say that in 1980 you needed 25% down. 00:04:36.700 --> 00:04:37.350 Let me just switch colors. 00:04:37.350 --> 00:04:39.130 That color is kind of ugly. 00:04:39.130 --> 00:04:40.380 You needed a steady job. 00:04:43.630 --> 00:04:45.200 And you needed a, I don't know. 00:04:45.200 --> 00:04:49.290 Let's say you needed a 700 credit score. 00:04:49.290 --> 00:04:51.620 And that was true from 1980 to, let's say, 2000. 00:04:51.620 --> 00:04:52.570 I'm exaggerating a bit. 00:04:52.570 --> 00:04:54.460 But this is just to give you the broad sense of what 00:04:54.460 --> 00:04:55.140 actually happened. 00:04:55.140 --> 00:04:58.310 But let's say then, in 2001 -- and I'll explain later why 00:04:58.310 --> 00:05:00.590 this might have happened -- the standards were lowered. 00:05:00.590 --> 00:05:02.780 That if you wanted to buy a house, all of a sudden you 00:05:02.780 --> 00:05:04.570 could actually find someone who was willing to give you a 00:05:04.570 --> 00:05:13.760 house for 10% down, maybe kind of a steady job, maybe just 00:05:13.760 --> 00:05:14.910 need a job. 00:05:14.910 --> 00:05:18.000 And maybe you had a 600 credit rating. 00:05:18.000 --> 00:05:20.650 So what happens when the standards on the mortgage go 00:05:20.650 --> 00:05:22.820 from this to this? 00:05:22.820 --> 00:05:24.900 Let's go back to these people who used to rent that house 00:05:24.900 --> 00:05:27.510 from us for $900. 00:05:27.510 --> 00:05:29.640 Maybe they didn't have $15,000, right? 00:05:29.640 --> 00:05:31.510 That would have been a 25% down payment. 00:05:31.510 --> 00:05:34.450 But maybe back then, they had 10%. 00:05:34.450 --> 00:05:35.810 Maybe they had $6,000. 00:05:35.810 --> 00:05:39.430 They just couldn't get up to $15,000 in savings. 00:05:39.430 --> 00:05:41.840 Back when they were doing this, back in the '80s, if the 00:05:41.840 --> 00:05:45.750 standards got a little bit freer, like they did in the 00:05:45.750 --> 00:05:48.390 early 2000s, those people could have bought a house. 00:05:48.390 --> 00:05:51.540 They would have said, man, we don't have to rent anymore. 00:05:51.540 --> 00:05:53.640 We saved up the 10% down payment. 00:05:53.640 --> 00:05:54.490 It's gotten a little easier. 00:05:54.490 --> 00:05:56.300 Our job now meets the requirements. 00:05:56.300 --> 00:05:57.820 Our credit now meets the requirements. 00:05:57.820 --> 00:05:59.250 We can go buy that house. 00:05:59.250 --> 00:06:00.720 So that would have increased the 00:06:00.720 --> 00:06:02.820 aggregate demand for housing. 00:06:02.820 --> 00:06:07.270 Even though, even if no one's incomes increased, even if the 00:06:07.270 --> 00:06:08.570 population didn't increase. 00:06:08.570 --> 00:06:11.370 All of a sudden, there's a new person who could get financing 00:06:11.370 --> 00:06:12.390 to buy a house. 00:06:12.390 --> 00:06:18.400 And then if we go to, let's say, 2003. 00:06:18.400 --> 00:06:20.290 They say, you know what, you don't even 00:06:20.290 --> 00:06:21.410 need any down payment. 00:06:21.410 --> 00:06:22.740 No down. 00:06:22.740 --> 00:06:24.260 No money down. 00:06:24.260 --> 00:06:27.200 So you can imagine, there's a whole set of people who maybe 00:06:27.200 --> 00:06:29.490 had a decent income, but they couldn't save any money. 00:06:29.490 --> 00:06:32.330 Now all of a sudden there was no down payment barrier to 00:06:32.330 --> 00:06:33.230 buying a house. 00:06:33.230 --> 00:06:35.785 Maybe you still needed a job. 00:06:35.785 --> 00:06:38.340 And maybe you just needed a 500 credit. 00:06:38.340 --> 00:06:38.880 Right? 00:06:38.880 --> 00:06:41.950 So all of a sudden, without people's incomes going up, 00:06:41.950 --> 00:06:46.000 without more jobs being available, without the 00:06:46.000 --> 00:06:48.230 population increase, there were more people 00:06:48.230 --> 00:06:49.870 who could get financing. 00:06:49.870 --> 00:06:52.200 Or more people who could bid up homes. 00:06:52.200 --> 00:06:54.010 And the situation actually got pretty bad. 00:06:54.010 --> 00:06:57.410 By 2004, 2005 -- and this isn't exact, but it gives you 00:06:57.410 --> 00:06:59.010 a sense of what happened. 00:06:59.010 --> 00:07:03.160 By 2004, 2005, you had a situation where they had these 00:07:03.160 --> 00:07:04.700 stated income -- they had these 00:07:04.700 --> 00:07:05.930 things called liar loans. 00:07:05.930 --> 00:07:07.610 Maybe I'll do videos on each of these. 00:07:07.610 --> 00:07:10.530 But these were essentially no down payment. 00:07:10.530 --> 00:07:13.260 If you had a job, you could kind of make it up. 00:07:13.260 --> 00:07:14.030 You just said, I have a job. 00:07:14.030 --> 00:07:15.060 They wouldn't validate it. 00:07:15.060 --> 00:07:17.140 These were stated income. 00:07:17.140 --> 00:07:18.970 You could just say what you made. 00:07:18.970 --> 00:07:22.530 So even though the mortgage might require an income of 00:07:22.530 --> 00:07:25.140 $10,000 a month, and your income is only $2,000 a month, 00:07:25.140 --> 00:07:27.300 you could say your income is $10,000 a month. 00:07:27.300 --> 00:07:33.065 So stated income, no down, maybe a job. 00:07:33.065 --> 00:07:35.260 And they didn't even do a credit check. 00:07:35.260 --> 00:07:39.660 So what happened from 2000 to 2004 is that credit just got 00:07:39.660 --> 00:07:41.320 easier and easier and easier. 00:07:41.320 --> 00:07:44.650 And every time credit got easier, there were more people 00:07:44.650 --> 00:07:46.620 who, despite the fact that they weren't making any more 00:07:46.620 --> 00:07:49.140 money, they were able to get financing. 00:07:49.140 --> 00:07:53.560 And so the pool of people who were able to bid on homes, or 00:07:53.560 --> 00:07:56.200 the demand for homes because now there was this financing, 00:07:56.200 --> 00:07:57.600 became larger and larger. 00:07:57.600 --> 00:08:00.630 And that's what increased the prices of homes. 00:08:00.630 --> 00:08:04.510 And now you know, the obvious question is, well, why did 00:08:04.510 --> 00:08:06.290 this happen? 00:08:06.290 --> 00:08:09.170 First of all, why did they get easier in 2001, get easier and 00:08:09.170 --> 00:08:10.510 easier as we went to 2004? 00:08:10.510 --> 00:08:14.510 And why did they get to this unbelievably absurd level, 00:08:14.510 --> 00:08:18.260 where by 2004 and 2005 -- you hear stories, especially in 00:08:18.260 --> 00:08:21.310 California and Florida, of people who were making maybe 00:08:21.310 --> 00:08:22.560 $40,000 a year. 00:08:22.560 --> 00:08:26.530 And they were able to buy houses with no money down. 00:08:26.530 --> 00:08:28.560 Some of these people were migrant laborers. 00:08:28.560 --> 00:08:32.260 And they were able to buy houses for $1 million. 00:08:32.260 --> 00:08:35.480 So in the next video, I will tell you why that happened. 00:08:35.480 --> 00:08:40.630 Why were people willing to give their cash to people to 00:08:40.630 --> 00:08:43.480 buy a house that had a very low likelihood of getting paid 00:08:43.480 --> 00:08:48.200 back, and for a house that had a very low likelihood of being 00:08:48.200 --> 00:08:50.750 able to retain its value. 00:08:50.750 --> 00:08:52.870 I'll see you in the next video.

The housing price conundrum

https://www.youtube.com/watch?v=8IR5LefXVPY

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https://www.youtube.com/api/timedtext?v=8IR5LefXVPY&ei=YmeUZeDpK-jMp-oPzcqSiAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=9A9F74DB849D19285D6C0BED06F8A37C62C486B6.60EA43E2C9E20627ED5DEA0AB8B9C695573F5B0B&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.960 --> 00:00:04.530 Until about 2006, if you talk to anyone, especially real 00:00:04.530 --> 00:00:07.150 estate agents, they'd always tell you that on average, 00:00:07.150 --> 00:00:10.660 nationwide, that housing always goes up in price. 00:00:10.660 --> 00:00:11.930 There could be layoffs. 00:00:11.930 --> 00:00:15.780 And maybe oil drops off and people have layoffs in Texas, 00:00:15.780 --> 00:00:17.210 so housing prices go down in Texas. 00:00:17.210 --> 00:00:20.250 Or they have layoffs in Michigan, so housing prices go 00:00:20.250 --> 00:00:20.710 down there. 00:00:20.710 --> 00:00:24.040 But nationwide, housing prices do nothing but go up. 00:00:24.040 --> 00:00:26.910 And that, for the most part, has been true since the Great 00:00:26.910 --> 00:00:27.410 Depression. 00:00:27.410 --> 00:00:30.980 Housing prices have been going up, maybe 1% or so per year. 00:00:30.980 --> 00:00:34.660 Actually a little bit less in real terms. But something 00:00:34.660 --> 00:00:38.810 fundamentally amazing happened in the beginning 00:00:38.810 --> 00:00:39.890 part of this decade. 00:00:39.890 --> 00:00:43.590 I have right here, this is the Case-Shiller index. 00:00:43.590 --> 00:00:47.300 And this is probably the best estimate of housing 00:00:47.300 --> 00:00:48.130 prices I can find. 00:00:48.130 --> 00:00:49.300 This is better than the median, because the 00:00:49.300 --> 00:00:53.300 Case-Shiller actually tries to compare the price you pay for 00:00:53.300 --> 00:00:54.200 the same house. 00:00:54.200 --> 00:00:56.090 And maybe I'll do another video later on how they 00:00:56.090 --> 00:00:57.210 exactly do that. 00:00:57.210 --> 00:00:59.350 But if we look at the Case-Shiller index. 00:00:59.350 --> 00:01:03.040 Let's see, in 2000 -- that's where they index it to -- a 00:01:03.040 --> 00:01:06.090 house that cost, you know, $100,000 in 2000. 00:01:06.090 --> 00:01:09.250 Or, the index was at $100,000 in 2000. 00:01:09.250 --> 00:01:14.310 By 2004 houses nationwide -- this is the national index 00:01:14.310 --> 00:01:19.290 right here -- nationwide, prices had increased by 46%. 00:01:19.290 --> 00:01:23.620 And by 2006, where they peak, they had increased by 88%. 00:01:23.620 --> 00:01:27.170 They had almost doubled since the price in 2000. 00:01:27.170 --> 00:01:30.450 And so the obvious question is, why did this happen? 00:01:30.450 --> 00:01:33.520 What drove prices to increase so fast? 00:01:33.520 --> 00:01:38.670 When really, for most of the history of America, housing 00:01:38.670 --> 00:01:42.830 prices have never increased this fast. Especially 00:01:42.830 --> 00:01:45.310 considering what was happening in the broader economy. 00:01:45.310 --> 00:01:46.010 What do I mean by that? 00:01:46.010 --> 00:01:48.720 Well for the price of anything to 00:01:48.720 --> 00:01:50.380 increase, what has to happen? 00:01:50.380 --> 00:01:53.110 Well the demand has to increase faster than the 00:01:53.110 --> 00:01:54.530 supply, right? 00:01:54.530 --> 00:01:56.170 So let's look at possible theories. 00:01:56.170 --> 00:01:58.960 What are demand drivers that could make 00:01:58.960 --> 00:02:00.100 housing prices go higher? 00:02:00.100 --> 00:02:02.290 Let me write that in green. 00:02:02.290 --> 00:02:04.950 Demand drivers. 00:02:04.950 --> 00:02:13.350 Well maybe the population grew faster than the housing stock? 00:02:13.350 --> 00:02:18.190 When I say the housing stock, I just mean that we're saying 00:02:18.190 --> 00:02:18.690 just demand. 00:02:18.690 --> 00:02:19.850 So let me just say population. 00:02:19.850 --> 00:02:21.200 Housing stock is supply. 00:02:21.200 --> 00:02:23.060 So population goes up. 00:02:23.060 --> 00:02:23.920 That's a demand driver. 00:02:23.920 --> 00:02:25.810 What's another demand driver? 00:02:25.810 --> 00:02:30.150 Incomes go up. 00:02:30.150 --> 00:02:30.400 Right? 00:02:30.400 --> 00:02:31.090 That's another reason. 00:02:31.090 --> 00:02:33.000 Maybe if a lot of people just become a lot richer, they're 00:02:33.000 --> 00:02:35.180 willing to pay for houses. 00:02:35.180 --> 00:02:38.160 And what are the supply drivers? 00:02:38.160 --> 00:02:40.850 Well these are just new homes built. 00:02:45.650 --> 00:02:50.640 So if you buy the classical supply-demand argument, why 00:02:50.640 --> 00:02:55.170 housing prices increased by 40% from 2000 to 2004, or why 00:02:55.170 --> 00:03:03.090 they increased by 80% from 2000 to 2006, these dynamics 00:03:03.090 --> 00:03:07.330 should have grown faster than these dynamics. 00:03:07.330 --> 00:03:10.380 So the population -- or maybe the total income, if you took 00:03:10.380 --> 00:03:12.130 the population and incomes-- grew faster than 00:03:12.130 --> 00:03:12.810 the new homes built. 00:03:12.810 --> 00:03:14.430 So let's see if that's true. 00:03:14.430 --> 00:03:16.720 So I found this New York Times article. 00:03:16.720 --> 00:03:19.600 And you could do some Google searches, and I'm sure you can 00:03:19.600 --> 00:03:21.670 find probably better data. 00:03:21.670 --> 00:03:26.470 This is just me doing a very fast search on this stuff. 00:03:26.470 --> 00:03:28.740 Let me see if I can get it up. 00:03:28.740 --> 00:03:31.510 OK, here it is. 00:03:31.510 --> 00:03:33.200 So this is from a New York Times article. 00:03:33.200 --> 00:03:34.610 This is a little graph. 00:03:34.610 --> 00:03:38.270 And this is showing the average of incomes reported on 00:03:38.270 --> 00:03:39.850 all tax returns. 00:03:39.850 --> 00:03:43.850 So notice, from 2000 to 2004 the average reported 00:03:43.850 --> 00:03:46.360 actually went down. 00:03:46.360 --> 00:03:49.290 It actually went down from 2000 to 2004. 00:03:49.290 --> 00:03:50.140 And this is interesting. 00:03:50.140 --> 00:03:52.860 Let me see if I can bring this in here. 00:03:52.860 --> 00:03:56.980 So here they say total reported income in 2004 00:03:56.980 --> 00:04:01.570 dollars -- so they adjusted for inflation -- fell 1.4%. 00:04:01.570 --> 00:04:04.910 But because the population grew during that period, 00:04:04.910 --> 00:04:08.330 average real incomes declined more than twice as much, 00:04:08.330 --> 00:04:12.650 falling by $1,641 a year, or 3%. 00:04:12.650 --> 00:04:13.680 So what are they saying? 00:04:13.680 --> 00:04:17.230 They're saying the total income fell by 1.4%, but the 00:04:17.230 --> 00:04:22.430 population must have grown by about 1.5%, and so the average 00:04:22.430 --> 00:04:23.900 per capita was 3%. 00:04:23.900 --> 00:04:26.100 So let me write that in summary. 00:04:26.100 --> 00:04:26.770 So what do we know? 00:04:26.770 --> 00:04:28.080 What happened? 00:04:28.080 --> 00:04:36.290 We know from 2000 to 2004 -- and this is nationwide -- we 00:04:36.290 --> 00:04:44.490 know that the population increased by roughly 1.5%. 00:04:50.220 --> 00:04:51.290 So not by much. 00:04:51.290 --> 00:04:52.750 I mean this is over a four-year period. 00:04:52.750 --> 00:04:57.200 So per year, it was growing by less than 1%. 00:04:57.200 --> 00:05:03.910 And then if you go to the income per person, or actually 00:05:03.910 --> 00:05:08.310 this is probably, well, this is income for tax filing. 00:05:08.310 --> 00:05:10.750 But that's a pretty good proxy. 00:05:10.750 --> 00:05:16.140 Income for tax filing, that declined by 3%. 00:05:16.140 --> 00:05:20.320 So the total money available, that New York Times article 00:05:20.320 --> 00:05:22.840 just showed us, actually declined. 00:05:22.840 --> 00:05:26.610 By, what did they say, by 1.4%. 00:05:26.610 --> 00:05:29.010 So the argument that somehow there's more money out there, 00:05:29.010 --> 00:05:32.010 chasing the same number of homes, or a slightly larger 00:05:32.010 --> 00:05:36.350 number of homes, doesn't really carry much weight. 00:05:36.350 --> 00:05:37.250 But let's just make sure. 00:05:37.250 --> 00:05:39.550 Maybe for some reason, maybe houses were destroyed. 00:05:39.550 --> 00:05:43.210 Or the number of homes built just didn't keep pace with 00:05:43.210 --> 00:05:44.450 this population increase. 00:05:44.450 --> 00:05:49.260 So let's see what data we can find on that. 00:05:49.260 --> 00:05:54.620 Well actually I found this thing. 00:05:54.620 --> 00:05:57.160 This says that -- this was in 1999 -- they say the 00:05:57.160 --> 00:06:00.550 composition of estimated 115 million housing units in the 00:06:00.550 --> 00:06:01.240 United States. 00:06:01.240 --> 00:06:05.170 So we can say, roughly, that in 2000 that there were 115 00:06:05.170 --> 00:06:06.433 million housing units. 00:06:15.060 --> 00:06:15.620 So let's see. 00:06:15.620 --> 00:06:18.300 Over this time period, roughly how many 00:06:18.300 --> 00:06:19.270 housing units were built? 00:06:19.270 --> 00:06:22.230 What percentage did the housing stock increase by? 00:06:22.230 --> 00:06:24.910 And I found this data here. 00:06:24.910 --> 00:06:28.840 And this is annualized new home builds by year. 00:06:28.840 --> 00:06:30.700 And I'm not going to go through all of the math. 00:06:30.700 --> 00:06:33.620 But if you see -- let's see, if I go back to 2000. 00:06:33.620 --> 00:06:35.740 I know this might be hard for you to see. 00:06:35.740 --> 00:06:40.570 But if we pick up pretty much any month from 2000, 2001. 00:06:40.570 --> 00:06:41.850 This is in thousands. 00:06:41.850 --> 00:06:46.280 So on an annualized basis, maybe 1.5 million homes. 00:06:46.280 --> 00:06:47.310 This was in 2000. 00:06:47.310 --> 00:06:50.950 But it started accelerating, all the way to 2004. 00:06:50.950 --> 00:06:52.900 By 2004, we were building roughly 2 00:06:52.900 --> 00:06:54.280 million homes a year. 00:06:54.280 --> 00:06:57.280 So over that time period, we can say, on average -- you can 00:06:57.280 --> 00:07:00.150 work the numbers to get an exact number, but it should 00:07:00.150 --> 00:07:02.410 work out -- we were building about 1.8 00:07:02.410 --> 00:07:03.800 million homes a year. 00:07:12.870 --> 00:07:16.550 And we can assume that homes destroyed were pretty 00:07:16.550 --> 00:07:17.700 negligible. 00:07:17.700 --> 00:07:19.610 I'm not aware of most neighborhoods where they were 00:07:19.610 --> 00:07:20.500 bulldozing homes. 00:07:20.500 --> 00:07:22.080 If anything, they were just renovating homes. 00:07:22.080 --> 00:07:23.650 But these are brand new homes. 00:07:23.650 --> 00:07:25.510 So over that four year period -- and I'm just going to focus 00:07:25.510 --> 00:07:27.400 there because that's where we got data from that New York 00:07:27.400 --> 00:07:29.330 Times article -- how many homes were built? 00:07:29.330 --> 00:07:32.945 Well, 1.8 times 4, that's what? 00:07:37.880 --> 00:07:41.850 So roughly 7.2 million homes, new homes, were built over 00:07:41.850 --> 00:07:42.910 that time period. 00:07:42.910 --> 00:07:45.140 And we started with a base of a 115 00:07:45.140 --> 00:07:50.710 million, roughly, in 2000. 00:07:50.710 --> 00:07:55.460 So over that time period, the housing stock increased by 6%. 00:07:55.460 --> 00:08:01.040 So the supply of homes went up by 6%. 00:08:01.040 --> 00:08:02.470 So what's going on here? 00:08:02.470 --> 00:08:06.510 From 2000 to 2004 we built a ton of houses. 00:08:06.510 --> 00:08:09.730 The supply of homes went up by 6%. 00:08:09.730 --> 00:08:11.310 People's incomes actually went down, because 00:08:11.310 --> 00:08:12.220 we were in a recession. 00:08:12.220 --> 00:08:14.150 People were getting laid off, or they were just willing to 00:08:14.150 --> 00:08:14.780 work for less. 00:08:14.780 --> 00:08:15.600 Income went down. 00:08:15.600 --> 00:08:17.320 And the population barely increased. 00:08:17.320 --> 00:08:19.470 And if we look at the total dollars that were being 00:08:19.470 --> 00:08:21.760 earned, that actually went down. 00:08:21.760 --> 00:08:25.590 So the actual money out there to pay for houses went down. 00:08:25.590 --> 00:08:27.110 And at the same time, the total number 00:08:27.110 --> 00:08:28.550 of houses went up. 00:08:28.550 --> 00:08:31.850 But at the same time, over this exact same period, the 00:08:31.850 --> 00:08:36.179 prices of houses went up by 46%. 00:08:36.179 --> 00:08:38.460 Or, I forgot the number, but it was 40-something percent. 00:08:38.460 --> 00:08:42.360 And it actually continued to race up until 2006, where it 00:08:42.360 --> 00:08:44.990 went up 80% relative to 2000. 00:08:44.990 --> 00:08:47.520 So this is bizarre. 00:08:50.240 --> 00:08:52.650 Basic economics would tell us that if the supply is 00:08:52.650 --> 00:08:57.610 increasing and the demand is decreasing, prices, if 00:08:57.610 --> 00:08:59.130 anything, should come down. 00:08:59.130 --> 00:09:00.520 So what happened? 00:09:00.520 --> 00:09:02.800 So I'm going to let you think about that a little bit. 00:09:02.800 --> 00:09:04.910 There you have the supply-demand thing that would 00:09:04.910 --> 00:09:05.840 tell you prices went down. 00:09:05.840 --> 00:09:08.590 But not only did they not go down, but they raced up faster 00:09:08.590 --> 00:09:11.240 than they've ever done in history, in the history of the 00:09:11.240 --> 00:09:12.250 United States. 00:09:12.250 --> 00:09:15.500 So in the next video I'm going to tell you, frankly, why I'm 00:09:15.500 --> 00:09:18.290 pretty sure housing prices did go up. 00:09:18.290 --> 00:09:19.710 See you soon.

What happens when housing depreciates

https://www.youtube.com/watch?v=QA2TBiIsdT0

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.510 --> 00:00:01.840 Welcome back. 00:00:01.840 --> 00:00:02.960 I now want to play a little bit of devil's 00:00:02.960 --> 00:00:04.190 advocate with myself. 00:00:04.190 --> 00:00:06.860 I made this argument where I show that for the exact 00:00:06.860 --> 00:00:08.590 identical house, if these are the numbers -- I mean you'd 00:00:08.590 --> 00:00:10.430 have to work it out based on your market, and what the 00:00:10.430 --> 00:00:11.530 numbers are at the time. 00:00:11.530 --> 00:00:14.600 But if this is the comparable rent for a $1 million house, I 00:00:14.600 --> 00:00:17.220 showed you that for the $1 million house you're burning 00:00:17.220 --> 00:00:19.070 $40,000 a year. 00:00:19.070 --> 00:00:21.110 This is not money that is going to build equity. 00:00:21.110 --> 00:00:23.400 This not money that's going to the principal of your house. 00:00:23.400 --> 00:00:25.390 This is money that just going out of your pocket, you'll 00:00:25.390 --> 00:00:26.320 never see again. 00:00:26.320 --> 00:00:29.130 In a way, and actually not in a way, in reality, you can 00:00:29.130 --> 00:00:33.390 view this $40,000 as rent on the money that you borrowed. 00:00:33.390 --> 00:00:35.460 Interest is nothing but rent. 00:00:35.460 --> 00:00:39.150 So when you have an asset, if the asset is cash, the rent on 00:00:39.150 --> 00:00:42.390 it is interest. If the asset is a house, the rent on it is 00:00:42.390 --> 00:00:44.120 your monthly rent payment. 00:00:44.120 --> 00:00:45.880 So when you think of it this way, when people say home 00:00:45.880 --> 00:00:48.010 ownership, they really aren't homeowners yet. 00:00:48.010 --> 00:00:51.000 You're not a homeowner until you don't have debt. 00:00:51.000 --> 00:00:54.870 You are a money renter. 00:00:54.870 --> 00:00:57.950 So your choice is either to be a money renter here, or to be 00:00:57.950 --> 00:00:59.220 a house renter here. 00:00:59.220 --> 00:01:01.200 And I show that you are burning 00:01:01.200 --> 00:01:02.590 almost double the money. 00:01:02.590 --> 00:01:06.530 But then there's the argument of well, there are advantages, 00:01:06.530 --> 00:01:07.830 still, to buying this house. 00:01:07.830 --> 00:01:09.090 And what are they? 00:01:09.090 --> 00:01:12.200 Well one example is, in this situation, if I did get a 00:01:12.200 --> 00:01:14.200 fixed-rate mortgage -- and we learned, when you look at all 00:01:14.200 --> 00:01:16.146 those adjustable-rate mortgages, we know that a lot 00:01:16.146 --> 00:01:17.040 of people didn't. 00:01:17.040 --> 00:01:19.550 But if I have a fixed-rate mortgage, I know what my 00:01:19.550 --> 00:01:22.920 payment is for the foreseeable future, for the next 30 years. 00:01:22.920 --> 00:01:26.470 While my landlord, in this case, they could 00:01:26.470 --> 00:01:27.960 keep raising my rent. 00:01:27.960 --> 00:01:33.260 So this might look good right now, but what if my landlord 00:01:33.260 --> 00:01:37.590 raised the rent to, I don't know, $3,500 a month. 00:01:37.590 --> 00:01:44.300 Well then, out of your pocket, 0.5 times 12, you'd be 00:01:44.300 --> 00:01:46.210 spending $42,000 a year. 00:01:46.210 --> 00:01:49.270 And then of course you get the interest from the money that 00:01:49.270 --> 00:01:50.050 you put in the bank. 00:01:50.050 --> 00:01:52.050 Plus 10. 00:01:52.050 --> 00:01:56.410 Oh, minus 10 actually, sorry. 00:01:56.410 --> 00:01:58.280 So in that case, if the rent goes up, then out of your 00:01:58.280 --> 00:02:00.250 pocket is $32,000 every year. 00:02:00.250 --> 00:02:00.910 Right? 00:02:00.910 --> 00:02:03.060 Or what if the interest that you get on your cash in the 00:02:03.060 --> 00:02:03.950 bank goes down? 00:02:03.950 --> 00:02:05.870 Then this $10,000 thousand will become lower. 00:02:05.870 --> 00:02:08.720 But as we can see, the rent would have to go up a lot to 00:02:08.720 --> 00:02:12.650 make up for $41,000, to make this a break-even situation. 00:02:12.650 --> 00:02:14.840 Let's figure out how much it would have to go up. 00:02:14.840 --> 00:02:20.130 So in this first scenario, in order for your net outflow to 00:02:20.130 --> 00:02:25.940 be $41,500, assuming you're getting $10,000 from the money 00:02:25.940 --> 00:02:31.560 in the bank, your rent would have to be $51,500. 00:02:31.560 --> 00:02:31.675 Right? 00:02:31.675 --> 00:02:35.130 Because you're getting $10,000 from the bank. 00:02:35.130 --> 00:02:45.400 And so divided by 12, your rent would have to be $4,300 00:02:45.400 --> 00:02:50.290 in this situation to make this a break-even proposition. 00:02:50.290 --> 00:02:51.180 This is another way to view it. 00:02:51.180 --> 00:02:53.540 If I were to buy the house, and if I were to move, how 00:02:53.540 --> 00:02:58.100 much would I have to rent this house out for, in order to not 00:02:58.100 --> 00:02:59.360 be losing money every month? 00:02:59.360 --> 00:03:02.750 Well I would have to rent it out for $4,300 a month, even 00:03:02.750 --> 00:03:06.130 though maybe the market rents are only at $3,000. 00:03:06.130 --> 00:03:09.440 And there is another devil's advocate argument. 00:03:09.440 --> 00:03:12.390 And that's, well, housing -- and this is something that you 00:03:12.390 --> 00:03:14.300 heard a lot about three years ago. 00:03:14.300 --> 00:03:16.730 And a lot of these people aren't talking as much now. 00:03:16.730 --> 00:03:19.780 But they would say, housing has never -- housing has done 00:03:19.780 --> 00:03:22.550 nothing but gone up, and I will build equity just from 00:03:22.550 --> 00:03:23.990 housing appreciation. 00:03:23.990 --> 00:03:27.190 So how much does my house have to appreciate every year? 00:03:27.190 --> 00:03:36.890 Well, to make up this difference-- $41,500 minus 00:03:36.890 --> 00:03:45.180 26-- so to make up that $15,500 difference every year, 00:03:45.180 --> 00:03:47.960 this is $15,500 favorable. 00:03:47.960 --> 00:03:50.360 My house would have to appreciate by a comparable 00:03:50.360 --> 00:03:51.570 amount, right? 00:03:51.570 --> 00:03:54.530 So how much appreciation is that on my house? 00:03:54.530 --> 00:03:56.270 Well that's a $1 million house, right? 00:03:56.270 --> 00:04:01.590 So $15,500 appreciation on a $1 million house. 00:04:01.590 --> 00:04:03.690 I'm doing everything in thousands, so 1,000 thousands 00:04:03.690 --> 00:04:04.705 is a million. 00:04:04.705 --> 00:04:07.760 So that's only 1.5% appreciation. 00:04:07.760 --> 00:04:12.450 So if my house appreciates by 1.5%, that's it-- 1.5%. 00:04:12.450 --> 00:04:17.440 If my house just appreciates by 1.5%, I'm going to make up 00:04:17.440 --> 00:04:18.899 this $15,500. 00:04:18.899 --> 00:04:21.019 And so it is worth it for me. 00:04:21.019 --> 00:04:24.110 It is worth it for me to blow this money by having kind of 00:04:24.110 --> 00:04:27.900 an increased -- by renting the money for more than I would 00:04:27.900 --> 00:04:29.970 have to pay to rent the house. 00:04:29.970 --> 00:04:32.330 And that might sound like a very reasonable proposition, 00:04:32.330 --> 00:04:34.780 that the house will appreciate by 1.5%. 00:04:34.780 --> 00:04:39.060 From 2001 to 2005, 2006, houses were appreciating like 00:04:39.060 --> 00:04:40.900 10%, 15% a year. 00:04:40.900 --> 00:04:44.350 So it seemed -- and a real estate agent would often do 00:04:44.350 --> 00:04:46.260 this very math with you, and say, well, you're definitely 00:04:46.260 --> 00:04:47.470 going to get 1.5%. 00:04:47.470 --> 00:04:49.410 In fact, you're probably going to get 10% appreciation. 00:04:49.410 --> 00:04:50.950 And you're going to make much more than this. 00:04:50.950 --> 00:04:53.590 But think about, in the presentation of the balance 00:04:53.590 --> 00:04:56.210 sheet and leverage, what happens if housing prices go 00:04:56.210 --> 00:04:58.290 down by 1.5%? 00:04:58.290 --> 00:05:01.240 What happens if it's minus 1.5%? 00:05:01.240 --> 00:05:04.740 Well, then you're going to spend this much to rent the 00:05:04.740 --> 00:05:06.390 money, right? 00:05:06.390 --> 00:05:07.870 And you're not going to gain this much. 00:05:07.870 --> 00:05:09.690 You're going to lose this much every year. 00:05:09.690 --> 00:05:12.220 And so the proposition becomes even worse. 00:05:12.220 --> 00:05:13.280 So this is a big deal. 00:05:13.280 --> 00:05:15.540 Now that, I think, on a nationwide basis, a lot of the 00:05:15.540 --> 00:05:18.470 housing indices show that housing prices have gone down, 00:05:18.470 --> 00:05:19.560 I think by 6%. 00:05:19.560 --> 00:05:22.280 That's what the Case-Shiller index says. 00:05:22.280 --> 00:05:23.600 6% is a lot. 00:05:23.600 --> 00:05:26.980 Especially on a $1 million house, that's $60,000 a year 00:05:26.980 --> 00:05:28.130 that's just evaporating. 00:05:28.130 --> 00:05:29.990 That's wealth that someone thought they had, that's just 00:05:29.990 --> 00:05:31.690 disappearing out of their equity. 00:05:31.690 --> 00:05:36.880 So this is rationale of pay more to rent the money for a 00:05:36.880 --> 00:05:40.140 house than to rent the house is justified if 00:05:40.140 --> 00:05:41.270 housing prices go up. 00:05:41.270 --> 00:05:44.610 It becomes 10 times worse when housing prices are flat. 00:05:44.610 --> 00:05:47.240 Or, God forbid, if housing prices actually go down. 00:05:47.240 --> 00:05:50.270 And now we see that housing prices actually go down. 00:05:50.270 --> 00:05:52.150 In the last couple of years especially, in the areas 00:05:52.150 --> 00:05:55.170 where, like the Bay Area, or Florida, or California, 00:05:55.170 --> 00:05:56.690 especially Southern California, 00:05:56.690 --> 00:05:57.760 where this is happening. 00:05:57.760 --> 00:06:00.560 And back even two or three years ago, when people used to 00:06:00.560 --> 00:06:02.020 make this argument. 00:06:02.020 --> 00:06:03.460 People used to make the argument, well you know, my 00:06:03.460 --> 00:06:06.290 house just has to go up 1% or 2% percent, and I'm going to 00:06:06.290 --> 00:06:07.500 make up the difference. 00:06:07.500 --> 00:06:09.000 I'd say well, why is your house going to 00:06:09.000 --> 00:06:11.430 go up 1% or 2% percent? 00:06:11.430 --> 00:06:13.750 I mean, there has to be some reason why next year someone's 00:06:13.750 --> 00:06:16.420 willing to pay 2% more for that house. 00:06:16.420 --> 00:06:18.880 Is it because rents are going up 2% a year, so the income 00:06:18.880 --> 00:06:20.650 stream is going to be 2% higher? 00:06:20.650 --> 00:06:25.150 And actually in the Bay Area, from 2001 to roughly 2003, 00:06:25.150 --> 00:06:26.440 rents were going down. 00:06:26.440 --> 00:06:27.760 And there were actually people moving out. 00:06:27.760 --> 00:06:29.610 All the tech workers were getting laid off. 00:06:29.610 --> 00:06:32.640 You had a lot of programming jobs being outsourced to India 00:06:32.640 --> 00:06:33.580 and wherever else. 00:06:33.580 --> 00:06:35.025 So you had this whole situation where the population 00:06:35.025 --> 00:06:36.570 was actually decreasing. 00:06:36.570 --> 00:06:39.500 Demand for housing was going down. 00:06:39.500 --> 00:06:42.280 But for some reason housing prices were going up. 00:06:42.280 --> 00:06:44.210 So people said well, they've been going up for the last 00:06:44.210 --> 00:06:45.260 five years, so they'll continue. 00:06:45.260 --> 00:06:46.760 And they've never gone down, et cetera, et cetera. 00:06:46.760 --> 00:06:49.450 But it didn't make an economic argument. 00:06:49.450 --> 00:06:51.960 And I'll show in a future video that the only reason why 00:06:51.960 --> 00:06:55.230 housing prices did go up is that it just became easier and 00:06:55.230 --> 00:06:59.300 easier and easier to buy a house. 00:06:59.300 --> 00:07:02.830 The standards that banks used for giving out a loan became 00:07:02.830 --> 00:07:03.730 lower and lower and lower. 00:07:03.730 --> 00:07:07.430 There are actually examples in Southern California, and in 00:07:07.430 --> 00:07:10.790 San Jose and some of the suburbs, where people who had 00:07:10.790 --> 00:07:12.930 incomes of $30,000 or $40,000 a year. 00:07:12.930 --> 00:07:16.360 The bank actually gave them a $1 million loan to buy a $1 00:07:16.360 --> 00:07:18.700 million house, based on stated income. 00:07:18.700 --> 00:07:20.900 There's things called stated income loans, where you just 00:07:20.900 --> 00:07:22.600 tell the bank what you earn. 00:07:22.600 --> 00:07:24.070 You don't have to prove it to them. 00:07:24.070 --> 00:07:26.540 And so every year that went by, it just became easier and 00:07:26.540 --> 00:07:27.560 easier and easier. 00:07:27.560 --> 00:07:29.910 More and more people just thought that housing always 00:07:29.910 --> 00:07:30.480 appreciates. 00:07:30.480 --> 00:07:33.610 So that's why they want to pay more and more to essentially 00:07:33.610 --> 00:07:35.380 rent the money for a house. 00:07:35.380 --> 00:07:37.360 And this became a self-fulfilling prophecy. 00:07:37.360 --> 00:07:39.720 But as we see on the way down, it works 00:07:39.720 --> 00:07:40.800 completely against you. 00:07:40.800 --> 00:07:43.240 So in the situation where we are now, where nationwide 00:07:43.240 --> 00:07:46.360 housing prices are actually declining-- and actually they 00:07:46.360 --> 00:07:49.530 will decline until this rent-versus-buy equation 00:07:49.530 --> 00:07:54.270 starts to make a little bit more sense-- it really hurts 00:07:54.270 --> 00:07:55.220 the home buyer. 00:07:55.220 --> 00:07:58.100 And what's even worse, and this is kind of adding insult 00:07:58.100 --> 00:08:03.330 to injury, is that this guy, if I bought this house, and 00:08:03.330 --> 00:08:06.250 all of a sudden I lose my job, and I can't pay the house 00:08:06.250 --> 00:08:10.260 back, I might lose my entire $250,000 down payment because 00:08:10.260 --> 00:08:11.750 maybe I can't sell the house, or the house 00:08:11.750 --> 00:08:13.470 is selling for less. 00:08:13.470 --> 00:08:15.770 Or maybe I want to move, and there's no one out there who 00:08:15.770 --> 00:08:19.070 can buy a house because the banks all of a sudden got 00:08:19.070 --> 00:08:20.930 smart again, and realized that they should become more 00:08:20.930 --> 00:08:23.170 serious in terms of who they give money to. 00:08:23.170 --> 00:08:26.170 And so I'm stuck holding this house, and my flexibility in 00:08:26.170 --> 00:08:27.950 terms of where I can move is limited. 00:08:27.950 --> 00:08:29.330 Actually a friend of mine was telling me that they've 00:08:29.330 --> 00:08:30.020 actually done studies. 00:08:30.020 --> 00:08:31.760 And there's a correlation between 00:08:31.760 --> 00:08:33.440 unemployment and home ownership. 00:08:33.440 --> 00:08:36.309 Because when you own a home, you have less flexibility in 00:08:36.309 --> 00:08:37.110 looking for a job. 00:08:37.110 --> 00:08:40.530 If I have a house in San Jose but there's a job in LA, I 00:08:40.530 --> 00:08:42.049 might not be able to take that job because I 00:08:42.049 --> 00:08:42.919 can't sell my house. 00:08:42.919 --> 00:08:44.910 Or I might not even want to look for a job in LA. 00:08:44.910 --> 00:08:49.340 While the renter, of course, my lease ends and I leave. 00:08:49.340 --> 00:08:52.870 So this is just a rough sense of the rent versus buy. 00:08:52.870 --> 00:08:54.490 And I know I get very impassioned about this. 00:08:54.490 --> 00:08:56.710 But that's just because I explain this a lot. 00:08:56.710 --> 00:08:59.850 And when I'm at parties and I start talking about the 00:08:59.850 --> 00:09:01.520 calculations, people's eyes glaze over. 00:09:01.520 --> 00:09:04.190 But I made this video now and I'll just tell 00:09:04.190 --> 00:09:06.010 people to watch it. 00:09:06.010 --> 00:09:07.950 See you in the next video.

Is buying a home always better?

https://www.youtube.com/watch?v=YL10H_EcB-E

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https://www.youtube.com/api/timedtext?v=YL10H_EcB-E&ei=YmeUZeHfK4u4vdIP9OCj8Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=34842F2EFCA8CA72B53D6C88E8424D4CA5AE0970.E6F2DA5F347C05D556F88DEA7EEE3FEF0D042B06&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.870 --> 00:00:01.660 Welcome back. 00:00:01.660 --> 00:00:03.960 I'm now going to take a slight tangent and cover a topic 00:00:03.960 --> 00:00:05.910 that, I think, this is probably the single most 00:00:05.910 --> 00:00:08.860 important video that really anyone can watch. 00:00:08.860 --> 00:00:11.080 I go to all of these parties where I go see family. 00:00:11.080 --> 00:00:13.980 And my wife and I right now, we live in Northern 00:00:13.980 --> 00:00:14.570 California. 00:00:14.570 --> 00:00:16.000 And we're renting. 00:00:16.000 --> 00:00:18.120 And I like to point out, by choice. 00:00:18.120 --> 00:00:20.360 And I have family members, why don't you buy? 00:00:20.360 --> 00:00:22.300 You're at that stage in life, that's a major 00:00:22.300 --> 00:00:23.150 milestone, all of this. 00:00:23.150 --> 00:00:24.380 There's a lot of pressure to buy. 00:00:24.380 --> 00:00:26.180 And when I tell friends, I tell them I'm 00:00:26.180 --> 00:00:26.720 not going to buy. 00:00:26.720 --> 00:00:31.450 Because I think I'm pretty convinced, almost 100% 00:00:31.450 --> 00:00:34.170 convinced, that housing prices are going to revert back. 00:00:34.170 --> 00:00:36.100 And I'm going to do a bunch of presentations to 00:00:36.100 --> 00:00:37.690 justify why they will. 00:00:37.690 --> 00:00:40.810 But then my friends, they'll just throw out the statement 00:00:40.810 --> 00:00:42.870 that I hear from them, that you hear from real estate 00:00:42.870 --> 00:00:45.030 agents, because obviously they want you to buy. 00:00:45.030 --> 00:00:48.850 Well, isn't buying always better than renting? 00:00:48.850 --> 00:00:51.530 And I think that kind of common wisdom comes out of the 00:00:51.530 --> 00:00:55.270 notion of, when you have a mortgage or when you borrow 00:00:55.270 --> 00:00:58.490 money to live in a house, every month that money that 00:00:58.490 --> 00:01:01.580 you give to the bank is kind of going into savings. 00:01:01.580 --> 00:01:02.760 That's the perception. 00:01:02.760 --> 00:01:04.590 While when you rent, that money's just 00:01:04.590 --> 00:01:06.160 disappearing into a vacuum. 00:01:06.160 --> 00:01:10.410 In this video I'm going to work through that assumption, 00:01:10.410 --> 00:01:13.360 and see if that actually is the case. 00:01:13.360 --> 00:01:15.050 So let's say I have a choice. 00:01:15.050 --> 00:01:16.300 Let's say there are two houses. 00:01:19.100 --> 00:01:20.940 This is house number one. 00:01:20.940 --> 00:01:23.870 And this is house number two. 00:01:23.870 --> 00:01:25.670 And let's say that they're identical houses. 00:01:25.670 --> 00:01:30.910 These are three bedroom, two bath, townhouses some place in 00:01:30.910 --> 00:01:32.960 Silicon Valley, which is where I live. 00:01:32.960 --> 00:01:36.090 And I want to live in one of these houses. 00:01:36.090 --> 00:01:37.970 I'm indifferent as to which house I live in, because they 00:01:37.970 --> 00:01:38.770 are identical. 00:01:38.770 --> 00:01:41.300 So living in them is the identical experience. 00:01:41.300 --> 00:01:54.320 I can rent this house for $3,000 a month. 00:01:54.320 --> 00:01:59.890 Or I could buy this house for $1 million. 00:01:59.890 --> 00:02:03.100 And let's say that in my bank account right now, let's say I 00:02:03.100 --> 00:02:06.970 have $250,000 cash. 00:02:06.970 --> 00:02:08.979 So let's see what happens in either scenario. 00:02:08.979 --> 00:02:13.470 Let's see how much money is being burned. 00:02:13.470 --> 00:02:16.760 So in this scenario what happens? 00:02:16.760 --> 00:02:17.400 I'm renting. 00:02:17.400 --> 00:02:20.060 So in a given year, let's just see how much money comes out 00:02:20.060 --> 00:02:20.910 of my pocket. 00:02:20.910 --> 00:02:23.790 So in a given year I pay $3,000. 00:02:23.790 --> 00:02:28.455 $3,000 times 12 months, so I lose $36,000. 00:02:28.455 --> 00:02:30.270 So I'll put a negative there, because that's 00:02:30.270 --> 00:02:31.460 what I spend in rent. 00:02:31.460 --> 00:02:37.550 $36,000 per year in rent. 00:02:37.550 --> 00:02:41.640 And then of course I have that $250,000. 00:02:41.640 --> 00:02:45.240 I'm going to put that into the bank, because I have nothing 00:02:45.240 --> 00:02:45.930 else to do with it. 00:02:45.930 --> 00:02:47.640 I didn't buy a house with it. 00:02:47.640 --> 00:02:50.480 And let's say that I can, in the bank, let's say 00:02:50.480 --> 00:02:51.210 I put it in a CD. 00:02:51.210 --> 00:02:52.800 And I get 4% on that. 00:02:52.800 --> 00:02:55.942 So let's see, 250, that's what? $10,000, I think. 00:02:55.942 --> 00:02:58.600 That's 0.04. 00:02:58.600 --> 00:03:01.800 Right, I get $10,000 in interest a year on that. 00:03:01.800 --> 00:03:02.890 So I get $10,000. 00:03:02.890 --> 00:03:07.840 So plus $10,000 a year in interest. 00:03:07.840 --> 00:03:11.890 So out of my pocket, for the privilege of living in this 00:03:11.890 --> 00:03:15.120 house, in Silicon Valley, with beautiful weather, out of my 00:03:15.120 --> 00:03:20.970 pocket every year goes $26,000. 00:03:20.970 --> 00:03:24.780 So that's scenario one. 00:03:24.780 --> 00:03:29.210 So what happens if I give in to the peer pressure of 00:03:29.210 --> 00:03:33.890 family, and realtors, and the mortgage industry, and I buy 00:03:33.890 --> 00:03:35.230 this house for $1 million? 00:03:35.230 --> 00:03:39.180 Well I only have $250,000, which is more, frankly, than 00:03:39.180 --> 00:03:42.110 most people who buy $1 million houses have. But I have 00:03:42.110 --> 00:03:45.330 $250,000 cash. 00:03:45.330 --> 00:03:47.550 So I need to borrow $750,000. 00:03:47.550 --> 00:03:57.970 So I take out a mortgage for $750,000. 00:03:57.970 --> 00:03:59.630 And I'm going to do a slight simplification. 00:03:59.630 --> 00:04:01.820 And maybe in a future presentation, I'll do kind of 00:04:01.820 --> 00:04:03.070 a more complicated one. 00:04:03.070 --> 00:04:05.830 In a lot of mortgages, when you pay your monthly payment, 00:04:05.830 --> 00:04:07.930 most of your monthly payment, at least initially, is the 00:04:07.930 --> 00:04:09.600 interest on the amount that you're borrowing. 00:04:09.600 --> 00:04:11.800 And you pay a little bit extra on that, to 00:04:11.800 --> 00:04:13.080 bring this value down. 00:04:13.080 --> 00:04:14.850 That's called paying off the principal. 00:04:14.850 --> 00:04:17.850 You can also take an interest-only loan, but the 00:04:17.850 --> 00:04:20.779 component of the interest is the same. 00:04:20.779 --> 00:04:22.710 Essentially, when you take a traditional mortgage, kind of 00:04:22.710 --> 00:04:25.640 a 30-year fixed, every month you're paying a little bit 00:04:25.640 --> 00:04:28.530 more than the interest, just to take down the balance. 00:04:28.530 --> 00:04:30.750 But for the simplicity of this argument, I'm just going to 00:04:30.750 --> 00:04:33.110 say that we're doing an interest-only mortgage. 00:04:33.110 --> 00:04:34.970 And then maybe with any extra savings, I can 00:04:34.970 --> 00:04:36.190 pay down the principal. 00:04:36.190 --> 00:04:37.440 And that's the same notion. 00:04:37.440 --> 00:04:40.090 And right now, if I do 25% down, and I'm buying a $1 00:04:40.090 --> 00:04:45.020 million house, I'll have to take a $750,000 mortgage. 00:04:45.020 --> 00:04:47.450 I don't know what a good rate is, 6%? 00:04:47.450 --> 00:04:55.040 So let's say at 6% interest. So to live in this house, how 00:04:55.040 --> 00:04:58.350 much am I paying just in interest? 00:04:58.350 --> 00:05:05.830 Well I'm paying $750,000 times 6% a year. 00:05:05.830 --> 00:05:18.510 So $750,000 times 0.06 is equal to $45,000 in interest. 00:05:18.510 --> 00:05:20.810 That's coming out of my pocket. 00:05:20.810 --> 00:05:23.240 And of course, on a monthly basis, that means in interest 00:05:23.240 --> 00:05:25.870 per month, I'm paying, just to get an idea. 00:05:25.870 --> 00:05:29.570 I'm paying about $3,700, $3,800 in interest a month. 00:05:29.570 --> 00:05:32.180 My mortgage actually might be something like $4,000 a month. 00:05:32.180 --> 00:05:36.150 So I pay the interest. And then I pay a little bit to 00:05:36.150 --> 00:05:39.480 chip away at the whole value of the loan. 00:05:39.480 --> 00:05:41.540 It takes 30 years to chip away at the whole thing. 00:05:41.540 --> 00:05:43.670 And over time, the interest component becomes less, and 00:05:43.670 --> 00:05:44.540 the principal becomes more. 00:05:44.540 --> 00:05:47.440 But for simplicity, this is the interest that I'm paying. 00:05:47.440 --> 00:05:49.420 $45,000 a year. 00:05:49.420 --> 00:05:51.570 And then of course at a party, when I start to explain this, 00:05:51.570 --> 00:05:52.740 it's like, ah-ha. 00:05:52.740 --> 00:05:56.220 But interest on a mortgage is tax deductible. 00:05:56.220 --> 00:05:59.750 And what tax deductible means, is that this amount of money 00:05:59.750 --> 00:06:03.400 that I spend on interest on my mortgage, I can 00:06:03.400 --> 00:06:04.870 deduct from my taxes. 00:06:04.870 --> 00:06:08.890 I can tell the IRS that I make $45,000 less 00:06:08.890 --> 00:06:10.860 than I actually did. 00:06:10.860 --> 00:06:14.700 So if I'm getting taxed at, let's say 30%, what is the 00:06:14.700 --> 00:06:16.060 actual cash savings? 00:06:16.060 --> 00:06:17.940 Well I'll save 30% of this. 00:06:17.940 --> 00:06:21.030 I'll have to pay $15,000 less in taxes. 00:06:21.030 --> 00:06:22.080 How does that work? 00:06:22.080 --> 00:06:22.720 Well, think about it. 00:06:22.720 --> 00:06:26.070 Let's say I earned $100,000 in a year. 00:06:26.070 --> 00:06:27.980 And I normally have to pay 30%. 00:06:27.980 --> 00:06:30.610 So I normally pay $30,000 in taxes. 00:06:30.610 --> 00:06:31.290 Right? 00:06:31.290 --> 00:06:33.800 This is, if I didn't have this great tax 00:06:33.800 --> 00:06:35.410 shelter with this house. 00:06:35.410 --> 00:06:37.720 Now I have this interest deduction. 00:06:37.720 --> 00:06:39.910 So now I tell the IRS that I'm actually 00:06:39.910 --> 00:06:46.800 making $55,000 a year. 00:06:46.800 --> 00:06:48.990 And let's say my tax rate is still 30%. 00:06:48.990 --> 00:06:51.420 it actually will probably go down since I'm -- but let's, 00:06:51.420 --> 00:06:54.280 just for simplicity, assume my tax rate is still $30,000. 00:06:54.280 --> 00:07:02.710 So now I'm going to pay $16,500 in taxes to the IRS. 00:07:02.710 --> 00:07:05.170 So how much did I save in taxes? 00:07:13.180 --> 00:07:18.640 So I saved $13,500 from taxes, from being able to deduct this 00:07:18.640 --> 00:07:20.030 $45,000 from my income. 00:07:20.030 --> 00:07:33.090 So let's say tax savings, plus $13,500. 00:07:33.090 --> 00:07:34.410 Now what else goes into this equation? 00:07:34.410 --> 00:07:36.290 Do I get any interest on my $250,000? 00:07:36.290 --> 00:07:36.650 Well, no. 00:07:36.650 --> 00:07:38.880 I had to use that as part of the down payment on my house. 00:07:38.880 --> 00:07:40.730 So I'm not getting interest there. 00:07:40.730 --> 00:07:42.390 But what I do have to do is, I have to 00:07:42.390 --> 00:07:46.160 pay taxes on my property. 00:07:46.160 --> 00:07:50.340 In California, out here we have to pay 1.25% in taxes, of 00:07:50.340 --> 00:07:51.890 the value of the house. 00:07:51.890 --> 00:07:52.910 So what's 1.25%? 00:07:52.910 --> 00:07:55.270 So, taxes, this is property tax. 00:07:55.270 --> 00:07:58.030 And that's actually tax deductible too, so it actually 00:07:58.030 --> 00:08:00.750 becomes more like 0.75% or 1%. 00:08:00.750 --> 00:08:03.540 So let's just say 1% just for simplicity. 00:08:03.540 --> 00:08:06.610 Property taxes. 00:08:06.610 --> 00:08:10.200 So 1% times $1 million. 00:08:10.200 --> 00:08:10.930 That equals what? 00:08:10.930 --> 00:08:14.100 1% of $1 million is another $10,000 a 00:08:14.100 --> 00:08:15.730 year in property taxes. 00:08:15.730 --> 00:08:18.310 And notice, I'm not talking about what percent of my 00:08:18.310 --> 00:08:19.490 mortgage goes to pay principal. 00:08:19.490 --> 00:08:21.980 I'm just talking about money that's being burned by owning 00:08:21.980 --> 00:08:23.210 this house. 00:08:23.210 --> 00:08:25.460 So what is the net effect? 00:08:25.460 --> 00:08:33.080 I have a $13,500 tax savings. 00:08:33.080 --> 00:08:35.049 I have to pay $10,000 -- actually I have to pay a 00:08:35.049 --> 00:08:37.080 little bit more than that, but we're getting a little bit of 00:08:37.080 --> 00:08:38.960 income tax savings on the deduction on 00:08:38.960 --> 00:08:40.900 the property taxes. 00:08:40.900 --> 00:08:45.170 And then I actually have to pay the $45,000 of interest 00:08:45.170 --> 00:08:46.270 that just goes out the door. 00:08:46.270 --> 00:08:52.320 So I'm paying $41,500. 00:08:52.320 --> 00:08:57.140 Notice, none of this $41,500 is building equity. 00:08:57.140 --> 00:08:58.560 None of it is getting saved. 00:08:58.560 --> 00:09:03.010 This is money that is just being burned. 00:09:03.010 --> 00:09:04.680 So this is a completely comparable 00:09:04.680 --> 00:09:06.970 value to this $26,000. 00:09:06.970 --> 00:09:09.400 So in this example -- this example is not that far off 00:09:09.400 --> 00:09:11.590 from real values. 00:09:11.590 --> 00:09:14.950 Out here in the Bay area, I can rent a $1 million house 00:09:14.950 --> 00:09:16.700 for about $3,000. 00:09:16.700 --> 00:09:21.020 But in this situation I am burning, every year $41,500, 00:09:21.020 --> 00:09:24.110 where I could just rent the same house for $26,000 out of 00:09:24.110 --> 00:09:26.360 my pocket, when I adjust for everything. 00:09:26.360 --> 00:09:29.140 And then people a couple of years ago said, oh, but houses 00:09:29.140 --> 00:09:29.750 appreciate. 00:09:29.750 --> 00:09:31.000 And that's what would make it up. 00:09:31.000 --> 00:09:32.940 But now you know, very recently -- we know that 00:09:32.940 --> 00:09:34.410 that's not the case. 00:09:34.410 --> 00:09:36.540 And in the next video, I'll delve into this, and 00:09:36.540 --> 00:09:37.480 a little bit more. 00:09:37.480 --> 00:09:39.080 I'll see you soon.

Housing equity loans

https://www.youtube.com/watch?v=7rrSuhFC7I0

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en

WEBVTT Kind: captions Language: en 00:00:00.740 --> 00:00:01.550 Welcome back. 00:00:01.550 --> 00:00:05.820 In the previous video we had this very positive scenario, 00:00:05.820 --> 00:00:08.770 where I had originally bought a house for $1.5 million. 00:00:08.770 --> 00:00:11.950 Then a year later, the value of the house, or at least my 00:00:11.950 --> 00:00:14.280 perceived value of the house, went up to $1.5 million, 00:00:14.280 --> 00:00:16.530 because my neighbors sold their identical 00:00:16.530 --> 00:00:19.210 house for $1.5 million. 00:00:19.210 --> 00:00:22.570 And so my initial equity investment went from $250,000 00:00:22.570 --> 00:00:23.500 to $750,000. 00:00:23.500 --> 00:00:24.110 And why is that? 00:00:24.110 --> 00:00:27.320 Well equity is nothing but, if I have an asset that's worth 00:00:27.320 --> 00:00:32.390 $1.5 million, and I owe $750,000-- that was my 00:00:32.390 --> 00:00:34.820 original mortgage on that asset-- then what I'm left 00:00:34.820 --> 00:00:35.480 with is the equity. 00:00:35.480 --> 00:00:36.810 So my equity just tripled. 00:00:36.810 --> 00:00:39.600 It went from $250,000 to $750,000. 00:00:39.600 --> 00:00:41.240 In this video, what I'm going to do is I'm going to show 00:00:41.240 --> 00:00:43.810 you, well, what can you do with that equity? 00:00:43.810 --> 00:00:44.950 I mean, it's not cash. 00:00:44.950 --> 00:00:48.000 It's kind of like this make believe amount of wealth that 00:00:48.000 --> 00:00:50.060 you have. You just feel richer. 00:00:50.060 --> 00:00:53.580 And I'll show you that you can actually turn it into cash 00:00:53.580 --> 00:00:56.220 using something called a home equity loan. 00:00:56.220 --> 00:00:59.350 And I'd argue that this is actually what drove our 00:00:59.350 --> 00:01:04.910 economy from about 2002 to probably still, to this day. 00:01:04.910 --> 00:01:06.520 Although I think we're in a recession now. 00:01:06.520 --> 00:01:08.840 In fact I'm about 100% sure we are. 00:01:08.840 --> 00:01:12.300 But definitely until about 2006. 00:01:12.300 --> 00:01:13.350 So what's a home equity loan? 00:01:13.350 --> 00:01:14.500 Well I go to the bank. 00:01:14.500 --> 00:01:18.100 I say, wow, bank, I have this $750,000 of equity. 00:01:18.100 --> 00:01:21.470 I wish -- I'm rich, but I don't have this in cash. 00:01:21.470 --> 00:01:23.540 I want to do something, though, with the equity. 00:01:23.540 --> 00:01:25.830 I would like to live like a rich person. 00:01:25.830 --> 00:01:28.940 Well the bank says, Sal, you know, you're right. 00:01:28.940 --> 00:01:33.970 Our only requirement is that you have $250,000-- or our 00:01:33.970 --> 00:01:36.260 only requirement is that you have 25% equity 00:01:36.260 --> 00:01:37.790 in your house, right? 00:01:37.790 --> 00:01:40.470 Because they want a cushion in case you can't pay and they 00:01:40.470 --> 00:01:42.870 get the house back, and they have to foreclose, and auction 00:01:42.870 --> 00:01:44.330 off the house, et cetera, et cetera. 00:01:44.330 --> 00:01:48.590 So they said, well, we're willing to lend you up to 75% 00:01:48.590 --> 00:01:50.220 of the value of your house. 00:01:50.220 --> 00:01:52.120 So what's 75% of the value of my house? 00:01:52.120 --> 00:01:58.390 So let's see, 1.5 times 75%, let's see that would be 00:01:58.390 --> 00:02:01.470 $750,000 plus half of $750,000. 00:02:01.470 --> 00:02:09.270 It'll be 1.075 million, I think. 00:02:09.270 --> 00:02:10.530 I did that in my head, it could be wrong. 00:02:10.530 --> 00:02:12.170 But it's roughly the right number. 00:02:12.170 --> 00:02:15.080 So the bank says, you know what, we're willing to lend 00:02:15.080 --> 00:02:18.820 you up to 75% of the value of your asset. 00:02:18.820 --> 00:02:22.400 And it's of course going to be guaranteed by this asset. 00:02:22.400 --> 00:02:25.850 So far, we lent you $750,000. 00:02:25.850 --> 00:02:28.660 So let's see how much you have more that you 00:02:28.660 --> 00:02:30.500 can borrow from us. 00:02:30.500 --> 00:02:34.420 Minus -- we're talking millions -- that's 0.075. 00:02:34.420 --> 00:02:34.940 So that's what? 00:02:34.940 --> 00:02:42.710 300, that's 250 plus 75, so up to $325,000 more that you 00:02:42.710 --> 00:02:44.380 could borrow. 00:02:44.380 --> 00:02:45.530 And what is this? 00:02:45.530 --> 00:02:47.180 Where am I taking this money out of? 00:02:47.180 --> 00:02:51.100 Well I'm essentially taking this money out of the equity 00:02:51.100 --> 00:02:51.860 of my house. 00:02:51.860 --> 00:02:53.190 And how does that make sense? 00:02:53.190 --> 00:02:54.440 Well, what's going to happen? 00:02:59.550 --> 00:03:00.560 Let's say I take this loan. 00:03:00.560 --> 00:03:02.380 Let's say I say, bank, great. 00:03:02.380 --> 00:03:08.180 I want $325,000 in cash. 00:03:08.180 --> 00:03:10.530 I want it right now. 00:03:10.530 --> 00:03:11.340 So what happens? 00:03:11.340 --> 00:03:16.990 Let me draw another series, another balance sheet. 00:03:16.990 --> 00:03:18.690 I stopped using the word balance sheet, even though 00:03:18.690 --> 00:03:21.690 that was the original purpose of this whole discussion. 00:03:21.690 --> 00:03:25.120 I'll draw it a little bit bigger. 00:03:25.120 --> 00:03:28.110 Remember liabilities plus equities are equal to assets. 00:03:32.500 --> 00:03:35.720 So what are my assets now? 00:03:35.720 --> 00:03:47.040 So now I have a $1.5 million house, and I also got $325,000 00:03:47.040 --> 00:03:55.280 cash from the bank, so we can call that 325K cash. 00:03:55.280 --> 00:03:56.860 Got it from the bank. 00:03:56.860 --> 00:03:58.640 Now what are my liabilities? 00:03:58.640 --> 00:04:01.050 Well I have the original mortgage on my house. 00:04:05.170 --> 00:04:10.080 The original mortgage is $750,000. 00:04:10.080 --> 00:04:12.680 This is liabilities on this side. 00:04:12.680 --> 00:04:13.696 Well not the whole side, we're going to 00:04:13.696 --> 00:04:14.390 have equity down here. 00:04:14.390 --> 00:04:16.500 So just this is liability, $750,000. 00:04:16.500 --> 00:04:21.060 And then I took a new loan to get this $325,000 of cash. 00:04:21.060 --> 00:04:27.480 So I have a new loan here, that amount is $325,000. 00:04:27.480 --> 00:04:30.150 And this was a home equity loan. 00:04:30.150 --> 00:04:32.690 I took a loan against the equity that 00:04:32.690 --> 00:04:33.420 I have in my house. 00:04:33.420 --> 00:04:35.030 This was the equity in my house. 00:04:35.030 --> 00:04:37.690 So what's the leftover equity? 00:04:37.690 --> 00:04:39.430 Let me just make everything clear. 00:04:39.430 --> 00:04:41.210 These are liabilities. 00:04:41.210 --> 00:04:42.442 These are assets. 00:04:42.442 --> 00:04:44.200 And equity is what you have leftover. 00:04:44.200 --> 00:04:45.640 So what are my assets? 00:04:45.640 --> 00:04:55.220 I have $1.825 million in assets, minus -- now what are 00:04:55.220 --> 00:04:56.440 my liabilities? 00:04:56.440 --> 00:05:05.230 Minus $1.075 -- that was the max that I could borrow -- 00:05:05.230 --> 00:05:06.110 liabilities. 00:05:06.110 --> 00:05:10.630 Assets minus liabilities is owners equity. 00:05:10.630 --> 00:05:15.710 So let's see, 825 minus 75. 00:05:15.710 --> 00:05:19.570 I still have $750,000 of equity. 00:05:19.570 --> 00:05:20.710 And that makes sense. 00:05:20.710 --> 00:05:23.500 If I just enter into some transaction where I get cash 00:05:23.500 --> 00:05:26.680 in exchange for debt, my equity shouldn't change. 00:05:26.680 --> 00:05:28.500 But now what does happen? 00:05:28.500 --> 00:05:31.960 Well I have this cash, and I'm feeling rich, because I've 00:05:31.960 --> 00:05:34.800 never seen numbers like $750,000. 00:05:34.800 --> 00:05:37.100 And that neighbor, that new neighbor that just bought that 00:05:37.100 --> 00:05:41.720 house right next door for $1.5 million, he just bought a 00:05:41.720 --> 00:05:44.060 beautiful new Hummer. 00:05:44.060 --> 00:05:48.990 And being a very down-to-earth person, I feel that I also 00:05:48.990 --> 00:05:51.660 deserve a Hummer, like my neighbor, because I am just as 00:05:51.660 --> 00:05:53.570 rich as they are. 00:05:53.570 --> 00:05:57.260 So I go decide to go out and I'm going to spend 00:05:57.260 --> 00:06:01.940 $100,000 on a Hummer. 00:06:01.940 --> 00:06:03.260 Actually, let's not do a Hummer, because a Hummer could 00:06:03.260 --> 00:06:04.890 actually be considered an asset. 00:06:04.890 --> 00:06:06.460 I want pure consumption. 00:06:06.460 --> 00:06:09.050 Although I think a Hummer is as pretty close as a car gets 00:06:09.050 --> 00:06:10.160 to pure consumption. 00:06:10.160 --> 00:06:13.430 Let's say that neighbor went on a round-the-world vacation 00:06:13.430 --> 00:06:15.050 for $100,000. 00:06:15.050 --> 00:06:17.830 And I too, because I did nothing but sit on my house 00:06:17.830 --> 00:06:21.710 and made $500,000 last year, I feel that I also deserve a 00:06:21.710 --> 00:06:23.730 $100,000 vacation. 00:06:23.730 --> 00:06:28.230 So what I do is I take $100,000 of this cash. 00:06:28.230 --> 00:06:33.930 So I'm now left with just $225,000, and I have the great 00:06:33.930 --> 00:06:35.940 experience of going on a vacation. 00:06:35.940 --> 00:06:37.980 But of course I didn't get any asset in return for that. 00:06:37.980 --> 00:06:40.820 Although maybe your happiness is an asset, I don't know. 00:06:40.820 --> 00:06:42.840 But it doesn't show up on your balance sheet. 00:06:42.840 --> 00:06:44.780 So we had $325,000 in cash. 00:06:44.780 --> 00:06:51.930 Now we have $225,000 in cash. 00:06:51.930 --> 00:06:54.410 So our total assets went down about $100,000. 00:06:54.410 --> 00:06:55.160 What are our assets now? 00:06:55.160 --> 00:06:58.550 It's $1.725 right? 00:06:58.550 --> 00:07:01.840 Because we spent $100,000 of our cash. 00:07:01.840 --> 00:07:03.780 So what's going to be the liabilities and equity? 00:07:03.780 --> 00:07:05.360 Well the liabilities won't change, right? 00:07:05.360 --> 00:07:06.885 Just because I went on vacation, the bank's not going 00:07:06.885 --> 00:07:08.890 to say, hey Sal, you owe us less money. 00:07:08.890 --> 00:07:13.300 I still owe the almost $1.075 million. 00:07:13.300 --> 00:07:17.380 The $100,000 is going to come all out of my equity. 00:07:17.380 --> 00:07:22.160 So now all of a sudden I don't have $750,000. 00:07:22.160 --> 00:07:28.170 I only have $650,000. 00:07:28.170 --> 00:07:30.060 And this isn't the balance sheet just for my house. 00:07:30.060 --> 00:07:32.960 This is kind of my whole personal balance sheet. 00:07:32.960 --> 00:07:34.255 And now my whole personal balance 00:07:34.255 --> 00:07:35.780 sheet, what just happened? 00:07:35.780 --> 00:07:39.440 I just took some of that original equity that I had. 00:07:39.440 --> 00:07:43.060 I took $100,000 of it, turned it into cash, and just went on 00:07:43.060 --> 00:07:45.610 a great one-year-long vacation. 00:07:45.610 --> 00:07:47.980 And this is what home equity loans are. 00:07:47.980 --> 00:07:51.840 And this is what, I would argue, drove the economy. 00:07:51.840 --> 00:07:54.310 Or at least took us into an expansionary stage 00:07:54.310 --> 00:07:56.560 from 2002 to 2003. 00:07:56.560 --> 00:07:59.150 Because if you remember, a lot of people were still getting 00:07:59.150 --> 00:08:01.780 laid off in 2002, 2003, but consumer 00:08:01.780 --> 00:08:02.910 spending kept going up. 00:08:02.910 --> 00:08:04.300 So people are earning less money, or they 00:08:04.300 --> 00:08:05.010 don't even have a job. 00:08:05.010 --> 00:08:06.330 How is spending going up? 00:08:06.330 --> 00:08:08.240 Well, the values of their house went up, and they 00:08:08.240 --> 00:08:10.190 borrowed against the value of their house. 00:08:10.190 --> 00:08:12.760 They took cash out of it, and they used that cash to buy 00:08:12.760 --> 00:08:18.240 their Hummers, to go on vacation, to buy fancy 00:08:18.240 --> 00:08:19.130 clothes, whatever. 00:08:19.130 --> 00:08:20.710 And that drove the economy. 00:08:20.710 --> 00:08:24.270 And in the next video I'll actually talk about, maybe, 00:08:24.270 --> 00:08:26.820 why those housing prices go up. 00:08:26.820 --> 00:08:29.590 Or why they went up, in particular, during this 00:08:29.590 --> 00:08:32.429 housing boom, this one that we're definitely in the 00:08:32.429 --> 00:08:34.340 process of getting out of. 00:08:34.340 --> 00:08:36.080 See you in the next video.

More on balance sheets and equity

https://www.youtube.com/watch?v=U2Nw5T44zvY

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en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:01.870 Welcome back. 00:00:01.870 --> 00:00:04.100 Where we left off in the last video, I had just purchased a 00:00:04.100 --> 00:00:05.290 $1 million house. 00:00:05.290 --> 00:00:08.220 To do it, I went to the bank and I said, bank, can you give 00:00:08.220 --> 00:00:09.910 me $750,000? 00:00:09.910 --> 00:00:13.020 They said, sure, Sal, you have an excellent credit rating, 00:00:13.020 --> 00:00:15.190 and you look like an all around great guy. 00:00:15.190 --> 00:00:17.420 So we'll give you $750,000. 00:00:17.420 --> 00:00:21.080 And so I took that $750,000 and the $250,000 that I had 00:00:21.080 --> 00:00:24.440 saved up through a lifetime of hard work, and I went and I 00:00:24.440 --> 00:00:25.540 bought that house. 00:00:25.540 --> 00:00:29.500 After that transaction, this is what my personal -- well, 00:00:29.500 --> 00:00:31.620 this might not involve everything, but it could be-- 00:00:31.620 --> 00:00:32.770 my personal balance sheet. 00:00:32.770 --> 00:00:34.660 But it looks like my whole world is this house. 00:00:34.660 --> 00:00:37.660 Which in a lot of cases, it is, for a lot of people. 00:00:37.660 --> 00:00:39.750 So in this situation, what are my assets? 00:00:39.750 --> 00:00:41.820 I have a $1 million house on my balance sheet. 00:00:41.820 --> 00:00:44.430 I have one asset in the world. 00:00:44.430 --> 00:00:47.210 I guess you can't quantify charisma and good looks. 00:00:47.210 --> 00:00:49.990 So the only real tangible asset I have is 00:00:49.990 --> 00:00:51.420 a $1 million house. 00:00:51.420 --> 00:00:53.050 And what are my liabilities? 00:00:53.050 --> 00:00:56.990 Well I owe $750,000 to the bank. 00:00:56.990 --> 00:01:00.030 And so we learned in the last video -- and you shouldn't 00:01:00.030 --> 00:01:01.460 view this as a formula. 00:01:01.460 --> 00:01:03.410 It should start to make a little bit of intuitive sense 00:01:03.410 --> 00:01:07.640 -- that assets are equal to liability plus equity. 00:01:07.640 --> 00:01:10.385 Or the other way to view it is, assets minus liabilities 00:01:10.385 --> 00:01:11.680 is equal to equity, right? 00:01:11.680 --> 00:01:14.000 Subtract the liability from both sides. 00:01:14.000 --> 00:01:17.190 And you know that if I have $1 million of assets, I owe 00:01:17.190 --> 00:01:20.530 $750,000, if I were to resolve everything, what I'd have left 00:01:20.530 --> 00:01:22.830 over at the end is $250,000. 00:01:22.830 --> 00:01:23.810 And I could make that happen. 00:01:23.810 --> 00:01:26.700 I could sell the house for $1 million, hopefully, and then 00:01:26.700 --> 00:01:27.430 pay the bank back. 00:01:27.430 --> 00:01:29.130 And I would have $250,000 left. 00:01:29.130 --> 00:01:31.220 So that's what equity is, just what you have left after you 00:01:31.220 --> 00:01:32.820 resolve everything. 00:01:32.820 --> 00:01:34.430 Or another way -- and this makes sense to you. 00:01:34.430 --> 00:01:36.710 If you talk about all the things you own minus all the 00:01:36.710 --> 00:01:38.040 things you owe to other people, equity 00:01:38.040 --> 00:01:39.040 is what's left over. 00:01:39.040 --> 00:01:40.490 Or that could be owner's equity. 00:01:40.490 --> 00:01:46.060 So now let's play with some scenarios of what happens, 00:01:46.060 --> 00:01:50.780 maybe, when the market value of the house changes. 00:01:50.780 --> 00:01:53.760 So let's say, what happens when -- oh, and one important 00:01:53.760 --> 00:01:57.280 thing to note, this bank, they're not just going to give 00:01:57.280 --> 00:02:01.080 me $750,000 just to do anything with it. 00:02:01.080 --> 00:02:04.610 They're not going to say, hey, Sal, here's $750,000. 00:02:04.610 --> 00:02:06.260 I know you'll pay it back to me, but you can go 00:02:06.260 --> 00:02:07.690 gamble it in Monaco. 00:02:07.690 --> 00:02:10.470 They want to know that they have a good chance of getting 00:02:10.470 --> 00:02:13.470 at least the money that they give, the loan amount, and 00:02:13.470 --> 00:02:16.100 that is often referred to as the principal. 00:02:16.100 --> 00:02:17.740 They want to know that they're going to be able to get that 00:02:17.740 --> 00:02:19.220 principal back one day. 00:02:19.220 --> 00:02:21.230 So what they say is, Sal, we're only going to give you 00:02:21.230 --> 00:02:24.260 this loan, but this loan has to be backed. 00:02:24.260 --> 00:02:27.960 Or it has to be collateralized by some asset. 00:02:27.960 --> 00:02:31.120 And so what I say is, OK, well, you know I'm taking this 00:02:31.120 --> 00:02:34.940 loan out to buy a house, a $1 million house. 00:02:34.940 --> 00:02:39.230 If for whatever reason, I lose my job, or I disappear 00:02:39.230 --> 00:02:41.850 somehow, or whatever happens. 00:02:41.850 --> 00:02:46.710 If I can't pay you the $750,000, you get the house. 00:02:46.710 --> 00:02:48.610 You'll get this $1 million house. 00:02:48.610 --> 00:02:50.800 And right now that looks like a pretty good deal to the 00:02:50.800 --> 00:02:51.710 bank, right? 00:02:51.710 --> 00:02:53.830 They almost hope that I'll default, because 00:02:53.830 --> 00:02:55.840 they gave me $750,000. 00:02:55.840 --> 00:02:58.680 If after a day I just say, you know what, bank, I can't pay 00:02:58.680 --> 00:03:01.330 this loan, I don't have the income, or I lost my job, I 00:03:01.330 --> 00:03:02.810 can't afford the mortgage. 00:03:02.810 --> 00:03:04.350 They get a $1 million house overnight. 00:03:04.350 --> 00:03:06.610 They would have made $250,000, right? 00:03:06.610 --> 00:03:09.700 They would have essentially gotten all my equity for free. 00:03:09.700 --> 00:03:12.220 So in that situation, the bank works out pretty good. 00:03:12.220 --> 00:03:14.620 And that's why they make sure that there's something that 00:03:14.620 --> 00:03:17.980 they can grab onto if you can't pay the loan. 00:03:17.980 --> 00:03:20.740 And that's why, back in the good old days, and I think the 00:03:20.740 --> 00:03:22.930 good old days are going to come back again, and I think 00:03:22.930 --> 00:03:25.590 they already are -- that the bank wants you to put some 00:03:25.590 --> 00:03:27.100 down payment in a house. 00:03:27.100 --> 00:03:29.500 Because there's a situation where, let's 00:03:29.500 --> 00:03:31.000 say that I do this. 00:03:31.000 --> 00:03:33.770 I borrow the money, and I buy the house. 00:03:33.770 --> 00:03:36.050 And I lose my job, or you know, whatever. 00:03:36.050 --> 00:03:38.850 I just drink away all of my money, whatever 00:03:38.850 --> 00:03:40.080 the case may be. 00:03:40.080 --> 00:03:42.210 And so the bank, they foreclose. 00:03:42.210 --> 00:03:46.020 Foreclose means that Sal isn't paying on his debt, so we're 00:03:46.020 --> 00:03:47.880 going to take the collateral back that he 00:03:47.880 --> 00:03:49.420 gave for the loan. 00:03:49.420 --> 00:03:52.080 So in that situation, the bank says, Sal can't pay, we're 00:03:52.080 --> 00:03:53.270 taking that house. 00:03:53.270 --> 00:03:55.090 Well when they take that house, there's a situation 00:03:55.090 --> 00:03:56.860 where maybe they're not going to get $1 00:03:56.860 --> 00:03:57.940 million for that house. 00:03:57.940 --> 00:04:00.850 They don't want to sit and wait for months and months and 00:04:00.850 --> 00:04:02.980 months while a real estate agent tries to sell it. 00:04:02.980 --> 00:04:05.410 So the bank might just auction off the house. 00:04:05.410 --> 00:04:08.090 And when it auctions off the house -- actually I think 00:04:08.090 --> 00:04:10.790 there are laws that it can't get more than the mortgage, or 00:04:10.790 --> 00:04:12.550 anything more than the mortgage it gets, it actually 00:04:12.550 --> 00:04:14.930 has to pay taxes, or -- we won't go into all of that. 00:04:14.930 --> 00:04:16.550 But it will auction off the house, and maybe it can only 00:04:16.550 --> 00:04:19.769 auction off the house for $800,000. 00:04:19.769 --> 00:04:20.399 Right? 00:04:20.399 --> 00:04:22.070 So the $1 million asset would really 00:04:22.070 --> 00:04:23.980 become an $800,000 asset. 00:04:23.980 --> 00:04:27.670 And so the bank keeps this equity cushion, right? 00:04:27.670 --> 00:04:32.350 That if they loan $750,000 for a $1 million house, and then 00:04:32.350 --> 00:04:35.260 the $1 million house only sells for $800,000, the bank 00:04:35.260 --> 00:04:37.190 still gets all of their money back. 00:04:37.190 --> 00:04:39.820 That's why, in the good old days, the banks wanted you to 00:04:39.820 --> 00:04:44.430 put 20% or 25% down, because they know even if the value of 00:04:44.430 --> 00:04:48.560 the house drops by 20% or 25%, it'll all 00:04:48.560 --> 00:04:50.690 come from your equity. 00:04:50.690 --> 00:04:53.810 And maybe I should draw a diagram to see that situation. 00:04:53.810 --> 00:04:59.680 Let's say that for whatever reason, I have to sell this 00:04:59.680 --> 00:05:00.840 house in a fire sale. 00:05:00.840 --> 00:05:04.190 Or let's say I can't sell the house and the bank is forcing 00:05:04.190 --> 00:05:05.790 me to liquidate my assets. 00:05:05.790 --> 00:05:08.210 The banks says well then, I want that house back. 00:05:08.210 --> 00:05:10.530 So in that situation -- well actually, that's not a good 00:05:10.530 --> 00:05:12.080 situation because the bank will just -- I'll 00:05:12.080 --> 00:05:13.110 just get wiped out. 00:05:13.110 --> 00:05:15.290 Let's just do the situation where let's say a neighbor's 00:05:15.290 --> 00:05:20.120 house sells for-- a neighbor's house that is identical. 00:05:20.120 --> 00:05:24.780 An identical neighbor's house, sells for $800,000, right? 00:05:24.780 --> 00:05:27.680 So in that situation, if I want to be honest with myself, 00:05:27.680 --> 00:05:29.360 and if I want to be honest with the balance sheet-- and 00:05:29.360 --> 00:05:32.340 actual real companies have to do this-- I'll say, you know 00:05:32.340 --> 00:05:35.450 what, this asset, I have to revalue it. 00:05:35.450 --> 00:05:38.840 I cannot in all honesty say that this is now worth, that 00:05:38.840 --> 00:05:41.030 this is a $1 million asset. 00:05:41.030 --> 00:05:42.860 So I would revalue the asset. 00:05:42.860 --> 00:05:45.000 And this is actually called marking to market. 00:05:45.000 --> 00:05:47.480 You probably heard of this concept. 00:05:47.480 --> 00:05:50.810 Marking to market means I have an asset, and every now and 00:05:50.810 --> 00:05:53.625 then, maybe every few months, every quarter -- a quarter is 00:05:53.625 --> 00:05:56.270 just a fourth of a year -- I have to figure out what that 00:05:56.270 --> 00:05:57.240 asset is worth. 00:05:57.240 --> 00:05:59.850 And the best way to figure out what that asset is worth is to 00:05:59.850 --> 00:06:02.000 see what identical assets like that are 00:06:02.000 --> 00:06:03.290 going for on the market. 00:06:03.290 --> 00:06:05.340 And very few houses are completely identical. 00:06:05.340 --> 00:06:07.630 Well there are, in a few suburbs. 00:06:07.630 --> 00:06:08.990 Very few assets are completely identical. 00:06:08.990 --> 00:06:11.290 But let's just say that I know for a fact that an identical 00:06:11.290 --> 00:06:13.730 house just sold for $800,000. 00:06:13.730 --> 00:06:16.210 So I have to be honest. And I have to mark it to market, and 00:06:16.210 --> 00:06:23.940 then say that my assets are now an $800,000 house. 00:06:23.940 --> 00:06:24.680 My same house. 00:06:24.680 --> 00:06:27.920 Nothing really happened, but the market value has dropped 00:06:27.920 --> 00:06:30.260 by $200,000 for whatever reason. 00:06:30.260 --> 00:06:35.520 Maybe the car factory nearby has gone out of business. 00:06:35.520 --> 00:06:36.860 So in this situation, what happens? 00:06:36.860 --> 00:06:38.440 What is my new balance sheet? 00:06:38.440 --> 00:06:41.490 Well has my liability changed, because my neighbor's house 00:06:41.490 --> 00:06:42.310 sold for less? 00:06:42.310 --> 00:06:45.650 Well, no, as far as the bank is concerned, I still owe 00:06:45.650 --> 00:06:51.030 $750,000 to the bank. 00:06:51.030 --> 00:06:52.440 This is a liability. 00:06:52.440 --> 00:06:54.220 I still owe $750,000. 00:06:54.220 --> 00:06:55.930 This is assets, of course. 00:06:55.930 --> 00:06:57.180 So what's leftover? 00:06:57.180 --> 00:06:59.240 What would be left over if I were to liquidate at the 00:06:59.240 --> 00:07:00.630 market price, if I were to sell the house 00:07:00.630 --> 00:07:01.950 at the market price? 00:07:01.950 --> 00:07:03.620 Well I would have $50,000 left over. 00:07:07.590 --> 00:07:11.090 Essentially when the market price of my asset dropped, all 00:07:11.090 --> 00:07:15.670 of that value came out of my equity. 00:07:15.670 --> 00:07:18.710 I'll do actually a whole other video on the benefits and the 00:07:18.710 --> 00:07:20.890 risks of leverage, because that's very relevant to what's 00:07:20.890 --> 00:07:23.200 happening in the world today. 00:07:23.200 --> 00:07:25.710 But I think you get a sense of what's happening. 00:07:25.710 --> 00:07:28.230 Equity kind of takes all of the risk. 00:07:28.230 --> 00:07:32.090 So in this situation, this is why the bank wants you to put 00:07:32.090 --> 00:07:33.020 some down payment. 00:07:33.020 --> 00:07:36.490 Because the bank, if you can't pay this loan right here, 00:07:36.490 --> 00:07:37.770 they're going to take your house. 00:07:37.770 --> 00:07:40.270 And even in the situation where the value of the house 00:07:40.270 --> 00:07:44.000 went down, if you can't pay the loan, the bank will still 00:07:44.000 --> 00:07:45.940 be able to get its $750,000, right? 00:07:45.940 --> 00:07:50.390 If you just leave town, or lose your job, and you just 00:07:50.390 --> 00:07:52.060 tell the bank I can't pay anymore, they're just going to 00:07:52.060 --> 00:07:54.540 take this house, sell it, hopefully for $800,000, 00:07:54.540 --> 00:07:56.260 because that's what your neighbor sold it for. 00:07:56.260 --> 00:07:58.020 And they're going to get the money back for their loan. 00:07:58.020 --> 00:08:01.840 So that's why the bank wants you to put some down payment. 00:08:01.840 --> 00:08:04.210 And then there's the other situation, which is maybe a 00:08:04.210 --> 00:08:05.180 more positive situation. 00:08:05.180 --> 00:08:08.120 And this is what happened in much of the world, and 00:08:08.120 --> 00:08:12.440 especially in areas like California and Florida and 00:08:12.440 --> 00:08:15.580 Nevada over the last five years or so. 00:08:15.580 --> 00:08:19.060 And I'll do a whole video on why it happened. 00:08:19.060 --> 00:08:20.950 But let's say your neighbor's house, a year later, didn't 00:08:20.950 --> 00:08:22.180 sell for $800,000. 00:08:22.180 --> 00:08:24.250 Let's say the identical neighbor's house 00:08:24.250 --> 00:08:27.150 sold for $1.5 million. 00:08:27.150 --> 00:08:28.230 And you say, gee whiz. 00:08:28.230 --> 00:08:29.250 That's great. 00:08:29.250 --> 00:08:32.020 Now my house is also worth $1.5 million because I'm 00:08:32.020 --> 00:08:34.020 marking to market. 00:08:34.020 --> 00:08:36.150 So now my asset -- nothing has really changed. 00:08:36.150 --> 00:08:37.110 It's still the same house. 00:08:37.110 --> 00:08:39.669 But I guess, since someone else sold it for $1.5 million, 00:08:39.669 --> 00:08:40.909 I guess I could, too. 00:08:40.909 --> 00:08:45.180 So my asset is now a $1.5 million house. 00:08:45.180 --> 00:08:45.920 What are my liabilities? 00:08:45.920 --> 00:08:48.320 Well your liabilities still haven't changed. 00:08:48.320 --> 00:08:55.810 I still owe $750,000 to bank. 00:08:55.810 --> 00:08:57.410 This is liabilities. 00:08:57.410 --> 00:08:58.280 So what's left over? 00:08:58.280 --> 00:08:59.530 What's my equity? 00:09:01.610 --> 00:09:03.200 Well, assets minus liability. 00:09:03.200 --> 00:09:06.860 So I have $750,000 of equity. 00:09:06.860 --> 00:09:08.200 That's awesome. 00:09:08.200 --> 00:09:10.820 Even though the house appreciated by 50%, right? 00:09:10.820 --> 00:09:14.990 It went from $1 million to $1.5 million, my equity grew 00:09:14.990 --> 00:09:15.900 three-fold. 00:09:15.900 --> 00:09:21.020 It appreciated by 200%. 00:09:21.020 --> 00:09:23.650 I think you're starting to get the benefits of what happens 00:09:23.650 --> 00:09:24.410 when you do leverage. 00:09:24.410 --> 00:09:27.750 Leverage is when you use debt to buy an asset. 00:09:27.750 --> 00:09:31.840 But when you use leverage, the return that you get on your 00:09:31.840 --> 00:09:34.550 asset gets multiplied when you get the return on your equity. 00:09:34.550 --> 00:09:36.070 I hope I'm not confusing you. 00:09:36.070 --> 00:09:37.150 But in this situation, all of a sudden I 00:09:37.150 --> 00:09:38.610 have a ton of equity. 00:09:38.610 --> 00:09:40.230 And I'm running out of time. 00:09:40.230 --> 00:09:41.760 But in the next video I'm going to talk 00:09:41.760 --> 00:09:42.820 about how this happened. 00:09:42.820 --> 00:09:44.200 Because you saw it in a lot of neighborhoods. 00:09:44.200 --> 00:09:50.370 A lot of houses appreciated from about 2001 to 2005. 00:09:50.370 --> 00:09:52.830 And people, all of a sudden, just sitting on their house, 00:09:52.830 --> 00:09:54.130 ended up with a lot of equity. 00:09:54.130 --> 00:09:57.380 And they felt that, wow, I just went from having $250,000 00:09:57.380 --> 00:10:00.050 of net wealth to $750,000 of wealth, 00:10:00.050 --> 00:10:00.900 without doing anything. 00:10:00.900 --> 00:10:02.940 Just by my neighbor's house selling for more. 00:10:02.940 --> 00:10:04.190 I'll see you in the next video.

Introduction to Balance Sheets

https://www.youtube.com/watch?v=mxsYHiDVNlk

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en

WEBVTT Kind: captions Language: en 00:00:00.760 --> 00:00:01.480 Welcome. 00:00:01.480 --> 00:00:04.160 Well there's been a lot of news lately about what's going 00:00:04.160 --> 00:00:07.850 on with Bear Stearns and Carlisle Capital. 00:00:07.850 --> 00:00:10.080 And I go to these parties, and I start explaining to people 00:00:10.080 --> 00:00:11.000 because it's very exciting. 00:00:11.000 --> 00:00:13.520 It's actually very important, to all of our collective 00:00:13.520 --> 00:00:15.890 futures and the whole health of the financial system, and I 00:00:15.890 --> 00:00:18.790 feel like people's eyes start to glaze over. 00:00:18.790 --> 00:00:21.330 So with that in mind, I decided to take a little bit 00:00:21.330 --> 00:00:23.820 of a hiatus from the core math and physics videos, and 00:00:23.820 --> 00:00:26.250 actually do some accounting and finance videos. 00:00:26.250 --> 00:00:27.700 Because I think what's happening in the world right 00:00:27.700 --> 00:00:30.560 now is extremely important. 00:00:30.560 --> 00:00:33.090 And I'm not just going to go straight into what's going 00:00:33.090 --> 00:00:37.770 into Carlisle and Thornburg and all of these characters. 00:00:37.770 --> 00:00:39.900 Because I think the newspapers do that, but a lot of people 00:00:39.900 --> 00:00:41.650 don't understand the basic accounting. 00:00:41.650 --> 00:00:43.490 What is a write-down, what does it mean when you don't 00:00:43.490 --> 00:00:45.630 have liquidity, in really tangible ways. 00:00:45.630 --> 00:00:48.500 So I'm going to use the same Khan Academy techniques to 00:00:48.500 --> 00:00:50.130 hopefully explain some of this. 00:00:50.130 --> 00:00:54.080 So I'm going to start with just a very basic accounting 00:00:54.080 --> 00:00:55.470 concept of the balance sheet. 00:00:55.470 --> 00:00:57.210 You might have a sense of what it is. 00:00:57.210 --> 00:00:57.980 So let's say a scenario. 00:00:57.980 --> 00:00:59.060 Let's say I want to buy a house. 00:00:59.060 --> 00:01:01.780 So this is, let me draw a house. 00:01:01.780 --> 00:01:05.650 So let's say this is the house I want to buy. 00:01:05.650 --> 00:01:07.540 And the owner of this house is asking for $1 00:01:07.540 --> 00:01:08.790 million for this house. 00:01:08.790 --> 00:01:11.420 And I like the house, and I think that's a fair price. 00:01:11.420 --> 00:01:13.092 Other houses in the neighborhood also went for $1 00:01:13.092 --> 00:01:14.180 million, whatever. 00:01:14.180 --> 00:01:15.230 Maybe they went for more, so I think it's 00:01:15.230 --> 00:01:16.620 actually a good deal. 00:01:16.620 --> 00:01:30.320 But all I have in my pocket is, let's say I have $250,000. 00:01:30.320 --> 00:01:33.680 So what I'm going to do is, I'm going to create my balance 00:01:33.680 --> 00:01:35.520 sheet before I do anything. 00:01:35.520 --> 00:01:36.720 Before I go to try to get the house. 00:01:36.720 --> 00:01:40.170 What is my before-house balance sheet? 00:01:40.170 --> 00:01:42.290 What are my assets? 00:01:42.290 --> 00:01:43.540 I'm going to write down Assets. 00:01:48.070 --> 00:01:49.700 Well before we know what my assets are, let me tell you 00:01:49.700 --> 00:01:50.510 what an asset is. 00:01:50.510 --> 00:01:55.040 An asset is something that's going to give you some future 00:01:55.040 --> 00:01:56.200 economic benefit. 00:01:56.200 --> 00:01:58.250 So for example, cash is an asset. 00:01:58.250 --> 00:01:59.220 Why is cash an asset? 00:01:59.220 --> 00:02:03.720 Because in the future you can use that cash to get stuff 00:02:03.720 --> 00:02:06.170 from people, or make them do things, or buy stuff. 00:02:06.170 --> 00:02:08.570 You can, in a month from now, you can use your cash. 00:02:08.570 --> 00:02:10.620 And you can make someone dance for you. 00:02:10.620 --> 00:02:14.290 Or you can buy a car, or you can go on vacation. 00:02:14.290 --> 00:02:16.430 So there's all sorts of things you can do. 00:02:16.430 --> 00:02:18.010 I don't know if someone dancing for you is an actual 00:02:18.010 --> 00:02:20.260 economic benefit, but you get the idea. 00:02:20.260 --> 00:02:21.820 So cash could be an asset. 00:02:21.820 --> 00:02:25.020 A house could be an asset, because the economic benefit 00:02:25.020 --> 00:02:26.730 you get in the future is, you get to live in it, and not 00:02:26.730 --> 00:02:29.190 freeze when it's freezing outside. 00:02:29.190 --> 00:02:30.690 So that's what an asset is. 00:02:30.690 --> 00:02:33.760 So what are my assets, before I buy the house, or get a 00:02:33.760 --> 00:02:35.650 loan, or all of the things that are about to happen? 00:02:35.650 --> 00:02:43.550 Well I have cash, I have $250,000 worth of cash. 00:02:43.550 --> 00:02:47.000 What are my liabilities? 00:02:47.000 --> 00:02:47.980 I'm going to write the liabilities on 00:02:47.980 --> 00:02:50.470 the left-hand side. 00:02:50.470 --> 00:02:52.180 I think that's the convention, but I forget. 00:02:52.180 --> 00:02:53.460 It doesn't matter. 00:02:53.460 --> 00:02:55.270 What are my liabilities? 00:02:55.270 --> 00:03:00.320 Well, a liability is something that's an economic obligation 00:03:00.320 --> 00:03:01.270 to someone else. 00:03:01.270 --> 00:03:04.370 So if I take a loan from someone, I owe them interest, 00:03:04.370 --> 00:03:06.610 or I have to pay them back the actual value of 00:03:06.610 --> 00:03:07.790 the loan one day. 00:03:07.790 --> 00:03:10.990 Say I have an IOU where I promise to dance for someone 00:03:10.990 --> 00:03:11.880 in the future. 00:03:11.880 --> 00:03:13.250 That could be a liability. 00:03:13.250 --> 00:03:15.140 It'd be hard to value, but that's something that I have 00:03:15.140 --> 00:03:16.550 to do in the future. 00:03:16.550 --> 00:03:17.690 But what are my liabilities here? 00:03:17.690 --> 00:03:21.420 Well in the example I gave, I'm just Sal, I have no debt, 00:03:21.420 --> 00:03:22.860 I paid off my college loans, everything. 00:03:22.860 --> 00:03:26.850 And I have $250,000 in cash. 00:03:26.850 --> 00:03:29.440 So what are my liabilities before I buy the house? 00:03:29.440 --> 00:03:30.290 Well, nothing. 00:03:30.290 --> 00:03:31.810 I don't have any liabilities. 00:03:31.810 --> 00:03:33.260 I don't owe anybody anything. 00:03:33.260 --> 00:03:35.320 And that's, actually, that to me is the 00:03:35.320 --> 00:03:37.360 definition of freedom. 00:03:37.360 --> 00:03:40.580 So I have zero liability. 00:03:40.580 --> 00:03:43.630 So what is my equity? 00:03:43.630 --> 00:03:45.630 And you've probably heard this word, people borrowing their 00:03:45.630 --> 00:03:47.010 equity, and all of these things. 00:03:47.010 --> 00:03:48.510 So I'm going to give you a little equation, actually, 00:03:48.510 --> 00:03:50.340 just to take a little bit of a tangent. 00:03:50.340 --> 00:03:56.020 That assets, A for assets, is equal to 00:03:56.020 --> 00:04:00.900 liabilities plus equity. 00:04:00.900 --> 00:04:05.890 So in this case, our assets are $250,000. 00:04:05.890 --> 00:04:06.810 My liabilities are what? 00:04:06.810 --> 00:04:10.130 I owe nothing to nobody. 00:04:10.130 --> 00:04:11.510 I don't know if that was correct, but anyway. 00:04:11.510 --> 00:04:13.200 I owe nothing to anyone. 00:04:13.200 --> 00:04:15.270 So my liabilities are zero. 00:04:15.270 --> 00:04:23.000 So my equity must be $250,000. 00:04:23.000 --> 00:04:25.450 So in this case, if I made a balance sheet before I enter 00:04:25.450 --> 00:04:27.430 into any transactions -- let me make it look a little bit 00:04:27.430 --> 00:04:30.380 like a balance sheet. 00:04:30.380 --> 00:04:32.100 My assets are $250,000. 00:04:32.100 --> 00:04:33.630 I have no liabilities. 00:04:33.630 --> 00:04:40.610 And then my equity would be $250,000. 00:04:40.610 --> 00:04:42.380 And if I were to draw this graphically-- actually, I 00:04:42.380 --> 00:04:43.210 should probably draw it like this. 00:04:43.210 --> 00:04:43.770 I have no liabilities. 00:04:43.770 --> 00:04:47.180 So let me draw another little mini balance sheet here. 00:04:47.180 --> 00:04:48.700 That's a neat square. 00:04:48.700 --> 00:04:50.200 You probably can't see that square. 00:04:53.940 --> 00:04:57.270 So I put my assets on the right-hand side. 00:04:57.270 --> 00:05:00.370 And I'll say, there, I have $250,000 of cash. 00:05:00.370 --> 00:05:02.100 And on the left-hand side, I have no liabilities. 00:05:02.100 --> 00:05:06.520 And I'll just say I have equity, I have $250,000. 00:05:06.520 --> 00:05:08.510 Now, equity might not make a lot of sense to you right now, 00:05:08.510 --> 00:05:11.010 because I'm just saying, well, my equity is equal to my cash. 00:05:11.010 --> 00:05:14.060 in general, equity is just what you own. 00:05:14.060 --> 00:05:16.890 After all of your assets and liabilities are kind of 00:05:16.890 --> 00:05:19.190 resolved, or they're cleared up, what do 00:05:19.190 --> 00:05:20.410 you have left over? 00:05:20.410 --> 00:05:21.270 That's equity. 00:05:21.270 --> 00:05:24.020 So in this situation, after I pay off all of my debts, what 00:05:24.020 --> 00:05:24.860 do I have left over? 00:05:24.860 --> 00:05:28.080 Well I have no debts, so I have $250,000 in cash, total. 00:05:28.080 --> 00:05:31.020 This will start to make sense when I go to the bank now to 00:05:31.020 --> 00:05:33.060 get a loan to buy this house. 00:05:33.060 --> 00:05:36.190 So this house is a $1 million house, right? 00:05:36.190 --> 00:05:37.770 So how much of a loan do I need? 00:05:37.770 --> 00:05:41.780 Well, I have $250,000 cash, so I'll go to the bank for a loan 00:05:41.780 --> 00:05:44.490 for the remainder, for $750,000. 00:05:44.490 --> 00:05:49.650 So let me draw the bank. 00:05:49.650 --> 00:05:50.650 This is the bank. 00:05:50.650 --> 00:05:53.610 The big dollar sign is made out of granite, to show you 00:05:53.610 --> 00:05:54.690 that it can never fail. 00:05:54.690 --> 00:05:57.700 It's going to be there forever, even if they do silly 00:05:57.700 --> 00:06:00.890 things, like-- well I won't go into all the silly things that 00:06:00.890 --> 00:06:02.410 they do, but they do many silly things. 00:06:02.410 --> 00:06:03.570 We'll go into that later. 00:06:03.570 --> 00:06:12.380 But the bank is going to give me another $750,000 in cash. 00:06:12.380 --> 00:06:17.700 And in return, I'm giving them essentially an IOU. 00:06:17.700 --> 00:06:20.090 And I'm going to pay interest. So they're going to hold this 00:06:20.090 --> 00:06:23.960 little security that says, Sal owes me $750,000. 00:06:23.960 --> 00:06:27.290 And he has to give me 10% interest every year. 00:06:27.290 --> 00:06:29.570 So $75,000 a year, or something like that. 00:06:29.570 --> 00:06:32.280 And in return I get $750,000 in cash. 00:06:32.280 --> 00:06:34.870 So what does my balance sheet look like now? 00:06:34.870 --> 00:06:36.780 Well, let me draw it. 00:06:36.780 --> 00:06:41.130 Let me make sure my balance sheet now looks, let me draw 00:06:41.130 --> 00:06:42.680 it like a square, because I think the visual 00:06:42.680 --> 00:06:51.000 representation is helpful, and then I will split it. 00:06:51.000 --> 00:06:53.060 So what are all my assets now? 00:06:53.060 --> 00:06:56.010 I had $250,000 and I got another 00:06:56.010 --> 00:07:00.260 $750,000 from the bank. 00:07:00.260 --> 00:07:01.480 So now, what are my assets? 00:07:01.480 --> 00:07:03.740 Well, $250,000 plus $750,000. 00:07:03.740 --> 00:07:06.530 I now have cash of $1 million. 00:07:10.060 --> 00:07:11.820 What are my liabilities? 00:07:11.820 --> 00:07:15.460 Well, my liability, that's something that I owe to 00:07:15.460 --> 00:07:16.000 someone else. 00:07:16.000 --> 00:07:19.700 I owe the bank $750,000. 00:07:19.700 --> 00:07:22.125 So liabilities, I'll just say L, L for liabilities, because 00:07:22.125 --> 00:07:23.560 I'm running out of space. 00:07:23.560 --> 00:07:26.240 My wife was complaining that I make these things very hard to 00:07:26.240 --> 00:07:28.060 read, but what can I do. 00:07:28.060 --> 00:07:28.390 Anyway. 00:07:28.390 --> 00:07:31.130 So my liabilities-- I owe the bank $750,000. 00:07:31.130 --> 00:07:33.700 So that's a liability. 00:07:33.700 --> 00:07:37.140 And then the equity is, essentially-- we would look at 00:07:37.140 --> 00:07:37.890 this formula. 00:07:37.890 --> 00:07:39.480 Assets equal liabilities plus equity. 00:07:39.480 --> 00:07:41.170 This is $1 million, this is $750,000. 00:07:41.170 --> 00:07:42.500 What do I have left over? 00:07:42.500 --> 00:07:44.670 Well, I have $250,000 left over. 00:07:44.670 --> 00:07:46.970 That's my equity. 00:07:46.970 --> 00:07:50.820 And I think hopefully the concept of equity is starting 00:07:50.820 --> 00:07:53.300 to make a little more sense. 00:07:53.300 --> 00:07:58.320 Now we have-- I could say that I have $1 million, and some 00:07:58.320 --> 00:07:59.120 people are like that. 00:07:59.120 --> 00:08:00.760 They think they're millionaires when they have $1 00:08:00.760 --> 00:08:02.000 million in assets. 00:08:02.000 --> 00:08:04.580 But they don't consider, well they might have $1 million of 00:08:04.580 --> 00:08:07.050 assets, but they might owe other people $900,000. 00:08:07.050 --> 00:08:09.040 So I wouldn't consider that person a millionaire. 00:08:09.040 --> 00:08:11.690 They're more of a hundred thousand-aire. 00:08:11.690 --> 00:08:14.140 Your assets might be $1 million, but you're not nearly 00:08:14.140 --> 00:08:15.620 a millionaire, because you still owe 00:08:15.620 --> 00:08:17.810 other people $750,000. 00:08:17.810 --> 00:08:21.490 What you have left over, that really is your net worth, or 00:08:21.490 --> 00:08:23.220 what you can have claim to. 00:08:23.220 --> 00:08:24.190 And that's your equity. 00:08:24.190 --> 00:08:25.810 Sometimes it's called owners' equity. 00:08:25.810 --> 00:08:27.676 Or if there was a bunch of people pitching together, it 00:08:27.676 --> 00:08:29.030 would be called shareholders' equity. 00:08:29.030 --> 00:08:31.960 And maybe I'll do a little bit more on that in the future. 00:08:31.960 --> 00:08:34.760 But hopefully now you can see that the balance sheet is 00:08:34.760 --> 00:08:38.330 starting to seem a little bit useful. 00:08:38.330 --> 00:08:42.620 I have the cash, and I took the loan from the bank, but 00:08:42.620 --> 00:08:44.410 now I still haven't bought the house yet. 00:08:44.410 --> 00:08:45.430 So what am I going to do? 00:08:45.430 --> 00:08:49.490 Well I'm going to give my cash to the old owner of the house. 00:08:49.490 --> 00:08:52.450 Or maybe this Toll Brothers, they just built this 00:08:52.450 --> 00:08:53.960 McMansion for me. 00:08:53.960 --> 00:08:58.060 So I give them $1 million, and in return they give me the 00:08:58.060 --> 00:08:59.140 deed to the house. 00:08:59.140 --> 00:09:01.140 I could just say they give me the house. 00:09:01.140 --> 00:09:03.270 The house is always there, but you know it's really just a 00:09:03.270 --> 00:09:05.440 contract and all the legal structure that I get around 00:09:05.440 --> 00:09:07.500 it, and all the property rights and all of that. 00:09:07.500 --> 00:09:09.910 But that's getting too philosophical. 00:09:09.910 --> 00:09:12.390 So now what does my balance sheet look like? 00:09:12.390 --> 00:09:15.530 Instead of cash-- I think I'm running out of space and time 00:09:15.530 --> 00:09:17.850 to draw another balance sheet-- I don't have cash 00:09:17.850 --> 00:09:19.460 worth $1 million. 00:09:19.460 --> 00:09:21.740 I now have a house worth $1 million. 00:09:21.740 --> 00:09:24.200 Assuming that it really is worth it, and that was the 00:09:24.200 --> 00:09:26.420 correct price, I didn't overpay, whatever. 00:09:26.420 --> 00:09:30.120 I now have, my assets are a $1 million house. 00:09:30.120 --> 00:09:33.500 And I owe the bank $750,000. 00:09:33.500 --> 00:09:39.000 So what's left over for me is $250,000 of equity. 00:09:39.000 --> 00:09:40.260 I'm about to run out of time. 00:09:40.260 --> 00:09:42.280 So I'm going to leave you from this video. 00:09:42.280 --> 00:09:43.830 In the next video, I'm going to start explaining what 00:09:43.830 --> 00:09:46.330 happens if the value of the house goes up or down, or you 00:09:46.330 --> 00:09:48.420 need cash, and all of these interesting things. 00:09:48.420 --> 00:09:51.240 And we'll start to learn a little bit more about what's 00:09:51.240 --> 00:09:52.210 going on in the world. 00:09:52.210 --> 00:09:53.640 See you soon.

Introduction to torque

https://www.youtube.com/watch?v=QhuJn8YBtmg

vtt

https://www.youtube.com/api/timedtext?v=QhuJn8YBtmg&ei=YmeUZdzXK4zwvdIPipmhiAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=3C1EC92E4B96F74F1EFB40B77E4EE408E0AFC218.19AD31CCBD163F5A17BB76B25BC542CDD891E259&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.880 --> 00:00:03.300 Welcome to the presentation on torque. 00:00:03.300 --> 00:00:06.060 So, if you watched the presentation on the center of 00:00:06.060 --> 00:00:08.690 mass, which you should have, you might have gotten a little 00:00:08.690 --> 00:00:11.850 bit of a glancing view of what torque is. 00:00:11.850 --> 00:00:14.350 And now we'll do some more in detail. 00:00:14.350 --> 00:00:19.680 So in general, from the center of mass video, we learned, if 00:00:19.680 --> 00:00:23.745 this is a ruler and this is the ruler's center of mass. 00:00:26.330 --> 00:00:32.259 And if I were to apply force at the center of mass, I would 00:00:32.259 --> 00:00:35.820 accelerate the whole ruler in the direction of the force. 00:00:35.820 --> 00:00:37.810 If I have the force applying at the center of mass there, 00:00:37.810 --> 00:00:41.240 the whole ruler would accelerate in that direction. 00:00:41.240 --> 00:00:43.510 And we'd figure it out by taking the force we're 00:00:43.510 --> 00:00:46.000 applying to it and dividing by the mass of the ruler. 00:00:46.000 --> 00:00:49.020 And in that center of mass video, I imply-- well, what 00:00:49.020 --> 00:00:52.040 happens if the force is applied here? 00:00:52.040 --> 00:00:54.050 Away from the center of mass? 00:00:54.050 --> 00:00:56.960 Well, in this situation, the object, assuming it's a free 00:00:56.960 --> 00:00:59.750 floating object on the Space Shuttle or something, it will 00:00:59.750 --> 00:01:02.450 rotate around the center of mass. 00:01:02.450 --> 00:01:06.410 And that's also true, if we didn't use the center of mass, 00:01:06.410 --> 00:01:07.810 but instead we fixed the point. 00:01:07.810 --> 00:01:14.110 Let's say we had another ruler. 00:01:14.110 --> 00:01:17.310 Although it has less height than the previous one. 00:01:17.310 --> 00:01:19.160 Instead of worrying about its center of mass, let's say that 00:01:19.160 --> 00:01:22.930 it's just fixed at a point here. 00:01:22.930 --> 00:01:23.980 Let's say it's fixed here. 00:01:23.980 --> 00:01:28.450 So if this could be the hand of a clock, and it's nailed 00:01:28.450 --> 00:01:30.510 down to the back of the clock right there. 00:01:30.510 --> 00:01:33.160 So if we were trying to rotate it, it would always rotate 00:01:33.160 --> 00:01:34.170 around this point. 00:01:34.170 --> 00:01:35.800 And the same thing would happen. 00:01:35.800 --> 00:01:38.830 If I were to apply a force at this point, maybe I could 00:01:38.830 --> 00:01:41.550 break the nail off the back of the clock, or something, but I 00:01:41.550 --> 00:01:44.830 won't rotate this needle or this ruler, or whatever you 00:01:44.830 --> 00:01:45.770 want to call it. 00:01:45.770 --> 00:01:51.440 But if I would apply a force here, I would rotate the ruler 00:01:51.440 --> 00:01:53.440 around the pivot point. 00:01:53.440 --> 00:01:58.010 And this force that's applied a distance away from the pivot 00:01:58.010 --> 00:02:01.080 point, or we could say from the axis of rotation, or the 00:02:01.080 --> 00:02:02.190 center of mass. 00:02:02.190 --> 00:02:03.770 That's called torque. 00:02:03.770 --> 00:02:08.949 And torque, the letter for torque is this Greek, I think 00:02:08.949 --> 00:02:11.860 that's tau, it's a curvy T. 00:02:11.860 --> 00:02:17.640 And torque is defined as force times distance. 00:02:17.640 --> 00:02:19.560 And what force and what distance is it? 00:02:19.560 --> 00:02:25.030 It's the force that's perpendicular to the object. 00:02:25.030 --> 00:02:26.960 I guess you could say to the distance vector. 00:02:26.960 --> 00:02:29.170 If this is the distance vector-- let me do it in a 00:02:29.170 --> 00:02:31.110 different color. 00:02:31.110 --> 00:02:37.790 If this is the distance vector, the component of the 00:02:37.790 --> 00:02:40.750 force is perpendicular to this distance vector. 00:02:40.750 --> 00:02:42.100 And this is torque. 00:02:42.100 --> 00:02:43.320 And so what are its units? 00:02:43.320 --> 00:02:46.810 Well, force is newtons, and distance is meters, so this is 00:02:46.810 --> 00:02:48.570 newton meters. 00:02:48.570 --> 00:02:51.010 And you're saying, hey Sal, newtons times meters, force 00:02:51.010 --> 00:02:54.440 times distance, that looks an awful lot like work. 00:02:54.440 --> 00:02:56.840 And it's very important to realize that this isn't work, 00:02:56.840 --> 00:02:59.300 and that's why we won't call this joules. 00:02:59.300 --> 00:03:01.130 Because in work, what are we doing? 00:03:01.130 --> 00:03:03.010 We are translating an object. 00:03:03.010 --> 00:03:06.960 If this is an object, and I'm applying a force, I'm taking 00:03:06.960 --> 00:03:09.610 the force over the distance in the same 00:03:09.610 --> 00:03:12.000 direction as the force. 00:03:12.000 --> 00:03:14.880 Here the distance and the force are 00:03:14.880 --> 00:03:15.750 parallel to each other. 00:03:15.750 --> 00:03:17.910 You could say the distance vector and the force vector 00:03:17.910 --> 00:03:20.850 are in the same direction. 00:03:20.850 --> 00:03:22.090 Of course, that's translational. 00:03:22.090 --> 00:03:23.350 The whole object is just moving. 00:03:23.350 --> 00:03:24.940 It's not rotating or anything. 00:03:24.940 --> 00:03:28.340 In the situation of torque, let me switch colors. 00:03:28.340 --> 00:03:32.000 The distance vector, this is the distance from the fulcrum 00:03:32.000 --> 00:03:33.770 or the pivot point of the center of mass, to where I'm 00:03:33.770 --> 00:03:34.840 applying the force. 00:03:34.840 --> 00:03:38.640 This distance vector is perpendicular to the force 00:03:38.640 --> 00:03:39.790 that's being applied. 00:03:39.790 --> 00:03:42.930 So torque and work are fundamentally two different 00:03:42.930 --> 00:03:45.940 things, even though their units are the same. 00:03:45.940 --> 00:03:48.660 And this is a little bit of notational. 00:03:48.660 --> 00:03:53.610 This distance is often called the moment arm distance. 00:03:53.610 --> 00:03:54.960 And I don't know where that came from. 00:03:54.960 --> 00:03:57.250 Maybe one of you all can write me a message saying where it 00:03:57.250 --> 00:03:58.180 did come from. 00:03:58.180 --> 00:04:01.200 And often in some of your physics classes they'll often 00:04:01.200 --> 00:04:03.400 call torque as a moment. 00:04:03.400 --> 00:04:04.970 But we'll deal with the term torque. 00:04:04.970 --> 00:04:08.280 And that's more fun, because eventually we can understand 00:04:08.280 --> 00:04:11.630 concepts like torque horsepower in cars. 00:04:11.630 --> 00:04:14.230 So let's do a little bit of math, hopefully I've given you 00:04:14.230 --> 00:04:16.930 a little bit of intuition. 00:04:16.930 --> 00:04:23.510 So let's say I had this ruler. 00:04:23.510 --> 00:04:28.780 And let's say that this is its pivot point right here. 00:04:28.780 --> 00:04:29.960 So it would rotate around that point. 00:04:29.960 --> 00:04:32.220 It's nailed to the wall or something. 00:04:32.220 --> 00:04:37.990 And let's say that I apply a force-- Let's say the moment 00:04:37.990 --> 00:04:39.760 arm distance. 00:04:39.760 --> 00:04:41.532 So let's say this distance, let me do it 00:04:41.532 --> 00:04:43.690 in different color. 00:04:43.690 --> 00:04:49.740 Let's say that this distance right here is 10 meters. 00:04:49.740 --> 00:04:57.300 And I were to apply a force of 5 newtons perpendicular to the 00:04:57.300 --> 00:05:00.710 distance vector, or to dimension of the moment arm, 00:05:00.710 --> 00:05:02.040 you could view it either way. 00:05:02.040 --> 00:05:04.300 So torque is pretty easy in this situation. 00:05:04.300 --> 00:05:11.580 Torque is going to be equal to the force, 5 newtons, times 00:05:11.580 --> 00:05:13.010 the distance, 10. 00:05:13.010 --> 00:05:16.816 So it would be 50 newton meters. 00:05:16.816 --> 00:05:18.910 And you're probably saying, well, Sal, how do I know if 00:05:18.910 --> 00:05:20.490 this torque is going to be positive or negative? 00:05:20.490 --> 00:05:22.880 And this is where there's just a general arbitrary convention 00:05:22.880 --> 00:05:23.550 in physics. 00:05:23.550 --> 00:05:25.190 And it's good to know. 00:05:25.190 --> 00:05:30.150 If you're rotating clockwise torque is negative. 00:05:30.150 --> 00:05:30.990 Let me go the other way. 00:05:30.990 --> 00:05:32.930 If you were rotating counterclockwise, like we were 00:05:32.930 --> 00:05:35.650 in this example, rotating counterclockwise, the opposite 00:05:35.650 --> 00:05:38.250 direction of which a clock would move in. 00:05:38.250 --> 00:05:39.560 Torque is positive. 00:05:39.560 --> 00:05:42.540 And if you rotate clockwise the other 00:05:42.540 --> 00:05:44.250 way, torque is negative. 00:05:44.250 --> 00:05:45.680 So clockwise is negative. 00:05:45.680 --> 00:05:50.390 And I'm not going to go into the whole cross product and 00:05:50.390 --> 00:05:52.210 the linear algebra of torque right now, because I think 00:05:52.210 --> 00:05:53.790 that's a little bit beyond the scope. 00:05:53.790 --> 00:05:55.450 But we'll do that once we do more 00:05:55.450 --> 00:05:58.160 mathematically intensive physics. 00:05:58.160 --> 00:06:00.190 But, so, good enough. 00:06:00.190 --> 00:06:02.790 There's a torque of 50 newton meters. 00:06:02.790 --> 00:06:04.390 And that's all of the torque that is acting 00:06:04.390 --> 00:06:05.040 on this object . 00:06:05.040 --> 00:06:06.330 So it's going to rotate in this direction. 00:06:06.330 --> 00:06:09.910 And we don't have the tools yet to figure out how quickly 00:06:09.910 --> 00:06:10.660 it will rotate. 00:06:10.660 --> 00:06:12.010 But we know it will rotate. 00:06:12.010 --> 00:06:14.510 And that's vaguely useful. 00:06:14.510 --> 00:06:17.100 But what if I said that the object is not rotating? 00:06:17.100 --> 00:06:24.940 And that I have another force acting here? 00:06:24.940 --> 00:06:35.180 And let's say that that force is-- I don't know, let me make 00:06:35.180 --> 00:06:37.570 up something, that's 5 meters to the left 00:06:37.570 --> 00:06:38.820 of the pivot point. 00:06:44.000 --> 00:06:48.130 If I were tell you that this object does not rotate. 00:06:48.130 --> 00:06:50.610 So if I tell you that the object is not rotating, that 00:06:50.610 --> 00:06:56.340 means the net torque on this ruler must be 0, because it's 00:06:56.340 --> 00:07:00.380 not-- its rate of change of rotation is not changing. 00:07:00.380 --> 00:07:01.860 I should be a little exact. 00:07:01.860 --> 00:07:07.600 If I'm applying some force here, and still not rotating, 00:07:07.600 --> 00:07:12.020 then we know that the net torque on this object is 0. 00:07:12.020 --> 00:07:14.730 So what is the force being applied here? 00:07:14.730 --> 00:07:16.750 Well, what is the net torque? 00:07:16.750 --> 00:07:19.170 Well, it's this torque, which we already figured out. 00:07:19.170 --> 00:07:20.740 It's going in the clockwise direction. 00:07:20.740 --> 00:07:24.340 So it's 5-- Let me do it in a brighter color. 00:07:24.340 --> 00:07:27.480 5 times 10. 00:07:27.480 --> 00:07:28.960 And then the net torque. 00:07:28.960 --> 00:07:31.580 The sum of all the torques have to be equal to 0. 00:07:31.580 --> 00:07:32.510 So what's this torque? 00:07:32.510 --> 00:07:34.480 So let's call this f. 00:07:34.480 --> 00:07:35.750 This is the force. 00:07:35.750 --> 00:07:40.790 So, plus-- Well, this force is acting in what direction? 00:07:40.790 --> 00:07:43.110 Clockwise or counterclockwise? 00:07:43.110 --> 00:07:44.810 Well, it's acting in the clockwise direction. 00:07:44.810 --> 00:07:47.710 This force wants to make the ruler rotate this way. 00:07:47.710 --> 00:07:50.040 So this is actually going to be a negative torque. 00:07:50.040 --> 00:07:55.570 So let's say, put a negative number here times f, times its 00:07:55.570 --> 00:07:59.630 moment arm distance, times 5, and all of this 00:07:59.630 --> 00:08:00.930 has to equal 0. 00:08:00.930 --> 00:08:05.300 The net torque is 0, because the object's rate of change of 00:08:05.300 --> 00:08:07.990 rotation isn't changing, or if it started off not rotating, 00:08:07.990 --> 00:08:09.660 it's still not rotating. 00:08:09.660 --> 00:08:16.300 So here we get 50 minus 5 f is equal to 0. 00:08:16.300 --> 00:08:20.440 That's 50 is equal to 5 f. 00:08:20.440 --> 00:08:22.290 f is equal to 10. 00:08:22.290 --> 00:08:25.700 If we follow the units all the way through, we would get that 00:08:25.700 --> 00:08:28.460 f is equal to 10 newtons. 00:08:28.460 --> 00:08:30.220 So that's interesting. 00:08:30.220 --> 00:08:34.289 I applied double the force at half the distance. 00:08:34.289 --> 00:08:38.409 And it offsetted half the force at twice the distance. 00:08:38.409 --> 00:08:41.059 And that should all connect, or start to connect, with what 00:08:41.059 --> 00:08:43.309 we talked about with mechanical advantage. 00:08:43.309 --> 00:08:45.270 You could view it the other way. 00:08:45.270 --> 00:08:48.160 Let's say these are people applying these forces. 00:08:48.160 --> 00:08:50.430 Say this guy over here is applying 10 newtons. 00:08:50.430 --> 00:08:51.410 He's much stronger. 00:08:51.410 --> 00:08:53.240 He's twice as strong as this guy over here. 00:08:53.240 --> 00:08:57.310 But because this guy is twice as far away from the pivot 00:08:57.310 --> 00:08:59.820 point, he balances the other guy. 00:08:59.820 --> 00:09:01.810 So you can kind of view it as this guy having some 00:09:01.810 --> 00:09:04.340 mechanical advantage or having a mechanical advantage of 2. 00:09:04.340 --> 00:09:06.330 And watch the mechanical advantage videos if that 00:09:06.330 --> 00:09:07.910 confuses you a little bit. 00:09:07.910 --> 00:09:09.550 But this is where to torque is useful. 00:09:09.550 --> 00:09:13.600 Because if an object's rate of rotation is not changing, you 00:09:13.600 --> 00:09:16.200 know that the net torque on that object is 0. 00:09:16.200 --> 00:09:19.910 And you can solve for the forces or the distances. 00:09:19.910 --> 00:09:21.460 I'm about to run out of time, so I will see 00:09:21.460 --> 00:09:23.380 you in the next video.

Center of mass

https://www.youtube.com/watch?v=VrflZifKIuw

vtt

https://www.youtube.com/api/timedtext?v=VrflZifKIuw&ei=YmeUZb28LIKpp-oP_dOQIA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=658A74076839716A556024D7003A96290F39B880.28E82D6927A291E5A8E537AD808DB683AE94B9D2&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:02.550 --> 00:00:05.840 I will now do a presentation on the center of mass. 00:00:05.840 --> 00:00:09.110 And the center mass, hopefully, is something that 00:00:09.110 --> 00:00:12.130 will be a little bit intuitive to you, and it actually has 00:00:12.130 --> 00:00:14.300 some very neat applications. 00:00:14.300 --> 00:00:18.530 So in very simple terms, the center of mass is a point. 00:00:18.530 --> 00:00:21.850 Let me draw an object. 00:00:21.850 --> 00:00:25.520 Let's say that this is my object. 00:00:25.520 --> 00:00:26.770 Let's say it's a ruler. 00:00:30.800 --> 00:00:33.270 This ruler, it exists, so it has some mass. 00:00:33.270 --> 00:00:35.310 And my question to you is what is the center mass? 00:00:35.310 --> 00:00:37.240 And you say, Sal, well, in order to know figure out the 00:00:37.240 --> 00:00:39.910 center mass, you have to tell me what the center of mass is. 00:00:39.910 --> 00:00:43.340 And what I tell you is the center mass is a point, and it 00:00:43.340 --> 00:00:46.750 actually doesn't have to even be a point in the object. 00:00:46.750 --> 00:00:48.850 I'll do an example soon where it won't be. 00:00:48.850 --> 00:00:49.820 But it's a point. 00:00:49.820 --> 00:00:53.705 And at that point, for dealing with this object as a whole or 00:00:53.705 --> 00:00:57.040 the mass of the object as a whole, we can pretend that the 00:00:57.040 --> 00:00:59.550 entire mass exists at that point. 00:00:59.550 --> 00:01:01.420 And what do I mean by saying that? 00:01:01.420 --> 00:01:05.540 Well, let's say that the center of mass is here. 00:01:05.540 --> 00:01:06.890 And I'll tell you why I picked this point. 00:01:06.890 --> 00:01:08.200 Because that is pretty close to where the center 00:01:08.200 --> 00:01:09.970 of mass will be. 00:01:09.970 --> 00:01:12.980 If the center of mass is there, and let's say the mass 00:01:12.980 --> 00:01:18.110 of this entire ruler is, I don't know, 10 kilograms. This 00:01:18.110 --> 00:01:25.820 ruler, if a force is applied at the center of mass, let's 00:01:25.820 --> 00:01:30.950 say 10 Newtons, so the mass of the whole ruler is 10 00:01:30.950 --> 00:01:37.430 kilograms. If a force is applied at the center of mass, 00:01:37.430 --> 00:01:42.250 this ruler will accelerate the same exact way as would a 00:01:42.250 --> 00:01:43.950 point mass. 00:01:43.950 --> 00:01:46.180 Let's say that we just had a little dot, but that little 00:01:46.180 --> 00:01:51.870 dot had the same mass, 10 kilograms, and we were to push 00:01:51.870 --> 00:01:54.500 on that dot with 10 Newtons. 00:01:54.500 --> 00:01:57.960 In either case, in the case of the ruler, we would accelerate 00:01:57.960 --> 00:01:58.860 upwards at what? 00:01:58.860 --> 00:02:02.460 Force divided by mass, so we would accelerate upwards at 1 00:02:02.460 --> 00:02:04.580 meter per second squared. 00:02:04.580 --> 00:02:07.580 And in this case of this point mass, we would 00:02:07.580 --> 00:02:09.009 accelerate that point. 00:02:09.009 --> 00:02:10.665 When I say point mass, I'm just saying something really, 00:02:10.665 --> 00:02:13.500 really small, but it has a mass of 10 kilograms, so it's 00:02:13.500 --> 00:02:15.470 much smaller, but it has the same mass as this ruler. 00:02:15.470 --> 00:02:20.270 This would also accelerate upwards with a magnitude of 1 00:02:20.270 --> 00:02:21.870 meters per second squared. 00:02:24.550 --> 00:02:26.660 So why is this useful to us? 00:02:26.660 --> 00:02:29.850 Well, sometimes we have some really crazy objects and we 00:02:29.850 --> 00:02:31.550 want to figure out exactly what it does. 00:02:31.550 --> 00:02:35.370 If we know its center of mass first, we can know how that 00:02:35.370 --> 00:02:37.440 object will behave without having to worry about the 00:02:37.440 --> 00:02:38.850 shape of that object. 00:02:38.850 --> 00:02:41.730 And I'll give you a really easy way of realizing where 00:02:41.730 --> 00:02:43.270 the center of mass is. 00:02:43.270 --> 00:02:47.710 If the object has a uniform distribution-- when I say 00:02:47.710 --> 00:02:51.160 that, it means, for simple purposes, if it's made out of 00:02:51.160 --> 00:02:53.710 the same thing and that thing that it's made out of, its 00:02:53.710 --> 00:02:56.970 density, doesn't really change throughout the object, the 00:02:56.970 --> 00:03:03.490 center of mass will be the object's geometric center. 00:03:03.490 --> 00:03:05.700 So in this case, this ruler's almost a 00:03:05.700 --> 00:03:06.660 one-dimensional object. 00:03:06.660 --> 00:03:07.840 We just went halfway. 00:03:07.840 --> 00:03:09.670 The distance from here to here and the distance from here to 00:03:09.670 --> 00:03:10.230 here are the same. 00:03:10.230 --> 00:03:11.240 This is the center of mass. 00:03:11.240 --> 00:03:13.900 If we had a two-dimensional object, let's say we had this 00:03:13.900 --> 00:03:18.080 triangle and we want to figure out its center of mass, it'll 00:03:18.080 --> 00:03:20.070 be the center in two dimensions. 00:03:20.070 --> 00:03:22.370 So it'll be something like that. 00:03:22.370 --> 00:03:28.730 Now, if I had another situation, let's say I have 00:03:28.730 --> 00:03:32.340 this square. 00:03:32.340 --> 00:03:34.800 I don't know if that's big enough for you to see. 00:03:34.800 --> 00:03:36.550 I need to draw it a little thicker. 00:03:36.550 --> 00:03:40.890 Let's say I have this square, but let's say that half of 00:03:40.890 --> 00:03:49.945 this square is made from lead. 00:03:55.780 --> 00:03:59.630 And let's say the other half of the square is made from 00:03:59.630 --> 00:04:01.120 something lighter than lead. 00:04:01.120 --> 00:04:02.920 It's made of styrofoam. 00:04:02.920 --> 00:04:05.450 That is lighter than lead. 00:04:05.450 --> 00:04:07.680 So in this situation, the center of mass isn't going to 00:04:07.680 --> 00:04:08.770 be the geographic center. 00:04:08.770 --> 00:04:12.080 I don't know how much denser lead is than styrofoam, but 00:04:12.080 --> 00:04:16.360 the center of mass is going to be someplace closer to the 00:04:16.360 --> 00:04:20.930 right because this object does not have a uniform density. 00:04:20.930 --> 00:04:25.065 It'll actually depend on how much denser the lead is than 00:04:25.065 --> 00:04:26.730 the styrofoam, which I don't know. 00:04:26.730 --> 00:04:29.100 But hopefully, that gives you a little intuition of what the 00:04:29.100 --> 00:04:31.100 center of mass is. 00:04:31.100 --> 00:04:34.130 And now I'll tell you something a little more 00:04:34.130 --> 00:04:35.650 interesting. 00:04:35.650 --> 00:04:40.530 Every problem we have done so far, we actually made the 00:04:40.530 --> 00:04:43.360 simplifying assumption that the force acts on 00:04:43.360 --> 00:04:44.930 the center of mass. 00:04:44.930 --> 00:04:48.510 So if I have an object, let's say the object that 00:04:48.510 --> 00:04:49.760 looks like a horse. 00:04:53.620 --> 00:04:55.590 Let's say that object. 00:04:55.590 --> 00:04:58.210 If this is the object's center of mass, I don't know where 00:04:58.210 --> 00:05:01.100 the horse's center of mass normally is, but let's say a 00:05:01.100 --> 00:05:03.810 horse's center of mass is here. 00:05:03.810 --> 00:05:10.990 If I apply a force directly on that center of mass, then the 00:05:10.990 --> 00:05:14.310 object will move in the direction of that force with 00:05:14.310 --> 00:05:15.660 the appropriate acceleration. 00:05:15.660 --> 00:05:19.390 We could divide the force by the mass of the entire horse 00:05:19.390 --> 00:05:19.990 and we would figure out the 00:05:19.990 --> 00:05:23.080 acceleration in that direction. 00:05:23.080 --> 00:05:25.480 But now I will throw in a twist. And actually, every 00:05:25.480 --> 00:05:28.200 problem we did, all of these Newton's Law's problems, we 00:05:28.200 --> 00:05:32.140 assumed that the force acted at the center of mass. 00:05:32.140 --> 00:05:35.820 But something more interesting happens if the force acts away 00:05:35.820 --> 00:05:37.070 from the center of mass. 00:05:40.350 --> 00:05:42.190 Let me actually take that ruler example. 00:05:42.190 --> 00:05:45.210 I don't know why I even drew the horse. 00:05:45.210 --> 00:05:52.570 If I have this ruler again and this is the center of mass, as 00:05:52.570 --> 00:05:56.120 we said, any force that we act on the center of mass, the 00:05:56.120 --> 00:05:59.040 whole object will move in the direction of the force. 00:05:59.040 --> 00:06:01.800 It'll be shifted by the force, essentially. 00:06:01.800 --> 00:06:02.810 Now, this is what's interesting. 00:06:02.810 --> 00:06:06.900 If that's the center of mass and if I were to apply a force 00:06:06.900 --> 00:06:11.736 someplace else away from the center of mass, let' say I 00:06:11.736 --> 00:06:15.880 apply a force here, I want you to think about for a second 00:06:15.880 --> 00:06:18.750 what will probably happen to the object. 00:06:18.750 --> 00:06:21.220 Well, it turns out that the object will rotate. 00:06:21.220 --> 00:06:23.240 And so think about if we're on the space shuttle or we're in 00:06:23.240 --> 00:06:25.540 deep space or something, and if I have a ruler, and if I 00:06:25.540 --> 00:06:27.980 just push at one end of the ruler, what's going to happen? 00:06:27.980 --> 00:06:30.240 Am I just going to push the whole ruler or is the whole 00:06:30.240 --> 00:06:31.300 ruler is going to rotate? 00:06:31.300 --> 00:06:33.430 And hopefully, your intuition is correct. 00:06:33.430 --> 00:06:36.770 The whole ruler will rotate around the center of mass. 00:06:36.770 --> 00:06:42.010 And in general, if you were to throw a monkey wrench at 00:06:42.010 --> 00:06:45.680 someone, and I don't recommend that you do, but if you did, 00:06:45.680 --> 00:06:48.890 and while the monkey wrench is spinning in the air, it's 00:06:48.890 --> 00:06:51.480 spinning around its center of mass. 00:06:51.480 --> 00:06:52.940 Same for a knife. 00:06:52.940 --> 00:06:54.900 If you're a knife catcher, that's something you should 00:06:54.900 --> 00:07:00.280 think about, that the object, when it's free, when it's not 00:07:00.280 --> 00:07:03.580 fixed to any point, it rotates around its center of mass, and 00:07:03.580 --> 00:07:04.570 that's very interesting. 00:07:04.570 --> 00:07:07.650 So you can actually throw random objects, and that point 00:07:07.650 --> 00:07:09.330 at which it rotates around, that's the 00:07:09.330 --> 00:07:10.340 object's center of mass. 00:07:10.340 --> 00:07:13.630 That's an experiment that you should do in an open field 00:07:13.630 --> 00:07:15.690 around no one else. 00:07:15.690 --> 00:07:18.970 Now, with all of this, and I'll actually in the next 00:07:18.970 --> 00:07:20.290 video tell you what this is. 00:07:20.290 --> 00:07:23.130 When you have a force that causes rotational motion as 00:07:23.130 --> 00:07:25.970 opposed to a shifting motion, that's torque, but we'll do 00:07:25.970 --> 00:07:26.850 that in the next video. 00:07:26.850 --> 00:07:30.610 But now I'll show you just a cool example of how the center 00:07:30.610 --> 00:07:34.880 of mass is relevant in everyday applications, like 00:07:34.880 --> 00:07:36.890 high jumping. 00:07:36.890 --> 00:07:40.340 So in general, let's say that this is a bar. 00:07:40.340 --> 00:07:42.690 This is a side view of a bar, and this is the 00:07:42.690 --> 00:07:43.530 thing holding the bar. 00:07:43.530 --> 00:07:45.440 And a guy wants to jump over the bar. 00:07:48.810 --> 00:07:51.880 His center of mass is-- most people's center of mass is 00:07:51.880 --> 00:07:53.170 around their gut. 00:07:53.170 --> 00:07:55.380 I think evolutionarily that's why our gut is there, because 00:07:55.380 --> 00:07:56.870 it's close to our center of mass. 00:07:56.870 --> 00:07:58.460 So there's two ways to jump. 00:07:58.460 --> 00:08:01.500 You could just jump straight over the bar, like a hurdle 00:08:01.500 --> 00:08:06.130 jump, in which case your center of mass would have to 00:08:06.130 --> 00:08:07.500 cross over the bar. 00:08:07.500 --> 00:08:09.510 And we could figure out this mass, and we can figure out 00:08:09.510 --> 00:08:12.657 how much energy and how much force is required to propel a 00:08:12.657 --> 00:08:16.630 mass that high because we know projectile motion and we know 00:08:16.630 --> 00:08:18.270 all of Newton's laws. 00:08:18.270 --> 00:08:21.410 But what you see a lot in the Olympics is people doing a 00:08:21.410 --> 00:08:24.810 very strange type of jump, where, when they're going over 00:08:24.810 --> 00:08:28.430 the bar, they look something like this. 00:08:28.430 --> 00:08:30.700 Their backs are arched over the bar. 00:08:34.600 --> 00:08:36.000 Not a good picture. 00:08:36.000 --> 00:08:38.520 But what happens when someone arches their back over 00:08:38.520 --> 00:08:39.340 the bar like this? 00:08:39.340 --> 00:08:40.370 I hope you get the point. 00:08:40.370 --> 00:08:42.440 This is the bar right here. 00:08:42.440 --> 00:08:44.350 Well, it's interesting. 00:08:44.350 --> 00:08:46.390 If you took the average of this person's density and 00:08:46.390 --> 00:08:48.940 figured out his geometric center and all of that, the 00:08:48.940 --> 00:08:51.500 center of mass in this situation, if someone jumps 00:08:51.500 --> 00:08:54.940 like that, actually travels below the bar. 00:08:54.940 --> 00:08:57.750 Because the person arches their back so much, if you 00:08:57.750 --> 00:09:00.590 took the average of the total mass of where the person is, 00:09:00.590 --> 00:09:03.770 their center of mass actually goes below the bar. 00:09:03.770 --> 00:09:06.320 And because of that, you can clear a bar without having 00:09:06.320 --> 00:09:08.820 your center of mass go as high as the bar and so you need 00:09:08.820 --> 00:09:09.960 less force to do it. 00:09:09.960 --> 00:09:13.370 Or another way to say it, with the same force, you could 00:09:13.370 --> 00:09:14.723 clear a higher bar. 00:09:14.723 --> 00:09:15.530 , 00:09:15.530 --> 00:09:18.720 Hopefully, I didn't confuse you, but that's exactly why 00:09:18.720 --> 00:09:23.037 these high jumpers arch their back, so that their center of 00:09:23.037 --> 00:09:26.910 mass is actually below the bar and they don't have to exert 00:09:26.910 --> 00:09:27.810 as much force. 00:09:27.810 --> 00:09:30.430 Anyway, hopefully you found that to be a vaguely useful 00:09:30.430 --> 00:09:32.610 introduction to the center of mass, and I'll see you in the 00:09:32.610 --> 00:09:35.100 next video on torque.

Mechanical advantage (part 3)

https://www.youtube.com/watch?v=vSsK7Rfa3yA

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WEBVTT Kind: captions Language: en 00:00:01.020 --> 00:00:02.000 Welcome back. 00:00:02.000 --> 00:00:04.050 Now let's do some more mechanical advantage problems. 00:00:04.050 --> 00:00:06.410 And in this video, we'll focus on pulleys, which is another 00:00:06.410 --> 00:00:08.260 form of a simple machine. 00:00:08.260 --> 00:00:10.105 And we've done some pulley problems in the past, but now 00:00:10.105 --> 00:00:12.200 we'll actually understand what the mechanical advantage 00:00:12.200 --> 00:00:14.350 inherent in these machines are. 00:00:14.350 --> 00:00:17.370 So let me start with a very simple pulley. 00:00:17.370 --> 00:00:19.190 So this is the ceiling up here. 00:00:23.000 --> 00:00:24.780 I don't know what they call that part of the pulley. 00:00:24.780 --> 00:00:26.540 I should learn my actual terminology. 00:00:26.540 --> 00:00:29.840 But let's say I have that little disk where the rope 00:00:29.840 --> 00:00:34.170 goes over and it rolls so that the rope can go over it and 00:00:34.170 --> 00:00:36.120 move without having a lot of friction. 00:00:36.120 --> 00:00:38.520 And let's say I have a rope going over that pulley. 00:00:42.600 --> 00:00:44.610 That's my rope. 00:00:44.610 --> 00:00:51.740 And at this end, let's say I have a weight, a 10-Newton 00:00:51.740 --> 00:00:57.730 weight, and I'm going to pull down on this end to make the 00:00:57.730 --> 00:00:59.160 weight to go up. 00:00:59.160 --> 00:01:01.410 So my question to you is what is the mechanical advantage of 00:01:01.410 --> 00:01:02.480 this system? 00:01:02.480 --> 00:01:05.740 What is the force that I have to pull down in order to lift 00:01:05.740 --> 00:01:09.770 this weight, this 10-Newton weight in order to produce 10 00:01:09.770 --> 00:01:11.620 Newtons of force upwards? 00:01:11.620 --> 00:01:14.985 Well, in any pulley situation-- and I don't know 00:01:14.985 --> 00:01:17.380 if textbooks cover it this way, but this is how I think 00:01:17.380 --> 00:01:19.610 about it, because you don't have to memorize formulas. 00:01:19.610 --> 00:01:21.920 I just think about, well, what happens to 00:01:21.920 --> 00:01:23.150 the lengths of rope? 00:01:23.150 --> 00:01:25.760 Or what is the total distance that the object you're trying 00:01:25.760 --> 00:01:27.000 to move travels? 00:01:27.000 --> 00:01:29.310 And if you know the distance that it travels versus the 00:01:29.310 --> 00:01:31.040 distance that you have to pull, you know 00:01:31.040 --> 00:01:32.590 the mechanical advantage. 00:01:32.590 --> 00:01:36.880 So in this situation, if I were to hold the rope at that 00:01:36.880 --> 00:01:41.670 point, and if I were to pull it down 10 feet or some 00:01:41.670 --> 00:01:44.350 arbitrary distance, what happens over here? 00:01:44.350 --> 00:01:46.300 Well, this weight is going to move up 00:01:46.300 --> 00:01:49.560 exactly the same amount. 00:01:49.560 --> 00:01:53.250 Whatever I pull, if I pull a foot down here, this weight 00:01:53.250 --> 00:01:57.360 will move up by a foot, so the distance that I pull here is 00:01:57.360 --> 00:02:00.130 equivalent to the distance that it pulls up here. 00:02:00.130 --> 00:02:02.880 And since we know that the work in has to equal the work 00:02:02.880 --> 00:02:06.670 out, we know that the force I'm pulling down has to be the 00:02:06.670 --> 00:02:10.080 same as the force or the tension that the rope is 00:02:10.080 --> 00:02:11.530 pulling up here. 00:02:11.530 --> 00:02:14.920 And we could have done that when we learned about tension, 00:02:14.920 --> 00:02:16.740 that the tension in the rope is constant. 00:02:16.740 --> 00:02:19.850 I'm producing tension in the rope when I pull here and 00:02:19.850 --> 00:02:22.480 that's the same pulling force of the tension on the weight. 00:02:22.480 --> 00:02:25.180 So this isn't too interesting of a machine. 00:02:25.180 --> 00:02:29.920 All it's doing is I pull down with a force of 10 Newtons and 00:02:29.920 --> 00:02:32.480 it will pull up with a force of 10 Newtons, and so the 00:02:32.480 --> 00:02:36.550 mechanical advantage is 1, no real mechanical advantage, 00:02:36.550 --> 00:02:37.940 although this could be useful. 00:02:37.940 --> 00:02:39.590 Maybe it's easier for me to pull down than 00:02:39.590 --> 00:02:40.880 for me to pull up. 00:02:40.880 --> 00:02:43.810 Or at some point, maybe I can't reach up here, so it's 00:02:43.810 --> 00:02:45.930 nice for me to pull down here where I can reach and the 00:02:45.930 --> 00:02:48.260 object will keep going up like in a flag pole or 00:02:48.260 --> 00:02:49.160 something like that. 00:02:49.160 --> 00:02:51.820 So this could still be useful even though its mechanical 00:02:51.820 --> 00:02:54.220 advantage is only 1. 00:02:54.220 --> 00:02:58.270 So let's see if we can construct a pulley situation 00:02:58.270 --> 00:03:03.170 where the mechanical advantage is more than 1. 00:03:03.170 --> 00:03:09.810 So let's say over here at the top, I still have the same 00:03:09.810 --> 00:03:14.080 pulley that's attached to the ceiling, but I'm going to add 00:03:14.080 --> 00:03:16.766 slight variation here. 00:03:16.766 --> 00:03:19.700 I have another pulley here. 00:03:19.700 --> 00:03:32.315 And now let me do the other pulley down here. 00:03:36.790 --> 00:03:40.470 And then let me see if I can draw my rope in a good way. 00:03:40.470 --> 00:03:46.010 So my rope starts up going up like that, then it comes back 00:03:46.010 --> 00:03:50.820 down, comes around the second pulley, and now this is 00:03:50.820 --> 00:03:54.330 attached to the ceiling up here. 00:03:54.330 --> 00:03:56.280 The second pulley is actually where the 00:03:56.280 --> 00:03:58.660 weight is attached to. 00:03:58.660 --> 00:04:00.750 And let's just call it a 10-Newton weight again, 00:04:00.750 --> 00:04:02.610 although it doesn't really matter what the weight is. 00:04:02.610 --> 00:04:05.030 Let's figure out what the mechanical advantage is. 00:04:05.030 --> 00:04:06.430 So the same question. 00:04:06.430 --> 00:04:07.930 And this is really the question you always have to 00:04:07.930 --> 00:04:08.850 ask yourself. 00:04:08.850 --> 00:04:15.150 If I were take a point on this rope and if I were to pull it 00:04:15.150 --> 00:04:18.399 2 feet down, so let's see I take this point and I move it 00:04:18.399 --> 00:04:23.970 2 feet down, what essentially happens to the rope? 00:04:23.970 --> 00:04:26.480 Well, every point on the rope's going to move 00:04:26.480 --> 00:04:29.720 2 feet to the right. 00:04:29.720 --> 00:04:31.640 I guess you can view it this way if you view that motion is 00:04:31.640 --> 00:04:33.330 to the right. 00:04:33.330 --> 00:04:37.780 But if this length of rope is getting 2 feet shorter, what 00:04:37.780 --> 00:04:39.740 is this length of rope getting? 00:04:39.740 --> 00:04:42.410 Well, this entire length of rope is also going to get 2 00:04:42.410 --> 00:04:46.530 feet shorter, this entire length of rope right here. 00:04:46.530 --> 00:04:49.940 But this entire length of rope is split between this side-- 00:04:49.940 --> 00:04:53.882 let me do it in different color-- between this 00:04:53.882 --> 00:04:58.050 side and this side. 00:04:58.050 --> 00:05:02.420 So if I make this side of the rope shorter-- I mean, the 00:05:02.420 --> 00:05:05.660 rope goes through the whole thing, but if I take this side 00:05:05.660 --> 00:05:08.080 of the rope and I pull down by 2 feet, 00:05:08.080 --> 00:05:10.080 what is going to happen? 00:05:10.080 --> 00:05:13.590 Well, this is going to get 1 foot shorter. 00:05:13.590 --> 00:05:15.840 This rope is going to get 1 foot shorter. 00:05:15.840 --> 00:05:18.960 This is going to go 1 foot shorter and this length of the 00:05:18.960 --> 00:05:20.590 rope is going to get 1 foot shorter. 00:05:20.590 --> 00:05:21.440 And how do I know that? 00:05:21.440 --> 00:05:23.070 Well, this is all the same rope. 00:05:23.070 --> 00:05:25.070 And if this is getting 1 foot shorter, and this is one 00:05:25.070 --> 00:05:26.950 getting 1 foot shorter, it makes sense this whole thing 00:05:26.950 --> 00:05:28.510 is getting 2 feet shorter. 00:05:28.510 --> 00:05:30.520 But the important thing to realize, if each of these are 00:05:30.520 --> 00:05:33.200 getting 1 foot shorter, then this weight is 00:05:33.200 --> 00:05:35.110 only moving up 1 foot. 00:05:38.050 --> 00:05:42.440 So when I pull the rope down 2 feet here, this weight only 00:05:42.440 --> 00:05:43.850 moves up 1 foot. 00:05:43.850 --> 00:05:46.320 So what is the work that I'm doing? 00:05:46.320 --> 00:05:48.560 Well, the work in is the same as the work out, and we know 00:05:48.560 --> 00:05:49.960 what the work out is. 00:05:49.960 --> 00:05:53.100 The work out is going to be the force that this 00:05:53.100 --> 00:05:55.530 contraption or this machine is pulling upwards with, and 00:05:55.530 --> 00:06:01.070 that's 10 Newtons, so the workout is equal to 10 Newtons 00:06:01.070 --> 00:06:03.255 times the distance that the force is 00:06:03.255 --> 00:06:05.730 pulling in, times 1 foot. 00:06:05.730 --> 00:06:06.750 Oh, why did I do feet? 00:06:06.750 --> 00:06:08.000 I should do meters. 00:06:10.920 --> 00:06:12.565 That's not a good thing for me to do. 00:06:12.565 --> 00:06:15.170 That should be meters. 00:06:15.170 --> 00:06:17.360 I shouldn't mix English and metric system. 00:06:17.360 --> 00:06:23.840 So 10 Newtons times 1 meter, so it equals 10 joules. 00:06:23.840 --> 00:06:25.310 And this has to be the work that I've put 00:06:25.310 --> 00:06:26.330 into it, too, right? 00:06:26.330 --> 00:06:32.350 So the work in also has to be 10 joules. 00:06:32.350 --> 00:06:33.890 Well, I know the distance that I pulled down. 00:06:33.890 --> 00:06:36.280 I know I pulled down 2 meters. 00:06:36.280 --> 00:06:39.100 So I pulled down 2 meters, so this has to equal the force 00:06:39.100 --> 00:06:39.890 times the distance. 00:06:39.890 --> 00:06:42.360 So the force, which I don't know, times the distance, 00:06:42.360 --> 00:06:45.420 which is 2 meters, is equal to 10 joules. 00:06:45.420 --> 00:06:48.090 So divide both sides by 2, so the force that I pulled down 00:06:48.090 --> 00:06:51.270 with is 5 Newtons. 00:06:51.270 --> 00:06:54.940 So I pulled down 5 Newtons for 2 meters, and it pulls up a 00:06:54.940 --> 00:06:57.100 10-Newton weight for 1 meter. 00:06:57.100 --> 00:06:59.780 Force times distance is equal to force times distance. 00:06:59.780 --> 00:07:01.800 So what was the input force? 00:07:01.800 --> 00:07:07.050 The input force is equal to 5 Newtons and the output force 00:07:07.050 --> 00:07:09.280 of this machine is equal to 10 Newtons. 00:07:09.280 --> 00:07:12.650 Mechanical advantage is the output over the input, so the 00:07:12.650 --> 00:07:16.110 mechanical advantage is equal to the force output by the 00:07:16.110 --> 00:07:20.400 force input, which equals 10/5, which equals 2. 00:07:20.400 --> 00:07:23.970 And that makes sense, because I have to pull twice as much 00:07:23.970 --> 00:07:32.280 for this thing to move up half of that distance. 00:07:32.280 --> 00:07:36.520 Let's see if we can do another mechanical advantage problem. 00:07:36.520 --> 00:07:39.410 Actually, let's do a really simple one that we've really 00:07:39.410 --> 00:07:41.690 been working with a long time. 00:07:41.690 --> 00:07:45.050 Let's say that I have a wedge. 00:07:45.050 --> 00:07:48.760 A wedge is actually considered a machine, which it took me a 00:07:48.760 --> 00:07:52.220 little while to get my mind around that, but 00:07:52.220 --> 00:07:53.940 a wedge is a machine. 00:07:53.940 --> 00:07:55.200 And why is a wedge a machine? 00:07:55.200 --> 00:07:57.670 Because it gives you mechanical advantage. 00:07:57.670 --> 00:07:59.770 So if I have this wedge here. 00:07:59.770 --> 00:08:05.580 And this is a 30-degree angle, if this distance up here, 00:08:05.580 --> 00:08:09.260 let's call this distance D, what is this 00:08:09.260 --> 00:08:11.630 distance going to be? 00:08:11.630 --> 00:08:13.530 Well, it's going to be D sine of 30. 00:08:13.530 --> 00:08:15.540 And we know that the sine of 30 degrees, hopefully by this 00:08:15.540 --> 00:08:19.520 point, is 1/2, so this is going to be 1/2D. 00:08:19.520 --> 00:08:21.190 You might want to review the trigonometry a little bit if 00:08:21.190 --> 00:08:23.800 that doesn't completely ring a bell for you. 00:08:23.800 --> 00:08:27.020 So if I take an object, if I take a box-- and let's assume 00:08:27.020 --> 00:08:28.280 it has no friction. 00:08:28.280 --> 00:08:29.970 We're not going to go into the whole normal 00:08:29.970 --> 00:08:31.240 force and all that. 00:08:31.240 --> 00:08:37.200 If I take a box, and I push it with some force all the way up 00:08:37.200 --> 00:08:42.200 here, what is the mechanical advantage of this system? 00:08:42.200 --> 00:08:44.670 Well, when the box is up here, we know what its 00:08:44.670 --> 00:08:46.220 potential energy is. 00:08:46.220 --> 00:08:49.820 Its potential energy is going to be the weight of the box. 00:08:49.820 --> 00:08:54.510 So let's say this is a 10-Newton box. 00:08:54.510 --> 00:08:56.790 The potential energy at this point is going to be 10 00:08:56.790 --> 00:08:59.750 Newtons times its height. 00:08:59.750 --> 00:09:03.540 So potential energy at this point has to equal 10 Newtons 00:09:03.540 --> 00:09:07.870 times the height, which is going to be 5 joules. 00:09:07.870 --> 00:09:11.380 And that's also the amount of work one has to put into the 00:09:11.380 --> 00:09:14.530 system in order to get it into this state, in order to get it 00:09:14.530 --> 00:09:16.380 this high in the air. 00:09:16.380 --> 00:09:20.090 So we know that we would have to put 5 joules of work in 00:09:20.090 --> 00:09:23.260 order to get the box up to this point. 00:09:23.260 --> 00:09:25.490 So what is the force that we had to apply? 00:09:25.490 --> 00:09:28.490 Well, it's that force, that input force, times this 00:09:28.490 --> 00:09:31.310 distance has to equal 5 joules. 00:09:33.850 --> 00:09:36.810 So this input force-- oh, sorry, this is going to be-- 00:09:36.810 --> 00:09:38.610 sorry, this isn't 5 joules. 00:09:38.610 --> 00:09:41.700 It's 10 times 1/2 times the distance. 00:09:41.700 --> 00:09:43.820 It's 5D joules. 00:09:43.820 --> 00:09:47.480 This isn't some kind of units. 00:09:47.480 --> 00:09:49.720 It's 10 Newtons times the distance that we're up, and 00:09:49.720 --> 00:09:52.610 that's 1/2D, so it's 5D joules. 00:09:52.610 --> 00:09:53.860 Sorry for confusing you. 00:09:56.510 --> 00:10:00.030 And so the force I'm pushing here times this distance has 00:10:00.030 --> 00:10:04.220 to also equal to 5D joules. 00:10:04.220 --> 00:10:05.500 I just remembered, I just used D as a 00:10:05.500 --> 00:10:06.820 variable the whole time. 00:10:06.820 --> 00:10:08.710 Dividing both sides by D, what do I get? 00:10:08.710 --> 00:10:14.780 The input force had to be equal to 5 Newtons. 00:10:14.780 --> 00:10:17.370 I'm dividing both sides by D meters. 00:10:17.370 --> 00:10:21.650 So I inputted 5 Newtons of force and I was able to lift 00:10:21.650 --> 00:10:24.370 essentially a 10-Newton object. 00:10:24.370 --> 00:10:26.410 So what is the mechanical advantage? 00:10:26.410 --> 00:10:29.390 Well, it's the force output, 10 Newtons, divided by the 00:10:29.390 --> 00:10:31.195 force input, 5 Newtons. 00:10:31.195 --> 00:10:34.120 The mechanical advantage is 2.

Mechanical advantage (part 2)

https://www.youtube.com/watch?v=DiBXxWBrV24

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en

WEBVTT Kind: captions Language: en 00:00:00.600 --> 00:00:01.240 Welcome back. 00:00:01.240 --> 00:00:03.200 When I left off, I was hurrying a little bit. 00:00:03.200 --> 00:00:06.510 But we'd hopefully come to the conclusion that if I have a 00:00:06.510 --> 00:00:09.780 simple lever, like I have here, and I know the distances 00:00:09.780 --> 00:00:12.280 from where I'm applying the force, to the 00:00:12.280 --> 00:00:13.360 fulcrum, to the pivot. 00:00:13.360 --> 00:00:16.379 And I know the distance from the pivot to where the machine 00:00:16.379 --> 00:00:18.745 is essentially applying the force, the machine being the 00:00:18.745 --> 00:00:21.710 lever in this situation, I know the relationship between 00:00:21.710 --> 00:00:23.380 the two forces I'm applying. 00:00:23.380 --> 00:00:26.005 The input force-- so actually I shouldn't call this a force 00:00:26.005 --> 00:00:28.010 too, I should call this an input force-- anyway, the 00:00:28.010 --> 00:00:31.010 input force times the distance from the input force to the 00:00:31.010 --> 00:00:36.980 fulcrum is equal to the output force times the distance from 00:00:36.980 --> 00:00:39.010 the output force to the fulcrum. 00:00:39.010 --> 00:00:41.860 And that all fell out of what we did in the last video. 00:00:41.860 --> 00:00:45.280 The conservation of energy and that the work in has to equal 00:00:45.280 --> 00:00:45.850 the work out. 00:00:45.850 --> 00:00:48.590 And all work is, is a transfer of energy, so the transfer of 00:00:48.590 --> 00:00:50.980 energy in has to be the transfer of energy out, 00:00:50.980 --> 00:00:53.050 assuming we have no friction and none of the energy is 00:00:53.050 --> 00:00:53.690 lost. 00:00:53.690 --> 00:00:55.520 And how is this useful? 00:00:55.520 --> 00:01:00.050 Well we could do a bunch of problems with this. 00:01:00.050 --> 00:01:07.930 Let's say that I have a 100 newton object 00:01:07.930 --> 00:01:10.820 right here, 100 newtons. 00:01:10.820 --> 00:01:15.810 And let's say that I know, no matter what I do, my maximum 00:01:15.810 --> 00:01:18.060 strength that I could push-- well let me draw this a little 00:01:18.060 --> 00:01:19.740 different-- let's say it's like this, cause of my goal is 00:01:19.740 --> 00:01:20.870 to lift the 100 newton object. 00:01:20.870 --> 00:01:23.050 So the 100 newton object is right here. 00:01:23.050 --> 00:01:24.150 That's a 100 newtons. 00:01:24.150 --> 00:01:27.180 And let's say I know that the maximum downward force that 00:01:27.180 --> 00:01:31.530 I'm capable of applying is only 10 newtons, right? 00:01:31.530 --> 00:01:34.940 So I want my force to be multiplied by 10 00:01:34.940 --> 00:01:36.380 to lift this force. 00:01:36.380 --> 00:01:37.510 So let's figure out what would happen. 00:01:37.510 --> 00:01:39.830 My input force is 10. 00:01:39.830 --> 00:01:42.530 And I want to figure out the distance. 00:01:42.530 --> 00:01:44.530 So let's say my input force is 10. 00:01:44.530 --> 00:01:47.220 And let's call this the input distance. 00:01:47.220 --> 00:01:50.920 And I want the output force to be 100, right? 00:01:50.920 --> 00:01:54.290 And let's call this the output distance. 00:01:54.290 --> 00:02:00.540 So if I have a fulcrum here, this is the input distance and 00:02:00.540 --> 00:02:01.770 this is the output distance. 00:02:01.770 --> 00:02:02.340 Let me switch colors. 00:02:02.340 --> 00:02:03.530 This is getting monotonous. 00:02:03.530 --> 00:02:05.670 This is the output distance, from here to here. 00:02:05.670 --> 00:02:09.680 And let's figure out what the ratio has to be, for the ratio 00:02:09.680 --> 00:02:11.970 of the input distance to the output distance. 00:02:11.970 --> 00:02:14.400 Well, if we just divide both sides by 10, we get the 00:02:14.400 --> 00:02:15.910 distance input. 00:02:15.910 --> 00:02:18.610 It has to be 10 times the distance output, right? 00:02:18.610 --> 00:02:20.680 100 divided by 10. 00:02:20.680 --> 00:02:24.030 So if the distance from the fulcrum to the weight is, I 00:02:24.030 --> 00:02:27.980 don't know, 5 meters, then the distance from where I'm 00:02:27.980 --> 00:02:30.920 applying the force to the fulcrum has 00:02:30.920 --> 00:02:31.750 to be 10 times that. 00:02:31.750 --> 00:02:34.060 It has to be 50 meters. 00:02:34.060 --> 00:02:37.390 So no matter what, the ratio of this length to this length 00:02:37.390 --> 00:02:38.350 has to be 10. 00:02:38.350 --> 00:02:40.090 And now what would happen? 00:02:40.090 --> 00:02:43.210 If I design this machine this way, I will be able to apply 00:02:43.210 --> 00:02:45.790 10 newtons here, which is my maximum strength, 10 newtons 00:02:45.790 --> 00:02:49.570 downwards, and I will lift a 100 newton object. 00:02:49.570 --> 00:02:50.960 And now what's the trade off though? 00:02:50.960 --> 00:02:52.560 Nothing just pops out of thin air. 00:02:52.560 --> 00:02:56.245 The trade off is, is that I am going to have to push down for 00:02:56.245 --> 00:02:59.300 a much longer distance, for actually 10 times the distance 00:02:59.300 --> 00:03:02.360 as this object is going to move up. 00:03:02.360 --> 00:03:05.210 And once again I know that because the work in has to 00:03:05.210 --> 00:03:06.000 equal the work out. 00:03:06.000 --> 00:03:10.200 I can't through some magical machine-- and if you were able 00:03:10.200 --> 00:03:12.330 to invent one, you shouldn't watch this video and you 00:03:12.330 --> 00:03:15.840 should go build it and become a trillionaire-- but a machine 00:03:15.840 --> 00:03:18.020 can never generate work out of thin air. 00:03:18.020 --> 00:03:19.660 Or it can never generate energy out of thin air. 00:03:19.660 --> 00:03:21.600 That energy has to come from some place. 00:03:21.600 --> 00:03:23.970 Most machines actually you lose energy to friction or 00:03:23.970 --> 00:03:25.660 whatever else. 00:03:25.660 --> 00:03:30.080 But in this situation, if I'm putting in 10 newtons of force 00:03:30.080 --> 00:03:33.660 times some distance, whatever that quantity is of work, the 00:03:33.660 --> 00:03:34.900 work cannot change. 00:03:34.900 --> 00:03:35.490 The total work. 00:03:35.490 --> 00:03:38.910 It can go down if there is some friction in the system. 00:03:38.910 --> 00:03:40.470 So let's do another problem. 00:03:45.160 --> 00:03:47.480 And really they're all kind of the same formula. 00:03:47.480 --> 00:03:55.670 And then I'll move into a few other types of simple systems. 00:03:55.670 --> 00:03:59.130 I should use the line tool. 00:03:59.130 --> 00:04:03.060 We'll make this up on the fly. 00:04:03.060 --> 00:04:05.180 And you could always create problems where you can 00:04:05.180 --> 00:04:07.800 compound it further and et cetera, et cetera, using some 00:04:07.800 --> 00:04:08.870 of the other concepts we've learned. 00:04:08.870 --> 00:04:10.600 But I won't worry about that right now. 00:04:16.680 --> 00:04:24.630 So let's say that I'm going to push up here. 00:04:24.630 --> 00:04:26.830 Well no let me see what I want to do. 00:04:26.830 --> 00:04:35.380 I want to push down here with a force of-- let's say that 00:04:35.380 --> 00:04:44.000 this distance right here is 35 meters, this distance is 5 00:04:44.000 --> 00:04:46.550 meters-- and let's say I'm going to push down with the 00:04:46.550 --> 00:04:49.730 force of 7 newtons, and what I want to figure out is how 00:04:49.730 --> 00:04:52.190 heavy of an object can I lift here. 00:04:52.190 --> 00:04:53.840 How heavy of an object. 00:04:53.840 --> 00:04:55.560 Well, all we have to do is use the same formula. 00:04:55.560 --> 00:04:58.240 But the moments-- and I know I used that word once before, so 00:04:58.240 --> 00:05:00.670 you might not know what it is-- but the moments on both 00:05:00.670 --> 00:05:02.980 sides of the fulcrum have to be the same. 00:05:02.980 --> 00:05:05.460 Or the input moment has to be the output moment. 00:05:05.460 --> 00:05:06.730 So what's the moment again? 00:05:06.730 --> 00:05:10.560 Well, the moment is just the force times the distance from 00:05:10.560 --> 00:05:12.680 the force to the fulcrum. 00:05:12.680 --> 00:05:19.360 So the input moment is 7 newtons times 35 meters. 00:05:19.360 --> 00:05:21.915 And realize that that does not work, because the distance 00:05:21.915 --> 00:05:24.740 this force is traveling is not 35 meters. 00:05:24.740 --> 00:05:26.240 The distance this force is traveling is 00:05:26.240 --> 00:05:28.470 something like, here. 00:05:28.470 --> 00:05:31.320 But this 35 meters is going to be proportional to the 00:05:31.320 --> 00:05:33.490 distance that this is traveling when you compare it 00:05:33.490 --> 00:05:34.910 to this other side. 00:05:34.910 --> 00:05:36.950 So this quantity, 7 newtons times 35 00:05:36.950 --> 00:05:38.390 meters, is the moment. 00:05:38.390 --> 00:05:41.990 And that is going to be equal to the moment on this side, 00:05:41.990 --> 00:05:43.500 the output moment. 00:05:43.500 --> 00:05:47.780 So that is equal to 5 meters times the force that I'm 00:05:47.780 --> 00:05:51.405 lifting, or the lifting force of the machine, times let's 00:05:51.405 --> 00:05:53.380 say the force out. 00:05:53.380 --> 00:05:56.230 So we can figure out the force out by just dividing both 00:05:56.230 --> 00:05:58.280 sides by 5. 00:05:58.280 --> 00:06:03.310 So let's see, 35 divided by 5 is 7, so you get 7 times 7 00:06:03.310 --> 00:06:07.720 equals the force out, or 49 newtons. 00:06:07.720 --> 00:06:10.280 And you can see that, because you can see that the length of 00:06:10.280 --> 00:06:13.320 this side of the lever is 7 times the length of this side 00:06:13.320 --> 00:06:14.710 of the lever. 00:06:14.710 --> 00:06:18.830 So when you input a force of 7, you output a force of 7 00:06:18.830 --> 00:06:19.650 times that. 00:06:19.650 --> 00:06:23.820 And of course, in order to move the block 1 meter up in 00:06:23.820 --> 00:06:24.760 this direction, you're going to have to 00:06:24.760 --> 00:06:27.210 push down for 7 meters. 00:06:27.210 --> 00:06:31.450 And that's where we know that the input work is equal to the 00:06:31.450 --> 00:06:32.580 output work. 00:06:32.580 --> 00:06:35.920 Well anyway hopefully I didn't confuse you and you have a 00:06:35.920 --> 00:06:38.470 reasonable sense of how levers work. 00:06:38.470 --> 00:06:41.050 In the next couple of videos, I'll introduce you to other 00:06:41.050 --> 00:06:43.680 machines, simple machines like a wedge-- I've always had 00:06:43.680 --> 00:06:45.140 trouble calling a wedge a machine, but it 00:06:45.140 --> 00:06:47.040 is one-- and pulleys. 00:06:47.040 --> 00:06:48.290 I'll see you in the next video.

Introduction to mechanical advantage

https://www.youtube.com/watch?v=pfzJ-z5Ij48

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https://www.youtube.com/api/timedtext?v=pfzJ-z5Ij48&ei=YmeUZZCgLKafxN8Pi-GDoAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5756C65F3FDF8B59A04BB4B626A8F85A24927136.3EA00F915E7715FFA343A44943590A22B8988E16&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.740 --> 00:00:01.530 Welcome back. 00:00:01.530 --> 00:00:04.080 We'll now use a little bit of what we've learned about work 00:00:04.080 --> 00:00:06.540 and energy and the conservation of energy and 00:00:06.540 --> 00:00:08.240 apply it to simple machines. 00:00:08.240 --> 00:00:10.650 And we'll learn a little bit about mechanical advantage. 00:00:10.650 --> 00:00:12.760 So I've drawn a simple lever here. 00:00:12.760 --> 00:00:14.830 And you've probably been exposed to 00:00:14.830 --> 00:00:15.890 simple levers before. 00:00:15.890 --> 00:00:18.040 They're really just kind of like a seesaw. 00:00:18.040 --> 00:00:19.910 This place where the lever pivots. 00:00:19.910 --> 00:00:20.970 This is called a fulcrum. 00:00:20.970 --> 00:00:22.090 Just really the pivot point. 00:00:22.090 --> 00:00:24.480 And you can kind of view this as either a seesaw or a big 00:00:24.480 --> 00:00:26.710 plank of wood on top of a triangle, which essentially is 00:00:26.710 --> 00:00:27.760 what I've drawn. 00:00:27.760 --> 00:00:30.750 So in this example, I have the big plank of wood. 00:00:30.750 --> 00:00:32.729 At one end I have this 10 newton weight, and I've 00:00:32.729 --> 00:00:33.720 written 10 in there. 00:00:33.720 --> 00:00:37.640 And what we're going to figure out is one, how much force-- 00:00:37.640 --> 00:00:39.480 well, we could figure out a couple of things. 00:00:39.480 --> 00:00:42.300 How much force do I have to apply here to 00:00:42.300 --> 00:00:45.430 just keep this level? 00:00:45.430 --> 00:00:49.800 Because this weight's going to be pushing downwards. 00:00:49.800 --> 00:00:52.370 So it would naturally want this whole 00:00:52.370 --> 00:00:54.200 lever to rotate clockwise. 00:00:54.200 --> 00:00:56.540 So what I want to figure out is, how much force do I have 00:00:56.540 --> 00:01:01.860 to apply to either keep the lever level or to actually 00:01:01.860 --> 00:01:04.450 rotate this lever counterclockwise? 00:01:04.450 --> 00:01:06.040 And when I rotate the lever 00:01:06.040 --> 00:01:07.310 counterclockwise, what's happening? 00:01:07.310 --> 00:01:09.810 I'm pushing down on this left-hand side, and I'm 00:01:09.810 --> 00:01:12.490 lifting this 10 newton block. 00:01:12.490 --> 00:01:14.950 So let's do a little thought experiment and see what 00:01:14.950 --> 00:01:18.200 happens after I rotate this lever a little bit. 00:01:18.200 --> 00:01:20.970 So let's say, what I've drawn here in mauve, that's our 00:01:20.970 --> 00:01:22.140 starting position. 00:01:22.140 --> 00:01:25.550 And in yellow, I'm going to draw the finishing position. 00:01:25.550 --> 00:01:27.040 So the finishing position is going to look 00:01:27.040 --> 00:01:28.290 something like this. 00:01:31.560 --> 00:01:34.091 I'll try my best to draw it. 00:01:34.091 --> 00:01:36.120 The finishing position is something like this. 00:01:36.120 --> 00:01:38.890 And also, one thing I want to figure out, that I wanted to 00:01:38.890 --> 00:01:43.000 write, is let's say that the distance, that this distance 00:01:43.000 --> 00:01:47.610 right here, from where I'm applying the force to the 00:01:47.610 --> 00:01:52.245 fulcrum, let's say that that distance is 2. 00:01:52.245 --> 00:01:54.990 And from the fulcrum to the weight that I'm lifting, that 00:01:54.990 --> 00:01:56.020 distance is 1. 00:01:56.020 --> 00:01:57.430 Let's just say that, just for the sake of argument. 00:01:57.430 --> 00:01:59.570 Let's say it's 2 meters and 1 meter, although it could be 2 00:01:59.570 --> 00:02:02.340 kilometers and 1 kilometer, we'll soon see. 00:02:02.340 --> 00:02:05.300 And what I did is I pressed down with some force, and I 00:02:05.300 --> 00:02:08.960 rotated it through an angle theta. 00:02:08.960 --> 00:02:12.540 So that's theta and this is also theta. 00:02:12.540 --> 00:02:14.540 So my question to you, and we'll have to take out a 00:02:14.540 --> 00:02:18.390 little bit of our trigonometry skills, is how much did this 00:02:18.390 --> 00:02:20.060 object move up? 00:02:20.060 --> 00:02:22.700 So essentially, what was this distance? 00:02:22.700 --> 00:02:24.850 What's its distance in the vertical direction? 00:02:24.850 --> 00:02:26.150 How much did it go up? 00:02:26.150 --> 00:02:29.550 And also, for what distance did I have to apply the force 00:02:29.550 --> 00:02:33.070 downwards here-- so that's this distance-- in order for 00:02:33.070 --> 00:02:35.770 this weight to move up this distance over here? 00:02:35.770 --> 00:02:37.680 So let's figure out either one. 00:02:37.680 --> 00:02:40.700 So this distance is what? 00:02:40.700 --> 00:02:41.830 Well, we have theta. 00:02:41.830 --> 00:02:42.870 This is the opposite. 00:02:42.870 --> 00:02:44.410 This is a 90 degree angle, because we 00:02:44.410 --> 00:02:46.060 started off at level. 00:02:46.060 --> 00:02:47.780 So this is opposite. 00:02:47.780 --> 00:02:49.250 And this is what? 00:02:49.250 --> 00:02:52.890 This is the adjacent angle. 00:02:52.890 --> 00:02:53.840 So what do we have there? 00:02:53.840 --> 00:02:54.900 Opposite over adjacent. 00:02:54.900 --> 00:02:59.920 Soh Cah Toa. 00:02:59.920 --> 00:03:01.760 Opposite over adjacent. 00:03:01.760 --> 00:03:03.750 Opposite over adjacent. 00:03:03.750 --> 00:03:05.490 That's Toa, or tangent. 00:03:05.490 --> 00:03:11.780 So in this situation, we know that the tangent of theta is 00:03:11.780 --> 00:03:19.070 equal to-- let's call this the distance 00:03:19.070 --> 00:03:20.702 that we move the weight. 00:03:20.702 --> 00:03:21.400 soon. 00:03:21.400 --> 00:03:25.480 So that equals opposite over adjacent, the distance that we 00:03:25.480 --> 00:03:29.050 moved the weight over 1. 00:03:29.050 --> 00:03:30.490 And then if we go on to this side, we 00:03:30.490 --> 00:03:32.070 can do the same thing. 00:03:32.070 --> 00:03:34.110 Tangent is opposite over adjacent. 00:03:34.110 --> 00:03:38.530 So let's call this the distance of the force. 00:03:38.530 --> 00:03:41.030 So here the opposite of the distance of the force and the 00:03:41.030 --> 00:03:45.240 adjacent is this 2 meters. 00:03:45.240 --> 00:03:47.340 Because this is the hypotenuse right here. 00:03:47.340 --> 00:03:52.530 So we also have the tangent of theta-- now you're using this 00:03:52.530 --> 00:03:55.030 triangle-- is equal to the opposite side. 00:03:55.030 --> 00:04:00.250 The distance of the force over 2 meters. 00:04:00.250 --> 00:04:01.020 So this is interesting. 00:04:01.020 --> 00:04:03.500 They're both equal to tangent of theta. 00:04:03.500 --> 00:04:04.530 We don't even have to figure out what the 00:04:04.530 --> 00:04:05.380 tangent of theta is. 00:04:05.380 --> 00:04:11.190 We know that this quantity is equal to this quantity. 00:04:11.190 --> 00:04:12.150 And we can write it here. 00:04:12.150 --> 00:04:16.250 We could write the distance of the force, that's the distance 00:04:16.250 --> 00:04:18.089 that we had to push down on the side of the lever 00:04:18.089 --> 00:04:22.460 downwards, over 2, is equal to the distance of the weight. 00:04:22.460 --> 00:04:25.070 The distance the weight traveled upwards is equal to 00:04:25.070 --> 00:04:30.350 the distance, the weight, divided by 1. 00:04:30.350 --> 00:04:33.110 Or we could say-- this 1 we can ignore. 00:04:33.110 --> 00:04:34.500 Something divided by 1 is just 1. 00:04:34.500 --> 00:04:37.130 Or we could say that the distance of the force is equal 00:04:37.130 --> 00:04:41.100 to 2 times the distance of the weight. 00:04:41.100 --> 00:04:44.270 And this is interesting, because now we can apply what 00:04:44.270 --> 00:04:47.870 we just learned here to figure out what the force was. 00:04:47.870 --> 00:04:48.750 And how do I do that? 00:04:48.750 --> 00:04:51.500 Well, when I'm applying a force here, over some 00:04:51.500 --> 00:04:53.950 distance, I'm putting energy into the system. 00:04:53.950 --> 00:04:54.710 I'm doing work. 00:04:54.710 --> 00:04:58.260 Work is just a transfer of energy into this machine. 00:04:58.260 --> 00:05:00.100 And when I do that, that machine is actually 00:05:00.100 --> 00:05:02.270 transferring that energy to this block. 00:05:02.270 --> 00:05:05.670 It's actually doing work on the block by lifting it up. 00:05:05.670 --> 00:05:08.640 So we know the law of conservation of energy, and 00:05:08.640 --> 00:05:11.280 we're assuming that this is a frictionless system, and that 00:05:11.280 --> 00:05:13.730 nothing is being lost to heat or whatever else. 00:05:13.730 --> 00:05:17.000 So the work in has to be equal to the work out. 00:05:17.000 --> 00:05:18.590 And so what's the work in? 00:05:18.590 --> 00:05:21.310 Well, it's the force that I'm applying downward times the 00:05:21.310 --> 00:05:22.430 distance of the force. 00:05:22.430 --> 00:05:24.670 So this is the work in. 00:05:24.670 --> 00:05:27.500 Force times the distance of the force. 00:05:27.500 --> 00:05:28.880 I'm going to switch colors just to keep things 00:05:28.880 --> 00:05:30.050 interesting. 00:05:30.050 --> 00:05:34.120 And that has to be the same thing as the work out. 00:05:34.120 --> 00:05:36.770 Well, what's the work out? 00:05:36.770 --> 00:05:41.190 It's the force of the weight pulling downwards. 00:05:41.190 --> 00:05:43.700 So we have to-- it's essentially the lifting force 00:05:43.700 --> 00:05:45.330 of the lever. 00:05:45.330 --> 00:05:47.320 It has to counteract the force of the weight pulling 00:05:47.320 --> 00:05:47.950 downwards actually. 00:05:47.950 --> 00:05:49.610 Sorry I mis-said it a little bit. 00:05:49.610 --> 00:05:52.030 But this lever is essentially going to be 00:05:52.030 --> 00:05:53.820 pushing up on this weight. 00:05:53.820 --> 00:05:55.190 The weight ends up here. 00:05:55.190 --> 00:05:57.200 So it pushes up with the force equal to the 00:05:57.200 --> 00:05:58.300 weight of the object. 00:05:58.300 --> 00:06:01.040 So that's the weight of the object, which is -- I said 00:06:01.040 --> 00:06:04.440 it's a 10 newton object -- So it's equal to 10 newtons. 00:06:04.440 --> 00:06:05.250 That's the force. 00:06:05.250 --> 00:06:06.190 The upward force here. 00:06:06.190 --> 00:06:08.630 And it does that for a distance of what? 00:06:08.630 --> 00:06:11.480 We figured out this object, this weight, moves up with a 00:06:11.480 --> 00:06:12.730 distance d sub w. 00:06:18.200 --> 00:06:22.380 And we know what the distance of the force is in terms of 00:06:22.380 --> 00:06:24.760 the distance of w. 00:06:24.760 --> 00:06:29.360 So we could rewrite this as force times, substitute here, 00:06:29.360 --> 00:06:35.930 2 d w is equal to 10 d w. 00:06:35.930 --> 00:06:43.830 Divide both sides by 2 you d w and you get force is equal to 00:06:43.830 --> 00:06:50.150 10 d w 2 two d w, which is equaled to, d w's cancel out, 00:06:50.150 --> 00:06:52.510 and you're just left with 5. 00:06:52.510 --> 00:06:54.090 So this is interesting. 00:06:54.090 --> 00:06:56.500 And I think you'll see where this is going, and we did it 00:06:56.500 --> 00:06:57.490 little complicated this time. 00:06:57.490 --> 00:07:01.490 But hopefully you'll realize a general theme. 00:07:01.490 --> 00:07:03.790 This was a 10 newton weight. 00:07:03.790 --> 00:07:06.700 And I only had to press down with 5 newtons in order to 00:07:06.700 --> 00:07:08.460 lift it up. 00:07:08.460 --> 00:07:11.500 But at the same time, I pressed down with 5 newtons, 00:07:11.500 --> 00:07:14.880 but I had to push down for twice as long. 00:07:14.880 --> 00:07:17.880 So my force was half as much, but my distance that I had to 00:07:17.880 --> 00:07:20.190 push was twice as much. 00:07:20.190 --> 00:07:25.470 And here the force is twice as much but the distance it 00:07:25.470 --> 00:07:27.480 traveled is half as much. 00:07:27.480 --> 00:07:29.350 So what essentially just happened here is, I 00:07:29.350 --> 00:07:30.640 multiplied my force. 00:07:30.640 --> 00:07:32.490 And because I multiplied my force, I 00:07:32.490 --> 00:07:35.100 essentially lost some distance. 00:07:35.100 --> 00:07:36.710 But I multiplied my force, because I 00:07:36.710 --> 00:07:38.320 inputted a 5 newton force. 00:07:38.320 --> 00:07:41.200 And I got a 10 newton force out, although the 10 newton 00:07:41.200 --> 00:07:43.030 force traveled for less distance. 00:07:43.030 --> 00:07:44.230 Because the work was constant. 00:07:44.230 --> 00:07:46.460 And this is called mechanical advantage. 00:07:46.460 --> 00:07:50.450 If I have an input force of 5, and I get an output force of 00:07:50.450 --> 00:07:53.250 10, the mechanical advantage is 2. 00:07:53.250 --> 00:07:57.660 So mechanical advantage is equal to output force over 00:07:57.660 --> 00:07:59.260 input force, and that should hopefully make a little 00:07:59.260 --> 00:08:00.910 intuitive sense to you. 00:08:00.910 --> 00:08:03.220 And another thing that maybe you're starting to realize 00:08:03.220 --> 00:08:07.110 now, is that proportion of the mechanical advantage was 00:08:07.110 --> 00:08:13.480 actually the ratio of this length to this length. 00:08:13.480 --> 00:08:15.660 And we figured that out by taking the tangent and doing 00:08:15.660 --> 00:08:16.670 these ratios. 00:08:16.670 --> 00:08:20.200 But in general, it makes sense, because this force 00:08:20.200 --> 00:08:23.340 times this distance has to be equal to this 00:08:23.340 --> 00:08:25.820 force times this distance. 00:08:25.820 --> 00:08:29.410 And we know that the distance this goes up is proportional 00:08:29.410 --> 00:08:34.169 to the length of from the fulcrum to the weight. 00:08:34.169 --> 00:08:36.340 And we know on this side the distance that you're pushing 00:08:36.340 --> 00:08:39.260 down, is proportional to the length from where you're 00:08:39.260 --> 00:08:42.020 applying the weight to the fulcrum. 00:08:42.020 --> 00:08:45.640 And now I'll introduce you to a concept of moments. 00:08:45.640 --> 00:08:46.890 In just a moment. 00:08:49.470 --> 00:08:52.240 So in general, if I have, and this is really all you have to 00:08:52.240 --> 00:08:54.350 learn, that last thought exercise was just 00:08:54.350 --> 00:08:55.370 to show it to you. 00:08:55.370 --> 00:09:03.740 If I have a fulcrum here, and if we call this distance d 1 00:09:03.740 --> 00:09:06.590 and we called this distance d 2. 00:09:06.590 --> 00:09:12.335 And if I want to apply an upward force here, 00:09:12.335 --> 00:09:15.770 let's call this f 1. 00:09:15.770 --> 00:09:20.830 And I have a downward force, f 2, in this machine. 00:09:20.830 --> 00:09:27.970 f 2 times d 2 is equal to d 1 times f 1. 00:09:27.970 --> 00:09:30.220 And this is really all you need to know. 00:09:30.220 --> 00:09:33.630 And this just all falls out of the work in is 00:09:33.630 --> 00:09:34.880 equal to the work out. 00:09:34.880 --> 00:09:37.540 Now, this quantity isn't exactly the work in. 00:09:37.540 --> 00:09:41.520 The work in was this force-- sorry, F2-- is this force 00:09:41.520 --> 00:09:43.130 times this distance. 00:09:43.130 --> 00:09:47.370 But this distance is proportional to this distance, 00:09:47.370 --> 00:09:48.610 and that's what you need to realize. 00:09:48.610 --> 00:09:51.610 And this quantity right here is actually called the moment. 00:09:51.610 --> 00:09:54.560 In the next video, which I'll start very soon because this 00:09:54.560 --> 00:09:55.730 video is about to end. 00:09:55.730 --> 00:09:56.660 I'm running out of time. 00:09:56.660 --> 00:09:59.330 I will use these quantities to solve a bunch of mechanical 00:09:59.330 --> 00:10:01.560 advantage problems. See

Work/energy problem with friction

https://www.youtube.com/watch?v=YvacYWgygaA

vtt

https://www.youtube.com/api/timedtext?v=YvacYWgygaA&ei=YmeUZYahLIqjmLAP5pm4uAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=49CFAAD88B1E0C208DA46D3F434DE65FE2481F64.529DAF1105597AE357BE9412777A5EE32EDE3603&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:01.580 Welcome back. 00:00:00.000 --> 00:00:01.580 Welcome back. Welcome back. 00:00:01.580 --> 00:00:03.540 I'll now do another conservation of energy 00:00:03.540 --> 00:00:06.110 problem, and this time I'll add another twist. So far, 00:00:06.110 --> 00:00:08.530 everything we've been doing, energy was conserved by the 00:00:08.530 --> 00:00:09.970 law of conservation. 00:00:09.970 --> 00:00:11.740 But that's because all of the forces that were acting in 00:00:11.740 --> 00:00:13.810 these systems were conservative forces. 00:00:13.810 --> 00:00:15.750 And now I'll introduce you to a problem that has a little 00:00:15.750 --> 00:00:17.810 bit of friction, and we'll see that some of that energy gets 00:00:17.810 --> 00:00:18.720 lost to friction. 00:00:18.720 --> 00:00:20.340 And we can think about it a little bit. 00:00:20.340 --> 00:00:22.470 Well where does that energy go? 00:00:22.470 --> 00:00:23.980 And I'm getting this problem from the University of 00:00:23.980 --> 00:00:25.230 Oregon's zebu.uoregon.edu. 00:00:27.990 --> 00:00:30.590 And they seem to have some nice physics problems, so I'll 00:00:30.590 --> 00:00:31.120 use theirs. 00:00:31.120 --> 00:00:32.530 And I just want to make sure they get credit. 00:00:32.530 --> 00:00:32.860 So let's see. 00:00:32.860 --> 00:00:35.910 They say a 90 kilogram bike and rider. 00:00:35.910 --> 00:00:38.725 So the bike and rider combined are 90 kilograms. So let's 00:00:38.725 --> 00:00:45.920 just say the mass is 90 kilograms. Start at rest from 00:00:45.920 --> 00:00:48.840 the top of a 500 meter long hill. 00:00:48.840 --> 00:00:50.860 OK, so I think they mean that the hill is 00:00:50.860 --> 00:00:51.710 something like this. 00:00:51.710 --> 00:00:57.550 So if this is the hill, that the hypotenuse here is 500 00:00:57.550 --> 00:00:58.270 hundred meters long. 00:00:58.270 --> 00:01:04.160 So the length of that, this is 500 meters. 00:01:04.160 --> 00:01:07.420 A 500 meter long hill with a 5 degree incline. 00:01:07.420 --> 00:01:08.670 So this is 5 degrees. 00:01:11.850 --> 00:01:14.370 And we can kind of just view it like a wedge, like we've 00:01:14.370 --> 00:01:18.070 done in other problems. There you go. 00:01:18.070 --> 00:01:20.890 That's pretty straight. 00:01:20.890 --> 00:01:22.550 OK. 00:01:22.550 --> 00:01:26.050 Assuming an average friction force of 60 newtons. 00:01:26.050 --> 00:01:28.490 OK, so they're not telling us the coefficient of friction 00:01:28.490 --> 00:01:29.580 and then we have to figure out the normal 00:01:29.580 --> 00:01:30.620 force and all of that. 00:01:30.620 --> 00:01:33.630 They're just telling us, what is the drag of friction? 00:01:33.630 --> 00:01:36.430 Or how much is actually friction acting against this 00:01:36.430 --> 00:01:38.750 rider's motion? 00:01:38.750 --> 00:01:40.360 We could think a little bit about where that friction is 00:01:40.360 --> 00:01:40.880 coming from. 00:01:40.880 --> 00:01:46.660 So the force of friction is equal to 60 newtons And of 00:01:46.660 --> 00:01:49.030 course, this is going to be going against his motion or 00:01:49.030 --> 00:01:50.040 her motion. 00:01:50.040 --> 00:01:53.150 And the question asks us, find the speed of the biker at the 00:01:53.150 --> 00:01:54.110 bottom of the hill. 00:01:54.110 --> 00:01:58.510 So the biker starts up here, stationary. 00:01:58.510 --> 00:01:59.240 That's the biker. 00:01:59.240 --> 00:02:01.560 My very artful rendition of the biker. 00:02:01.560 --> 00:02:05.230 And we need to figure out the velocity at the bottom. 00:02:05.230 --> 00:02:07.920 This to some degree is a potential energy problem. 00:02:11.380 --> 00:02:14.790 It's definitely a conservation of mechanical energy problem. 00:02:14.790 --> 00:02:17.610 So let's figure out what the energy of the system is when 00:02:17.610 --> 00:02:19.550 the rider starts off. 00:02:19.550 --> 00:02:21.360 So the rider starts off at the top of this hill. 00:02:21.360 --> 00:02:23.930 So definitely some potential energy. 00:02:23.930 --> 00:02:26.850 And is stationary, so there's no kinetic energy. 00:02:26.850 --> 00:02:28.760 So all of the energy is potential, and what is the 00:02:28.760 --> 00:02:29.600 potential energy? 00:02:29.600 --> 00:02:35.030 Well potential energy is equal to mass times the acceleration 00:02:35.030 --> 00:02:37.950 of gravity times height, right? 00:02:37.950 --> 00:02:41.300 Well that's equal to, if the mass is 90, the acceleration 00:02:41.300 --> 00:02:43.830 of gravity is 9.8 meters per second squared. 00:02:43.830 --> 00:02:44.540 And then what's the height? 00:02:44.540 --> 00:02:45.960 Well here we're going to have to break out a little 00:02:45.960 --> 00:02:46.810 trigonometry. 00:02:46.810 --> 00:02:51.010 We need to figure out this side of this triangle, if you 00:02:51.010 --> 00:02:53.270 consider this whole thing a triangle. 00:02:53.270 --> 00:02:53.630 Let's see. 00:02:53.630 --> 00:02:55.080 We want to figure out the opposite. 00:02:55.080 --> 00:02:58.630 We know the hypotenuse and we know this angle here. 00:02:58.630 --> 00:03:01.920 So the sine of this angle is equal to opposite over 00:03:01.920 --> 00:03:03.440 hypotenuse. 00:03:03.440 --> 00:03:04.680 So, SOH. 00:03:04.680 --> 00:03:06.910 Sine is opposite over hypotenuse. 00:03:06.910 --> 00:03:10.490 So we know that the height-- so let me do a little work 00:03:10.490 --> 00:03:15.010 here-- we know that sine of 5 degrees is equal to 00:03:15.010 --> 00:03:17.510 the height over 500. 00:03:17.510 --> 00:03:25.420 Or that the height is equal to 500 sine of 5 degrees. 00:03:25.420 --> 00:03:28.470 And I calculated the sine of 5 degrees ahead of time. 00:03:28.470 --> 00:03:30.380 Let me make sure I still have it. 00:03:30.380 --> 00:03:32.820 That's cause I didn't have my calculator with me today. 00:03:32.820 --> 00:03:34.290 But you could do this on your own. 00:03:34.290 --> 00:03:38.400 So this is equal to 500, and the sine of 00:03:38.400 --> 00:03:42.806 5 degrees is 0.087. 00:03:42.806 --> 00:03:46.870 So when you multiply these out, what do I get? 00:03:46.870 --> 00:03:49.220 I'm using the calculator on Google actually. 00:03:49.220 --> 00:03:52.400 500 times sine. 00:03:52.400 --> 00:03:54.670 You get 43.6. 00:03:54.670 --> 00:03:58.900 So this is equal to 43.6. 00:03:58.900 --> 00:04:03.970 So the height of the hill is 43.6 meters. 00:04:03.970 --> 00:04:06.390 So going back to the potential energy, we have the mass times 00:04:06.390 --> 00:04:07.850 the acceleration of gravity times the height. 00:04:07.850 --> 00:04:09.740 Times 43.6. 00:04:09.740 --> 00:04:12.290 And this is equal to, and then I can use just my regular 00:04:12.290 --> 00:04:13.640 calculator since I don't have to figure out 00:04:13.640 --> 00:04:15.090 trig functions anymore. 00:04:15.090 --> 00:04:25.380 So 90-- so you can see the whole thing-- times 9.8 times 00:04:25.380 --> 00:04:34.860 43.6 is equal to, let's see, roughly 38,455. 00:04:34.860 --> 00:04:42.180 So this is equal to 38,455 joules or newton meters. 00:04:42.180 --> 00:04:44.270 And that's a lot of potential energy. 00:04:44.270 --> 00:04:44.980 So what happens? 00:04:44.980 --> 00:04:47.690 At the bottom of the hill-- sorry, I have to readjust my 00:04:47.690 --> 00:04:52.070 chair-- at the bottom of the hill, all of this gets 00:04:52.070 --> 00:04:53.590 converted to, or maybe I should 00:04:53.590 --> 00:04:54.260 pose that as a question. 00:04:54.260 --> 00:04:57.450 Does all of it get converted to kinetic energy? 00:04:57.450 --> 00:05:00.560 Almost. We have a force of friction here. 00:05:00.560 --> 00:05:03.880 And friction, you can kind of view friction as something 00:05:03.880 --> 00:05:08.120 that eats up mechanical energy. 00:05:08.120 --> 00:05:10.250 These are also called nonconservative forces because 00:05:10.250 --> 00:05:12.590 when you have these forces at play, all of the 00:05:12.590 --> 00:05:14.580 force is not conserved. 00:05:14.580 --> 00:05:19.160 So a way to think about it is, is that the energy, let's just 00:05:19.160 --> 00:05:20.430 call it total energy. 00:05:20.430 --> 00:05:30.670 So let's say total energy initial, well let me just 00:05:30.670 --> 00:05:40.740 write initial energy is equal to the energy wasted in 00:05:40.740 --> 00:05:51.870 friction-- I should have written just letters-- from 00:05:51.870 --> 00:06:00.170 friction plus final energy. 00:06:00.170 --> 00:06:03.670 So we know what the initial energy is in this system. 00:06:03.670 --> 00:06:07.120 That's the potential energy of this bicyclist and this 00:06:07.120 --> 00:06:12.730 roughly 38 and 1/2 kilojoules or 38,500 joules, roughly. 00:06:12.730 --> 00:06:17.530 And now let's figure out the energy wasted from friction, 00:06:17.530 --> 00:06:20.110 and the energy wasted from friction is the negative work 00:06:20.110 --> 00:06:20.850 that friction does. 00:06:20.850 --> 00:06:22.470 And what does negative work mean? 00:06:22.470 --> 00:06:29.000 Well the bicyclist is moving 500 meters in this direction. 00:06:29.000 --> 00:06:30.740 So distance is 500 meters. 00:06:30.740 --> 00:06:33.050 But friction isn't acting along the same 00:06:33.050 --> 00:06:34.290 direction as distance. 00:06:34.290 --> 00:06:37.470 The whole time, friction is acting against the distance. 00:06:37.470 --> 00:06:40.690 So when the force is going in the opposite direction as the 00:06:40.690 --> 00:06:42.430 distance, your work is negative. 00:06:45.440 --> 00:06:49.500 So another way of thinking of this problem is energy initial 00:06:49.500 --> 00:06:56.420 is equal to, or you could say the energy initial plus the 00:06:56.420 --> 00:06:58.370 negative work of friction, right? 00:06:58.370 --> 00:07:02.090 If we say that this is a negative quantity, then this 00:07:02.090 --> 00:07:05.730 is equal to the final energy. 00:07:05.730 --> 00:07:07.920 And here, I took the friction and put it on the other side 00:07:07.920 --> 00:07:10.410 because I said this is going to be a negative quantity in 00:07:10.410 --> 00:07:11.290 the system. 00:07:11.290 --> 00:07:13.680 And so you should always just make sure that if you have 00:07:13.680 --> 00:07:15.770 friction in the system, just as a reality check, that your 00:07:15.770 --> 00:07:18.100 final energy is less than your initial energy. 00:07:18.100 --> 00:07:25.500 Our initial energy is, let's just say 38.5 kilojoules. 00:07:25.500 --> 00:07:28.290 What is the negative work that friction is doing? 00:07:28.290 --> 00:07:29.530 Well it's 500 meters. 00:07:29.530 --> 00:07:33.600 And the entire 500 meters, it's always pushing back on 00:07:33.600 --> 00:07:36.420 the rider with a force of 60 newtons. 00:07:36.420 --> 00:07:37.960 So force times distance. 00:07:37.960 --> 00:07:40.690 So it's minus 60 newtons, cause it's going in the 00:07:40.690 --> 00:07:44.530 opposite direction of the motion, times 500. 00:07:44.530 --> 00:07:49.000 And this is going to equal the ending, oh, no. 00:07:49.000 --> 00:07:53.460 This is going to equal the final energy, right? 00:07:53.460 --> 00:07:54.030 And what is this? 00:07:54.030 --> 00:08:00.100 60 times 500, that's 3,000. 00:08:00.100 --> 00:08:01.190 No, 30,000, right. 00:08:01.190 --> 00:08:04.975 So let's subtract 30,000 from 38,500. 00:08:04.975 --> 00:08:06.490 So let's see. 00:08:06.490 --> 00:08:09.580 Minus 30. 00:08:09.580 --> 00:08:10.460 I didn't have to do that. 00:08:10.460 --> 00:08:11.780 I could have done that in my head. 00:08:11.780 --> 00:08:19.990 So that gives us 8,455 joules is equal to the final energy. 00:08:19.990 --> 00:08:21.820 And what is all the final energy? 00:08:21.820 --> 00:08:24.530 Well by this time, the rider's gotten back to, I guess we 00:08:24.530 --> 00:08:25.410 could call the sea level. 00:08:25.410 --> 00:08:26.890 So all of the energy is now going to be 00:08:26.890 --> 00:08:28.930 kinetic energy, right? 00:08:28.930 --> 00:08:30.510 What's the formula for kinetic energy? 00:08:30.510 --> 00:08:35.059 It's 1/2 mv squared. 00:08:35.059 --> 00:08:36.580 And we know what m is. 00:08:36.580 --> 00:08:38.126 The mass of the rider is 90. 00:08:38.126 --> 00:08:42.480 So we have this is 90. 00:08:42.480 --> 00:08:44.610 So if we divide both sides. 00:08:44.610 --> 00:08:45.850 So the 1/2 times 90. 00:08:45.850 --> 00:08:48.160 That's 45. 00:08:48.160 --> 00:08:50.140 So 8,455 divided by 45. 00:08:50.140 --> 00:08:59.540 So we get v squared is equal to 187.9. 00:08:59.540 --> 00:09:01.500 And let's take the square root of that and we get the 00:09:01.500 --> 00:09:06.690 velocity, 13.7. 00:09:06.690 --> 00:09:09.340 So if we take the square root of both sides of this, so the 00:09:09.340 --> 00:09:12.900 final velocity is 13.7. 00:09:12.900 --> 00:09:14.270 I know you can't read that. 00:09:14.270 --> 00:09:17.510 13.7 meters per second. 00:09:17.510 --> 00:09:20.960 And this was a slightly more interesting problem because 00:09:20.960 --> 00:09:24.480 here we had the energy wasn't completely conserved. 00:09:24.480 --> 00:09:27.460 Some of the energy, you could say, was eaten by friction. 00:09:27.460 --> 00:09:28.700 And actually that energy just didn't 00:09:28.700 --> 00:09:30.080 disappear into a vacuum. 00:09:30.080 --> 00:09:32.860 It was actually generated into heat. 00:09:32.860 --> 00:09:33.490 And it makes sense. 00:09:33.490 --> 00:09:37.930 If you slid down a slide of sandpaper, your pants would 00:09:37.930 --> 00:09:40.750 feel very warm by the time you got to the bottom of that. 00:09:40.750 --> 00:09:43.400 But the friction of this, they weren't specific on where the 00:09:43.400 --> 00:09:45.420 friction came from, but it could have come from the 00:09:45.420 --> 00:09:46.420 gearing within the bike. 00:09:46.420 --> 00:09:48.550 It could have come from the wind. 00:09:48.550 --> 00:09:50.470 Maybe the bike actually skidded a little 00:09:50.470 --> 00:09:51.280 bit on the way down. 00:09:51.280 --> 00:09:52.190 I don't know. 00:09:52.190 --> 00:09:54.290 But hopefully you found that a little bit interesting. 00:09:54.290 --> 00:09:57.230 And now you can not only work with conservation of 00:09:57.230 --> 00:09:59.950 mechanical energy, but you can work problems where there's a 00:09:59.950 --> 00:10:01.820 little bit of friction involved as well. 00:10:01.820 --> 00:10:03.380 Anyway, I'll see you in the next video.

Conservation of energy

https://www.youtube.com/watch?v=kw_4Loo1HR4

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https://www.youtube.com/api/timedtext?v=kw_4Loo1HR4&ei=YmeUZeXsLLfjxN8P1aC5kAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=AF0EE7E32DDDBB3309DE91E3397969FF01398310.E19E31F330A39C9A60942EFE0FCC542595B90538&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:01.550 Welcome back. 00:00:01.550 --> 00:00:02.935 At the end of the last video, I left you 00:00:02.935 --> 00:00:03.990 with a bit of a question. 00:00:03.990 --> 00:00:08.370 We had a situation where we had a 1 kilogram object. 00:00:08.370 --> 00:00:11.890 This is the 1 kilogram object, which I've drawn neater in 00:00:11.890 --> 00:00:12.955 this video. 00:00:12.955 --> 00:00:15.910 That is 1 kilogram. 00:00:15.910 --> 00:00:20.110 And we're on earth, and I need to mention that because 00:00:20.110 --> 00:00:22.380 gravity is different from planet to planet. 00:00:22.380 --> 00:00:24.670 But as I mentioned, I'm holding it. 00:00:24.670 --> 00:00:27.200 Let's say I'm holding it 10 meters above the ground. 00:00:27.200 --> 00:00:34.590 So this distance or this height is 10 meters. 00:00:34.590 --> 00:00:39.600 And we're assuming the acceleration of gravity, which 00:00:39.600 --> 00:00:42.480 we also write as just g, let's assume it's just 10 meters per 00:00:42.480 --> 00:00:44.680 second squared just for the simplicity of the math instead 00:00:44.680 --> 00:00:45.880 of the 9.8. 00:00:45.880 --> 00:00:48.560 So what we learned in the last video is that the potential 00:00:48.560 --> 00:00:53.690 energy in this situation, the potential energy, which equals 00:00:53.690 --> 00:00:59.740 m times g times h is equal to the mass is 1 kilogram times 00:00:59.740 --> 00:01:02.300 the acceleration of gravity, which is 10 00:01:02.300 --> 00:01:03.490 meters per second squared. 00:01:03.490 --> 00:01:05.349 I'm not going to write the units down just to save space, 00:01:05.349 --> 00:01:08.890 although you should do this when you do it on your test. 00:01:08.890 --> 00:01:11.920 And then the height is 10 meters. 00:01:11.920 --> 00:01:13.910 And the units, if you work them all out, it's in newton 00:01:13.910 --> 00:01:17.960 meters or joules and so it's equal to 100 joules. 00:01:17.960 --> 00:01:20.400 That's the potential energy when I'm holding it up there. 00:01:20.400 --> 00:01:22.290 And I asked you, well when I let go, what happens? 00:01:22.290 --> 00:01:24.490 Well the block obviously will start falling. 00:01:24.490 --> 00:01:26.370 And not only falling, it will start accelerating to the 00:01:26.370 --> 00:01:29.870 ground at 10 meters per second squared roughly. 00:01:29.870 --> 00:01:33.940 And right before it hits the ground-- let me draw that in 00:01:33.940 --> 00:01:38.640 brown for ground-- right before the object hits the 00:01:38.640 --> 00:01:43.130 ground or actually right when it hits the ground, what will 00:01:43.130 --> 00:01:46.490 be the potential energy of the object? 00:01:46.490 --> 00:01:48.070 Well it has no height, right? 00:01:48.070 --> 00:01:49.790 Potential energy is mgh. 00:01:49.790 --> 00:01:51.810 The mass and the acceleration of gravity stay the same, but 00:01:51.810 --> 00:01:52.470 the height is 0. 00:01:52.470 --> 00:01:54.460 So they're all multiplied by each other. 00:01:54.460 --> 00:01:56.540 So down here, the potential energy is going 00:01:56.540 --> 00:01:58.360 to be equal to 0. 00:01:58.360 --> 00:02:00.120 And I told you in the last video that we have the law of 00:02:00.120 --> 00:02:01.160 conservation of energy. 00:02:01.160 --> 00:02:02.950 That energy is conserved. 00:02:02.950 --> 00:02:04.510 It cannot be created or destroyed. 00:02:04.510 --> 00:02:06.920 It can just be converted from one form to another. 00:02:06.920 --> 00:02:10.919 But I'm just showing you, this object had 100 joules of 00:02:10.919 --> 00:02:12.010 energy or, in this case, 00:02:12.010 --> 00:02:13.920 gravitational potential energy. 00:02:13.920 --> 00:02:16.820 And down here, it has no energy. 00:02:16.820 --> 00:02:18.780 Or at least it has no gravitational potential 00:02:18.780 --> 00:02:20.130 energy, and that's the key. 00:02:20.130 --> 00:02:22.800 That gravitational potential energy was converted into 00:02:22.800 --> 00:02:23.920 something else. 00:02:23.920 --> 00:02:25.480 And that something else it was converted 00:02:25.480 --> 00:02:28.070 into is kinetic energy. 00:02:28.070 --> 00:02:32.800 And in this case, since it has no potential energy, all of 00:02:32.800 --> 00:02:35.580 that previous potential energy, all of this 100 joules 00:02:35.580 --> 00:02:39.880 that it has up here is now going to be converted into 00:02:39.880 --> 00:02:40.800 kinetic energy. 00:02:40.800 --> 00:02:43.620 And we can use that information to figure out its 00:02:43.620 --> 00:02:46.980 velocity right before it hits the ground. 00:02:46.980 --> 00:02:47.760 So how do we do that? 00:02:47.760 --> 00:02:49.770 Well what's the formula for kinetic energy? 00:02:49.770 --> 00:02:52.615 And we solved it two videos ago, and hopefully it 00:02:52.615 --> 00:02:53.677 shouldn't be too much of a mystery to you. 00:02:53.677 --> 00:02:55.840 It's something good to memorize, but it's also good 00:02:55.840 --> 00:02:58.870 to know how we got it and go back two videos if you forgot. 00:03:01.430 --> 00:03:05.440 So first we know that all the potential energy was converted 00:03:05.440 --> 00:03:07.450 into kinetic energy. 00:03:07.450 --> 00:03:10.330 We had 100 joules of potential energy, so we're still going 00:03:10.330 --> 00:03:12.210 to have 100 joules, but now all of it's going to be 00:03:12.210 --> 00:03:13.350 kinetic energy. 00:03:13.350 --> 00:03:17.810 And kinetic energy is 1/2 mv squared. 00:03:17.810 --> 00:03:21.080 So we know that 1/2 mv squared, or the kinetic 00:03:21.080 --> 00:03:24.990 energy, is now going to equal 100 joules. 00:03:24.990 --> 00:03:25.710 What's the mass? 00:03:25.710 --> 00:03:27.960 The mass is 1. 00:03:27.960 --> 00:03:29.930 And we can solve for v now. 00:03:29.930 --> 00:03:31.910 1/2 v squared equals 100 joules, and v 00:03:31.910 --> 00:03:36.050 squared is equal to 200. 00:03:36.050 --> 00:03:39.310 And then we get v is equal to square root of 200, which is 00:03:39.310 --> 00:03:40.560 something over 14. 00:03:40.560 --> 00:03:41.860 We can get the exact number. 00:03:41.860 --> 00:03:46.725 Let's see, 200 square root, 14.1 roughly. 00:03:46.725 --> 00:03:51.360 The velocity is going to be 14.1 meters per 00:03:51.360 --> 00:03:52.870 second squared downwards. 00:03:52.870 --> 00:03:55.500 Right before the object touches the ground. 00:03:55.500 --> 00:03:56.930 Right before it touches the ground. 00:03:56.930 --> 00:03:59.180 And you might say, well Sal that's nice and everything. 00:03:59.180 --> 00:04:01.170 We learned a little bit about energy. 00:04:01.170 --> 00:04:02.890 I could have solved that or hopefully you could have 00:04:02.890 --> 00:04:04.990 solved that problem just using your kinematics formula. 00:04:04.990 --> 00:04:07.550 So what's the whole point of introducing 00:04:07.550 --> 00:04:09.590 these concepts of energy? 00:04:09.590 --> 00:04:11.280 And I will now show you. 00:04:11.280 --> 00:04:14.640 So let's say they have the same 1 kilogram object up here 00:04:14.640 --> 00:04:17.529 and it's 10 meters in the air, but I'm going to change things 00:04:17.529 --> 00:04:18.779 a little bit. 00:04:20.660 --> 00:04:24.970 Let me see if I can competently erase all of this. 00:04:24.970 --> 00:04:26.940 Nope, that's not what I wanted to do. 00:04:29.920 --> 00:04:31.900 OK, there you go. 00:04:31.900 --> 00:04:37.150 I'm trying my best to erase this, all of this stuff. 00:04:37.150 --> 00:04:38.570 OK. 00:04:38.570 --> 00:04:40.770 So I have the same object. 00:04:40.770 --> 00:04:43.025 It's still 10 meters in the air and I'll 00:04:43.025 --> 00:04:45.390 write that in a second. 00:04:45.390 --> 00:04:47.160 And I'm just holding it there and I'm still going to drop 00:04:47.160 --> 00:04:50.130 it, but something interesting is going to happen. 00:04:50.130 --> 00:04:51.880 Instead of it going straight down, it's actually going to 00:04:51.880 --> 00:04:54.610 drop on this ramp of ice. 00:04:57.180 --> 00:04:58.710 The ice has lumps on it. 00:05:01.600 --> 00:05:02.660 And then this is the bottom. 00:05:02.660 --> 00:05:04.510 This is the ground down here. 00:05:04.510 --> 00:05:07.310 This is the ground. 00:05:07.310 --> 00:05:09.070 So what's going to happen this time? 00:05:09.070 --> 00:05:11.130 I'm still 10 meters in the air, so let me draw that. 00:05:11.130 --> 00:05:12.130 That's still 10 meters. 00:05:12.130 --> 00:05:15.360 I should switch colors just so not everything is ice. 00:05:15.360 --> 00:05:18.190 So that's still 10 meters, but instead of the object going 00:05:18.190 --> 00:05:21.090 straight down now, it's going to go down here and then start 00:05:21.090 --> 00:05:21.840 sliding, right? 00:05:21.840 --> 00:05:24.080 It's going to go sliding along this hill. 00:05:24.080 --> 00:05:27.870 And then at this point it's going to be going really fast 00:05:27.870 --> 00:05:29.010 in the horizontal direction. 00:05:29.010 --> 00:05:31.760 And right now we don't know how fast. 00:05:31.760 --> 00:05:34.630 And just using our kinematics formula, this would have been 00:05:34.630 --> 00:05:35.880 a really tough formula. 00:05:35.880 --> 00:05:38.630 This would have been difficult. 00:05:38.630 --> 00:05:40.400 I mean you could have attempted it and it actually 00:05:40.400 --> 00:05:42.990 would have taken calculus because the angle of the slope 00:05:42.990 --> 00:05:44.150 changes continuously. 00:05:44.150 --> 00:05:46.990 We don't even know the formula for the angle of the slope. 00:05:46.990 --> 00:05:48.410 You would have had to break it out into vectors. 00:05:48.410 --> 00:05:49.690 You would have to do all sorts of complicated things. 00:05:49.690 --> 00:05:52.050 This would have been a nearly impossible problem. 00:05:52.050 --> 00:05:55.050 But using energy, we can actually figure out what the 00:05:55.050 --> 00:05:58.680 velocity of this object is at this point. 00:05:58.680 --> 00:06:00.880 And we use the same idea. 00:06:00.880 --> 00:06:03.330 Here we have 100 joules of potential energy. 00:06:03.330 --> 00:06:05.070 We just figured that out. 00:06:05.070 --> 00:06:07.190 Down here, what's the height above the ground? 00:06:07.190 --> 00:06:08.570 Well the height is 0. 00:06:08.570 --> 00:06:10.580 So all the potential energy has disappeared. 00:06:10.580 --> 00:06:13.270 And just like in the previous situation, all of the 00:06:13.270 --> 00:06:16.530 potential energy is now converted into kinetic energy. 00:06:16.530 --> 00:06:18.520 And so what is that kinetic energy going to equal? 00:06:18.520 --> 00:06:21.440 It's going to be equal to the initial potential energy. 00:06:21.440 --> 00:06:27.150 So here the kinetic energy is equal to 100 joules. 00:06:27.150 --> 00:06:30.090 And that equals 1/2 mv squared, just 00:06:30.090 --> 00:06:31.300 like we just solved. 00:06:31.300 --> 00:06:34.380 And if you solve for v, the mass is 1 kilogram. 00:06:34.380 --> 00:06:39.690 So the velocity in the horizontal direction will be, 00:06:39.690 --> 00:06:42.400 if you solve for it, 14.1 meters per second. 00:06:42.400 --> 00:06:44.070 Instead of going straight down, now it's going to be 00:06:44.070 --> 00:06:47.300 going in the horizontal to the right. 00:06:47.300 --> 00:06:49.270 And the reason why I said it was ice is because I wanted 00:06:49.270 --> 00:06:52.360 this to be frictionless and I didn't want any energy lost to 00:06:52.360 --> 00:06:53.760 heat or anything like that. 00:06:53.760 --> 00:06:56.320 And you might say OK Sal, that's kind of interesting. 00:06:56.320 --> 00:06:59.990 And you kind of got the same number for the velocity than 00:06:59.990 --> 00:07:01.870 if I just dropped the object straight down. 00:07:01.870 --> 00:07:02.660 And that's interesting. 00:07:02.660 --> 00:07:07.910 But what else can this do for me? 00:07:07.910 --> 00:07:10.100 And this is where it's really cool. 00:07:10.100 --> 00:07:15.080 Not only can I figure out the velocity when all of the 00:07:15.080 --> 00:07:17.100 potential energy has disappeared, but I can figure 00:07:17.100 --> 00:07:19.180 out the velocity of any point-- and this is 00:07:19.180 --> 00:07:21.460 fascinating-- along this slide. 00:07:21.460 --> 00:07:25.200 So let's say when the box is sliding down here, so let's 00:07:25.200 --> 00:07:29.600 say the box is at this point. 00:07:29.600 --> 00:07:31.630 It changes colors too as it falls. 00:07:31.630 --> 00:07:34.510 So this is the 1 kilogram box, right? 00:07:34.510 --> 00:07:35.840 It falls and it slides down here. 00:07:35.840 --> 00:07:40.480 And let's say at this point it's height above the ground 00:07:40.480 --> 00:07:42.750 is 5 meters. 00:07:42.750 --> 00:07:44.860 So what's its potential energy here? 00:07:44.860 --> 00:07:45.770 So let's just write something. 00:07:45.770 --> 00:07:47.810 All of the energy is conserved, right? 00:07:47.810 --> 00:07:51.080 So the initial potential energy plus the initial 00:07:51.080 --> 00:07:57.360 kinetic energy is equal to the final potential energy plus 00:07:57.360 --> 00:07:59.960 the final kinetic energy. 00:07:59.960 --> 00:08:02.100 I'm just saying energy is conserved here. 00:08:02.100 --> 00:08:05.580 Up here, what's the initial total energy in the system? 00:08:05.580 --> 00:08:08.400 Well the potential energy is 100 and the kinetic energy is 00:08:08.400 --> 00:08:10.420 0 because it's stationary. 00:08:10.420 --> 00:08:11.880 I haven't dropped it. 00:08:11.880 --> 00:08:13.090 I haven't let go of it yet. 00:08:13.090 --> 00:08:14.360 It's just stationary. 00:08:14.360 --> 00:08:18.530 So the initial energy is going to be equal to 100 joules. 00:08:18.530 --> 00:08:20.830 That's cause this is 0 and this is 100. 00:08:20.830 --> 00:08:22.840 So the initial energy is 100 joules. 00:08:22.840 --> 00:08:27.500 At this point right here, what's the potential energy? 00:08:27.500 --> 00:08:32.270 Well we're 5 meters up, so mass times 00:08:32.270 --> 00:08:32.990 gravity times height. 00:08:32.990 --> 00:08:36.900 Mass is 1, times gravity, 10 meters per second squared. 00:08:36.900 --> 00:08:39.049 Times height, times 5. 00:08:39.049 --> 00:08:40.630 So it's 50 joules. 00:08:40.630 --> 00:08:43.409 That's our potential energy at this point. 00:08:43.409 --> 00:08:47.130 And then we must have some kinetic energy with the 00:08:47.130 --> 00:08:48.250 velocity going roughly in that direction. 00:08:48.250 --> 00:08:51.950 Plus our kinetic energy at this point. 00:08:51.950 --> 00:08:54.560 And we know that no energy was destroyed. 00:08:54.560 --> 00:08:55.330 It's just converted. 00:08:55.330 --> 00:08:59.080 So we know the total energy still has to be 100 joules. 00:08:59.080 --> 00:09:01.410 So essentially what happened, and if we solve for this-- 00:09:01.410 --> 00:09:03.430 it's very easy, subtract 50 from both sides-- we know that 00:09:03.430 --> 00:09:06.120 the kinetic energy is now also going to 00:09:06.120 --> 00:09:07.100 be equal to 50 joules. 00:09:07.100 --> 00:09:07.890 So what happened? 00:09:07.890 --> 00:09:12.370 Halfway down, essentially half of the potential energy got 00:09:12.370 --> 00:09:14.120 converted to kinetic energy. 00:09:14.120 --> 00:09:16.120 And we can use this information that the kinetic 00:09:16.120 --> 00:09:18.230 energy is 50 joules to figure out the 00:09:18.230 --> 00:09:20.170 velocity at this point. 00:09:20.170 --> 00:09:24.840 1/2 mv squared is equal to 50. 00:09:24.840 --> 00:09:26.080 The mass is 1. 00:09:26.080 --> 00:09:27.530 Multiply both sides by 2. 00:09:27.530 --> 00:09:30.370 You get v squared is equal to 100. 00:09:30.370 --> 00:09:34.570 The velocity is 10 meters per second along 00:09:34.570 --> 00:09:37.390 this crazy, icy slide. 00:09:37.390 --> 00:09:39.970 And that is something that I would have challenged you to 00:09:39.970 --> 00:09:42.670 solve using traditional kinematics formulas, 00:09:42.670 --> 00:09:46.500 especially considering that we don't know really much about 00:09:46.500 --> 00:09:49.490 the surface of this slide. 00:09:49.490 --> 00:09:51.930 And even if we did, that would have been a million times 00:09:51.930 --> 00:09:55.440 harder than just using the law of conservation of energy and 00:09:55.440 --> 00:09:58.380 realizing that at this point, half the potential energy is 00:09:58.380 --> 00:10:00.620 now kinetic energy and it's going along the 00:10:00.620 --> 00:10:02.620 direction of the slide. 00:10:02.620 --> 00:10:03.870 I will see you in the next video.

Introduction to work and energy

https://www.youtube.com/watch?v=2WS1sG9fhOk

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https://www.youtube.com/api/timedtext?v=2WS1sG9fhOk&ei=YmeUZfr5LNyZhcIPsYCUmAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=EFEF250032B91C17A5C689347542F822E65139FD.B55A88A544CECCF5079029F50A01C879BEBF33FB&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.790 --> 00:00:01.760 Welcome back. 00:00:01.760 --> 00:00:02.980 I'm now going to introduce you to the 00:00:02.980 --> 00:00:05.310 concepts of work and energy. 00:00:05.310 --> 00:00:08.490 And these are two words that are-- I'm sure you use in your 00:00:08.490 --> 00:00:10.220 everyday life already and you have some notion 00:00:10.220 --> 00:00:11.760 of what they mean. 00:00:11.760 --> 00:00:13.690 But maybe just not in the physics context, although 00:00:13.690 --> 00:00:15.340 they're not completely unrelated. 00:00:15.340 --> 00:00:18.400 So work, you know what work is. 00:00:18.400 --> 00:00:20.280 Work is when you do something. 00:00:20.280 --> 00:00:21.870 You go to work, you make a living. 00:00:21.870 --> 00:00:25.480 In physics, work is-- and I'm going to use a lot of words 00:00:25.480 --> 00:00:28.500 and they actually end up being kind of circular in their 00:00:28.500 --> 00:00:29.350 definitions. 00:00:29.350 --> 00:00:31.220 But I think when we start doing the math, you'll start 00:00:31.220 --> 00:00:34.450 to get at least a slightly more intuitive notion of what 00:00:34.450 --> 00:00:35.250 they all are. 00:00:35.250 --> 00:00:38.580 So work is energy transferred by a force. 00:00:38.580 --> 00:00:43.910 So I'll write that down, energy transferred-- and I got 00:00:43.910 --> 00:00:47.480 this from Wikipedia because I wanted a good, I guess, 00:00:47.480 --> 00:00:49.780 relatively intuitive definition. 00:00:49.780 --> 00:00:51.880 Energy transferred by a force. 00:00:51.880 --> 00:00:54.360 And that makes reasonable sense to me. 00:00:54.360 --> 00:00:56.590 But then you're wondering, well, I know what a force is, 00:00:56.590 --> 00:00:58.400 you know, force is mass times acceleration. 00:00:58.400 --> 00:00:59.670 But what is energy? 00:00:59.670 --> 00:01:04.069 And then I looked up energy on Wikipedia and I found this, 00:01:04.069 --> 00:01:05.190 well, entertaining. 00:01:05.190 --> 00:01:08.560 But it also I think tells you something that these are just 00:01:08.560 --> 00:01:12.770 concepts that we use to, I guess, work with what we 00:01:12.770 --> 00:01:16.900 perceive as motion and force and work and all of these 00:01:16.900 --> 00:01:17.630 types of things. 00:01:17.630 --> 00:01:22.160 But they really aren't independent notions. 00:01:22.160 --> 00:01:23.380 They're related. 00:01:23.380 --> 00:01:27.150 So Wikipedia defines energy as the ability to do work. 00:01:27.150 --> 00:01:29.430 So they kind of use each other to define each other. 00:01:29.430 --> 00:01:33.090 Ability to do work. 00:01:33.090 --> 00:01:37.360 Which is frankly, as good of a definition as I could find. 00:01:37.360 --> 00:01:40.520 And so, with just the words, these kind of don't give you 00:01:40.520 --> 00:01:41.390 much information. 00:01:41.390 --> 00:01:43.690 So what I'm going to do is move onto the equations, and 00:01:43.690 --> 00:01:46.000 this'll give you a more quantitative feel of what 00:01:46.000 --> 00:01:47.540 these words mean. 00:01:47.540 --> 00:01:55.470 So the definition of work in mechanics, work is equal to 00:01:55.470 --> 00:01:58.665 force times distance. 00:02:01.890 --> 00:02:04.030 So let's say that I have a block and-- let me do it in a 00:02:04.030 --> 00:02:05.720 different color just because this yellow 00:02:05.720 --> 00:02:07.540 might be getting tedious. 00:02:07.540 --> 00:02:11.840 And I apply a force of-- let's say I apply 00:02:11.840 --> 00:02:17.740 a force of 10 Newtons. 00:02:17.740 --> 00:02:22.390 And I move that block by applying 00:02:22.390 --> 00:02:24.210 a force of 10 Newtons. 00:02:24.210 --> 00:02:27.770 I move that block, let's say I move it-- I 00:02:27.770 --> 00:02:31.960 don't know-- 7 meters. 00:02:31.960 --> 00:02:35.930 So the work that I applied to that block, or the energy that 00:02:35.930 --> 00:02:42.650 I've transferred to that block, the work is equal to 00:02:42.650 --> 00:02:46.880 the force, which is 10 Newtons, times the distance, 00:02:46.880 --> 00:02:49.300 times 7 meters. 00:02:49.300 --> 00:02:54.150 And that would equal 70-- 10 times 7-- Newton meters. 00:02:54.150 --> 00:03:00.450 So Newton meters is one, I guess, way of describing work. 00:03:00.450 --> 00:03:03.720 And this is also defined as one joule. 00:03:03.720 --> 00:03:05.340 And I'll do another presentation on all of the 00:03:05.340 --> 00:03:06.230 things that soon. 00:03:06.230 --> 00:03:06.920 Joule did. 00:03:06.920 --> 00:03:09.170 But joule is the unit of work and it's 00:03:09.170 --> 00:03:10.540 also the unit of energy. 00:03:10.540 --> 00:03:12.650 And they're kind of transferrable. 00:03:12.650 --> 00:03:14.380 Because if you look at the definitions that Wikipedia 00:03:14.380 --> 00:03:17.620 gave us, work is energy transferred by a force and 00:03:17.620 --> 00:03:19.880 energy is the ability to work. 00:03:19.880 --> 00:03:23.900 So I'll leave this relatively circular definition alone now. 00:03:23.900 --> 00:03:26.680 But we'll use this definition, which I think helps us a 00:03:26.680 --> 00:03:30.710 little bit more to understand the types of work we can do. 00:03:30.710 --> 00:03:34.680 And then, what kind of energy we actually are transferring 00:03:34.680 --> 00:03:37.380 to an object when we do that type of work. 00:03:37.380 --> 00:03:40.025 So let me do some examples. 00:03:42.960 --> 00:03:44.225 Let's say I have a block. 00:03:49.120 --> 00:03:53.290 I have a block of mass m. 00:03:53.290 --> 00:03:57.610 I have a block of mass m and it starts at rest. And then I 00:03:57.610 --> 00:04:00.200 apply force. 00:04:00.200 --> 00:04:09.070 Let's say I apply a force, F, for a distance of, I think, 00:04:09.070 --> 00:04:10.850 you can guess what the distance I'm going to apply it 00:04:10.850 --> 00:04:13.380 is, for a distance of d. 00:04:13.380 --> 00:04:17.500 So I'm pushing on this block with a force of F for a 00:04:17.500 --> 00:04:18.540 distance of d. 00:04:18.540 --> 00:04:21.290 And what I want to figure out is-- well, we know 00:04:21.290 --> 00:04:22.610 what the work is. 00:04:22.610 --> 00:04:27.630 I mean, by definition, work is equal to this force times this 00:04:27.630 --> 00:04:30.480 distance that I'm applying the block-- that 00:04:30.480 --> 00:04:31.930 I'm pushing the block. 00:04:31.930 --> 00:04:36.030 But what is the velocity going to be of this block over here? 00:04:36.030 --> 00:04:36.620 Right? 00:04:36.620 --> 00:04:39.320 It's going to be something somewhat faster. 00:04:39.320 --> 00:04:42.270 Because force isn't-- and I'm assuming that this is 00:04:42.270 --> 00:04:43.570 frictionless on here. 00:04:43.570 --> 00:04:48.040 So force isn't just moving the block with a constant 00:04:48.040 --> 00:04:50.420 velocity, force is equal to mass times acceleration. 00:04:50.420 --> 00:04:52.430 So I'm actually going to be accelerating the block. 00:04:52.430 --> 00:04:55.390 So even though it's stationary here, by the time we get to 00:04:55.390 --> 00:04:57.960 this point over here, that block is going 00:04:57.960 --> 00:05:00.290 to have some velocity. 00:05:00.290 --> 00:05:02.200 We don't know what it is because we're using all 00:05:02.200 --> 00:05:03.550 variables, we're not using numbers. 00:05:03.550 --> 00:05:06.650 But let's figure out what it is in terms of v. 00:05:06.650 --> 00:05:10.980 So if you remember your kinematics equations, and if 00:05:10.980 --> 00:05:12.120 you don't, you might want to go back. 00:05:12.120 --> 00:05:14.060 Or if you've never seen the videos, there's a whole set of 00:05:14.060 --> 00:05:17.380 videos on projectile motion and kinematics. 00:05:17.380 --> 00:05:20.170 But we figured out that when we're accelerating an object 00:05:20.170 --> 00:05:23.400 over a distance, that the final velocity-- let me change 00:05:23.400 --> 00:05:27.910 colors just for variety-- the final velocity squared is 00:05:27.910 --> 00:05:31.850 equal to the initial velocity squared plus 2 times the 00:05:31.850 --> 00:05:33.530 acceleration times the distance. 00:05:33.530 --> 00:05:36.210 And we proved this back then, so I won't redo it now. 00:05:36.210 --> 00:05:38.640 But in this situation, what's the initial velocity? 00:05:38.640 --> 00:05:40.525 Well the initial velocity was 0. 00:05:43.250 --> 00:05:44.160 Right? 00:05:44.160 --> 00:05:50.026 So the equation becomes vf squared is equal to 2 times 00:05:50.026 --> 00:05:54.990 the acceleration times the distance. 00:05:54.990 --> 00:05:57.830 And then, we could rewrite the acceleration 00:05:57.830 --> 00:05:59.180 in terms of, what? 00:05:59.180 --> 00:06:01.050 The force and the mass, right? 00:06:01.050 --> 00:06:03.010 So what is the acceleration? 00:06:03.010 --> 00:06:04.260 Well F equals ma. 00:06:07.310 --> 00:06:12.450 Or, acceleration is equal to force divided by you mass. 00:06:12.450 --> 00:06:18.910 So we get vf squared is equal to 2 times the force divided 00:06:18.910 --> 00:06:22.360 by the mass times the distance. 00:06:22.360 --> 00:06:23.930 And then we could take the square root of both sides if 00:06:23.930 --> 00:06:26.930 we want, and we get the final velocity of this block, at 00:06:26.930 --> 00:06:37.140 this point, is going to be equal to the square root of 2 00:06:37.140 --> 00:06:41.780 times force times distance divided by mass. 00:06:41.780 --> 00:06:43.760 And so that's how we could figure it out. 00:06:43.760 --> 00:06:46.000 And there's something interesting going on here. 00:06:46.000 --> 00:06:49.060 There's something interesting in what we did just now. 00:06:49.060 --> 00:06:52.160 Do you see something that looks a little bit like work? 00:06:52.160 --> 00:06:52.910 Well sure. 00:06:52.910 --> 00:06:54.570 You have this force times distance 00:06:54.570 --> 00:06:56.090 expression right here. 00:06:56.090 --> 00:06:58.660 Force times distance right here. 00:06:58.660 --> 00:07:01.290 So let's write another equation. 00:07:01.290 --> 00:07:07.380 If we know the given amount of velocity something has, if we 00:07:07.380 --> 00:07:09.590 can figure out how much work needed to be put into the 00:07:09.590 --> 00:07:12.580 system to get to that velocity. 00:07:12.580 --> 00:07:15.135 Well we can just replace force times distance with work. 00:07:15.135 --> 00:07:15.690 Right? 00:07:15.690 --> 00:07:17.370 Because work is equal to force times distance. 00:07:17.370 --> 00:07:20.905 So let's go straight from this equation because we don't have 00:07:20.905 --> 00:07:22.200 to re-square it. 00:07:22.200 --> 00:07:27.670 So we get vf squared is equal to 2 00:07:27.670 --> 00:07:29.370 times force times distance. 00:07:29.370 --> 00:07:31.150 That's work. 00:07:31.150 --> 00:07:33.230 Took that definition right here. 00:07:33.230 --> 00:07:37.630 2 times work divided by the mass. 00:07:37.630 --> 00:07:40.870 Let's multiply both sides of this equation times the mass. 00:07:40.870 --> 00:07:44.150 So you get mass times the velocity. 00:07:44.150 --> 00:07:46.290 And we don't have to write-- I'm going to get rid of this f 00:07:46.290 --> 00:07:48.530 because we know that we started at rest and that the 00:07:48.530 --> 00:07:51.180 velocity is going to be-- let's just call it v. 00:07:51.180 --> 00:07:56.390 So m times V squared is equal to 2 times the work. 00:07:56.390 --> 00:07:58.090 Divide both sides by 2. 00:07:58.090 --> 00:08:03.380 Or that the work is equal to mv squared over 2. 00:08:03.380 --> 00:08:05.860 Just divided both sides by 2. 00:08:05.860 --> 00:08:07.880 And of course, the unit here is joules. 00:08:07.880 --> 00:08:09.570 So this is interesting. 00:08:09.570 --> 00:08:16.840 Now if I know the velocity of an object, I can figure out, 00:08:16.840 --> 00:08:19.590 using this formula, which hopefully wasn't too 00:08:19.590 --> 00:08:21.200 complicated to derive. 00:08:21.200 --> 00:08:25.110 I can figure out how much work was imputed into that object 00:08:25.110 --> 00:08:26.790 to get it to that velocity. 00:08:26.790 --> 00:08:31.270 And this, by definition, is called kinetic energy. 00:08:31.270 --> 00:08:32.500 This is kinetic energy. 00:08:32.500 --> 00:08:35.760 And once again, the definition that Wikipedia gives us is the 00:08:35.760 --> 00:08:41.190 energy due to motion, or the work needed to accelerate from 00:08:41.190 --> 00:08:43.179 an object from being stationary 00:08:43.179 --> 00:08:44.980 to its current velocity. 00:08:44.980 --> 00:08:48.490 And I'm actually almost out of time, but what I will do is I 00:08:48.490 --> 00:08:51.550 will leave you with this formula, that kinetic energy 00:08:51.550 --> 00:08:53.600 is mass times velocity squared divided by 00:08:53.600 --> 00:08:55.780 2, or 1/2 mv squared. 00:08:55.780 --> 00:08:57.150 It's a very common formula. 00:08:57.150 --> 00:08:59.030 And I'll leave you with that and that 00:08:59.030 --> 00:09:00.660 is one form of energy. 00:09:00.660 --> 00:09:02.680 And I'll leave you with that idea. 00:09:02.680 --> 00:09:04.390 And in the next video, I will show you 00:09:04.390 --> 00:09:05.690 another form of energy. 00:09:05.690 --> 00:09:07.560 And then, I will introduce you to the law of 00:09:07.560 --> 00:09:08.590 conservation of energy. 00:09:08.590 --> 00:09:11.220 And that's where things become useful, because you can see 00:09:11.220 --> 00:09:13.730 how one form of energy can be converted to another to figure 00:09:13.730 --> 00:09:15.190 out what happens to an object. 00:09:15.190 --> 00:09:16.640 I'll see

Work and energy (part 2)

https://www.youtube.com/watch?v=3mier94pbnU

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https://www.youtube.com/api/timedtext?v=3mier94pbnU&ei=YmeUZcaiLeXGp-oPz4-XiAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=196E2CD39DA9A17B7068155835B24C1292782DAE.7E6BC0040FFECFA844EC754D523A09EF2505BE28&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.750 --> 00:00:01.650 Welcome back. 00:00:01.650 --> 00:00:05.850 In the last video, I showed you or hopefully, I did show 00:00:05.850 --> 00:00:10.230 you that if I apply a force of F to a stationary, an 00:00:10.230 --> 00:00:14.990 initially stationary object with mass m, and I apply that 00:00:14.990 --> 00:00:20.920 force for distance d, that that force times distance, the 00:00:20.920 --> 00:00:24.460 force times the distance that I'm pushing the object is 00:00:24.460 --> 00:00:30.980 equal to 1/2 mv squared, where m is the mass of the object, 00:00:30.980 --> 00:00:34.430 and v is the velocity of the object after pushing it for a 00:00:34.430 --> 00:00:35.690 distance of d. 00:00:35.690 --> 00:00:37.505 And we defined in that last video, we just 00:00:37.505 --> 00:00:38.810 said this is work. 00:00:38.810 --> 00:00:42.830 Force times distance by definition, is work. 00:00:42.830 --> 00:00:47.340 And 1/2 mv squared, I said this is called kinetic energy. 00:00:51.680 --> 00:00:56.110 And so, by definition, kinetic energy is the amount of work-- 00:00:56.110 --> 00:00:57.870 and I mean this is the definition right here. 00:00:57.870 --> 00:01:00.725 It's the amount of work you need to put into an object or 00:01:00.725 --> 00:01:03.820 apply to an object to get it from rest 00:01:03.820 --> 00:01:05.550 to its current velocity. 00:01:05.550 --> 00:01:07.100 So its velocity over here. 00:01:07.100 --> 00:01:10.740 So let's just say I looked at an object here with mass m and 00:01:10.740 --> 00:01:14.250 it was moving with the velocity v. 00:01:14.250 --> 00:01:16.960 I would say well, this has a kinetic 00:01:16.960 --> 00:01:19.660 energy of 1/2 mv squared. 00:01:19.660 --> 00:01:21.365 And if the numbers are confusing you, let's say the 00:01:21.365 --> 00:01:22.760 mass was-- I don't know. 00:01:22.760 --> 00:01:25.640 Let's say this was a 5 kilogram object and it's 00:01:25.640 --> 00:01:29.330 moving at 7 meters per second. 00:01:29.330 --> 00:01:33.325 So I would say the kinetic energy of this object is going 00:01:33.325 --> 00:01:40.190 to be 5-- 1/2 times the mass times 5 times 7 squared, times 00:01:40.190 --> 00:01:40.850 velocity squared. 00:01:40.850 --> 00:01:42.890 It's times 49. 00:01:42.890 --> 00:01:43.220 So let's see. 00:01:43.220 --> 00:01:46.270 1/2 times 49, that's a little under 25. 00:01:46.270 --> 00:01:53.170 So it'll be approximately 125 Newton meters, which is 00:01:53.170 --> 00:01:55.500 approximately-- and Newton meter is just 00:01:55.500 --> 00:01:58.220 a joule-- 125 joules. 00:01:58.220 --> 00:02:00.420 So this is if we actually put numbers in. 00:02:00.420 --> 00:02:02.770 And so when we immediately know this, even if we didn't 00:02:02.770 --> 00:02:05.770 know what happened, how did this object get to this speed? 00:02:05.770 --> 00:02:08.560 Let's say we didn't know that someone else had applied a 00:02:08.560 --> 00:02:13.090 force of m for a distance of d to this object, just by 00:02:13.090 --> 00:02:16.710 calculating its kinetic energy as 125 joules, we immediately 00:02:16.710 --> 00:02:20.120 know that that's the amount of work that was necessary. 00:02:20.120 --> 00:02:22.500 And we don't know if this is exactly how this object got to 00:02:22.500 --> 00:02:24.920 this velocity, but we know that that is the amount of 00:02:24.920 --> 00:02:28.530 work that was necessary to accelerate the object to this 00:02:28.530 --> 00:02:32.620 velocity of 7 meters per second. 00:02:32.620 --> 00:02:35.330 So let's give another example. 00:02:35.330 --> 00:02:39.420 And instead of this time just pushing something in a 00:02:39.420 --> 00:02:41.640 horizontal direction and accelerating it, I'm going to 00:02:41.640 --> 00:02:43.940 show you an example we're going to push something up, 00:02:43.940 --> 00:02:46.550 but its velocity really isn't going to change. 00:02:49.430 --> 00:02:51.280 Invert. 00:02:51.280 --> 00:02:54.140 Let's say I have a different situation, and we're on this 00:02:54.140 --> 00:02:55.990 planet, we're not in deep space. 00:02:55.990 --> 00:03:01.090 And I have a mass of m and I were to apply a force. 00:03:01.090 --> 00:03:06.470 So let's say the force that I apply is equal to mass times 00:03:06.470 --> 00:03:08.930 the acceleration of gravity. 00:03:08.930 --> 00:03:11.460 Mass times-- let's just call that gravity, right? 00:03:11.460 --> 00:03:13.140 9.8 meters per second squared. 00:03:13.140 --> 00:03:18.740 And I were to apply this force for a distance of d upwards. 00:03:18.740 --> 00:03:19.500 Right? 00:03:19.500 --> 00:03:20.810 Or instead of d, let's say h. 00:03:20.810 --> 00:03:23.050 H for height. 00:03:23.050 --> 00:03:27.030 So in this case, the force times the distance is equal 00:03:27.030 --> 00:03:31.490 to-- well the force is mass times the acceleration of 00:03:31.490 --> 00:03:33.240 gravity, right? 00:03:33.240 --> 00:03:35.880 And remember, I'm pushing with the acceleration of gravity 00:03:35.880 --> 00:03:38.410 upwards, while the acceleration of gravity is 00:03:38.410 --> 00:03:41.100 pulling downwards. 00:03:41.100 --> 00:03:45.045 So the force is mass times gravity, and I'm applying that 00:03:45.045 --> 00:03:47.390 for a distance of h, right? 00:03:47.390 --> 00:03:48.050 d is h. 00:03:48.050 --> 00:03:50.810 So the force is this. 00:03:50.810 --> 00:03:52.580 This is the force. 00:03:52.580 --> 00:03:56.730 And then the distance I'm applying is going to be h. 00:03:56.730 --> 00:04:00.900 And what's interesting is-- I mean if you want to think of 00:04:00.900 --> 00:04:05.910 an exact situation, imagine an elevator that is already 00:04:05.910 --> 00:04:08.140 moving because you would actually have to apply a force 00:04:08.140 --> 00:04:10.330 slightly larger than the acceleration of gravity just 00:04:10.330 --> 00:04:11.270 to get the object moving. 00:04:11.270 --> 00:04:12.280 But let's say that the object is 00:04:12.280 --> 00:04:14.860 already at constant velocity. 00:04:14.860 --> 00:04:17.500 Let's say it's an elevator. 00:04:17.500 --> 00:04:20.950 And it is just going up with a constant velocity. 00:04:20.950 --> 00:04:23.520 And let's say the mass of the elevator is-- I don't know-- 00:04:23.520 --> 00:04:29.230 10 kilograms. And it moves up with a constant velocity. 00:04:32.270 --> 00:04:35.380 It moves up 100 meters. 00:04:35.380 --> 00:04:38.280 So we know that the work done by whatever was pulling on 00:04:38.280 --> 00:04:41.370 this elevator, it probably was the tension in this wire that 00:04:41.370 --> 00:04:43.750 was pulling up on the elevator, but we know that the 00:04:43.750 --> 00:04:47.210 work done is the force necessary to pull up on it. 00:04:47.210 --> 00:04:49.330 Well that's just going to be the force of gravity. 00:04:49.330 --> 00:04:50.780 So we're assuming that the elevator's not 00:04:50.780 --> 00:04:52.040 accelerating, right? 00:04:52.040 --> 00:04:55.690 Because if the elevator was accelerating upwards, then the 00:04:55.690 --> 00:04:57.690 force applied to it would be more than 00:04:57.690 --> 00:04:59.080 the force of gravity. 00:04:59.080 --> 00:05:01.840 And if the elevator was accelerating downwards, or if 00:05:01.840 --> 00:05:05.250 it was slowing down upwards, then the force being applied 00:05:05.250 --> 00:05:06.850 would be less than the acceleration of gravity. 00:05:06.850 --> 00:05:10.360 But since the elevator is at a constant velocity moving up, 00:05:10.360 --> 00:05:14.760 we know that the force pulling upwards is completely equal to 00:05:14.760 --> 00:05:16.330 the force pulling downwards, right? 00:05:16.330 --> 00:05:17.250 No net force. 00:05:17.250 --> 00:05:20.520 Because gravity and this force are at the same level, so 00:05:20.520 --> 00:05:22.050 there's no change in velocity. 00:05:22.050 --> 00:05:24.020 I think I said that two times. 00:05:24.020 --> 00:05:26.840 So we know that this upward force is equal to 00:05:26.840 --> 00:05:28.720 the force of gravity. 00:05:28.720 --> 00:05:31.045 At least in magnitude in the opposite direction. 00:05:31.045 --> 00:05:35.310 So this is mg. 00:05:35.310 --> 00:05:38.500 So what's m? m is 10 kilograms. Times the 00:05:38.500 --> 00:05:39.320 acceleration of gravity. 00:05:39.320 --> 00:05:41.900 Let's say that's 9.8 meters per second squared. 00:05:41.900 --> 00:05:43.750 I'm not writing the units here, but we're all assuming 00:05:43.750 --> 00:05:45.490 kilograms and meters per second squared. 00:05:45.490 --> 00:05:50.760 And we're moving that for a distance of 100 meters. 00:05:50.760 --> 00:05:55.490 So how much work was put into this elevator, or into this 00:05:55.490 --> 00:05:57.980 object-- it doesn't have to be an elevator-- by whatever 00:05:57.980 --> 00:06:00.210 force that was essentially pushing up on it or 00:06:00.210 --> 00:06:01.770 pulling up on it? 00:06:01.770 --> 00:06:02.410 And so, let's see. 00:06:02.410 --> 00:06:05.160 This would be 98 times 100. 00:06:05.160 --> 00:06:13.820 So it's 9,800 Newton meters or 9,800 joules. 00:06:13.820 --> 00:06:17.270 After we've moved up 100 meters, notice there's no 00:06:17.270 --> 00:06:18.760 change in velocity. 00:06:18.760 --> 00:06:22.220 So the question is, where did all that work get 00:06:22.220 --> 00:06:24.140 put into the object? 00:06:24.140 --> 00:06:26.780 And the answer here is, is that the work got transferred 00:06:26.780 --> 00:06:29.940 to something called potential energy. 00:06:29.940 --> 00:06:33.150 And potential energy is defined as-- well, 00:06:33.150 --> 00:06:34.780 gravitational potential energy. 00:06:34.780 --> 00:06:37.150 We'll work with other types of potential energy later with 00:06:37.150 --> 00:06:38.810 springs and things. 00:06:38.810 --> 00:06:42.580 Potential energy is defined as mass times the force of 00:06:42.580 --> 00:06:45.760 gravity times the height that the object is at. 00:06:45.760 --> 00:06:47.840 And why is this called potential energy? 00:06:47.840 --> 00:06:51.430 Because at this point, the energy-- work had to be put 00:06:51.430 --> 00:06:53.920 into the object to get it to this-- in the case of 00:06:53.920 --> 00:06:57.670 gravitational potential energy, work had to be put 00:06:57.670 --> 00:07:00.090 into the object to get it to this height. 00:07:00.090 --> 00:07:02.720 But the object now, it's not moving or anything, so it 00:07:02.720 --> 00:07:04.110 doesn't have any kinetic energy. 00:07:04.110 --> 00:07:06.510 But it now has a lot of potential to do work. 00:07:06.510 --> 00:07:08.860 And what do I mean by potential to do work? 00:07:08.860 --> 00:07:12.430 Well after I move an object up 100 meters into the air, 00:07:12.430 --> 00:07:14.420 what's its potential to do work? 00:07:14.420 --> 00:07:19.520 Well, I could just let go of it and have no outside force 00:07:19.520 --> 00:07:20.360 other than gravity. 00:07:20.360 --> 00:07:22.080 The gravity will still be there. 00:07:22.080 --> 00:07:25.190 And because of gravity, the object will come down and be 00:07:25.190 --> 00:07:27.490 at a very, very fast velocity when it lands. 00:07:27.490 --> 00:07:31.230 And maybe I could apply this to some machine or something, 00:07:31.230 --> 00:07:33.150 and this thing could do work. 00:07:33.150 --> 00:07:34.400 And if that's a little confusing, let 00:07:34.400 --> 00:07:38.000 me give you an example. 00:07:38.000 --> 00:07:41.300 It all works together with our-- 00:07:41.300 --> 00:07:47.882 So let's say I have an object that is-- oh, I don't know-- a 00:07:47.882 --> 00:07:53.060 1 kilogram object and we're on earth. 00:07:53.060 --> 00:07:55.230 And let's say that is 10 meters above the ground. 00:07:58.700 --> 00:08:06.070 So we know that its potential energy is equal to mass times 00:08:06.070 --> 00:08:09.510 gravitational acceleration times height. 00:08:09.510 --> 00:08:11.240 So mass is 1. 00:08:11.240 --> 00:08:13.300 Let's just say gravitational acceleration is 10 meters per 00:08:13.300 --> 00:08:15.660 second squared. 00:08:15.660 --> 00:08:17.435 Times 10 meters per second squared. 00:08:17.435 --> 00:08:19.820 Times 10 meters, which is the height. 00:08:19.820 --> 00:08:25.720 So it's approximately equal to 100 Newton meters, which is 00:08:25.720 --> 00:08:28.020 the same thing is 100 joules. 00:08:28.020 --> 00:08:28.580 Fair enough. 00:08:28.580 --> 00:08:29.700 And what do we know about this? 00:08:29.700 --> 00:08:33.500 We know that it would take about 100-- or exactly-- 100 00:08:33.500 --> 00:08:38.799 joules of work to get this object from the ground to this 00:08:38.799 --> 00:08:40.809 point up here. 00:08:40.809 --> 00:08:44.730 Now what we can do now is use our traditional kinematics 00:08:44.730 --> 00:08:47.480 formulas to figure out, well, if I just let this object go, 00:08:47.480 --> 00:08:50.750 how fast will it be when it hits the ground? 00:08:50.750 --> 00:08:52.645 And we could do that, but what I'll show you is 00:08:52.645 --> 00:08:53.380 even a faster way. 00:08:53.380 --> 00:08:56.070 And this is where all of the work and energy 00:08:56.070 --> 00:08:57.440 really becomes useful. 00:08:57.440 --> 00:08:59.840 We have something called the law of conservation of energy. 00:08:59.840 --> 00:09:02.550 It's that energy cannot be created or destroyed, it just 00:09:02.550 --> 00:09:04.600 gets transferred from one form to another. 00:09:04.600 --> 00:09:06.770 And there's some minor caveats to that. 00:09:06.770 --> 00:09:09.090 But for our purposes, we'll just stick with that. 00:09:09.090 --> 00:09:12.110 So in the situation where I just take the object and I let 00:09:12.110 --> 00:09:16.120 go up here, up here it has a ton of potential energy. 00:09:16.120 --> 00:09:18.560 And by the time it's down here, it has no potential 00:09:18.560 --> 00:09:21.530 energy because the height becomes 0, right? 00:09:21.530 --> 00:09:27.310 So here, potential energy is equal to 100 and here, 00:09:27.310 --> 00:09:29.870 potential energy is equal to 0. 00:09:29.870 --> 00:09:31.796 And so the natural question is-- I just told you the law 00:09:31.796 --> 00:09:34.930 of conservation of energy, but if you look at this example, 00:09:34.930 --> 00:09:37.330 all the potential energy just disappeared. 00:09:37.330 --> 00:09:39.160 And it looks like I'm running out of time, but what I'll 00:09:39.160 --> 00:09:41.130 show you in the next video is that that potential energy 00:09:41.130 --> 00:09:42.780 gets converted into another type of energy. 00:09:42.780 --> 00:09:44.980 And I think you might be able to guess what type that is 00:09:44.980 --> 00:09:48.030 because this object is going to be moving really fast right 00:09:48.030 --> 00:09:49.150 before it hits the ground. 00:09:49.150 --> 00:09:51.030 I'll see you in the next video.

2-dimensional momentum problem (part 2)

https://www.youtube.com/watch?v=leudxqivIJI

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https://www.youtube.com/api/timedtext?v=leudxqivIJI&ei=Y2eUZcXQD93BmLAPqMiHYA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249811&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=84EF4428EE51DA150758F5B84F447B4928F352C7.079A38BA7162EB03361DEBA4007603E44EECA209&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.630 --> 00:00:01.440 Welcome back. 00:00:01.440 --> 00:00:03.440 When I left off I was rushing at the end of this problem 00:00:03.440 --> 00:00:06.510 because I tend to rush at the end of problems when I am 00:00:06.510 --> 00:00:08.160 getting close to the YouTube 10 minute limit. 00:00:08.160 --> 00:00:10.680 But I just wanted to review the end of it because I feel 00:00:10.680 --> 00:00:11.520 like I rushed it. 00:00:11.520 --> 00:00:14.660 And then, actually continue with it and actually solve for 00:00:14.660 --> 00:00:16.970 the angle and then, introduce a little bit of-- a little 00:00:16.970 --> 00:00:18.280 more trigonometry. 00:00:18.280 --> 00:00:21.240 So just to review what we did, we said momentum is conserved 00:00:21.240 --> 00:00:24.270 and in two dimensions that means momentum is conserved in 00:00:24.270 --> 00:00:25.650 each of the dimensions. 00:00:25.650 --> 00:00:28.040 So we figured out what the initial momentum of the entire 00:00:28.040 --> 00:00:30.920 system was and we said, well, in the x direction, the 00:00:30.920 --> 00:00:33.070 initial momentum-- and all the momentum was coming from the 00:00:33.070 --> 00:00:34.440 ball A right. 00:00:34.440 --> 00:00:37.110 Because ball B wasn't moving, so its velocity was 0. 00:00:37.110 --> 00:00:38.570 So its momentum was 0. 00:00:38.570 --> 00:00:41.300 So ball A in the x direction and it was only moving in the 00:00:41.300 --> 00:00:42.190 x direction. 00:00:42.190 --> 00:00:44.390 So it's momentum in the x direction was 3 meters per 00:00:44.390 --> 00:00:46.610 second times 10 kilogram meters per second. 00:00:46.610 --> 00:00:49.050 And we got 30 kilogram meters per second. 00:00:49.050 --> 00:00:51.960 And then there was no momentum in the y direction. 00:00:51.960 --> 00:00:54.680 And then we knew that well after they hit each other, 00:00:54.680 --> 00:00:56.910 ball A kind of ricochets off at a 30 degree angle at 2 00:00:56.910 --> 00:00:58.110 meters per second. 00:00:58.110 --> 00:01:00.430 We used that information to figure out the x and y 00:01:00.430 --> 00:01:02.700 components of A's velocity. 00:01:02.700 --> 00:01:05.840 So A's velocity in the y direction was 1 meter per 00:01:05.840 --> 00:01:10.420 second and A's velocity in the x direction was 00:01:10.420 --> 00:01:11.700 square root of 3. 00:01:11.700 --> 00:01:14.090 And we used that information to figure out A's momentum in 00:01:14.090 --> 00:01:15.060 each direction. 00:01:15.060 --> 00:01:18.010 We said well, the momentum in the y direction must be 1 00:01:18.010 --> 00:01:21.600 meter per second times A's mass, which is 10 kilogram 00:01:21.600 --> 00:01:22.370 meters per second. 00:01:22.370 --> 00:01:27.170 Which I wrote-- what I wrote here. 00:01:27.170 --> 00:01:31.800 And then we figured out A's momentum in the B direction 00:01:31.800 --> 00:01:33.550 and we said well, that's just going to be square 00:01:33.550 --> 00:01:35.700 root of 3 times 10. 00:01:35.700 --> 00:01:37.070 And that's 10 square root of 3. 00:01:37.070 --> 00:01:39.040 And then we used that information to 00:01:39.040 --> 00:01:40.680 solve for B's momentum. 00:01:40.680 --> 00:01:43.070 Because we said well, B's momentum plus A's momentum in 00:01:43.070 --> 00:01:45.350 the x direction has to add up to 30. 00:01:45.350 --> 00:01:47.680 This was the x direction before. 00:01:47.680 --> 00:01:52.000 And we knew that B's momentum plus A's momentum in the y 00:01:52.000 --> 00:01:54.220 direction had to add up to 0, right? 00:01:54.220 --> 00:02:00.430 And so, since y's momentum going upwards was 10 kilogram 00:02:00.430 --> 00:02:04.580 meters per second, we knew that B's momentum going 00:02:04.580 --> 00:02:06.550 downwards would also have to be 10 00:02:06.550 --> 00:02:07.350 kilogram meters per second. 00:02:07.350 --> 00:02:09.190 Or you could even say it's negative 10. 00:02:09.190 --> 00:02:11.870 And we figure that out based on the fact that B 00:02:11.870 --> 00:02:13.450 had half the mass. 00:02:13.450 --> 00:02:15.910 That its velocity going down was 2 meters per second. 00:02:15.910 --> 00:02:21.140 And similarly, we knew that A's momentum in the x 00:02:21.140 --> 00:02:24.770 direction, which was 10 square root of 3 kilogram meters per 00:02:24.770 --> 00:02:26.830 second, plus B's momentum in the x 00:02:26.830 --> 00:02:29.730 direction is equal to 30. 00:02:29.730 --> 00:02:32.800 And then we just subtracted out and we got B's momentum in 00:02:32.800 --> 00:02:33.630 the x direction. 00:02:33.630 --> 00:02:36.270 And then we divided by B's mass to get its velocity. 00:02:36.270 --> 00:02:39.150 Which we got as 2.54. 00:02:39.150 --> 00:02:41.340 So that's where I left off and we were rushing. 00:02:41.340 --> 00:02:46.140 And already, this gives you a sense of what B is doing. 00:02:46.140 --> 00:02:48.770 Although it's broken up into the x and y direction. 00:02:48.770 --> 00:02:51.800 Now if we wanted to simplify this, if we wanted to kind of 00:02:51.800 --> 00:02:54.470 write B's new velocity the same way that the problem gave 00:02:54.470 --> 00:02:55.490 us A's velocity, right? 00:02:55.490 --> 00:02:58.090 They told us A's velocity was 2 meters per second at an 00:02:58.090 --> 00:02:59.390 angle of 30 degrees. 00:02:59.390 --> 00:03:04.010 We now have to use this information to figure out B's 00:03:04.010 --> 00:03:06.450 velocity and the angle of it. 00:03:06.450 --> 00:03:07.150 And how do we do that? 00:03:07.150 --> 00:03:09.890 Well this is just straight up trigonometry at this point, or 00:03:09.890 --> 00:03:13.360 really just straight up geometry. 00:03:13.360 --> 00:03:14.400 Let me clear all of this. 00:03:14.400 --> 00:03:21.680 And let's remember these two numbers, 2.54 and minus 2. 00:03:21.680 --> 00:03:30.590 So B, we learned that in the x direction its velocity-- this 00:03:30.590 --> 00:03:37.260 is all for B-- is equal to 2.54 meters per second and 00:03:37.260 --> 00:03:39.900 then y direction, it was moving down. 00:03:39.900 --> 00:03:41.910 We could write this as minus 2. 00:03:45.970 --> 00:03:51.830 But I'll just write this as 2 meters per second downwards. 00:03:51.830 --> 00:03:52.480 Right? 00:03:52.480 --> 00:03:52.970 Same thing. 00:03:52.970 --> 00:03:55.820 Minus 2 up is the same thing as 2 meters per second down. 00:03:55.820 --> 00:03:57.410 So the resulting vector's going to look 00:03:57.410 --> 00:03:59.240 something like this. 00:03:59.240 --> 00:04:02.725 When you add two vectors you just put them-- put the one's 00:04:02.725 --> 00:04:07.350 end at the beginning of the other-- put them front to end, 00:04:07.350 --> 00:04:08.320 like we did here. 00:04:08.320 --> 00:04:09.760 And then you add them together and this is 00:04:09.760 --> 00:04:10.730 the resulting vector. 00:04:10.730 --> 00:04:14.230 And I think you're used to that at this point. 00:04:14.230 --> 00:04:18.510 And now we have to figure out this angle and this side. 00:04:18.510 --> 00:04:21.100 Well this side is easy because this is a right angle, so we 00:04:21.100 --> 00:04:21.810 use Pythagorean theorem. 00:04:21.810 --> 00:04:27.530 So this is going to be the square root of 2.54 squared 00:04:27.530 --> 00:04:29.460 plus 2 squared. 00:04:29.460 --> 00:04:31.860 And what's 2.54 squared? 00:04:31.860 --> 00:04:40.880 2.54 times-- whoops. 00:04:40.880 --> 00:04:48.840 2.54 times 2.54 is equal to 6.45. 00:04:48.840 --> 00:04:55.530 So that's the square root of 6.45 plus 4, which equals the 00:04:55.530 --> 00:05:02.070 square root of 10.45. 00:05:02.070 --> 00:05:03.550 And take the square root of that. 00:05:03.550 --> 00:05:07.800 So that's 3.2, roughly. 00:05:07.800 --> 00:05:11.200 So the resulting velocity in this direction, whatever angle 00:05:11.200 --> 00:05:16.550 this is, is 3.2 meters per second. 00:05:16.550 --> 00:05:18.250 And I just used Pythagorean theorem. 00:05:18.250 --> 00:05:22.440 So now all we have to do is figure out the angle. 00:05:22.440 --> 00:05:25.540 We could use really any of the trig ratios because we know 00:05:25.540 --> 00:05:26.730 all of the sides. 00:05:26.730 --> 00:05:29.270 So I don't know, let's use one that you 00:05:29.270 --> 00:05:30.220 feel comfortable with. 00:05:30.220 --> 00:05:32.920 Well let's use sine. 00:05:32.920 --> 00:05:39.500 So sine of theta is equal to what? 00:05:39.500 --> 00:05:40.680 SOH CAH TOA. 00:05:40.680 --> 00:05:42.820 Sine is opposite over hypotenuse. 00:05:42.820 --> 00:05:47.020 So the opposite side is the y direction, so that's 2, over 00:05:47.020 --> 00:05:50.530 the hypotenuse, 3.2. 00:05:50.530 --> 00:05:58.790 So 2 divided by 2 divided by 3.2 is equal to 0.625, which 00:05:58.790 --> 00:06:01.265 equals 0.625. 00:06:01.265 --> 00:06:03.350 So sine of theta equals 0.625. 00:06:03.350 --> 00:06:05.510 And maybe you're not familiar with arcsine yet because I 00:06:05.510 --> 00:06:07.200 don't think I actually have covered yet in the trig 00:06:07.200 --> 00:06:09.780 modules, although I will eventually. 00:06:09.780 --> 00:06:13.090 So we know it's just the inverse function of sine. 00:06:13.090 --> 00:06:18.170 So sine of theta is equal to 0.625. 00:06:18.170 --> 00:06:25.920 Then we know that theta is equal to the arcsine of 0.625. 00:06:25.920 --> 00:06:29.235 This is essentially saying, when you say arcsine, this 00:06:29.235 --> 00:06:32.090 says, tell me the angle whose sine is this number? 00:06:32.090 --> 00:06:33.540 That's what arcsine is. 00:06:33.540 --> 00:06:38.460 And we can take out Google because it actually happens 00:06:38.460 --> 00:06:44.310 that Google has a-- let's see. 00:06:44.310 --> 00:06:46.680 Google actually-- it's an automatic calculator. 00:06:46.680 --> 00:06:52.590 So you could type in arcsine on Google of 0.625. 00:06:52.590 --> 00:06:55.460 Although I think the answer they give 00:06:55.460 --> 00:06:57.720 you will be in radians. 00:06:57.720 --> 00:07:00.230 So I'll take that answer that will be in radians and I want 00:07:00.230 --> 00:07:03.300 to convert to degrees, so I multiply it times 180 over pi. 00:07:03.300 --> 00:07:05.850 That's just how I convert from radians to degrees. 00:07:05.850 --> 00:07:07.350 And let's see what I get. 00:07:07.350 --> 00:07:12.670 So Google, you see, Google says 38.68 degrees. 00:07:12.670 --> 00:07:14.260 They multiplied the whole thing times 180 and then 00:07:14.260 --> 00:07:16.630 divided by pi, but that should be the same thing. 00:07:16.630 --> 00:07:20.225 So roughly 38.7 degrees is theta. 00:07:20.225 --> 00:07:22.090 Hope you understand that. 00:07:22.090 --> 00:07:24.430 You could pause it here if you don't, but let me 00:07:24.430 --> 00:07:25.300 just write that down. 00:07:25.300 --> 00:07:30.070 So it's 38 degrees. 00:07:30.070 --> 00:07:35.080 So theta is equal to 38.7 degrees. 00:07:35.080 --> 00:07:36.360 So then we're done. 00:07:36.360 --> 00:07:39.250 We figured out that ball B gets hit. 00:07:39.250 --> 00:07:40.950 This is ball B and it got hit by ball A. 00:07:40.950 --> 00:07:43.900 Ball A went off in that direction at a 30 degree 00:07:43.900 --> 00:07:47.880 angle, at a 30 degree angle at 2 meters per second. 00:07:47.880 --> 00:07:52.040 And now ball B goes at 38.-- or we could say roughly 39 00:07:52.040 --> 00:07:56.320 degrees below the horizontal at a velocity of 3.2 meters 00:07:56.320 --> 00:07:57.700 per second. 00:07:57.700 --> 00:08:01.270 And does this intuitively make sense to you? 00:08:01.270 --> 00:08:03.140 Well if you remember the problem from before-- and I 00:08:03.140 --> 00:08:03.860 know I erased everything. 00:08:03.860 --> 00:08:07.540 Ball A had a mass of 10 kilograms while ball B had a 00:08:07.540 --> 00:08:10.780 mass of 5 kilograms. So it makes sense. 00:08:10.780 --> 00:08:12.210 So let's think about just the y direction. 00:08:12.210 --> 00:08:16.680 Ball A, we figured out, the y component of its velocity was 00:08:16.680 --> 00:08:19.260 1 meter per second. 00:08:19.260 --> 00:08:23.100 And ball B's y component is 2 meters per second downwards. 00:08:23.100 --> 00:08:24.060 And does that makes sense? 00:08:24.060 --> 00:08:25.000 Well sure. 00:08:25.000 --> 00:08:27.030 Because their momentums have to add up to 0. 00:08:27.030 --> 00:08:31.470 There was no y component of the momentum before they hit 00:08:31.470 --> 00:08:32.440 each other. 00:08:32.440 --> 00:08:37.539 And in order for B to have the same momentum going downwards 00:08:37.539 --> 00:08:41.049 in the y direction as A going upwards, its velocity has to 00:08:41.049 --> 00:08:44.690 be essentially double, because its mass is half. 00:08:44.690 --> 00:08:48.680 And a similar logic, although the cosine-- it doesn't work 00:08:48.680 --> 00:08:50.550 out exactly like that. 00:08:50.550 --> 00:08:53.450 But a similar logic would mean that its overall velocity is 00:08:53.450 --> 00:08:59.670 going to be faster than the- than A's velocity. 00:08:59.670 --> 00:09:03.680 And so what was I just-- oh yeah. 00:09:03.680 --> 00:09:07.550 My phone was ringing and I got caught up. 00:09:07.550 --> 00:09:09.730 My brain starts to malfunction. 00:09:09.730 --> 00:09:11.100 But anyway, as I was saying, so just 00:09:11.100 --> 00:09:12.220 intuitively it makes sense. 00:09:12.220 --> 00:09:16.330 B has a smaller mass than A, so it makes sense that-- one, 00:09:16.330 --> 00:09:18.460 B will be going faster and that it gets deflected a 00:09:18.460 --> 00:09:20.350 little bit more as well. 00:09:20.350 --> 00:09:21.800 The reason why it seems like it gets deflected more is 00:09:21.800 --> 00:09:23.095 because its y component is more. 00:09:23.095 --> 00:09:26.200 But anyway, that last piece is just to kind of hopefully give 00:09:26.200 --> 00:09:29.510 you a sense of what's happening and I will see you 00:09:29.510 --> 00:09:31.120 in the next video.

2-dimensional momentum problem

https://www.youtube.com/watch?v=CFygKiTB-4A

vtt

https://www.youtube.com/api/timedtext?v=CFygKiTB-4A&ei=YmeUZbSkLpOnp-oP-f-5uAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A5F4972FD7FD89FFA74AEF0579225CE722176A6C.5143AC5AF0C304D291858C8364F5E794CC3F201D&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.740 --> 00:00:01.690 Welcome back. 00:00:01.690 --> 00:00:05.950 We will now do a momentum problem in two dimensions. 00:00:05.950 --> 00:00:07.170 So let's see what we have here. 00:00:07.170 --> 00:00:10.070 So we have this ball A and we could maybe even think of it 00:00:10.070 --> 00:00:13.470 as maybe what's going on on the surface of a pool table. 00:00:13.470 --> 00:00:16.730 We have ball A and it's moving with its 10 kilograms. So 00:00:16.730 --> 00:00:19.070 these numbers are the mass of the balls. 00:00:19.070 --> 00:00:22.300 This is a 10 kilogram ball and it's moving to the right at 3 00:00:22.300 --> 00:00:23.710 meters per second. 00:00:23.710 --> 00:00:27.720 And then it hits this ball, B, which is a 5 kilogram ball. 00:00:27.720 --> 00:00:31.920 And then we know that ball A, ball A kind of ricochets off 00:00:31.920 --> 00:00:36.740 of ball B and gets set onto this new trajectory. 00:00:36.740 --> 00:00:40.010 Now, instead of going due right, it's going at a 30 00:00:40.010 --> 00:00:43.250 degree angle to, I guess we could say, horizontal. 00:00:43.250 --> 00:00:46.610 It's going in a 30 degree angle at 2 meters per second. 00:00:46.610 --> 00:00:49.650 And the question is, what is the velocity of ball B? 00:00:49.650 --> 00:00:51.840 So velocity is both magnitude and direction. 00:00:51.840 --> 00:00:53.970 So we need to figure out essentially, 00:00:53.970 --> 00:00:55.190 what is ball B doing? 00:00:55.190 --> 00:00:59.190 Ball B is going to be going-- We can just think about it. 00:00:59.190 --> 00:01:01.130 If you ever played pool, we could guess that ball B is 00:01:01.130 --> 00:01:03.055 going to go roughly in that direction. 00:01:03.055 --> 00:01:05.990 But we need to figure out exactly what the angle is and 00:01:05.990 --> 00:01:09.410 exactly what its velocity is. 00:01:09.410 --> 00:01:11.910 So let's do this problem. 00:01:11.910 --> 00:01:14.600 So at first you're saying, oh, Sal, this looks confusing. 00:01:14.600 --> 00:01:15.900 You know, I know momentum should be 00:01:15.900 --> 00:01:16.880 conserved and all that. 00:01:16.880 --> 00:01:19.590 But now we have these vectors and there's two dimensions and 00:01:19.590 --> 00:01:20.570 how do I do that? 00:01:20.570 --> 00:01:23.540 And the key here is that there's just really one more 00:01:23.540 --> 00:01:25.710 step when you're working on it in two dimensions or really 00:01:25.710 --> 00:01:28.240 three dimensions or an arbitrary number of 00:01:28.240 --> 00:01:29.340 dimensions. 00:01:29.340 --> 00:01:31.770 When we did one dimension, you made sure that momentum was 00:01:31.770 --> 00:01:33.590 conserved in that one dimension. 00:01:33.590 --> 00:01:36.070 So when you do two dimensions, what you do is you figure out 00:01:36.070 --> 00:01:38.800 the initial momentum in each of the dimensions. 00:01:38.800 --> 00:01:41.270 So you break it up into the x and y components. 00:01:41.270 --> 00:01:44.710 And then you say the final momentum of both objects are 00:01:44.710 --> 00:01:47.680 going to equal the initial x momentum and are going to 00:01:47.680 --> 00:01:49.570 equal the initial y momentum. 00:01:49.570 --> 00:01:53.980 So let's figure out the initial x momentum. 00:01:53.980 --> 00:01:55.230 So P for momentum. 00:01:58.180 --> 00:02:00.390 Because the m is for mass. 00:02:00.390 --> 00:02:03.660 So let's say the initial momentum in the x direction-- 00:02:03.660 --> 00:02:05.730 And we don't have to write initial or final because 00:02:05.730 --> 00:02:08.080 really, the total momentum in the x direction is always 00:02:08.080 --> 00:02:08.729 going to be the same. 00:02:08.729 --> 00:02:10.400 So let's say what the initial-- Actually, let me 00:02:10.400 --> 00:02:13.130 write initial just so it hits the point home that initial 00:02:13.130 --> 00:02:14.630 and final don't change. 00:02:14.630 --> 00:02:17.260 So the initial momentum in the x direction. 00:02:17.260 --> 00:02:20.600 So i for initial, x-- I should do something better than keep 00:02:20.600 --> 00:02:24.780 writing these subscripts --is equal to what? 00:02:24.780 --> 00:02:26.770 Well ball B has no initial velocity, 00:02:26.770 --> 00:02:28.280 so it has no momentum. 00:02:28.280 --> 00:02:32.600 Ball A is 10 kilograms. 00:02:32.600 --> 00:02:34.660 And what is its velocity in the x direction? 00:02:34.660 --> 00:02:37.590 Well all of its velocity is in the x direction. 00:02:37.590 --> 00:02:39.980 So it's 3. 00:02:39.980 --> 00:02:42.550 I mean, this is only moving in the x direction. 00:02:42.550 --> 00:02:46.580 So the momentum in the x direction is 30 kilogram 00:02:46.580 --> 00:02:48.350 meters per second. 00:02:48.350 --> 00:02:51.120 Mass times velocity, kilogram meters per second. 00:02:51.120 --> 00:02:53.710 And what's the initial momentum in the y direction? 00:02:56.880 --> 00:02:58.420 Well B isn't moving at all, so it has no 00:02:58.420 --> 00:02:59.480 momentum in any direction. 00:02:59.480 --> 00:03:02.390 And A, all of A's movement is in the x direction. 00:03:02.390 --> 00:03:04.980 It's not moving at an angle or up at all, so it has no 00:03:04.980 --> 00:03:07.630 momentum in the y direction. 00:03:07.630 --> 00:03:11.030 So we immediately know that after the collision, the 00:03:11.030 --> 00:03:14.590 combined momentum of both of these balls in the x direction 00:03:14.590 --> 00:03:17.670 has to be 30, and the combined momentum of both of these 00:03:17.670 --> 00:03:21.040 balls in the y direction has to be 0. 00:03:21.040 --> 00:03:24.580 So let's figure out what A's momentum in the x and y 00:03:24.580 --> 00:03:26.840 directions are. 00:03:26.840 --> 00:03:29.080 So to figure out what A's momentum is, we just have to 00:03:29.080 --> 00:03:32.040 figure out what A's velocity in the x and y directions are 00:03:32.040 --> 00:03:33.720 and then multiply that times the mass. 00:03:33.720 --> 00:03:35.240 Because mass doesn't have any direction. 00:03:35.240 --> 00:03:39.400 So let's figure out the x and y components of this velocity. 00:03:39.400 --> 00:03:42.860 Let's do the x component of the vector first. 00:03:42.860 --> 00:03:44.610 So the x is just this vector. 00:03:49.720 --> 00:03:52.450 Change colors to keep things interesting. 00:03:52.450 --> 00:03:54.230 The y is this vector. 00:03:57.710 --> 00:04:01.490 That is the y component. 00:04:01.490 --> 00:04:03.140 And so, what are those? 00:04:03.140 --> 00:04:06.090 And this hopefully, is going to be almost second nature to 00:04:06.090 --> 00:04:08.810 you if you've been watching all of the other videos on 00:04:08.810 --> 00:04:09.880 Newton's laws. 00:04:09.880 --> 00:04:12.620 This is just our trigonometry and we can write out our 00:04:12.620 --> 00:04:16.160 SOH-CAH-TOA again. 00:04:16.160 --> 00:04:19.260 And I reassure you, this is the hardest part of any of 00:04:19.260 --> 00:04:22.019 these multi-dimensional trig problems-- Multi-dimensional 00:04:22.019 --> 00:04:24.500 physics problems, which really are just trig problems. 00:04:24.500 --> 00:04:27.870 So if we want to figure out the x component, so the 00:04:27.870 --> 00:04:31.880 velocity of A in the x direction, 00:04:31.880 --> 00:04:32.740 what is it equal to? 00:04:32.740 --> 00:04:35.240 Well this is adjacent to the angle. 00:04:35.240 --> 00:04:36.830 We know the hypotenuse. 00:04:36.830 --> 00:04:43.592 So we know VA sub x or the velocity of A in the x 00:04:43.592 --> 00:04:47.470 direction over the hypotenuse, over 2 meters per second, is 00:04:47.470 --> 00:04:48.770 equal to what? 00:04:48.770 --> 00:04:51.520 Adjacent over hypotenuse. 00:04:51.520 --> 00:04:52.620 Cosine. 00:04:52.620 --> 00:04:56.880 Is equal to cosine of 30 degrees. 00:04:56.880 --> 00:05:02.720 Or the velocity of A in the x direction is equal to 2 cosine 00:05:02.720 --> 00:05:04.420 of 30 degrees. 00:05:04.420 --> 00:05:07.300 What's cosine of 30 degrees? 00:05:07.300 --> 00:05:09.000 It's square root of 3 over 2. 00:05:09.000 --> 00:05:11.580 This is square root of 3 over 2. 00:05:11.580 --> 00:05:17.600 And square root of 3 over 2 times 2 is equal to 00:05:17.600 --> 00:05:19.120 square root of 3. 00:05:19.120 --> 00:05:24.650 So this is equal to the square root of 3 meters per second. 00:05:24.650 --> 00:05:29.045 And what is the velocity of A in the y direction? 00:05:29.045 --> 00:05:31.100 Well hopefully, this second nature to you as well. 00:05:31.100 --> 00:05:33.260 But since opposite over hypotenuse is equal to the 00:05:33.260 --> 00:05:34.730 sine of 30. 00:05:34.730 --> 00:05:42.370 So VA in the y direction is equal to 2 times the sine of 00:05:42.370 --> 00:05:43.605 30 degrees. 00:05:43.605 --> 00:05:46.730 The sine of 30 degrees is 1/2. 00:05:46.730 --> 00:05:49.330 So this is 1/2. 00:05:49.330 --> 00:05:52.360 1/2 times 2 is equal to 1 meter per second. 00:05:52.360 --> 00:05:54.490 So after the collision, A is moving at 1 00:05:54.490 --> 00:05:57.950 meter per second up. 00:05:57.950 --> 00:06:01.100 One meter per second in the upwards direction. 00:06:01.100 --> 00:06:05.100 And it's moving at square root of 3 meters per second in the 00:06:05.100 --> 00:06:06.830 rightwards direction. 00:06:06.830 --> 00:06:10.350 So what is going to be A's momentum in each of the 00:06:10.350 --> 00:06:11.280 directions? 00:06:11.280 --> 00:06:13.810 Well, we figured out its velocity, so we just multiply 00:06:13.810 --> 00:06:16.160 each of the velocities times the mass. 00:06:16.160 --> 00:06:20.920 So A has a mass of 10 kilograms. And this is going 00:06:20.920 --> 00:06:23.720 to be the final momentum. 00:06:23.720 --> 00:06:28.430 Momentum of A in the x direction is going to equal 00:06:28.430 --> 00:06:30.630 square root of 3 times 10. 00:06:30.630 --> 00:06:33.330 Square root of 3 is the velocity, 10 is the mass. 00:06:33.330 --> 00:06:39.350 So it's 10 square roots of 3 kilogram meters per second. 00:06:39.350 --> 00:06:45.640 And the momentum of A in the y direction is going to be-- and 00:06:45.640 --> 00:06:49.450 since it's going up, we'll say its positive --it's 1 meters 00:06:49.450 --> 00:06:51.210 per second is the velocity times the mass. 00:06:51.210 --> 00:06:56.860 So 10 times 1 is 10 kilogram meter per second. 00:06:56.860 --> 00:06:58.730 So now let's figure out B. 00:06:58.730 --> 00:07:01.950 Let's do the y direction first because they add up to 0. 00:07:01.950 --> 00:07:03.720 I'm going to switch colors. 00:07:03.720 --> 00:07:07.230 We know that the momentum of-- and this 00:07:07.230 --> 00:07:07.995 is after the collision. 00:07:07.995 --> 00:07:12.840 The momentum of A in the y direction plus momentum of B 00:07:12.840 --> 00:07:14.780 in the y direction have to equal what? 00:07:14.780 --> 00:07:17.150 What was the initial momentum in the y direction? 00:07:17.150 --> 00:07:19.450 Right, it was 0. 00:07:19.450 --> 00:07:22.040 There was no movement in the y direction initially. 00:07:22.040 --> 00:07:23.940 We know the momentum of A in the y direction. 00:07:23.940 --> 00:07:25.760 It's 10. 00:07:25.760 --> 00:07:29.790 10 kilogram meters per second plus the momentum of B in the 00:07:29.790 --> 00:07:32.040 y direction is equal to 0. 00:07:32.040 --> 00:07:35.050 So solving for this, just subtract 10 from both sides. 00:07:35.050 --> 00:07:41.080 So the momentum of B in the y direction is equal to 10 00:07:41.080 --> 00:07:42.360 kilogram meters per second. 00:07:45.430 --> 00:07:46.700 You know the units. 00:07:46.700 --> 00:07:50.900 So if its momentum is 10 in the y direction, what is its 00:07:50.900 --> 00:07:53.140 velocity in the y direction? 00:07:53.140 --> 00:07:56.810 Well, momentum is equal to mass times velocity. 00:07:56.810 --> 00:08:02.520 So we know that 5 times the velocity in the y direction-- 00:08:02.520 --> 00:08:05.660 that's its mass --is equal to 10. 00:08:05.660 --> 00:08:07.410 10 is its momentum. 00:08:07.410 --> 00:08:10.970 So the velocity of the y direction of B must be 2 00:08:10.970 --> 00:08:12.570 meters per second. 00:08:12.570 --> 00:08:13.350 So there we go. 00:08:13.350 --> 00:08:15.160 We figured out B's velocity. 00:08:15.160 --> 00:08:19.110 And so let's say this is B's velocity vector in the y 00:08:19.110 --> 00:08:22.740 direction is-- And this is a minus because this is 00:08:22.740 --> 00:08:26.935 equal to minus 10. 00:08:26.935 --> 00:08:30.380 So it's going down. 00:08:30.380 --> 00:08:32.740 It was a velocity of positive 1 going up and then the 00:08:32.740 --> 00:08:35.380 minuses carry through and this is a velocity of minus 2 00:08:35.380 --> 00:08:40.059 meters per second for B in the y direction. 00:08:40.059 --> 00:08:41.340 So now let's figure out the velocity 00:08:41.340 --> 00:08:42.808 of B in the x direction. 00:08:42.808 --> 00:08:45.540 And I'm running out of space and it's getting messy. 00:08:45.540 --> 00:08:49.860 But we just have to remember that the momentum of B in 00:08:49.860 --> 00:08:54.480 the-- The momentum of A in the x direction, which is 10 00:08:54.480 --> 00:09:02.810 square root of 3, plus momentum of B in the x 00:09:02.810 --> 00:09:05.770 direction has to equal what? 00:09:05.770 --> 00:09:08.090 It has to equal the initial momentum in the x direction, 00:09:08.090 --> 00:09:10.240 which is 30. 00:09:10.240 --> 00:09:12.590 So to figure out the momentum of B in the x direction, we 00:09:12.590 --> 00:09:15.010 just subtract 10 square root of 3 from 30. 00:09:15.010 --> 00:09:17.760 And let's do that. 00:09:17.760 --> 00:09:24.010 So let's figure out 3 square root times 10 equals. 00:09:24.010 --> 00:09:29.630 And then subtract that from 30. 00:09:29.630 --> 00:09:34.300 And we get let's just say 12.7. 00:09:34.300 --> 00:09:38.380 So we know that the momentum of B in the x direction is 00:09:38.380 --> 00:09:43.420 equal to 12.7, 12.7 kilogram meters per second. 00:09:43.420 --> 00:09:45.680 And we know the momentum, so we just divide by the mass and 00:09:45.680 --> 00:09:47.030 we get its velocity in the x direction. 00:09:47.030 --> 00:09:48.950 So 12.7 divided by 5. 00:09:48.950 --> 00:09:51.630 So velocity of B in the x direction is 00:09:51.630 --> 00:09:54.970 12.7 divided by 5. 00:09:54.970 --> 00:10:04.460 12.7 divided by 5 is equal to 2.54 meters per second. 00:10:04.460 --> 00:10:07.020 So its velocity in the x direction is 00:10:07.020 --> 00:10:10.080 2.54 meters per second. 00:10:10.080 --> 00:10:14.250 So it's going faster in both directions. 00:10:14.250 --> 00:10:15.970 I'm not going to figure out the angle here because I've 00:10:15.970 --> 00:10:17.650 actually run out of time. 00:10:17.650 --> 00:10:20.070 But if you were to add these two vectors, you'd get an 00:10:20.070 --> 00:10:21.340 angle something like this. 00:10:21.340 --> 00:10:24.700 And you could figure out the angle by taking the arc tan. 00:10:24.700 --> 00:10:27.080 Well, I won't go into the-- that's a complexity right now. 00:10:27.080 --> 00:10:29.050 Actually, I'll do that in the next video just so I won't 00:10:29.050 --> 00:10:29.620 leave you hanging. 00:10:29.620 --> 00:10:33.660 But we know what the x and y components of B's velocity is. 00:10:33.660 --> 00:10:35.380 See you in the next video.

Momentum: Ice skater throws a ball

https://www.youtube.com/watch?v=vPkkCOlGND4

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https://www.youtube.com/api/timedtext?v=vPkkCOlGND4&ei=YmeUZeONMJ29mLAPxoSJmAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B2AF49B13F28A416448ED75727F31DA006C4E879.BBE99D3546BA436753F8498C941AE33C148F93B0&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:01.440 Welcome back. 00:00:01.440 --> 00:00:03.950 I'll now do a couple of more momentum problems. 00:00:03.950 --> 00:00:07.060 So this first problem, I have this ice skater and she's on 00:00:07.060 --> 00:00:08.630 an ice skating rink. 00:00:08.630 --> 00:00:10.360 And what she's doing is she's holding a ball. 00:00:10.360 --> 00:00:14.740 And this ball-- let me draw the ball-- this is a 0.15 00:00:14.740 --> 00:00:15.990 kilogram ball. 00:00:18.610 --> 00:00:20.580 And she throws it. 00:00:20.580 --> 00:00:23.640 Let's just say she throws it directly straight forward in 00:00:23.640 --> 00:00:25.200 front of her, although she's staring at us. 00:00:25.200 --> 00:00:27.230 She's actually forward for her body. 00:00:27.230 --> 00:00:32.790 So she throws it exactly straight forward. 00:00:32.790 --> 00:00:35.075 And I understand it is hard to throw something straight 00:00:35.075 --> 00:00:38.490 forward, but let's assume that she can. 00:00:38.490 --> 00:00:41.510 So she throws it exactly straight forward with a 00:00:41.510 --> 00:00:44.280 speed-- or since we're going to give the direction as well, 00:00:44.280 --> 00:00:48.000 it's a velocity, right, cause speed is just a magnitude 00:00:48.000 --> 00:00:51.200 while a velocity is a magnitude and a direction-- so 00:00:51.200 --> 00:00:58.160 she throws the ball at 35 meters per second, and this 00:00:58.160 --> 00:01:03.160 ball is 0.15 kilograms. 00:01:03.160 --> 00:01:08.560 Now, what the problem says is that their combined mass, her 00:01:08.560 --> 00:01:17.520 plus the ball, is 50 kilograms. So they're both 00:01:17.520 --> 00:01:20.130 stationary before she does anything, and then she throws 00:01:20.130 --> 00:01:22.990 this ball, and the question is, after throwing this ball, 00:01:22.990 --> 00:01:25.000 what is her recoil velocity? 00:01:25.000 --> 00:01:28.930 Or essentially, well how much, by throwing the ball, does she 00:01:28.930 --> 00:01:30.230 push herself backwards? 00:01:30.230 --> 00:01:33.060 So what is her velocity in the backward direction? 00:01:33.060 --> 00:01:36.340 And if you're not familiar with the term recoil, it's 00:01:36.340 --> 00:01:39.600 often applied to when someone, I guess, not that we want to 00:01:39.600 --> 00:01:42.250 think about violent things, but if you shoot a gun, your 00:01:42.250 --> 00:01:44.830 shoulder recoils back, because once 00:01:44.830 --> 00:01:45.900 again momentum is conserved. 00:01:45.900 --> 00:01:48.270 So there's a certain amount of momentum going into that 00:01:48.270 --> 00:01:51.020 bullet, which is very light and fast going forward. 00:01:51.020 --> 00:01:54.940 But since momentum is conserved, your shoulder has 00:01:54.940 --> 00:01:55.780 velocity backwards. 00:01:55.780 --> 00:01:57.250 But we'll do another problem with that. 00:01:57.250 --> 00:01:58.960 So let's get back to this problem. 00:01:58.960 --> 00:02:02.410 So like I just said, momentum is conserved. 00:02:02.410 --> 00:02:05.760 So what's the momentum at the start of the problem, the 00:02:05.760 --> 00:02:08.289 initial momentum? 00:02:08.289 --> 00:02:09.690 Let me do a different color. 00:02:09.690 --> 00:02:11.730 So this is the initial momentum. 00:02:11.730 --> 00:02:18.060 Initially, the mass is 50 kilograms, right, cause her 00:02:18.060 --> 00:02:22.110 and the ball combined are 50 kilograms, times the velocity. 00:02:22.110 --> 00:02:23.810 Well the velocity is 0. 00:02:23.810 --> 00:02:29.800 So initially, there is 0 velocity in the system. 00:02:29.800 --> 00:02:34.060 So the momentum is 0. 00:02:34.060 --> 00:02:37.430 The P initial is equal to 0. 00:02:37.430 --> 00:02:41.560 And since we start with a net 0 momentum, we have to finish 00:02:41.560 --> 00:02:42.880 with a net 0 momentum. 00:02:42.880 --> 00:02:44.030 So what's momentum later? 00:02:44.030 --> 00:02:47.730 Well we have a ball moving at 35 meters per second and the 00:02:47.730 --> 00:02:58.040 ball has a mass of 0.15 kilograms. I'll ignore the 00:02:58.040 --> 00:02:59.710 units for now just to save space. 00:02:59.710 --> 00:03:01.930 Times the velocity of the ball. 00:03:01.930 --> 00:03:05.060 Times 35 meters per second. 00:03:05.060 --> 00:03:08.930 So this is the momentum of the ball plus the new momentum of 00:03:08.930 --> 00:03:10.020 the figure skater. 00:03:10.020 --> 00:03:12.060 So what's her mass? 00:03:12.060 --> 00:03:14.440 Well her mass is going to be 50 minus this. 00:03:14.440 --> 00:03:21.550 It actually won't matter a ton, but let's say it's 49-- 00:03:21.550 --> 00:03:25.330 what is that-- 49.85 kilograms, 00:03:25.330 --> 00:03:28.180 times her new velocity. 00:03:28.180 --> 00:03:29.040 Times velocity. 00:03:29.040 --> 00:03:31.410 Let's call that the velocity of the skater. 00:03:31.410 --> 00:03:34.890 So let me get my trusty calculator out. 00:03:37.910 --> 00:03:40.640 OK, so let's see. 00:03:40.640 --> 00:03:50.780 0.15 times 35 is equal to 5.25. 00:03:50.780 --> 00:03:56.260 So that equals 5.25. 00:03:56.260 --> 00:04:02.350 plus 49.85 times the skater's velocity, the final velocity. 00:04:02.350 --> 00:04:04.550 And of course, this equals 0 because the initial 00:04:04.550 --> 00:04:05.930 velocity was 0. 00:04:05.930 --> 00:04:10.000 So let's, I don't know, subtract 5.25 from both sides 00:04:10.000 --> 00:04:18.200 and then the equation becomes minus 5.25 is equal to 49.85 00:04:18.200 --> 00:04:20.279 times the velocity of the skater. 00:04:20.279 --> 00:04:23.480 So we're essentially saying that the momentum of just the 00:04:23.480 --> 00:04:25.380 ball is 5.25. 00:04:25.380 --> 00:04:29.480 And since the combined system has to have 0 net momentum, 00:04:29.480 --> 00:04:32.660 we're saying that the momentum of the skater has to be 5.25 00:04:32.660 --> 00:04:35.960 in the other direction, going backwards, or has a momentum 00:04:35.960 --> 00:04:39.230 of minus 5.25. 00:04:39.230 --> 00:04:41.480 And to figure out the velocity, we just divide her 00:04:41.480 --> 00:04:43.780 momentum by her mass. 00:04:43.780 --> 00:04:48.380 And so divide both sides by 49.85 and you get the velocity 00:04:48.380 --> 00:04:49.695 of the skater. 00:04:49.695 --> 00:04:50.725 So let's see. 00:04:50.725 --> 00:05:01.520 Let's make this a negative number divided by 49.85 equals 00:05:01.520 --> 00:05:05.370 minus 0.105. 00:05:05.370 --> 00:05:15.520 So minus 0.105 meters per second. 00:05:15.520 --> 00:05:16.270 So that's interesting. 00:05:16.270 --> 00:05:20.370 When she throws this ball out at 35 meters per second, which 00:05:20.370 --> 00:05:24.670 is pretty fast, she will recoil back at about 10 00:05:24.670 --> 00:05:28.440 centimeters, yeah, roughly 10 centimeters per second. 00:05:28.440 --> 00:05:30.530 So she will recoil a lot slower, although 00:05:30.530 --> 00:05:31.740 she will move back. 00:05:31.740 --> 00:05:34.350 And if you think about it, this is a form of propulsion. 00:05:34.350 --> 00:05:35.790 This is how rockets work. 00:05:35.790 --> 00:05:40.120 They eject something that maybe has less mass, but super 00:05:40.120 --> 00:05:44.500 fast. And that, since we have a conservation of momentum, it 00:05:44.500 --> 00:05:47.740 makes the rocket move in the other direction. 00:05:47.740 --> 00:05:51.550 Well anyway, let's see if we could fit another problem in. 00:05:51.550 --> 00:05:54.600 Actually, it's probably better to leave this problem done and 00:05:54.600 --> 00:05:56.760 then I'll have more time for the next problem, which will 00:05:56.760 --> 00:05:58.515 be slightly more difficult. 00:05:58.515 --> 00:05:59.765 See you soon.

Introduction to momentum

https://www.youtube.com/watch?v=XFhntPxow0U

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https://www.youtube.com/api/timedtext?v=XFhntPxow0U&ei=YmeUZZn4Muu4p-oPnbaCyAc&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=C9BCD9D0AAC0B163E0E64B512A19E2256CC0E237.E4911A13B909B29EE44A6AA10235AAF71952DAC3&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.810 --> 00:00:01.580 Welcome back. 00:00:01.580 --> 00:00:05.930 I will now introduce you to the concept of momentum. 00:00:05.930 --> 00:00:08.800 And the letter for momentum is, in physics, or at least in 00:00:08.800 --> 00:00:10.230 mechanics, it's the letter P. 00:00:10.230 --> 00:00:11.890 P for momentum. 00:00:11.890 --> 00:00:15.010 And I assume that's because the letter M has already been 00:00:15.010 --> 00:00:17.200 used for mass, which is I guess an even 00:00:17.200 --> 00:00:18.400 more fundamental idea. 00:00:18.400 --> 00:00:20.280 So P for momentum. 00:00:20.280 --> 00:00:21.180 So what is momentum? 00:00:21.180 --> 00:00:25.030 Well, you probably have a general idea of it. 00:00:25.030 --> 00:00:27.740 If you see a big guy running really fast, they'll say, he 00:00:27.740 --> 00:00:28.990 has a lot of momentum. 00:00:28.990 --> 00:00:31.440 And if there's a big guy running really fast and a 00:00:31.440 --> 00:00:33.940 small guy running really fast, most people would say, well, 00:00:33.940 --> 00:00:36.530 the big guy has more momentum. 00:00:36.530 --> 00:00:37.980 Maybe they don't have a quantitative sense of why 00:00:37.980 --> 00:00:40.040 they're saying that, but they just feel that 00:00:40.040 --> 00:00:40.910 that must be true. 00:00:40.910 --> 00:00:43.070 And if we look at the definition of momentum, it'll 00:00:43.070 --> 00:00:43.730 make sense. 00:00:43.730 --> 00:00:50.450 The definition of momentum is equal to mass times velocity. 00:00:50.450 --> 00:00:53.060 So something with, say, a medium mass and a huge 00:00:53.060 --> 00:00:55.400 velocity is going to have a big momentum. 00:00:55.400 --> 00:00:59.340 Or something with maybe a medium mass, but-- the other 00:00:59.340 --> 00:00:59.770 way around. 00:00:59.770 --> 00:01:00.860 I forgot what I just said. 00:01:00.860 --> 00:01:03.230 So medium mass and big velocity, huge momentum, or 00:01:03.230 --> 00:01:03.940 the other way around. 00:01:03.940 --> 00:01:06.280 Huge mass, medium velocity would have maybe the same 00:01:06.280 --> 00:01:08.230 momentum, but it would still have a big momentum. 00:01:08.230 --> 00:01:11.910 Or another way of doing momentum is how little you 00:01:11.910 --> 00:01:16.050 would like to be in the way of that object as it passes by. 00:01:18.850 --> 00:01:21.610 How unpleasant would it be to be hit by that object? 00:01:21.610 --> 00:01:25.060 That's a good way of thinking about momentum. 00:01:25.060 --> 00:01:27.603 So momentum is mass times velocity. 00:01:30.660 --> 00:01:32.480 So how does it relate to everything we've 00:01:32.480 --> 00:01:33.420 been learning so far? 00:01:33.420 --> 00:01:41.270 So we know that force is equal to mass times acceleration. 00:01:41.270 --> 00:01:42.030 And what's acceleration? 00:01:42.030 --> 00:01:44.420 Well acceleration is just change in velocity. 00:01:44.420 --> 00:01:52.580 So we also know that force is equal to mass times change in 00:01:52.580 --> 00:01:57.850 velocity per unit of time, right? 00:01:57.850 --> 00:02:00.890 Per change in time. 00:02:00.890 --> 00:02:02.720 T for time. 00:02:02.720 --> 00:02:06.550 So force is also equal to-- well, mass 00:02:06.550 --> 00:02:07.290 times change in velocity. 00:02:07.290 --> 00:02:10.039 Mass, let's assume that mass doesn't change. 00:02:10.039 --> 00:02:15.000 So that could also be viewed as the change in mass times 00:02:15.000 --> 00:02:18.300 velocity in the unit amount of time. 00:02:18.300 --> 00:02:20.130 And this is a little tricky here, I said, you know, the 00:02:20.130 --> 00:02:21.960 mass times the change in velocity, that's the same 00:02:21.960 --> 00:02:24.230 thing as the change in the mass times the velocity, 00:02:24.230 --> 00:02:26.160 assuming the mass doesn't change. 00:02:26.160 --> 00:02:28.960 And here we have mass times velocity, which is momentum. 00:02:28.960 --> 00:02:34.310 So force can also be viewed as change in 00:02:34.310 --> 00:02:39.026 momentum per unit of time. 00:02:39.026 --> 00:02:40.420 And I'll introduce you to another 00:02:40.420 --> 00:02:41.810 concept called impulse. 00:02:41.810 --> 00:02:43.990 And impulse kind of means that you think it means. 00:02:43.990 --> 00:02:46.640 An impulse is defined as force times time. 00:02:46.640 --> 00:02:48.980 And I just want to introduce this to you just in case you 00:02:48.980 --> 00:02:51.760 see it on the exam or whatever, show you it's not a 00:02:51.760 --> 00:02:53.010 difficult concept. 00:02:53.010 --> 00:02:57.120 So force times change in time, or time, if you assume time 00:02:57.120 --> 00:02:58.670 starts at time 0. 00:02:58.670 --> 00:03:01.805 But force times change in time is equal to impulse. 00:03:01.805 --> 00:03:04.430 I actually don't know-- I should look up what letters 00:03:04.430 --> 00:03:05.720 they use for impulse. 00:03:05.720 --> 00:03:08.490 But another way of viewing impulse is force 00:03:08.490 --> 00:03:09.360 times change in time. 00:03:09.360 --> 00:03:12.960 Well that's the same thing as change in momentum over change 00:03:12.960 --> 00:03:15.840 in time times change in time. 00:03:15.840 --> 00:03:16.110 Right? 00:03:16.110 --> 00:03:19.160 Because this is just the same thing as force. 00:03:19.160 --> 00:03:21.210 And that's just change in momentum, so 00:03:21.210 --> 00:03:22.460 that's impulse as well. 00:03:26.280 --> 00:03:28.940 And the unit of impulse is the joule. 00:03:28.940 --> 00:03:30.450 And we'll go more into the joule when we do 00:03:30.450 --> 00:03:31.200 work in all of that. 00:03:31.200 --> 00:03:33.500 And if this confuses you, don't worry about it too much. 00:03:33.500 --> 00:03:37.050 The main thing about momentum is that you realize it's mass 00:03:37.050 --> 00:03:39.100 times velocity. 00:03:39.100 --> 00:03:42.700 And since force is change in momentum per unit of time, if 00:03:42.700 --> 00:03:47.220 you don't have any external forces on a system or, on say, 00:03:47.220 --> 00:03:51.960 on a set of objects, their combined, or their net 00:03:51.960 --> 00:03:53.070 momentum won't change. 00:03:53.070 --> 00:03:54.550 And that comes from Newton's Laws. 00:03:54.550 --> 00:03:58.260 The only way you can get a combined change in momentum is 00:03:58.260 --> 00:04:02.150 if you have some type of net force acting on the system. 00:04:02.150 --> 00:04:03.770 So with that in mind, let's do some 00:04:03.770 --> 00:04:08.340 momentum problems. Whoops. 00:04:08.340 --> 00:04:10.330 Invert colors. 00:04:10.330 --> 00:04:11.450 OK. 00:04:11.450 --> 00:04:14.470 So let's say we have a car. 00:04:14.470 --> 00:04:17.029 Say it's a car. 00:04:17.029 --> 00:04:19.860 Let me do some more interesting colors. 00:04:19.860 --> 00:04:23.340 A car with a magenta bottom. 00:04:23.340 --> 00:04:25.390 And it is, let's see, what does this problem say? 00:04:25.390 --> 00:04:31.100 It's 1,000 kilograms. So a little over a ton. 00:04:31.100 --> 00:04:35.410 And it's moving at 9 meters per second east. So its 00:04:35.410 --> 00:04:41.550 velocity is equal to 9 meters per second east, or to the 00:04:41.550 --> 00:04:43.290 right in this example. 00:04:43.290 --> 00:04:47.460 And it strikes a stationary 2, 000 kilogram truck. 00:04:47.460 --> 00:04:48.710 So here's my truck. 00:04:53.420 --> 00:04:58.450 Here's my truck and this is a 2,000 kilogram truck. 00:04:58.450 --> 00:05:02.580 And it's stationary, so the velocity is 0. 00:05:02.580 --> 00:05:06.330 And when the car hits the truck, let's just say that it 00:05:06.330 --> 00:05:08.600 somehow gets stuck in the truck and they just both keep 00:05:08.600 --> 00:05:09.740 moving together. 00:05:09.740 --> 00:05:12.340 So they get stuck together. 00:05:12.340 --> 00:05:17.640 The question is, what is the resulting speed of the 00:05:17.640 --> 00:05:22.050 combination truck and car after the collision? 00:05:22.050 --> 00:05:24.190 Well, all we have to do is think about what is the 00:05:24.190 --> 00:05:27.120 combined momentum before the collision? 00:05:27.120 --> 00:05:27.610 Well let's see. 00:05:27.610 --> 00:05:30.800 The momentum of the car is going to be the mass times the 00:05:30.800 --> 00:05:33.850 car-- mass of the car. 00:05:33.850 --> 00:05:36.530 Well the total momentum is going to the mass of the car 00:05:36.530 --> 00:05:43.180 times the velocity of the car plus the mass of the truck 00:05:43.180 --> 00:05:47.250 times the velocity of the truck. 00:05:47.250 --> 00:05:49.220 And this is before they hit each other. 00:05:49.220 --> 00:05:50.370 So what's the mass of the car? 00:05:50.370 --> 00:05:52.660 That's 1,000. 00:05:52.660 --> 00:05:53.760 What's the velocity of the car? 00:05:53.760 --> 00:05:56.370 It's 9 meters per second. 00:05:56.370 --> 00:05:59.110 So as you can imagine, a unit of momentum would be kilogram 00:05:59.110 --> 00:06:00.130 meters per second. 00:06:00.130 --> 00:06:02.990 So it's 1,000 times 9 kilogram meters per second, but I won't 00:06:02.990 --> 00:06:05.950 write that right now just to keep things simple, or so I 00:06:05.950 --> 00:06:06.960 save space. 00:06:06.960 --> 00:06:09.575 And then the mass of the truck is 2,000. 00:06:09.575 --> 00:06:10.735 And what's its velocity? 00:06:10.735 --> 00:06:11.540 Well, it's 0. 00:06:11.540 --> 00:06:13.480 It's stationary initially. 00:06:13.480 --> 00:06:16.380 So the initial momentum of the system-- this is 2,000 times 00:06:16.380 --> 00:06:24.080 0-- is 9,000 plus 0, which equals 9,000 kilogram meters 00:06:24.080 --> 00:06:25.090 per second. 00:06:25.090 --> 00:06:28.440 That's the momentum before the car hits 00:06:28.440 --> 00:06:29.970 the back of the truck. 00:06:29.970 --> 00:06:32.170 Now what happens after the car hits the back of the truck? 00:06:32.170 --> 00:06:33.520 So let's go to that situation. 00:06:33.520 --> 00:06:36.230 So we have the truck. 00:06:36.230 --> 00:06:37.860 I'll draw it a little less neatly. 00:06:37.860 --> 00:06:41.050 And then you have the car and it's probably a little bit-- 00:06:41.050 --> 00:06:44.040 well, I won't go into whether it's banged up and whether it 00:06:44.040 --> 00:06:45.360 released heat and all of that. 00:06:45.360 --> 00:06:50.060 Let's assume that there was nothing-- if this is a simple 00:06:50.060 --> 00:06:51.830 problem that we can do. 00:06:51.830 --> 00:06:53.700 So if we assume that, there would be 00:06:53.700 --> 00:06:54.990 no change in momentum. 00:06:54.990 --> 00:06:57.410 Because we're saying that there's no net forces acting 00:06:57.410 --> 00:06:58.010 on the system. 00:06:58.010 --> 00:06:59.910 And when I say system, I mean the combination of 00:06:59.910 --> 00:07:01.410 the car and the truck. 00:07:01.410 --> 00:07:04.580 So what we're saying is, is this combination, this new 00:07:04.580 --> 00:07:09.100 vehicle called a car truck, its momentum will have to be 00:07:09.100 --> 00:07:13.070 the same as the car and the truck's momentum when they 00:07:13.070 --> 00:07:14.600 were separate. 00:07:14.600 --> 00:07:16.780 So what do we know about this car truck object? 00:07:16.780 --> 00:07:18.330 Well we know its new mass. 00:07:18.330 --> 00:07:21.050 The car truck object, it will be the 00:07:21.050 --> 00:07:22.290 combined mass of the two. 00:07:22.290 --> 00:07:26.550 So it's 1,000 kilograms plus 2,000 kilograms. So it's 3,000 00:07:26.550 --> 00:07:30.690 kilograms. And now we can use that information to figure out 00:07:30.690 --> 00:07:31.510 its velocity. 00:07:31.510 --> 00:07:32.190 How? 00:07:32.190 --> 00:07:36.300 Well, its momentum-- this 3,000 kilogram object's 00:07:36.300 --> 00:07:40.440 momentum-- has to be the same as the momentum of the two 00:07:40.440 --> 00:07:42.510 objects before the collision. 00:07:42.510 --> 00:07:46.230 So it still has to be 9,000 kilogram meters per second. 00:07:46.230 --> 00:07:49.300 So once again, mass times velocity. 00:07:49.300 --> 00:07:54.120 So mass is 3,000 times the new velocity. 00:07:54.120 --> 00:07:57.580 So we could call that, I don't know, new velocity, v sub n. 00:07:57.580 --> 00:08:01.200 That will equal 9,000. 00:08:01.200 --> 00:08:02.870 Because momentum is conserved. 00:08:02.870 --> 00:08:04.390 That's what you always have to remember. 00:08:04.390 --> 00:08:07.630 Momentum doesn't change unless there's a net force acting on 00:08:07.630 --> 00:08:08.030 the system. 00:08:08.030 --> 00:08:13.110 Because we saw a force is change in momentum per time. 00:08:13.110 --> 00:08:14.670 So if you have no force in it, you have 00:08:14.670 --> 00:08:16.626 no change in momentum. 00:08:16.626 --> 00:08:17.870 So let's just solve. 00:08:17.870 --> 00:08:22.990 Divide both sides of this by 3,000 and you get the new 00:08:22.990 --> 00:08:28.460 velocity is 3 meters per second. 00:08:28.460 --> 00:08:29.490 And that kind of makes sense. 00:08:29.490 --> 00:08:32.840 You have a relatively light car moving at 9 meters per 00:08:32.840 --> 00:08:34.080 second and a stationary truck. 00:08:34.080 --> 00:08:36.299 Then it smacks the truck and they move together. 00:08:36.299 --> 00:08:41.690 The combined object-- and it's going to be to the east. And 00:08:41.690 --> 00:08:44.970 we'll do more later, but we assume that a positive 00:08:44.970 --> 00:08:47.310 velocity is east. If somehow we ended up with a negative, 00:08:47.310 --> 00:08:50.040 it would have been west. But it makes sense because we have 00:08:50.040 --> 00:08:53.590 a light object and a stationery, heavy object. 00:08:53.590 --> 00:08:56.590 And when the light object hits the stationery, heavy object, 00:08:56.590 --> 00:08:59.510 the combined objects still keeps moving to the right, but 00:08:59.510 --> 00:09:03.010 it moves at a relatively slower speed. 00:09:03.010 --> 00:09:04.660 So hopefully that gives you a little bit of intuition for 00:09:04.660 --> 00:09:08.350 momentum, and that was not too confusing of a problem. 00:09:08.350 --> 00:09:11.170 And in the next couple of videos, I'll do more momentum 00:09:11.170 --> 00:09:13.430 problems and then I'll introduce you to momentum 00:09:13.430 --> 00:09:15.350 problems in two dimensions. 00:09:15.350 --> 00:09:17.450 I will see you soon.

Tension in an accelerating system and pie in the face

https://www.youtube.com/watch?v=52wxpYnS64U

vtt

https://www.youtube.com/api/timedtext?v=52wxpYnS64U&ei=YmeUZcv2Mpe5mLAPsu6TyA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8EACCB56E982574458B8031062E9236A8CBC79F4.3AFC4BE7E7EC35A7D7FF435731732EA5FDE1D839&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.680 --> 00:00:01.360 Welcome back. 00:00:01.360 --> 00:00:03.820 We just finished this problem with the pulleys and the 00:00:03.820 --> 00:00:04.490 inclined plane. 00:00:04.490 --> 00:00:06.890 And I just wanted to do one final thing on this problem 00:00:06.890 --> 00:00:08.430 just because I think it's interesting. 00:00:08.430 --> 00:00:11.150 And then we can move onto what seems like 00:00:11.150 --> 00:00:12.810 a pretty fun problem. 00:00:12.810 --> 00:00:14.780 So the last thing I want to figure out is, we figured out 00:00:14.780 --> 00:00:17.720 that this 20 kilo-- actually, the whole system will 00:00:17.720 --> 00:00:20.860 accelerate up and to the right at 4.13 00:00:20.860 --> 00:00:22.260 meters per second squared. 00:00:22.260 --> 00:00:24.580 And then the second part of this question is, what is the 00:00:24.580 --> 00:00:28.485 tension in this rope or this wire? 00:00:28.485 --> 00:00:30.990 And at first you might say, this is complicated. 00:00:30.990 --> 00:00:32.940 You know, this thing isn't static anymore. 00:00:32.940 --> 00:00:34.390 The thing is actually accelerating. 00:00:34.390 --> 00:00:35.370 How do I do it? 00:00:35.370 --> 00:00:36.900 Well this is how you think about it. 00:00:36.900 --> 00:00:39.350 Just pick one part of the system. 00:00:39.350 --> 00:00:43.430 Let's say that all we could see was this 20 kilogram mass. 00:00:43.430 --> 00:00:48.340 So let's say all we could see was this 20 kilogram mass. 00:00:48.340 --> 00:00:50.760 And we know it's suspended from a wire. 00:00:50.760 --> 00:00:54.780 And we also know that this 20 kilogram mass is not 00:00:54.780 --> 00:00:56.840 accelerating as fast as it would if 00:00:56.840 --> 00:00:57.980 the wire wasn't there. 00:00:57.980 --> 00:01:01.210 It's accelerating only at 4.13 meters per second. 00:01:01.210 --> 00:01:04.510 If the wire wasn't there, it'd be accelerating at 9.8 meters 00:01:04.510 --> 00:01:06.430 per second, the acceleration of gravity. 00:01:06.430 --> 00:01:09.140 So the wire must be exerting some upward 00:01:09.140 --> 00:01:10.520 force on the object. 00:01:10.520 --> 00:01:13.306 And that is the force of tension. 00:01:13.306 --> 00:01:17.060 That is what's slowing-- that's what's moderating its 00:01:17.060 --> 00:01:21.090 acceleration from being 9.8 meters per second squared to 00:01:21.090 --> 00:01:24.180 being 4.13 meters per second squared. 00:01:24.180 --> 00:01:27.290 So essentially, what is the net force on this object? 00:01:27.290 --> 00:01:29.550 On just this object? 00:01:29.550 --> 00:01:32.310 Well the net force is-- and you can ignore what I said 00:01:32.310 --> 00:01:36.100 before about the net force in all the other places. 00:01:36.100 --> 00:01:42.050 But we know that the object is accelerating downwards. 00:01:42.050 --> 00:01:44.400 Well, we know it's 20 kilograms. So that's its mass. 00:01:44.400 --> 00:01:46.980 And we know that it's accelerating downwards at 4.13 00:01:46.980 --> 00:01:48.230 meters per second squared. 00:01:51.220 --> 00:02:00.460 So the net force, 20 times-- see, times 20 is 82-- let's 00:02:00.460 --> 00:02:03.120 just say 83 Newtons. 00:02:03.120 --> 00:02:05.400 83 Newtons down. 00:02:05.400 --> 00:02:08.990 We know that the net force is 83 Newtons down. 00:02:08.990 --> 00:02:16.520 We also know that the tension force plus the force of 00:02:16.520 --> 00:02:18.200 gravity-- and what's the force of gravity? 00:02:18.200 --> 00:02:20.780 The force of gravity is just the weight of the object. 00:02:20.780 --> 00:02:24.680 So the force of tension, which goes up, plus the weight of -- 00:02:24.680 --> 00:02:29.030 the force of gravity is equal to the net force. 00:02:29.030 --> 00:02:31.010 And the way I set this up, tension's going to be a 00:02:31.010 --> 00:02:33.270 negative number. 00:02:33.270 --> 00:02:36.530 Just because I'm saying positive numbers are 00:02:36.530 --> 00:02:39.060 downwards, so a negative number would be upwards. 00:02:39.060 --> 00:02:46.870 So tension will be what is 83 minus 196? 00:02:46.870 --> 00:02:54.800 Minus 196 is equal to minus 113 Newtons. 00:02:54.800 --> 00:02:57.040 And the only reason why I got a negative number is because I 00:02:57.040 --> 00:02:59.150 used positive numbers for downwards. 00:02:59.150 --> 00:03:02.520 So minus 113 Newtons downwards, which is the same 00:03:02.520 --> 00:03:06.160 thing as 113 Newtons upwards. 00:03:06.160 --> 00:03:09.530 And so that is the tension in the rope. 00:03:09.530 --> 00:03:11.970 And you could have done the same thing on this side of the 00:03:11.970 --> 00:03:14.120 problem, although it would have been-- well, yeah. 00:03:14.120 --> 00:03:15.290 You could have done the exact same thing on 00:03:15.290 --> 00:03:16.150 this side of the problem. 00:03:16.150 --> 00:03:17.750 You would've said, well what would it have accelerated 00:03:17.750 --> 00:03:20.870 naturally if there wasn't some force of tension on this rope 00:03:20.870 --> 00:03:22.230 going backwards? 00:03:22.230 --> 00:03:24.090 And then you're saying, oh, well, we know it would have 00:03:24.090 --> 00:03:26.350 gone in this direction at some acceleration, but instead it's 00:03:26.350 --> 00:03:27.570 going in the other direction. 00:03:27.570 --> 00:03:29.090 So you use that. 00:03:29.090 --> 00:03:31.460 You figure out the net force, and then you say the tension 00:03:31.460 --> 00:03:33.840 plus all of these forces have to equal the net force. 00:03:33.840 --> 00:03:36.200 And then you should solve for the tension. 00:03:36.200 --> 00:03:38.510 And it would be the same tension. 00:03:38.510 --> 00:03:44.760 Now we will do a fun and somewhat simple, but maybe 00:03:44.760 --> 00:03:46.780 instructive problem. 00:03:46.780 --> 00:03:49.860 So I have a pie. 00:03:49.860 --> 00:03:51.110 This is the pie. 00:03:54.140 --> 00:03:55.580 This is parallel. 00:03:55.580 --> 00:03:58.270 And I have my hand. 00:03:58.270 --> 00:04:02.070 You can tell that my destiny was really to be a great 00:04:02.070 --> 00:04:04.170 artist. This is my hand. 00:04:04.170 --> 00:04:10.100 And I'm holding a pie, and I'm looking to smash this pie into 00:04:10.100 --> 00:04:12.670 this individual's face. 00:04:19.382 --> 00:04:24.790 I actually was a, I was the newspaper cartoonist in high 00:04:24.790 --> 00:04:28.360 school, so I have some minor-- but anyway. 00:04:28.360 --> 00:04:30.140 Let's make it a bald man. 00:04:30.140 --> 00:04:32.950 Well anyway, I shouldn't be focusing on the drawing. 00:04:36.400 --> 00:04:37.650 He has a moustache. 00:04:41.280 --> 00:04:45.720 Anyway, I'm looking to throw this pie into this guy's face. 00:04:45.720 --> 00:04:49.090 And the problem is, I need to figure out how fast do I need 00:04:49.090 --> 00:04:52.250 to accelerate this pie for it to not fall down? 00:04:52.250 --> 00:04:52.490 Right? 00:04:52.490 --> 00:04:53.190 Because what's happening? 00:04:53.190 --> 00:04:56.290 Well there's the force of gravity on this pie. 00:04:56.290 --> 00:04:58.500 There's a force of gravity on this pie and if I don't 00:04:58.500 --> 00:05:01.220 accelerate it fast enough, it's just going to slide down. 00:05:01.220 --> 00:05:03.000 And I'll never be able to, It'll never 00:05:03.000 --> 00:05:04.200 reach the guy's face. 00:05:04.200 --> 00:05:06.440 So I don't want this pie to slide down at all. 00:05:06.440 --> 00:05:09.070 How fast do I have to push on it? 00:05:09.070 --> 00:05:11.640 Well, we know that the coefficient of friction-- you 00:05:11.640 --> 00:05:13.400 don't know this, but I know that the coefficient of 00:05:13.400 --> 00:05:17.700 friction between my hand and the pie, the coefficient of 00:05:17.700 --> 00:05:23.170 friction is equal to 0.8. 00:05:23.170 --> 00:05:26.290 So given that, how fast do I have to accelerate it? 00:05:26.290 --> 00:05:28.650 Well let's see what's happening. 00:05:28.650 --> 00:05:30.720 So we have the force of gravity pulling down. 00:05:30.720 --> 00:05:35.550 So let's say that the mass of the pie is m. 00:05:35.550 --> 00:05:38.860 m equals mass. 00:05:38.860 --> 00:05:40.400 So what is the force of gravity 00:05:40.400 --> 00:05:41.630 pulling down on the pie? 00:05:41.630 --> 00:05:47.350 Well the force of gravity is just equal to m times 9.8. 00:05:47.350 --> 00:05:48.490 Right? 00:05:48.490 --> 00:05:51.420 The force of gravity is equal to m times 9.8. 00:05:51.420 --> 00:05:55.810 In order for this pie to not move down, what do we know 00:05:55.810 --> 00:05:57.950 about the net forces on that pie? 00:05:57.950 --> 00:06:02.330 Well we know the net forces on that pie have to be 0. 00:06:02.330 --> 00:06:04.010 So what would be the offsetting force? 00:06:04.010 --> 00:06:05.630 Well, it would be the force of friction. 00:06:05.630 --> 00:06:08.106 So we would have a force of friction acting upwards. 00:06:08.106 --> 00:06:08.690 Right? 00:06:08.690 --> 00:06:11.140 Because the force of friction always acts opposite to the 00:06:11.140 --> 00:06:14.540 direction that the thing would move otherwise. 00:06:14.540 --> 00:06:20.190 So essentially, our force of friction has to be greater 00:06:20.190 --> 00:06:23.160 than, roughly, greater than or equal to. 00:06:23.160 --> 00:06:24.710 Because if it's greater than, it's not like the pie is going 00:06:24.710 --> 00:06:25.400 to move up. 00:06:25.400 --> 00:06:28.410 Friction by itself will never move something, it'll just 00:06:28.410 --> 00:06:29.940 keep something from being moved. 00:06:29.940 --> 00:06:31.250 But let's just figure out the minimum. 00:06:31.250 --> 00:06:33.210 I won't do the whole inequalities. 00:06:33.210 --> 00:06:37.950 The force of friction has to be equal similarly, to 9.8 00:06:37.950 --> 00:06:42.390 times the mass of the pie. 00:06:42.390 --> 00:06:46.680 So if the coefficient of friction is 0.8, what is the 00:06:46.680 --> 00:06:50.380 force that I have to apply? 00:06:50.380 --> 00:06:52.830 Well, the force I have to apply in this case is going to 00:06:52.830 --> 00:06:53.970 be the normal force, right? 00:06:53.970 --> 00:06:58.600 That's normal to the bottom of the pie. 00:06:58.600 --> 00:07:00.170 Right? 00:07:00.170 --> 00:07:03.270 My hand is now like the surface of the ramp. 00:07:03.270 --> 00:07:05.830 So this is the normal force. 00:07:05.830 --> 00:07:09.030 And we know that the force of friction is equal to the 00:07:09.030 --> 00:07:11.550 coefficient of friction times the normal force. 00:07:11.550 --> 00:07:12.960 I'm going to switch colors because this is getting 00:07:12.960 --> 00:07:15.300 monotonous. 00:07:15.300 --> 00:07:17.120 And the force of friction, we know has to be 00:07:17.120 --> 00:07:20.030 9.8 times the mass. 00:07:20.030 --> 00:07:23.010 So 9.8 meters per second times the mass. 00:07:23.010 --> 00:07:25.420 9.8m is the force of friction. 00:07:25.420 --> 00:07:27.830 And that has to equal to coefficient of friction times 00:07:27.830 --> 00:07:28.865 the normal force. 00:07:28.865 --> 00:07:31.310 And remember, the normal force is essentially the force that 00:07:31.310 --> 00:07:33.472 I'm pushing the pie with. 00:07:33.472 --> 00:07:38.270 And we know this is 0.8, so we have 9.8 times the mass-- 00:07:38.270 --> 00:07:43.100 that's not meters, that's the mass-- is equal to 0.8 times 00:07:43.100 --> 00:07:45.720 the normal force. 00:07:45.720 --> 00:07:51.960 So you have the normal force is equal to 9.8 times the mass 00:07:51.960 --> 00:07:54.500 divided by 0.8. 00:07:54.500 --> 00:07:56.320 What's 9.8 divided by 0.8? 00:07:56.320 --> 00:08:05.200 9.8 divided by 0.8 is equal to 12.25. 00:08:05.200 --> 00:08:08.580 So the normal force that I have to apply is 00:08:08.580 --> 00:08:12.890 12.25 times the mass. 00:08:12.890 --> 00:08:14.860 So that's the force I'm applying. 00:08:14.860 --> 00:08:15.540 It's time the mass. 00:08:15.540 --> 00:08:17.040 We don't know the mass of the pie. 00:08:17.040 --> 00:08:19.570 So how fast am I accelerating the pie? 00:08:19.570 --> 00:08:22.590 Well, force is equal to mass times acceleration. 00:08:22.590 --> 00:08:29.540 This is the force, 12.25m-- that's the force-- is equal to 00:08:29.540 --> 00:08:31.170 the mass times the 00:08:31.170 --> 00:08:32.720 acceleration of the pie, right? 00:08:32.720 --> 00:08:34.799 And it's the same pie that we're dealing with the whole 00:08:34.799 --> 00:08:35.970 time, so it's still m. 00:08:35.970 --> 00:08:39.049 And you can take out m from both sides of the equation. 00:08:39.049 --> 00:08:42.380 So the acceleration, the rate at which I have to change the 00:08:42.380 --> 00:08:45.450 velocity, or the acceleration that I have to apply to the 00:08:45.450 --> 00:08:53.860 pie is 12.25 meters per second squared. 00:08:53.860 --> 00:08:58.090 And so actually, I have to apply more than 1g, right? 00:08:58.090 --> 00:09:01.330 Because g is the force of gravity. 00:09:01.330 --> 00:09:05.430 And gravity accelerates something at 9.8 seconds-- 9.8 00:09:05.430 --> 00:09:06.750 meters per second squared. 00:09:06.750 --> 00:09:09.870 So I have to do something at 12-- I have to push and 00:09:09.870 --> 00:09:13.270 accelerate the pie at 12.25 meters per second squared. 00:09:13.270 --> 00:09:16.720 So it's something a little over 1g in order for that pie 00:09:16.720 --> 00:09:19.910 to not fall and in order for my normal force to provide a 00:09:19.910 --> 00:09:22.630 force of friction so that the pie can reach 00:09:22.630 --> 00:09:25.360 this bald man's face. 00:09:25.360 --> 00:09:26.720 I will see you in the next video.

Introduction to tension (part 2)

https://www.youtube.com/watch?v=zwDJ1wVr7Is

vtt

https://www.youtube.com/api/timedtext?v=zwDJ1wVr7Is&ei=YmeUZbyZMtK2mLAPgcKLwAM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=0A387E124D47E4955FCE053B10D85601D7D56D14.10F0C3DEE59E1B0AAAECEA54FB8559B7AD3A9550&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.790 --> 00:00:01.450 Welcome back. 00:00:01.450 --> 00:00:04.520 We'll now do another tension problem and this one is just a 00:00:04.520 --> 00:00:06.340 slight increment harder than the previous one just because 00:00:06.340 --> 00:00:10.390 we have to take out slightly more sophisticated algebra 00:00:10.390 --> 00:00:11.490 tools than we did in the last one. 00:00:11.490 --> 00:00:14.000 But it's not really any harder. 00:00:14.000 --> 00:00:15.990 But you should actually see this type of problem because 00:00:15.990 --> 00:00:17.810 you'll probably see it on an exam. 00:00:17.810 --> 00:00:19.440 So let's figure out the tension in the wire. 00:00:19.440 --> 00:00:22.950 So first of all, we know that this point 00:00:22.950 --> 00:00:24.380 right here isn't moving. 00:00:24.380 --> 00:00:26.700 So the tension in this little small wire right here is easy. 00:00:26.700 --> 00:00:28.170 It's trivial. 00:00:28.170 --> 00:00:30.850 The force of gravity is pulling down at this point 00:00:30.850 --> 00:00:33.170 with 10 Newtons because you have this weight here. 00:00:33.170 --> 00:00:35.200 And of course, since this point is stationary, the 00:00:35.200 --> 00:00:37.750 tension in this wire has to be 10 Newtons upward. 00:00:37.750 --> 00:00:38.300 That's an easy one. 00:00:38.300 --> 00:00:41.600 So let's just figure out the tension in these two slightly 00:00:41.600 --> 00:00:44.550 more difficult wires to figure out the tensions of. 00:00:44.550 --> 00:00:47.500 So once again, we know that this point right here, this 00:00:47.500 --> 00:00:49.760 point is not accelerating in any direction. 00:00:49.760 --> 00:00:52.900 It's not accelerating in the x direction, nor is it 00:00:52.900 --> 00:00:55.340 accelerating in the vertical direction or the y direction. 00:00:55.340 --> 00:00:57.970 So we know that the net forces in the x direction need to be 00:00:57.970 --> 00:01:00.800 0 on it and we know the net forces in the y 00:01:00.800 --> 00:01:03.970 direction need to be 0. 00:01:03.970 --> 00:01:06.410 So what are the net forces in the x direction? 00:01:06.410 --> 00:01:10.120 Well they're going to be the x components of these two-- of 00:01:10.120 --> 00:01:12.750 the tension vectors of both of these wires. 00:01:12.750 --> 00:01:15.820 I guess let's draw the tension vectors of the two wires. 00:01:15.820 --> 00:01:18.065 So this T1, it's pulling. 00:01:18.065 --> 00:01:21.870 The tension vector pulls in the direction of the wire 00:01:21.870 --> 00:01:23.240 along the same line. 00:01:23.240 --> 00:01:26.740 So let's say that this is the tension vector of T1. 00:01:26.740 --> 00:01:30.620 If that's the tension vector, its x component will be this. 00:01:30.620 --> 00:01:33.690 Let me see how good I can draw this. 00:01:33.690 --> 00:01:35.440 It's intended to be a straight line, but that 00:01:35.440 --> 00:01:36.920 would be its x component. 00:01:36.920 --> 00:01:42.570 And its x component, let's see, this is 30 degrees. 00:01:42.570 --> 00:01:44.440 This is 30 degrees right here. 00:01:44.440 --> 00:01:47.070 And hopefully this is a bit second nature to you. 00:01:47.070 --> 00:01:49.320 If this value up here is T1, what is the 00:01:49.320 --> 00:01:51.590 value of the x component? 00:01:51.590 --> 00:01:59.860 It's T1 cosine of 30 degrees. 00:01:59.860 --> 00:02:01.200 And you could do your SOH-CAH-TOA. 00:02:01.200 --> 00:02:04.810 You know, cosine is adjacent over hypotenuse. 00:02:04.810 --> 00:02:09.430 So the cosine of 30 degrees is equal to-- This over T1 one is 00:02:09.430 --> 00:02:10.800 equal to the x component over T1. 00:02:10.800 --> 00:02:14.010 And if you multiply both sides by T1, you get this. 00:02:14.010 --> 00:02:15.690 This should be a little bit of second nature right now. 00:02:15.690 --> 00:02:20.050 That the x component is going to be the cosine of the angle 00:02:20.050 --> 00:02:23.350 between the hypotenuse and the x component times the 00:02:23.350 --> 00:02:24.440 hypotenuse. 00:02:24.440 --> 00:02:27.485 And similarly, the x component here-- Let me 00:02:27.485 --> 00:02:28.860 draw this force vector. 00:02:28.860 --> 00:02:34.675 So if this is T2, this would be its x component. 00:02:38.260 --> 00:02:44.320 And very similarly, this is 60 degrees, so this would be T2 00:02:44.320 --> 00:02:48.130 cosine of 60. 00:02:48.130 --> 00:02:50.920 Now what do we know about these two vectors? 00:02:50.920 --> 00:02:55.150 We know that their net force is 0. 00:02:55.150 --> 00:02:57.500 Or that you also know that the magnitude of these two vectors 00:02:57.500 --> 00:02:59.340 should cancel each other out or that they're equal. 00:02:59.340 --> 00:03:01.570 I mean, they're pulling in opposite directions. 00:03:01.570 --> 00:03:04.030 That's pretty obvious. 00:03:04.030 --> 00:03:05.900 And so you know that their magnitudes need to be equal. 00:03:05.900 --> 00:03:08.950 So we know that T1 cosine of 30 is going to equal 00:03:08.950 --> 00:03:10.810 T2 cosine of 60. 00:03:10.810 --> 00:03:12.990 So let's write that down. 00:03:12.990 --> 00:03:27.070 T1 cosine of 30 degrees is equal to T2 cosine of 60. 00:03:27.070 --> 00:03:29.890 And then we could bring the T2 on to this side. 00:03:29.890 --> 00:03:31.900 And actually, let's also-- I'm trying to save as much space 00:03:31.900 --> 00:03:33.950 as possible because I'm guessing this is going to take 00:03:33.950 --> 00:03:35.550 up a lot of room, this problem. 00:03:35.550 --> 00:03:38.070 What's the cosine of 30 degrees? 00:03:38.070 --> 00:03:39.410 If you haven't memorized it already, it's square 00:03:39.410 --> 00:03:40.040 root of 3 over 2. 00:03:40.040 --> 00:03:45.640 So this becomes square root of 3 over 2 times T1. 00:03:45.640 --> 00:03:47.510 That's the cosine of 30 degrees. 00:03:47.510 --> 00:03:49.290 And then I'm going to bring this on to this side. 00:03:49.290 --> 00:03:54.590 So the cosine of 60 is actually 1/2. 00:03:54.590 --> 00:03:56.520 You could use your calculator if you forgot that. 00:03:56.520 --> 00:03:58.320 So this is 1/2 T2. 00:03:58.320 --> 00:04:01.590 Bring it on this side so it becomes minus 1/2. 00:04:01.590 --> 00:04:04.940 I'm skipping more steps than normal just because I don't 00:04:04.940 --> 00:04:07.020 want to waste too much space. 00:04:07.020 --> 00:04:08.160 And this equals 0. 00:04:08.160 --> 00:04:11.370 But if you seen the other videos, hopefully I'm not 00:04:11.370 --> 00:04:12.410 creating too many gaps. 00:04:12.410 --> 00:04:14.960 And this is relatively easy to follow. 00:04:14.960 --> 00:04:19.269 So we have the square root of 3 times T1 minus 1/2 T2 is 00:04:19.269 --> 00:04:19.800 equal to 0. 00:04:19.800 --> 00:04:21.700 So that gives us an equation. 00:04:21.700 --> 00:04:24.320 One equation with two unknowns, so it doesn't help 00:04:24.320 --> 00:04:25.050 us much so far. 00:04:25.050 --> 00:04:29.250 But let's square that away because I have a feeling this 00:04:29.250 --> 00:04:32.330 will be useful. 00:04:32.330 --> 00:04:34.330 Now what's going to be happening on the y components? 00:04:34.330 --> 00:04:39.580 So let's say that this is the y component of T1 and this is 00:04:39.580 --> 00:04:42.230 the y component of T2. 00:04:42.230 --> 00:04:42.860 What do we know? 00:04:42.860 --> 00:04:44.450 What what do we know about the two y components? 00:04:44.450 --> 00:04:46.890 I could've drawn them here too and then just shift them over 00:04:46.890 --> 00:04:48.240 to the left and the right. 00:04:48.240 --> 00:04:52.210 We know that their combined pull upwards, the combined 00:04:52.210 --> 00:04:54.500 pull of the two vertical tension components has to 00:04:54.500 --> 00:04:57.770 offset the force of gravity pulling down because this 00:04:57.770 --> 00:04:59.130 point is stationary. 00:04:59.130 --> 00:05:02.310 So we know these two y components, when you add them 00:05:02.310 --> 00:05:05.210 together, the combined tension in the vertical direction has 00:05:05.210 --> 00:05:06.680 to be 10 Newtons. 00:05:06.680 --> 00:05:09.240 Because it's offsetting this force of gravity. 00:05:09.240 --> 00:05:10.990 So what's this y component? 00:05:10.990 --> 00:05:12.970 Well, this was T1 of cosine of 30. 00:05:12.970 --> 00:05:15.140 This should start to become a little second nature to you 00:05:15.140 --> 00:05:19.180 that this is T1 sine of 30, this y component right here. 00:05:19.180 --> 00:05:23.570 So T1-- Let me write it here. 00:05:23.570 --> 00:05:34.875 T1 sine of 30 degrees plus this vector, which is T2 sine 00:05:34.875 --> 00:05:38.820 of 60 degrees. 00:05:38.820 --> 00:05:39.910 You could review your trigonometry and your 00:05:39.910 --> 00:05:41.650 SOH-CAH-TOA. 00:05:41.650 --> 00:05:44.830 Frankly, I think, just seeing what people get confused on is 00:05:44.830 --> 00:05:45.680 the trigonometry. 00:05:45.680 --> 00:05:48.340 But you can review the trig modules and maybe some of the 00:05:48.340 --> 00:05:50.370 earlier force vector modules that we did. 00:05:50.370 --> 00:05:52.100 And hopefully, these will make sense. 00:05:52.100 --> 00:05:53.460 I'm skipping a few steps. 00:05:53.460 --> 00:05:58.870 And these will equal 10 Newtons. 00:05:58.870 --> 00:06:02.370 And let's rewrite this up here where I substitute the values. 00:06:02.370 --> 00:06:03.650 So what's the sine of 30? 00:06:03.650 --> 00:06:06.180 Actually, let me do it right here. 00:06:06.180 --> 00:06:07.400 What's the sine of 30 degrees? 00:06:07.400 --> 00:06:15.000 The sine of 30 degrees is 1/2 so we get 1/2 T1 plus the sine 00:06:15.000 --> 00:06:17.420 of 60 degrees, which is square root of 3 over 2. 00:06:17.420 --> 00:06:22.150 Square root of 3 over 2 T2 is equal to 10. 00:06:22.150 --> 00:06:23.850 And then I don't like this, all these 2's 00:06:23.850 --> 00:06:25.780 and this 1/2 here. 00:06:25.780 --> 00:06:27.850 So let's multiply this whole equation by 2. 00:06:27.850 --> 00:06:30.170 So 2 times 1/2, that's 1. 00:06:30.170 --> 00:06:37.940 So you get T1 plus the square root of 3 T2 is equal to, 2 00:06:37.940 --> 00:06:40.400 times 10 , is 20. 00:06:40.400 --> 00:06:42.330 Similarly, let's take this equation up here and let's 00:06:42.330 --> 00:06:46.520 multiply this equation by 2 and bring it down here. 00:06:46.520 --> 00:06:47.980 So this is the original one that we got. 00:06:47.980 --> 00:06:50.630 So if we multiply this whole thing by 2-- I'll do it in 00:06:50.630 --> 00:06:52.130 this color so that you know that 00:06:52.130 --> 00:06:54.230 it's a different equation. 00:06:54.230 --> 00:06:56.410 So if you multiply square root of 3 over 2 times 2-- I'm just 00:06:56.410 --> 00:06:58.350 doing this to get rid of the 2's in the denominator. 00:06:58.350 --> 00:07:10.360 So you get square root of 3 T1 minus T2 is equal to 0 because 00:07:10.360 --> 00:07:12.080 0 times 2 is 0. 00:07:12.080 --> 00:07:13.410 And let's see what we could do. 00:07:13.410 --> 00:07:15.960 What if we take this top equation because we want to 00:07:15.960 --> 00:07:18.790 start canceling out some terms. Let's take this top 00:07:18.790 --> 00:07:24.880 equation and let's multiply it by-- oh, I don't know. 00:07:24.880 --> 00:07:28.620 Let's multiply it by the square root of 3. 00:07:28.620 --> 00:07:36.740 So you get the square root of 3 T1. 00:07:36.740 --> 00:07:37.950 I'm taking this top equation multiplied by the 00:07:37.950 --> 00:07:38.320 square root of 3. 00:07:38.320 --> 00:07:39.680 This is just a system of equations 00:07:39.680 --> 00:07:41.100 that I'm solving for. 00:07:41.100 --> 00:07:43.990 And the square root of 3 times this right here. 00:07:43.990 --> 00:07:46.030 Square root of 3 times square root of 3 is 3. 00:07:46.030 --> 00:07:54.200 So plus 3 T2 is equal to 20 square root of 3. 00:07:54.200 --> 00:07:59.600 And now what I want to do is let's-- I know I'm doing a lot 00:07:59.600 --> 00:08:01.160 of equation manipulation here. 00:08:01.160 --> 00:08:04.380 But this is just hopefully, a review of algebra for you. 00:08:04.380 --> 00:08:09.590 Let's subtract this equation from this equation. 00:08:09.590 --> 00:08:11.960 So you can also view it as multiplying it by negative 1 00:08:11.960 --> 00:08:13.300 and then adding the 2. 00:08:13.300 --> 00:08:15.930 So when you subtract this from this, these two terms cancel 00:08:15.930 --> 00:08:17.620 out because they're the same. 00:08:17.620 --> 00:08:22.010 And so then you're left with minus T2 from here. 00:08:22.010 --> 00:08:31.170 Minus this, minus 3 T2 is equal to 0 minus 20 square 00:08:31.170 --> 00:08:33.490 roots of 3. 00:08:33.490 --> 00:08:40.460 And so this becomes minus 4 T2 is equal to minus 20 square 00:08:40.460 --> 00:08:42.480 roots of 3. 00:08:42.480 --> 00:08:46.930 And then, divide both sides by minus 4 and you get T2 is 00:08:46.930 --> 00:08:52.320 equal to 5 square roots of 3 Newtons. 00:08:52.320 --> 00:08:54.150 So that's the tension in this wire. 00:08:54.150 --> 00:08:58.090 And now we can substitute and figure out T1. 00:08:58.090 --> 00:08:59.460 Let's use this formula right here because it 00:08:59.460 --> 00:09:02.120 looks suitably simple. 00:09:02.120 --> 00:09:08.160 So we have the square root of 3 times T1 minus T2. 00:09:08.160 --> 00:09:10.960 Well T2 is 5 square roots of 3. 00:09:10.960 --> 00:09:14.950 5 square roots of 3 is equal to 0. 00:09:14.950 --> 00:09:19.860 So we have the square root of 3 T1 is equal to five square 00:09:19.860 --> 00:09:21.060 roots of 3. 00:09:21.060 --> 00:09:23.370 Divide both sides by square root of 3 and you get the 00:09:23.370 --> 00:09:28.010 tension in the first wire is equal to 5 Newtons. 00:09:28.010 --> 00:09:31.880 So this is pulling with a force or tension of 5 Newtons. 00:09:31.880 --> 00:09:33.150 Or a force. 00:09:33.150 --> 00:09:36.460 And this is pulling-- the second wire --with a tension 00:09:36.460 --> 00:09:39.990 of 5 square roots of 3 Newtons. 00:09:39.990 --> 00:09:43.430 So this wire right here is actually 00:09:43.430 --> 00:09:45.410 doing more of the pulling. 00:09:45.410 --> 00:09:48.670 It's actually more of the force of gravity is ending up 00:09:48.670 --> 00:09:49.650 on this wire. 00:09:49.650 --> 00:09:51.820 That makes sense because it's steeper. 00:09:51.820 --> 00:09:54.800 So since it's steeper, it's contributing 00:09:54.800 --> 00:09:56.970 more to the y component. 00:09:56.970 --> 00:09:58.600 It's good whenever you do these problems to kind of do a 00:09:58.600 --> 00:10:01.520 reality check just to make sure your numbers make sense. 00:10:01.520 --> 00:10:05.800 And if you think about it, their combined tension is 00:10:05.800 --> 00:10:07.490 something more than 10 Newtons. 00:10:07.490 --> 00:10:10.750 And that makes sense because some of the force that they're 00:10:10.750 --> 00:10:13.940 pulling with is wasted against pulling each other in the 00:10:13.940 --> 00:10:15.520 horizontal direction. 00:10:15.520 --> 00:10:18.150 Anyway, I'll see you all in the next video.

Introduction to tension

https://www.youtube.com/watch?v=_UrfHFEBIpU

vtt

https://www.youtube.com/api/timedtext?v=_UrfHFEBIpU&ei=YmeUZYCoMonFmLAP5_SKmAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2602F2CE63E10B80654577B2A4E6C17F7AC62C48.E896A166A2998911744C069725A801F5FBEAF199&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.780 --> 00:00:04.070 I will now introduce you to the concept of tension. 00:00:04.070 --> 00:00:07.390 So tension is really just the force that exists either 00:00:07.390 --> 00:00:10.760 within or applied by a string or wire. 00:00:10.760 --> 00:00:13.280 It's usually lifting something or pulling on something. 00:00:13.280 --> 00:00:16.040 So let's say I had a weight. 00:00:16.040 --> 00:00:21.080 Let's say I have a weight here. 00:00:21.080 --> 00:00:27.450 And let's say it's 100 Newtons. 00:00:27.450 --> 00:00:31.575 And it's suspended from this wire, which is right here. 00:00:31.575 --> 00:00:34.920 Let's say it's attached to the ceiling right there. 00:00:34.920 --> 00:00:38.010 Well we already know that the force-- if we're on this 00:00:38.010 --> 00:00:41.390 planet that this weight is being pull down by gravity. 00:00:41.390 --> 00:00:44.370 So we already know that there's a downward force on 00:00:44.370 --> 00:00:47.730 this weight, which is a force of gravity. 00:00:47.730 --> 00:00:51.130 And that equals 100 Newtons. 00:00:51.130 --> 00:00:54.090 But we also know that this weight isn't accelerating, 00:00:54.090 --> 00:00:54.850 it's actually stationary. 00:00:54.850 --> 00:00:56.040 It also has no velocity. 00:00:56.040 --> 00:00:59.570 But the important thing is it's not accelerating. 00:00:59.570 --> 00:01:05.080 But given that, we know that the net force on it must be 0 00:01:05.080 --> 00:01:07.950 by Newton's laws. 00:01:07.950 --> 00:01:10.820 So what is the counteracting force? 00:01:10.820 --> 00:01:12.840 You didn't have to know about tension to say well, the 00:01:12.840 --> 00:01:14.300 string's pulling on it. 00:01:14.300 --> 00:01:17.490 The string is what's keeping the weight from falling. 00:01:17.490 --> 00:01:20.920 So the force that the string or this wire applies on this 00:01:20.920 --> 00:01:24.460 weight you can view as the force of tension. 00:01:24.460 --> 00:01:27.400 Another way to think about it is that's also the force 00:01:27.400 --> 00:01:30.350 that's within the wire. 00:01:33.020 --> 00:01:37.320 And that is going to exactly offset the force of gravity on 00:01:37.320 --> 00:01:37.700 this weight. 00:01:37.700 --> 00:01:43.190 And that's what keeps this point right here stationery 00:01:43.190 --> 00:01:45.800 and keeps it from accelerating. 00:01:45.800 --> 00:01:47.040 That's pretty straightforward. 00:01:47.040 --> 00:01:49.640 Tension, it's just the force of a string. 00:01:49.640 --> 00:01:53.270 And just so you can conceptualize it, on a guitar, 00:01:53.270 --> 00:01:58.830 the more you pull on some of those higher-- what was it? 00:01:58.830 --> 00:02:01.820 The really thin strings that sound higher pitched. 00:02:01.820 --> 00:02:04.040 The more you pull on it, the higher the tension. 00:02:04.040 --> 00:02:06.780 It actually creates a higher pitched note. 00:02:06.780 --> 00:02:08.210 So you've dealt with tension a lot. 00:02:08.210 --> 00:02:11.260 I think actually when they sell wires or strings they'll 00:02:11.260 --> 00:02:13.730 probably tell you the tension that that wire or string can 00:02:13.730 --> 00:02:15.650 support, which is important if you're going to build a bridge 00:02:15.650 --> 00:02:16.980 or a swing or something. 00:02:16.980 --> 00:02:20.170 So tension is something that should be hopefully, a little 00:02:20.170 --> 00:02:21.600 bit intuitive to you. 00:02:21.600 --> 00:02:24.710 So let's, with that fairly simple example done, let's 00:02:24.710 --> 00:02:27.980 create a slightly more complicated example. 00:02:27.980 --> 00:02:29.900 So let's take the same weight. 00:02:29.900 --> 00:02:31.720 Instead of making the ceiling here, let's 00:02:31.720 --> 00:02:34.510 add two more strings. 00:02:34.510 --> 00:02:37.660 Let's add this green string. 00:02:40.770 --> 00:02:43.010 Green string there. 00:02:43.010 --> 00:02:46.610 And it's attached to the ceiling up here. 00:02:46.610 --> 00:02:48.970 That's the ceiling now. 00:02:48.970 --> 00:02:49.530 And let's see. 00:02:49.530 --> 00:02:52.630 This is the wall. 00:02:52.630 --> 00:02:54.710 And let's say there's another string right here 00:02:54.710 --> 00:02:57.240 attached to the wall. 00:02:57.240 --> 00:03:00.600 So my question to you is, what is the tension in these two 00:03:00.600 --> 00:03:08.620 strings So let's call this T1 and T2. 00:03:08.620 --> 00:03:12.470 Well like the first problem, this point right here, this 00:03:12.470 --> 00:03:15.430 red point, is stationary. 00:03:15.430 --> 00:03:17.820 It's not accelerating in either the left/right 00:03:17.820 --> 00:03:20.570 directions and it's not accelerating in the up/down 00:03:20.570 --> 00:03:21.260 directions. 00:03:21.260 --> 00:03:24.400 So we know that the net forces in both the x and y 00:03:24.400 --> 00:03:27.560 dimensions must be 0. 00:03:27.560 --> 00:03:30.950 My second question to you is, what is 00:03:30.950 --> 00:03:31.950 going to be the offset? 00:03:31.950 --> 00:03:34.770 Because we know already that at this point right here, 00:03:34.770 --> 00:03:37.330 there's going to be a downward force, which is the force of 00:03:37.330 --> 00:03:39.270 gravity again. 00:03:39.270 --> 00:03:40.220 The weight of this whole thing. 00:03:40.220 --> 00:03:43.490 We can assume that the wires have no weight for simplicity. 00:03:43.490 --> 00:03:46.300 So we know that there's going to be a downward force here, 00:03:46.300 --> 00:03:47.920 this is the force of gravity, right? 00:03:47.920 --> 00:03:50.560 The whole weight of this entire object of weight plus 00:03:50.560 --> 00:03:52.200 wire is pulling down. 00:03:52.200 --> 00:03:55.470 So what is going to be the upward force here? 00:03:55.470 --> 00:03:57.880 Well let's look at each of the wires. 00:03:57.880 --> 00:04:02.200 This second wire, T2, or we could call it w2, I guess. 00:04:02.200 --> 00:04:05.070 The second wire is just pulling to the left. 00:04:05.070 --> 00:04:06.335 It has no y components. 00:04:06.335 --> 00:04:08.680 It's not lifting up at all. 00:04:08.680 --> 00:04:10.590 So it's just pulling to the left. 00:04:10.590 --> 00:04:13.910 So all of the upward lifting, all of that's going to occur 00:04:13.910 --> 00:04:17.500 from this first wire, from T1. 00:04:17.500 --> 00:04:22.430 So we know that the y component of T1, so let's 00:04:22.430 --> 00:04:25.395 call-- so if we say that this vector here. 00:04:25.395 --> 00:04:28.510 Let me do it in a different color. 00:04:28.510 --> 00:04:30.470 Because I know when I draw these diagrams it starts to 00:04:30.470 --> 00:04:31.720 get confusing. 00:04:34.460 --> 00:04:36.940 Let me actually use the line tool. 00:04:36.940 --> 00:04:39.480 So I have this. 00:04:39.480 --> 00:04:42.920 Let me make a thicker line. 00:04:42.920 --> 00:04:45.120 So we have this vector here, which is T1. 00:04:49.110 --> 00:04:51.070 And we would need to figure out what that is. 00:04:51.070 --> 00:04:53.200 And then we have the other vector, which is its y 00:04:53.200 --> 00:04:55.890 component, and I'll draw that like here. 00:04:59.640 --> 00:05:00.890 This is its y component. 00:05:05.780 --> 00:05:09.900 We could call this T1 sub y. 00:05:09.900 --> 00:05:11.920 And then of course, it has an x component too, and I'll do 00:05:11.920 --> 00:05:15.360 that in-- let's see. 00:05:15.360 --> 00:05:19.160 I'll do that in red. 00:05:19.160 --> 00:05:21.170 Once again, this is just breaking up a force into its 00:05:21.170 --> 00:05:25.250 component vectors like we've-- a vector force into its x and 00:05:25.250 --> 00:05:27.350 y components like we've been doing in the last several 00:05:27.350 --> 00:05:30.100 problems. And these are just trigonometry problems, right? 00:05:32.990 --> 00:05:35.920 We could actually now, visually see that this is T 00:05:35.920 --> 00:05:38.620 sub 1 x and this is T sub 1 sub y. 00:05:38.620 --> 00:05:41.130 Oh, and I forgot to give you an important property of this 00:05:41.130 --> 00:05:44.780 problem that you needed to know before solving it. 00:05:44.780 --> 00:05:47.820 Is that the angle that the first wire forms with the 00:05:47.820 --> 00:05:51.370 ceiling, this is 30 degrees. 00:05:51.370 --> 00:05:58.470 So if that is 30 degrees, we also know that this is a 00:05:58.470 --> 00:06:01.240 parallel line to this. 00:06:01.240 --> 00:06:03.880 So if this is 30 degrees, this is also 00:06:03.880 --> 00:06:07.290 going to be 30 degrees. 00:06:07.290 --> 00:06:11.240 So this angle right here is also going to be 30 degrees. 00:06:11.240 --> 00:06:13.370 And that's from our-- you know, we know about parallel 00:06:13.370 --> 00:06:15.690 lines and alternate interior angles. 00:06:15.690 --> 00:06:17.870 We could have done it the other way. 00:06:17.870 --> 00:06:21.745 We could have said that if this angle is 30 degrees, this 00:06:21.745 --> 00:06:22.900 angle is 60 degrees. 00:06:22.900 --> 00:06:24.620 This is a right angle, so this is also 30. 00:06:24.620 --> 00:06:26.620 But that's just review of geometry 00:06:26.620 --> 00:06:27.290 that you already know. 00:06:27.290 --> 00:06:30.280 But anyway, we know that this angle is 30 degrees, so what's 00:06:30.280 --> 00:06:31.600 its y component? 00:06:31.600 --> 00:06:33.460 Well the y component, let's see. 00:06:33.460 --> 00:06:36.090 What involves the hypotenuse and the opposite side? 00:06:36.090 --> 00:06:38.640 Let me write soh cah toa at the top because this is really 00:06:38.640 --> 00:06:39.920 just trigonometry. 00:06:39.920 --> 00:06:42.670 soh cah toa in blood red. 00:06:42.670 --> 00:06:45.730 So what involves the opposite and the hypotenuse? 00:06:45.730 --> 00:06:47.770 So opposite over hypotenuse. 00:06:47.770 --> 00:06:55.310 So that we know the sine-- let me switch to the sine of 30 00:06:55.310 --> 00:07:05.630 degrees is equal to T1 sub y over the tension in the string 00:07:05.630 --> 00:07:07.610 going in this direction. 00:07:07.610 --> 00:07:15.730 So if we solve for T1 sub y we get T1 sine of 30 degrees is 00:07:15.730 --> 00:07:20.870 equal to T1 sub y. 00:07:20.870 --> 00:07:23.210 And what did we just say before we kind of 00:07:23.210 --> 00:07:24.990 dived into the math? 00:07:24.990 --> 00:07:30.600 We said all of the lifting on this point is being done by 00:07:30.600 --> 00:07:32.550 the y component of T1. 00:07:32.550 --> 00:07:36.310 Because T2 is not doing any lifting up or down, it's only 00:07:36.310 --> 00:07:38.920 pulling to the left. 00:07:38.920 --> 00:07:44.590 So the entire component that's keeping this object up, 00:07:44.590 --> 00:07:46.770 keeping it from falling is the y component of 00:07:46.770 --> 00:07:48.040 this tension vector. 00:07:48.040 --> 00:07:51.930 So that has to equal the force of gravity pulling down. 00:07:51.930 --> 00:07:54.840 This has to equal the force of gravity. 00:07:54.840 --> 00:07:58.710 That has to equal this or this point. 00:07:58.710 --> 00:08:01.260 So that's 100 Newtons. 00:08:01.260 --> 00:08:04.185 And I really want to hit this point home because it might be 00:08:04.185 --> 00:08:06.140 a little confusing to you. 00:08:06.140 --> 00:08:07.810 We just said, this point is stationery. 00:08:07.810 --> 00:08:09.000 It's not moving up or down. 00:08:09.000 --> 00:08:10.720 It's not accelerating up or down. 00:08:10.720 --> 00:08:14.540 And so we know that there's a downward force of 100 Newtons, 00:08:14.540 --> 00:08:17.480 so there must be an upward force that's being provided by 00:08:17.480 --> 00:08:18.800 these two wires. 00:08:18.800 --> 00:08:21.150 This wire is providing no upward force. 00:08:21.150 --> 00:08:24.140 So all of the upward force must be the y component or the 00:08:24.140 --> 00:08:29.350 upward component of this force vector on the first wire. 00:08:29.350 --> 00:08:33.309 So given that, we can now solve for the tension in this 00:08:33.309 --> 00:08:39.409 first wire because we have T1-- what's sine of 30? 00:08:39.409 --> 00:08:43.050 Sine of 30 degrees, in case you haven't memorized it, sine 00:08:43.050 --> 00:08:44.990 of 30 degrees is 1/2. 00:08:44.990 --> 00:08:52.640 So T1 times 1/2 is equal to 100 Newtons. 00:08:52.640 --> 00:08:56.200 Divide both sides by 1/2 and you get T1 is 00:08:56.200 --> 00:09:03.350 equal to 200 Newtons. 00:09:03.350 --> 00:09:06.800 So now we've got to figure out what the tension in this 00:09:06.800 --> 00:09:08.870 second wire is. 00:09:08.870 --> 00:09:10.790 And we also, there's another clue here. 00:09:10.790 --> 00:09:14.640 This point isn't moving left or right, it's stationary. 00:09:14.640 --> 00:09:19.970 So we know that whatever the tension in this wire must be, 00:09:19.970 --> 00:09:24.120 it must be being offset by a tension or some other force in 00:09:24.120 --> 00:09:25.260 the opposite direction. 00:09:25.260 --> 00:09:29.150 And that force in the opposite direction is the x component 00:09:29.150 --> 00:09:31.220 of the first wire's tension. 00:09:31.220 --> 00:09:34.140 So it's this. 00:09:34.140 --> 00:09:39.200 So T2 is equal to the x component of the 00:09:39.200 --> 00:09:40.990 first wire's tension. 00:09:40.990 --> 00:09:42.410 And what's the x component? 00:09:42.410 --> 00:09:45.520 Well, it's going to be the tension in the first wire, 200 00:09:45.520 --> 00:09:51.250 Newtons times the cosine of 30 degrees. 00:09:51.250 --> 00:09:53.900 It's adjacent over hypotenuse. 00:09:53.900 --> 00:09:55.320 And that's square root of 3 over 2. 00:09:55.320 --> 00:10:00.410 So it's 200 times the square root of 3 over 2, which equals 00:10:00.410 --> 00:10:03.590 100 square root of 3. 00:10:03.590 --> 00:10:07.940 So the tension in this wire is 100 square root of 3, which 00:10:07.940 --> 00:10:12.590 completely offsets to the left and the x component of this 00:10:12.590 --> 00:10:16.540 wire is 100 square root of 3 Newtons to the right. 00:10:16.540 --> 00:10:17.446 Hopefully I didn't confuse you. 00:10:17.446 --> 00:10:19.250 See you in the next video.

Newton's Laws

https://www.youtube.com/watch?v=16StQAx83kA

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:03.440 Let's do some more problems involving Newton's laws. 00:00:03.440 --> 00:00:06.770 OK, so this problem that I am picking from-- I think it's 00:00:06.770 --> 00:00:08.210 from Oregon University. 00:00:08.210 --> 00:00:10.890 It's zebu.uoregan.edu. 00:00:10.890 --> 00:00:13.260 I want to give them credit for their problem. 00:00:13.260 --> 00:00:16.585 Let's see, let me draw-- this is going to be the ground. 00:00:20.050 --> 00:00:23.080 And then the problem says that I have a train-- 00:00:23.080 --> 00:00:27.020 so this is a train. 00:00:27.020 --> 00:00:29.730 Try my best to draw a train. 00:00:29.730 --> 00:00:31.460 Something like that. 00:00:31.460 --> 00:00:32.810 And there's smoke coming out. 00:00:32.810 --> 00:00:36.540 And it says that the train of mass 6.84. 00:00:36.540 --> 00:00:44.910 So let's say the mass of the train is 6.84 times 10 to the 00:00:44.910 --> 00:00:49.310 sixth kilograms. That's the mass of this train. 00:00:49.310 --> 00:00:52.650 Is moving at a speed of 80 kilometers per hour. 00:00:52.650 --> 00:00:55.360 So it's velocity, we could say it's initial velocity because 00:00:55.360 --> 00:00:57.270 I think it's going to ask me what happens to its velocity. 00:00:57.270 --> 00:01:03.630 So its initial velocity is equal to 80 00:01:03.630 --> 00:01:08.395 kilometers per hour. 00:01:08.395 --> 00:01:13.860 So it's going this way at 80 kilometers per hour. 00:01:13.860 --> 00:01:17.380 The brakes, which produce a net backward force of-- so the 00:01:17.380 --> 00:01:21.900 brakes are going to pull backwards-- a net backward 00:01:21.900 --> 00:01:29.640 force of-- so force is equal to 1.93 times 10 00:01:29.640 --> 00:01:31.200 to the sixth Newtons. 00:01:31.200 --> 00:01:33.490 That's 1.93 times 10 to the sixth Newtons. 00:01:33.490 --> 00:01:36.760 I know it looks a little bit messy. 00:01:36.760 --> 00:01:41.520 The brakes are going to be applied for 25 seconds. 00:01:41.520 --> 00:01:45.460 I'll say force sub b or the force of the brakes is 1.93 00:01:45.460 --> 00:01:48.110 times 10 to the sixth Newtons. 00:01:48.110 --> 00:01:53.760 And then, they will be applied for 25 seconds. 00:01:53.760 --> 00:01:56.330 So the first question to ask is, what is the new 00:01:56.330 --> 00:01:57.100 speed of the train? 00:01:57.100 --> 00:01:59.025 So what is the speed of the train after the brakes are 00:01:59.025 --> 00:02:00.820 applied for 25 seconds? 00:02:00.820 --> 00:02:05.440 Well, all we have to figure out is how fast is the train 00:02:05.440 --> 00:02:06.900 decelerating? 00:02:06.900 --> 00:02:09.060 Or what rate of deceleration does this 00:02:09.060 --> 00:02:10.090 backward force create? 00:02:10.090 --> 00:02:12.340 So we just go back to our basics. 00:02:12.340 --> 00:02:15.640 Let me switch colors because this red's getting a little 00:02:15.640 --> 00:02:16.800 monotonous. 00:02:16.800 --> 00:02:20.480 Force is equal to mass times acceleration. 00:02:20.480 --> 00:02:23.530 Probably the easiest formula ever to memorize. 00:02:23.530 --> 00:02:25.300 And what's the backward force? 00:02:25.300 --> 00:02:34.960 The backward force is 1.93 times 10 to the sixth. 00:02:34.960 --> 00:02:38.370 And that equals mass times acceleration. 00:02:38.370 --> 00:02:39.130 What's the mass? 00:02:39.130 --> 00:02:40.380 Mass is right here. 00:02:43.920 --> 00:02:44.570 That's a decimal. 00:02:44.570 --> 00:02:51.200 6.84 times 10 to the sixth times the acceleration. 00:02:51.200 --> 00:02:52.520 And acceleration is what we're trying to figure out. 00:02:52.520 --> 00:02:55.400 And remember, if the force is pulling back, then of course, 00:02:55.400 --> 00:02:57.090 the acceleration is also going to be backwards. 00:02:57.090 --> 00:02:59.960 Or it's going to essentially, slow down the train. 00:02:59.960 --> 00:03:02.830 I know this is very messy, but let's just divide-- well, we 00:03:02.830 --> 00:03:04.840 could divide both sides by 10 to the sixth just to get that 00:03:04.840 --> 00:03:05.260 out of the way. 00:03:05.260 --> 00:03:08.550 That simplifies things a little bit. 00:03:08.550 --> 00:03:10.540 And then we're left with the acceleration. 00:03:10.540 --> 00:03:12.810 I'm just flipping both sides of this equation. 00:03:12.810 --> 00:03:20.890 Acceleration is equal to 1.93 divided by 6.84. soon. 00:03:20.890 --> 00:03:25.110 And this is going to be given in meters per second squared. 00:03:25.110 --> 00:03:27.660 And I just want to make sure I got my units right. 00:03:27.660 --> 00:03:31.390 Because this number here, the mass was in kilograms. And 00:03:31.390 --> 00:03:32.820 then the force was given in units. 00:03:32.820 --> 00:03:35.660 So the acceleration will be in meters per second squared. 00:03:35.660 --> 00:03:37.640 And let's get the handy calculator here because this 00:03:37.640 --> 00:03:40.330 looks-- I don't want to waste your time reviewing how to 00:03:40.330 --> 00:03:41.350 divide decimals. 00:03:41.350 --> 00:03:42.340 Although, that might not be a bad 00:03:42.340 --> 00:03:43.960 review; most people forget. 00:03:43.960 --> 00:03:50.980 1.93 divided by 6.-- whoops. 00:03:50.980 --> 00:03:52.270 I think I messed up. 00:03:52.270 --> 00:03:52.830 Let me see. 00:03:52.830 --> 00:04:02.850 1.93 divided by 6.84 is 0.282. 00:04:02.850 --> 00:04:09.040 So its acceleration equals-- is equal to 0.28-- let's just 00:04:09.040 --> 00:04:13.560 say 282 meters per second squared. 00:04:13.560 --> 00:04:15.070 And so how fast is the train going to be 00:04:15.070 --> 00:04:17.800 going after 25 seconds? 00:04:17.800 --> 00:04:19.329 So what's the change in velocity? 00:04:19.329 --> 00:04:23.630 Well the change in velocity is equal to 00:04:23.630 --> 00:04:26.050 acceleration times time. 00:04:26.050 --> 00:04:28.240 And if we take its velocity as a positive number, then its 00:04:28.240 --> 00:04:29.760 acceleration will be a negative number, right? 00:04:29.760 --> 00:04:32.920 Because it's going to be going in the opposite direction. 00:04:32.920 --> 00:04:34.400 Could have done it the other way around. 00:04:34.400 --> 00:04:37.950 So the change in velocity is going to be equal to this 00:04:37.950 --> 00:04:42.080 minus 0.282 meters per second squared. 00:04:42.080 --> 00:04:43.510 I won't write the units here because I'm 00:04:43.510 --> 00:04:44.750 running out of space. 00:04:44.750 --> 00:04:51.200 Minus 0.282 meters per second squared times 25 seconds. 00:04:51.200 --> 00:04:52.950 Let's get the calculator back here. 00:04:52.950 --> 00:04:53.700 Where'd it go? 00:04:53.700 --> 00:04:54.140 There it is. 00:04:54.140 --> 00:04:55.430 OK. 00:04:55.430 --> 00:04:57.510 So now we want to multiply that number we just figured 00:04:57.510 --> 00:05:01.970 out times 25. 00:05:01.970 --> 00:05:09.020 Equals 7.05. 00:05:09.020 --> 00:05:13.560 So the change in velocity is equal to minus 00:05:13.560 --> 00:05:21.560 7.05 meters per second. 00:05:21.560 --> 00:05:24.020 OK, so let's see what's going on. 00:05:24.020 --> 00:05:29.920 The initial speed of the train is 80 kilometers per hour. 00:05:29.920 --> 00:05:32.000 But we have the change in velocity given 00:05:32.000 --> 00:05:34.590 in meters per second. 00:05:34.590 --> 00:05:37.900 So we can either change this change in velocity to 00:05:37.900 --> 00:05:40.740 kilometers per hour, or we could go the other way around 00:05:40.740 --> 00:05:44.220 and do the- or change the kilometers per hour into 00:05:44.220 --> 00:05:46.200 meters per second. 00:05:46.200 --> 00:05:48.810 I don't know, let's change the meters per second into 00:05:48.810 --> 00:05:50.360 kilometers per hour. 00:05:50.360 --> 00:05:56.790 So there are how many meters-- so it'll be minus 7.05. 00:05:56.790 --> 00:05:59.920 And how many meters are there in a kilometer? 00:05:59.920 --> 00:06:02.010 Well there are a thousand meters in a kilometer. 00:06:02.010 --> 00:06:06.450 So it's going to be going that number divided by a thousand 00:06:06.450 --> 00:06:06.960 kilometers. 00:06:06.960 --> 00:06:08.890 And if this is a little confusing, I'm skipping a 00:06:08.890 --> 00:06:09.720 couple of steps. 00:06:09.720 --> 00:06:12.500 You might want to review the video on unit conversion. 00:06:12.500 --> 00:06:14.910 And then, this is meters per second, but we want to get 00:06:14.910 --> 00:06:17.150 kilometers per hour. 00:06:17.150 --> 00:06:18.260 How may seconds are there in an hour? 00:06:18.260 --> 00:06:19.390 There are 3,600. 00:06:19.390 --> 00:06:20.120 right? 00:06:20.120 --> 00:06:23.220 So it'll go 3,600 times as far in an hour as it 00:06:23.220 --> 00:06:24.740 does it in a second. 00:06:24.740 --> 00:06:28.800 Times 3,600. 00:06:28.800 --> 00:06:31.930 So we can get rid of two 0's here and two 0's here. 00:06:31.930 --> 00:06:41.120 So it's essentially going to be 7.05 you times 36. 00:06:41.120 --> 00:06:42.540 I just got rid of the 0's. 00:06:42.540 --> 00:06:44.640 I'll write the 0's there, just so you don't get confused. 00:06:44.640 --> 00:06:47.640 Divided by a thousand. 00:06:47.640 --> 00:06:49.770 1, 2, 3. 00:06:49.770 --> 00:06:55.830 Equals 25.39 kilometers per hour. 00:06:55.830 --> 00:06:57.690 So I'll just round. 00:06:57.690 --> 00:06:59.730 25 kilometers per hour. 00:06:59.730 --> 00:07:07.870 So the change in velocity is minus 25 kilometers per hour. 00:07:07.870 --> 00:07:09.330 What's the change velocity? 00:07:09.330 --> 00:07:12.430 If it was starting off at 80 kilometers and then the change 00:07:12.430 --> 00:07:16.090 in velocity is minus 25, what's its new velocity? 00:07:16.090 --> 00:07:20.120 Well it's going to be 80 minus 25, the initial velocity minus 00:07:20.120 --> 00:07:21.740 the change in velocity. 00:07:21.740 --> 00:07:24.270 Or plus the change in velocity depending on how you view that 00:07:24.270 --> 00:07:25.290 negative number. 00:07:25.290 --> 00:07:26.660 So 80 minus 25. 00:07:26.660 --> 00:07:32.920 So we could say the final velocity is going to be 80 00:07:32.920 --> 00:07:40.040 minus 25 is 55 kilometers per hour. 00:07:40.040 --> 00:07:42.940 Now the second part of this question is, how far has the 00:07:42.940 --> 00:07:44.280 train traveled in this time? 00:07:44.280 --> 00:07:48.960 So essentially, how long-- how far does the train go as it 00:07:48.960 --> 00:07:51.660 brakes from 80 kilometers per hour to 55 00:07:51.660 --> 00:07:53.170 kilometers per hour? 00:07:53.170 --> 00:07:55.300 Well, all we have to figure out is the average velocity 00:07:55.300 --> 00:07:57.750 and then multiply that times the time. 00:07:57.750 --> 00:07:59.690 So what's the average velocity here? 00:08:02.740 --> 00:08:04.720 Well actually, let me just clear this because I think 00:08:04.720 --> 00:08:07.736 it's getting-- you don't need a drawing of a train. 00:08:07.736 --> 00:08:11.900 Let's see if I remember my-- so the initial velocity was 80 00:08:11.900 --> 00:08:14.110 kilometers per hour. 00:08:14.110 --> 00:08:18.950 Final velocity is 55 kilometers per hour. 00:08:18.950 --> 00:08:25.120 So the average velocity is going to be 80 plus 55 over 2, 00:08:25.120 --> 00:08:25.710 which is, what? 00:08:25.710 --> 00:08:28.590 67 and 1/2, something like that. 00:08:28.590 --> 00:08:35.730 67 and 1/2 kilometers per hour. 00:08:35.730 --> 00:08:37.549 And then if we want to figure out the total distance it 00:08:37.549 --> 00:08:41.870 travels, distance is equal to average velocity times time. 00:08:41.870 --> 00:08:42.929 The average velocity? 00:08:42.929 --> 00:08:45.700 Well, we have the time in seconds, right? 00:08:45.700 --> 00:08:48.280 So we have 25 seconds here. 00:08:48.280 --> 00:08:50.810 But this is given in kilometers per hour. 00:08:50.810 --> 00:08:54.770 So we want to convert this into, maybe meters per second. 00:08:54.770 --> 00:08:58.310 So what we do is take the 67.5, it's going 00:08:58.310 --> 00:08:59.870 to go times a thousand. 00:08:59.870 --> 00:09:00.040 that's. 00:09:00.040 --> 00:09:01.540 How many meters it's going to go. 00:09:01.540 --> 00:09:04.535 And then in a second it's going to go 1/3,600 of that. 00:09:04.535 --> 00:09:07.530 So let's see what that gives us. 00:09:07.530 --> 00:09:17.305 67.5 times 1,000 divided by 3,600. 00:09:17.305 --> 00:09:23.380 18.75. 00:09:23.380 --> 00:09:25.340 That's the meters per second. 00:09:25.340 --> 00:09:28.180 That's the average velocity in meters per second. 00:09:28.180 --> 00:09:31.800 And we go for 25 seconds, so what does that give us? 00:09:31.800 --> 00:09:38.625 Times 25 equals 468 meters. 00:09:38.625 --> 00:09:42.790 468.75 meters. 00:09:42.790 --> 00:09:46.370 That's how far it traveled just to brake from 80 00:09:46.370 --> 00:09:49.750 kilometers per hour to 55 kilometers per hour. 00:09:49.750 --> 00:09:51.670 And it took 25 seconds. 00:09:51.670 --> 00:09:53.320 Hopefully I haven't confused you too much. 00:09:53.320 --> 00:09:54.570 See

Newton's Laws Examples (part 2)

https://www.youtube.com/watch?v=x5Bz0ManOuc

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https://www.youtube.com/api/timedtext?v=x5Bz0ManOuc&ei=YmeUZfn-MJGnp-oP5bybGA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B8B7FFB4EAAF4E63C35E5D0D6950CCCD2EE0E083.17D9B5C4B44DBFF3CBF7493DACB5C1982166CF4F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.680 --> 00:00:01.620 Welcome back. 00:00:01.620 --> 00:00:05.710 And I will now do that same problem in a much easier way. 00:00:05.710 --> 00:00:07.400 Let me clear it a little bit. 00:00:10.300 --> 00:00:12.810 So the original problem we said, you know, we apply some 00:00:12.810 --> 00:00:19.330 force to m sub 0 and that gives some 00:00:19.330 --> 00:00:21.340 acceleration, a sub 0. 00:00:21.340 --> 00:00:25.090 And then we said when we apply the same force to a 00:00:25.090 --> 00:00:32.310 combination of m sub 0 and m sub 1, we get 1/5 the 00:00:32.310 --> 00:00:33.560 acceleration. 00:00:35.780 --> 00:00:37.900 And we worked it through with all the variables. 00:00:37.900 --> 00:00:39.390 What I'll show you is you can actually do this type of 00:00:39.390 --> 00:00:41.680 problem just by substituting numbers. 00:00:41.680 --> 00:00:43.610 This is kind of quick and dirty, but it's good to do a 00:00:43.610 --> 00:00:44.740 reality check. 00:00:44.740 --> 00:00:47.720 And oftentimes, you can solve the problem without having to 00:00:47.720 --> 00:00:49.360 go through all the variable mess. 00:00:49.360 --> 00:00:50.930 So I can just pick some numbers here. 00:00:50.930 --> 00:00:56.060 So I could say, well what if F sub 0 is equal to 10 Newtons, 00:00:56.060 --> 00:01:02.070 m sub 0 is equal to-- I don't know-- 2 kilograms. Than a sub 00:01:02.070 --> 00:01:06.320 0 is equal to-- well, it would be 10 divided by 2. 00:01:06.320 --> 00:01:08.200 Because force is equal to mass times acceleration. 00:01:08.200 --> 00:01:12.940 So it'd be 5 meters per second squared. 00:01:12.940 --> 00:01:18.810 And then in this case, this would be 10 Newtons. 00:01:18.810 --> 00:01:20.250 1/5 a sub 0? 00:01:20.250 --> 00:01:21.540 Well that will be 1/5 this. 00:01:21.540 --> 00:01:24.940 So it would be 1 meters per second squared. 00:01:24.940 --> 00:01:28.030 And then we could solve for what the new mass is. 00:01:28.030 --> 00:01:28.860 How would we do that? 00:01:28.860 --> 00:01:33.030 Well we have force, which is 10 Newtons, is equal to the 00:01:33.030 --> 00:01:34.120 sum of the masses. 00:01:34.120 --> 00:01:37.600 So m1 plus m sub 0. 00:01:37.600 --> 00:01:42.060 But m sub 0 we already learned is 2 kilograms. Times the 00:01:42.060 --> 00:01:42.810 acceleration. 00:01:42.810 --> 00:01:46.110 Times 1/5 a sub 0, which is 1 meter per second squared. 00:01:48.910 --> 00:01:53.650 So then we have-- this 1, we could ignore it essentially. 00:01:53.650 --> 00:01:57.200 So then we essentially have that 10-- and since all of our 00:01:57.200 --> 00:01:59.490 units are right, we can kind of drop the units because we 00:01:59.490 --> 00:02:00.530 know they work out. 00:02:00.530 --> 00:02:04.220 10 is equal to m1 plus 2. 00:02:04.220 --> 00:02:07.550 So you get m1 is equal to 8. 00:02:07.550 --> 00:02:10.550 And then once again, if we want to know the ratio of m 00:02:10.550 --> 00:02:14.290 sub 0 to m sub 1, we can just substitute the numbers. 00:02:14.290 --> 00:02:18.440 m sub 0 is 2 kilograms. m sub 1 is 8 kilograms. So 00:02:18.440 --> 00:02:19.970 the ratio is 1:4. 00:02:19.970 --> 00:02:22.450 You probably find that a little bit easier. 00:02:22.450 --> 00:02:23.700 Let's do another problem. 00:02:28.570 --> 00:02:29.610 Whoops. 00:02:29.610 --> 00:02:30.570 Invert colors. 00:02:30.570 --> 00:02:31.820 OK. 00:02:33.650 --> 00:02:38.230 This next problem I think you'll find interesting. 00:02:38.230 --> 00:02:42.860 So let's say I have a sky diver and he's in his sky 00:02:42.860 --> 00:02:47.820 diver position, falling towards the ground. 00:02:47.820 --> 00:02:51.120 And let's say he weighs 70 kilograms. So his mass is 00:02:51.120 --> 00:02:57.340 equal to 70 kilograms. Let's say that the terminal velocity 00:02:57.340 --> 00:02:59.080 is 120 miles per hour. 00:02:59.080 --> 00:03:03.330 So he's moving downward at 120 miles per hour, which is 00:03:03.330 --> 00:03:04.260 actually accurate. 00:03:04.260 --> 00:03:07.450 I've gone sky diving. 00:03:07.450 --> 00:03:09.560 And if we convert that-- you could convert that for fun 00:03:09.560 --> 00:03:10.690 into the metric system. 00:03:10.690 --> 00:03:12.270 But I'll do that for you. 00:03:12.270 --> 00:03:14.660 But it's good to know just so you have a sense of how fast 00:03:14.660 --> 00:03:18.140 you fall when you sky dive before the parachute opens. 00:03:18.140 --> 00:03:22.910 This translates to about 53.6 meters per second. 00:03:22.910 --> 00:03:24.790 I'm reading this from a problem from a website at the 00:03:24.790 --> 00:03:27.090 University of Oregon. 00:03:27.090 --> 00:03:31.030 But anyway, they are asking us, what force does air 00:03:31.030 --> 00:03:33.870 resistance exert on the sky diver? 00:03:33.870 --> 00:03:35.350 So let's be clear a couple things. 00:03:35.350 --> 00:03:37.960 This 120 miles per hour, this is the sky 00:03:37.960 --> 00:03:40.500 diver's terminal velocity. 00:03:40.500 --> 00:03:42.880 And if you're not familiar with what terminal velocity 00:03:42.880 --> 00:03:46.010 is, I will now explain it to you. 00:03:46.010 --> 00:03:50.610 So when you fall from a plane, you have a bunch of wind 00:03:50.610 --> 00:03:51.510 pushing on you. 00:03:51.510 --> 00:03:54.100 You have a lot of wind resistance. 00:03:54.100 --> 00:03:56.480 It causes friction; it slows you down as you can imagine. 00:03:56.480 --> 00:03:57.900 I mean that's how a parachute works. 00:04:02.610 --> 00:04:04.600 It creates a lot more resistance from the wind and 00:04:04.600 --> 00:04:05.600 then you slow down. 00:04:05.600 --> 00:04:11.760 So the terminal velocity is the velocity at which you no 00:04:11.760 --> 00:04:13.230 longer go faster than. 00:04:13.230 --> 00:04:17.420 So it's the velocity at which you stop accelerating or it's 00:04:17.420 --> 00:04:20.690 the velocity you reach and you don't go any faster than that. 00:04:20.690 --> 00:04:23.940 It's basically based on your wind resistance. 00:04:23.940 --> 00:04:26.560 So at the terminal velocity your acceleration is 0. 00:04:29.580 --> 00:04:34.180 So what we know is, is that the force of the air-- we 00:04:34.180 --> 00:04:36.720 could call that the air force. 00:04:36.720 --> 00:04:41.460 So we know that the force of the air is exactly equal to 00:04:41.460 --> 00:04:42.710 the force of gravity. 00:04:46.260 --> 00:04:47.620 And how do we know that? 00:04:47.620 --> 00:04:50.370 Because the guy's not accelerating. 00:04:50.370 --> 00:04:51.660 It's his terminal velocity. 00:04:51.660 --> 00:04:54.070 He's at a very high speed. 00:04:54.070 --> 00:04:56.060 He had accelerated all the way to this point. 00:04:56.060 --> 00:04:59.160 But the more he accelerated and the faster he got, the 00:04:59.160 --> 00:05:03.250 more resistance the wind provided up to a point where 00:05:03.250 --> 00:05:05.810 the wind provided so much resistance that he stopped 00:05:05.810 --> 00:05:06.790 going any faster. 00:05:06.790 --> 00:05:08.010 And that's the terminal velocity. 00:05:08.010 --> 00:05:11.680 So at terminal velocity, the force of the air is equal to 00:05:11.680 --> 00:05:14.120 the force of gravity. 00:05:14.120 --> 00:05:15.640 What's the force of gravity? 00:05:15.640 --> 00:05:18.250 Well the force of gravity is just the guy's weight. 00:05:18.250 --> 00:05:23.490 So the force of gravity is equal to the guy's mass, 70 00:05:23.490 --> 00:05:25.240 kilograms. And we have our units right. 00:05:25.240 --> 00:05:29.580 70 kilograms. Times the acceleration of gravity. 00:05:29.580 --> 00:05:32.940 Well the acceleration of gravity is 9.8 roughly meters 00:05:32.940 --> 00:05:34.580 per second squared. 00:05:34.580 --> 00:05:36.940 We could use a calculator to calculate this. 00:05:40.730 --> 00:05:41.540 I feel cheap now. 00:05:41.540 --> 00:05:43.790 I could have done it by hand anyway. 00:05:43.790 --> 00:05:45.840 686. 00:05:45.840 --> 00:05:52.760 So it equals 686 Newtons. 00:05:52.760 --> 00:05:55.140 The second part of the question-- and this is 00:05:55.140 --> 00:05:56.650 interesting. 00:05:56.650 --> 00:06:00.060 If a sky diver pulls in their arm and aims their body 00:06:00.060 --> 00:06:04.010 downward, so now the sky diver looks more like this and he 00:06:04.010 --> 00:06:06.500 pulled in his arms and he aimed his body down. 00:06:06.500 --> 00:06:09.770 So he's diving, really. 00:06:09.770 --> 00:06:13.810 The terminal velocity can be increased to about 180 miles 00:06:13.810 --> 00:06:17.070 per hour or 80.5 meters per second. 00:06:17.070 --> 00:06:17.680 They give us this. 00:06:17.680 --> 00:06:19.310 We could've figured it out though. 00:06:19.310 --> 00:06:22.400 So now he's going a lot faster. 00:06:22.400 --> 00:06:25.850 Roughly 50% faster than he was, or maybe-- well, 30% 00:06:25.850 --> 00:06:27.830 faster than he was going before. 00:06:27.830 --> 00:06:30.040 He's going a lot faster and why is that? 00:06:30.040 --> 00:06:33.960 Because he's more aerodynamic now. 00:06:33.960 --> 00:06:36.760 We'll do more on pressure later, but I want you to get 00:06:36.760 --> 00:06:40.110 the intuition that when you're laying flat there's just a lot 00:06:40.110 --> 00:06:41.470 of wind pressing against your body. 00:06:41.470 --> 00:06:44.560 You have a lot of surface area exposed to the wind. 00:06:44.560 --> 00:06:46.970 But when you're diving in this situation, like the sky diver 00:06:46.970 --> 00:06:49.660 is, he has a lot less exposed to the wind. 00:06:49.660 --> 00:06:50.450 Really just his head. 00:06:50.450 --> 00:06:53.420 His head is breaking the wind and nothing else. 00:06:53.420 --> 00:07:00.760 And that's why it takes a lot more speed for the force of 00:07:00.760 --> 00:07:04.420 the wind resistance to match the force of gravity. 00:07:04.420 --> 00:07:07.270 So the question is asking, if the sky diver pulls in their 00:07:07.270 --> 00:07:10.310 arms and aims their body downward, the terminal 00:07:10.310 --> 00:07:14.510 velocity can be increased to about 80.5 meters per second 00:07:14.510 --> 00:07:15.880 or 180 miles per hour. 00:07:15.880 --> 00:07:17.140 So he's going very fast. 00:07:17.140 --> 00:07:22.720 What force does air resistance now exert on the sky diver? 00:07:22.720 --> 00:07:24.460 And I'll let you think about that for a second. 00:07:24.460 --> 00:07:27.580 Maybe you want to pause it and think of it yourself. 00:07:27.580 --> 00:07:30.120 And now that you've unpaused it, I'll tell you that this is 00:07:30.120 --> 00:07:31.290 a trick question. 00:07:31.290 --> 00:07:33.810 Because once again, the sky diver has reached a new 00:07:33.810 --> 00:07:35.270 terminal velocity. 00:07:35.270 --> 00:07:39.500 By definition, at the terminal velocity, the sky diver is no 00:07:39.500 --> 00:07:40.960 longer accelerating. 00:07:40.960 --> 00:07:43.940 The sky diver is not going any faster because the wind 00:07:43.940 --> 00:07:46.713 resistance is so strong that it completely matches the 00:07:46.713 --> 00:07:47.920 force of gravity. 00:07:47.920 --> 00:07:50.950 So once again, the wind resistance, the force of the 00:07:50.950 --> 00:07:53.430 air or the force of the wind, is equal to 00:07:53.430 --> 00:07:55.250 the force of gravity. 00:07:55.250 --> 00:07:56.620 And what is the force of gravity? 00:07:56.620 --> 00:07:57.770 Well that's his weight. 00:07:57.770 --> 00:07:58.890 And we already figured that out. 00:07:58.890 --> 00:08:01.720 That was 686 Newtons. 00:08:01.720 --> 00:08:02.640 And I know what you're thinking. 00:08:02.640 --> 00:08:05.560 You're saying, Sal, this doesn't make sense. 00:08:05.560 --> 00:08:11.070 He's now going so much faster, doesn't the air exert more 00:08:11.070 --> 00:08:12.410 force on him? 00:08:12.410 --> 00:08:13.810 Well no. 00:08:13.810 --> 00:08:16.640 The air is exerting the same force. 00:08:16.640 --> 00:08:18.990 If he was going that same speed, but he was flattened 00:08:18.990 --> 00:08:21.940 out, I would agree with you. 00:08:21.940 --> 00:08:24.680 The air would be exerting more force on him. 00:08:24.680 --> 00:08:27.820 But what's happened now is that he's once again reached a 00:08:27.820 --> 00:08:29.970 state where his acceleration is 0. 00:08:29.970 --> 00:08:32.870 It's at a higher velocity and I want you 00:08:32.870 --> 00:08:33.980 think a lot about this. 00:08:33.980 --> 00:08:36.250 He's at a much higher velocity now. 00:08:38.950 --> 00:08:41.330 On his head, for example, there's a lot 00:08:41.330 --> 00:08:42.360 more wind going by. 00:08:42.360 --> 00:08:46.200 But it's pressing on a smaller surface area. 00:08:46.200 --> 00:08:48.520 And I'm not going to go too much into detail of 00:08:48.520 --> 00:08:49.560 pressure right now. 00:08:49.560 --> 00:08:51.730 But I want you to get that intuition. 00:08:51.730 --> 00:08:56.360 So although the wind is going a lot faster, it's going a lot 00:08:56.360 --> 00:08:59.460 faster on a smaller area. 00:08:59.460 --> 00:09:02.390 And its actual force is the exact same thing. 00:09:02.390 --> 00:09:04.850 And we know that because he's not accelerating anymore. 00:09:04.850 --> 00:09:07.060 Because he's at his terminal velocity. 00:09:07.060 --> 00:09:08.070 So think about that a bit. 00:09:08.070 --> 00:09:10.000 It's a bit of a trick question, but I think it gives 00:09:10.000 --> 00:09:12.890 you a good intuition on what acceleration means, what 00:09:12.890 --> 00:09:15.370 terminal velocity means, and it'll start to give you a 00:09:15.370 --> 00:09:18.430 little bit of an intuition on even wind 00:09:18.430 --> 00:09:19.820 resistance, on pressure. 00:09:19.820 --> 00:09:21.860 I'll see you in the next video.

Newton's Laws Problems (part 1)

https://www.youtube.com/watch?v=wGKXIq-gdok

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https://www.youtube.com/api/timedtext?v=wGKXIq-gdok&ei=ZWeUZfq_D5m4vdIPhvC5sAs&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249813&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=173B1D581B95FCD5D0400CC7AC1954EFADDC3974.93383F83EDBE19265ABDBAABE0BD3D4E30EA0532&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:01.820 Welcome back. 00:00:01.820 --> 00:00:04.340 Now that we've hopefully, learned a little bit about 00:00:04.340 --> 00:00:07.670 Newton's law, let's apply them to solve some problems. 00:00:07.670 --> 00:00:10.630 Let's say I have a-- I don't know-- some kind of vehicle, a 00:00:10.630 --> 00:00:12.390 car, a motorcycle or something. 00:00:12.390 --> 00:00:18.150 And let's say its mass is 500 grams. And let's say that I 00:00:18.150 --> 00:00:23.150 can accelerate this vehicle at an acceleration of-- I don't 00:00:23.150 --> 00:00:33.410 know-- 3 centimeters per second squared. 00:00:33.410 --> 00:00:37.500 What's the force that I need to apply to this mass to 00:00:37.500 --> 00:00:39.210 accelerate it at this speed? 00:00:39.210 --> 00:00:41.180 And we want the answer in Newton's. 00:00:41.180 --> 00:00:42.976 So what's the force? 00:00:42.976 --> 00:00:45.130 So we're just going to use Newton's second law and 00:00:45.130 --> 00:00:48.820 Newton's second law tells us force is equal to mass times 00:00:48.820 --> 00:00:49.810 acceleration. 00:00:49.810 --> 00:00:53.410 So you might want to, or you might be tempted just to 00:00:53.410 --> 00:00:56.550 multiply mass times acceleration and you'd get 00:00:56.550 --> 00:01:04.530 force is equal to-- let's see, 500 grams times 3 centimeters 00:01:04.530 --> 00:01:05.700 per second squared. 00:01:05.700 --> 00:01:12.440 And you would get force is equal to 1,500 gram 00:01:12.440 --> 00:01:16.320 centimeters per second squared. 00:01:16.320 --> 00:01:18.880 And if you did this, you would be right. 00:01:18.880 --> 00:01:21.250 Although your answer would not be in Newtons and now you 00:01:21.250 --> 00:01:23.670 would have to somehow, try to convert this 00:01:23.670 --> 00:01:25.990 set of units to Newtons. 00:01:25.990 --> 00:01:27.020 And what are Newtons? 00:01:27.020 --> 00:01:30.850 Well we learned when we did Newton's laws that a Newton, 00:01:30.850 --> 00:01:35.360 so 1 Newton is equal to 1 kilogram 00:01:35.360 --> 00:01:37.900 meter per second squared. 00:01:37.900 --> 00:01:40.250 So somehow we have to convert the gram to kilograms and we 00:01:40.250 --> 00:01:42.260 have to convert the centimeters to meters. 00:01:42.260 --> 00:01:45.050 We could do it after the fact here, or what I find it easier 00:01:45.050 --> 00:01:48.150 to do is actually to convert the mass and the acceleration 00:01:48.150 --> 00:01:51.200 units first and then just do the F equals ma. 00:01:51.200 --> 00:01:55.360 So what's 500 grams in kilograms? 00:01:55.360 --> 00:02:00.940 Well 500 grams is half-- well, a kilogram as a thousand 00:02:00.940 --> 00:02:04.010 grams. So 500 is going to be half a kilogram. 00:02:07.440 --> 00:02:11.430 1 kilogram is a thousand grams, so 0.5 kilograms is 500 00:02:11.430 --> 00:02:13.040 grams. 00:02:13.040 --> 00:02:18.000 Similarly, 3 centimeters is how many meters? 00:02:18.000 --> 00:02:24.370 Well, 1 meter is 300-- sorry. 00:02:24.370 --> 00:02:25.510 I think I'm dehydrated. 00:02:25.510 --> 00:02:28.150 1 meter is a hundred centimeters, right? 00:02:28.150 --> 00:02:36.280 So 3 centimeters is 0.03 meters per second squared. 00:02:36.280 --> 00:02:38.530 Hopefully this make sense to you. 00:02:38.530 --> 00:02:40.740 3 centimeters is 0.03 meters. 00:02:40.740 --> 00:02:41.870 And now we already. 00:02:41.870 --> 00:02:44.770 We have our mass in kilograms and we have our acceleration 00:02:44.770 --> 00:02:46.130 in meters per second squared. 00:02:46.130 --> 00:02:49.130 And if this is confusing, you should watch the unit videos 00:02:49.130 --> 00:02:50.710 because this is all I'm doing; I'm just doing unit 00:02:50.710 --> 00:02:51.660 conversion. 00:02:51.660 --> 00:02:53.880 So let's go back to force equals mass times 00:02:53.880 --> 00:02:55.280 acceleration. 00:02:55.280 --> 00:03:04.850 So the force was equal to 0.5 kilograms times the 00:03:04.850 --> 00:03:12.970 acceleration, which is 0.03 meters per second squared. 00:03:12.970 --> 00:03:13.700 And this equals-- 00:03:13.700 --> 00:03:15.490 What's 0.5 times 0.03? 00:03:15.490 --> 00:03:18.400 I'll do it down here just because multiplying decimals 00:03:18.400 --> 00:03:21.370 seems to be a problem for a lot of people, including 00:03:21.370 --> 00:03:22.980 myself, many times. 00:03:22.980 --> 00:03:26.000 So what you do, you just multiply the numbers. 00:03:26.000 --> 00:03:29.510 5 times 03 or 5 times 3 is 15. 00:03:29.510 --> 00:03:31.640 And then, how many points do we have behind the decimal? 00:03:31.640 --> 00:03:33.210 How many digits behind the decimal? 00:03:33.210 --> 00:03:33.740 Let's see. 00:03:33.740 --> 00:03:35.130 We have 1, 2, 3. 00:03:35.130 --> 00:03:37.660 So 1, 2, 3. 00:03:37.660 --> 00:03:40.240 We have to add the 0 because we need three spaces behind 00:03:40.240 --> 00:03:41.730 the decimal point. 00:03:41.730 --> 00:03:50.190 So we get the force is equal to 0.015 kilogram meters per 00:03:50.190 --> 00:03:52.640 second squared. 00:03:52.640 --> 00:03:54.400 And this is a Newton. 00:03:54.400 --> 00:04:01.066 So the force is equal to 0.015 Newtons. 00:04:01.066 --> 00:04:03.470 Let's do another problem. 00:04:03.470 --> 00:04:05.280 And this one's going to be-- and actually, I think you'll 00:04:05.280 --> 00:04:09.600 find most of the difficult Newton's laws problems or 00:04:09.600 --> 00:04:12.120 force problems, they're just some combination of making 00:04:12.120 --> 00:04:15.120 sure you get the units right when we're talking about in 00:04:15.120 --> 00:04:16.680 one dimension. 00:04:16.680 --> 00:04:20.959 The difficult part is usually getting the units right or 00:04:20.959 --> 00:04:22.710 just the math, just the algebra. 00:04:22.710 --> 00:04:24.610 So if you have trouble with this it's usually because you 00:04:24.610 --> 00:04:26.890 have to just brush up a little bit on the algebra. 00:04:26.890 --> 00:04:29.100 The physics itself is just force equals mass times 00:04:29.100 --> 00:04:32.130 acceleration as we will see in this problem. 00:04:32.130 --> 00:04:35.950 So let's say that when I apply some force, some particular 00:04:35.950 --> 00:04:40.700 force, I use this little 0 here. 00:04:40.700 --> 00:04:43.360 So I call that force F sub 0. 00:04:43.360 --> 00:04:44.820 So this means a particular force. 00:04:44.820 --> 00:04:46.010 This is some value. 00:04:46.010 --> 00:04:50.600 When I apply that force to some mass, let's call that m 00:04:50.600 --> 00:04:54.070 sub 0, I get some acceleration. 00:04:54.070 --> 00:04:56.810 I get acceleration a sub 0. 00:04:56.810 --> 00:04:57.720 We could've put numbers here. 00:04:57.720 --> 00:05:02.810 We could've said, well, if I apply a force of 10 Newtons to 00:05:02.810 --> 00:05:07.940 a mass of-- I don't know-- to a mass of let's say 2 00:05:07.940 --> 00:05:14.330 kilograms, I have an acceleration of 5 meters per 00:05:14.330 --> 00:05:15.170 second squared. 00:05:15.170 --> 00:05:16.890 But I'm just doing this because this could be any 00:05:16.890 --> 00:05:17.790 relationship. 00:05:17.790 --> 00:05:20.380 And let's say the problem tells us that when I put 00:05:20.380 --> 00:05:23.680 another mass with this first mass, so let's say that, you 00:05:23.680 --> 00:05:26.400 know-- let me draw this diagram. 00:05:26.400 --> 00:05:29.480 So here's my mass, m sub 0. 00:05:29.480 --> 00:05:34.610 When I apply a force of f sub 0 to it, I get an acceleration 00:05:34.610 --> 00:05:36.250 of a sub 0. 00:05:36.250 --> 00:05:38.390 Now the problem tell us when I add another mass-- so let's 00:05:38.390 --> 00:05:39.880 says I stack it up and we're like in an ice 00:05:39.880 --> 00:05:41.670 skating ring or something. 00:05:41.670 --> 00:05:44.196 And I stack another mass up here, and let's 00:05:44.196 --> 00:05:46.990 call this mass m1. 00:05:46.990 --> 00:05:49.890 When I stack another mass on here-- so let me redraw it 00:05:49.890 --> 00:05:51.250 actually down here. 00:05:51.250 --> 00:05:53.420 Because it's a different case. 00:05:53.420 --> 00:05:57.720 And I apply the same force, and now I have 00:05:57.720 --> 00:05:59.020 this new mass on here. 00:05:59.020 --> 00:06:01.926 I'll do it in red. 00:06:01.926 --> 00:06:03.920 This is m1. 00:06:03.920 --> 00:06:08.540 The problem tells us that my new acceleration is 1/5 of the 00:06:08.540 --> 00:06:09.650 original acceleration. 00:06:09.650 --> 00:06:12.070 So it's 1/5 of whatever this was. 00:06:12.070 --> 00:06:16.390 So it's 1/5 a sub 0. 00:06:16.390 --> 00:06:21.620 So the question is, what is the ratio of m 00:06:21.620 --> 00:06:23.610 sub 0 to m sub 1? 00:06:23.610 --> 00:06:28.590 So m sub 0 to m sub 1 is equal to what? 00:06:28.590 --> 00:06:30.990 And I'm going to keep it in abstract variables just to 00:06:30.990 --> 00:06:31.710 confuse you. 00:06:31.710 --> 00:06:33.200 And then I'll show you that you can actually substitute 00:06:33.200 --> 00:06:35.720 numbers and the problem becomes a little easier. 00:06:35.720 --> 00:06:38.210 And you might want to pause it and try it for yourself. 00:06:38.210 --> 00:06:39.670 So let's work it through. 00:06:39.670 --> 00:06:41.590 So we know we have this relationship 00:06:41.590 --> 00:06:43.250 to start off with. 00:06:43.250 --> 00:06:48.760 And just for simplicity, let's write what m sub 0 is in terms 00:06:48.760 --> 00:06:49.720 of F and a. 00:06:49.720 --> 00:06:54.890 So we just divide both sides by a sub 0 and you get F sub 0 00:06:54.890 --> 00:06:58.250 divided by a sub 0 is equal to m sub 0. 00:06:58.250 --> 00:06:58.580 Good. 00:06:58.580 --> 00:07:00.970 So let's just put that aside for a second. 00:07:00.970 --> 00:07:04.340 And let's do that same relationship here with this. 00:07:04.340 --> 00:07:09.160 So here, this relationship tells us that F sub 0 is equal 00:07:09.160 --> 00:07:19.130 to m1 plus m0 times this new acceleration, which 00:07:19.130 --> 00:07:20.660 is 1/5 a sub 0. 00:07:24.570 --> 00:07:28.390 And so, if we divide both sides by this term right here, 00:07:28.390 --> 00:07:32.820 we get-- dividing by 1/5 is the same thing as 00:07:32.820 --> 00:07:34.260 multiplying by 5. 00:07:34.260 --> 00:07:44.060 So you get 5 F sub 0 over a sub 0 is equal to m1 plus m0. 00:07:44.060 --> 00:07:47.370 I just divided both sides by this term right here. 00:07:47.370 --> 00:07:48.910 Well, what's F sub 0? 00:07:48.910 --> 00:07:49.490 What's this? 00:07:49.490 --> 00:07:50.340 Let me switch colors again. 00:07:50.340 --> 00:07:53.910 What's F sub 0 divided by a sub 0? 00:07:53.910 --> 00:07:54.570 Well it's here's. 00:07:54.570 --> 00:07:55.860 It's what we solved for in the beginning. 00:07:55.860 --> 00:07:57.770 We just got it from this relationship. 00:07:57.770 --> 00:07:59.310 So we could substitute. 00:07:59.310 --> 00:08:02.780 5 times F sub 0 divided by a sub 0 is the same 00:08:02.780 --> 00:08:05.425 thing as 5 times m0. 00:08:05.425 --> 00:08:06.790 Draw a line here. 00:08:06.790 --> 00:08:07.930 So we have a new relationship. 00:08:07.930 --> 00:08:13.330 5m0 is equal to m1 plus m0. 00:08:13.330 --> 00:08:15.960 All I did is I substituted this for 00:08:15.960 --> 00:08:17.980 this, or this for that. 00:08:17.980 --> 00:08:20.150 And I used this relationship, which we got in the beginning 00:08:20.150 --> 00:08:21.370 to do that. 00:08:21.370 --> 00:08:22.830 And now what do we have? 00:08:22.830 --> 00:08:25.660 We have 5 m sub 0 is equal to m1 plus m0. 00:08:25.660 --> 00:08:28.460 We could subtract m0 from both sides. 00:08:28.460 --> 00:08:32.270 You get 4 m0 is equal to m1. 00:08:35.520 --> 00:08:39.460 You could divide both sides by m0 and you got 4 is equal to 00:08:39.460 --> 00:08:42.730 m1 over m0. 00:08:42.730 --> 00:08:44.760 And you could invert this relationship and you can get 00:08:44.760 --> 00:08:49.210 m0 over m sub 1 is equal to 1/4. 00:08:49.210 --> 00:08:54.060 So what we learned is the ratio of the old mass to the 00:08:54.060 --> 00:08:56.400 new mass is 1 to 4. 00:08:56.400 --> 00:08:57.410 And that's a problem. 00:08:57.410 --> 00:08:59.750 And actually, I will leave it for you as an exercise to 00:08:59.750 --> 00:09:01.410 figure out-- to just do the same 00:09:01.410 --> 00:09:04.500 problem using the numbers. 00:09:04.500 --> 00:09:06.650 I will do that actually, in the next video just to show 00:09:06.650 --> 00:09:07.750 you that that actually would've been a 00:09:07.750 --> 00:09:08.860 simpler way to do it. 00:09:08.860 --> 00:09:12.160 But it's good to get used to this just so you can solve 00:09:12.160 --> 00:09:15.710 things in general terms. I'll see you in the next video.

Newton's Second Law of Motion

https://www.youtube.com/watch?v=3FQ58lVtbCg

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https://www.youtube.com/api/timedtext?v=3FQ58lVtbCg&ei=YmeUZc7VMtW3mLAP04aNqA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=077976D868BE0BE4BA29F260FDF3C0E05B5B3899.E7CDD8294FE48F34C238BDDD712CBD8F99E3D7AA&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.970 --> 00:00:01.550 Welcome back. 00:00:01.550 --> 00:00:03.460 We're now ready for Newton's second law. 00:00:03.460 --> 00:00:05.970 And Newton's second law can simply be stated-- and you've 00:00:05.970 --> 00:00:12.310 probably seen this before as force is equal to mass times 00:00:12.310 --> 00:00:13.100 acceleration. 00:00:13.100 --> 00:00:18.480 This is probably, if not the most famous formula in all of 00:00:18.480 --> 00:00:20.730 time or all of physics, it's up there. 00:00:20.730 --> 00:00:23.190 It's probably up there with E equals mc squared. 00:00:23.190 --> 00:00:24.810 But that one's a little bit more complicated. 00:00:24.810 --> 00:00:26.030 So what does this tell us? 00:00:26.030 --> 00:00:29.550 This tells us that the force, the net force upon an object, 00:00:29.550 --> 00:00:33.520 is equal to the object's mass times its acceleration. 00:00:33.520 --> 00:00:35.490 So let's stay in the metric system because most of what 00:00:35.490 --> 00:00:37.660 you'll do in physic class is in the metric system, and that 00:00:37.660 --> 00:00:40.160 tends to be because the metric system makes more sense. 00:00:40.160 --> 00:00:42.390 So let's say that I have a 1 kilogram object. 00:00:47.330 --> 00:00:50.430 So its mass is 1 kilogram. 00:00:50.430 --> 00:00:53.450 And it's being pulled down at-- let's say its 00:00:53.450 --> 00:00:54.850 acceleration. 00:00:54.850 --> 00:01:01.040 It's being accelerated downward at 9.8 meters per 00:01:01.040 --> 00:01:02.290 second squared. 00:01:05.209 --> 00:01:06.850 These kind of units should be familiar with you from all the 00:01:06.850 --> 00:01:08.380 projectile motion problems. 00:01:08.380 --> 00:01:11.380 So the force applied on that object in order to get this 00:01:11.380 --> 00:01:16.820 type of acceleration would be-- you just multiply mass 00:01:16.820 --> 00:01:17.570 times acceleration. 00:01:17.570 --> 00:01:25.610 The force would have had to be 9.8 kilogram times the meter. 00:01:25.610 --> 00:01:26.250 kilogram. 00:01:26.250 --> 00:01:29.540 times meter over second square. 00:01:29.540 --> 00:01:31.240 That's the force applied on the object. 00:01:31.240 --> 00:01:33.950 And you're saying, sal, this is very messy. 00:01:33.950 --> 00:01:36.340 I don't like writing kilogram meters per second squared. 00:01:36.340 --> 00:01:39.315 And you are in luck because there is a unit and that unit 00:01:39.315 --> 00:01:40.450 is the Newton. 00:01:40.450 --> 00:01:46.490 1 Newton is equal to 1 kilogram 00:01:46.490 --> 00:01:49.400 meter per second squared. 00:01:49.400 --> 00:01:57.740 So if I'm pulling down on an object at 9.8 Newtons, that's 00:01:57.740 --> 00:01:58.440 just this, right? 00:01:58.440 --> 00:02:00.340 This is 1 Newton. 00:02:00.340 --> 00:02:05.050 If I'm pulling down at 9.8 Newtons on an object that is 1 00:02:05.050 --> 00:02:12.230 kilogram, its acceleration is going to be 9.8 meters per 00:02:12.230 --> 00:02:14.340 second squared down. 00:02:14.340 --> 00:02:17.200 And notice I said the word down, but I didn't write it 00:02:17.200 --> 00:02:19.300 anywhere in the formula. 00:02:19.300 --> 00:02:22.690 And I guess we can imply that both force and acceleration 00:02:22.690 --> 00:02:28.700 have direction by writing this in the formula. 00:02:28.700 --> 00:02:32.620 That force is a vector and acceleration is a vector. 00:02:32.620 --> 00:02:36.500 And so we could have written 9.8 Newtons-- I don't know. 00:02:36.500 --> 00:02:38.250 You'll never see this convention. 00:02:38.250 --> 00:02:43.990 We could say Newtons down is equal to 1 kilogram times 9.8 00:02:43.990 --> 00:02:47.340 meters per second down. 00:02:47.340 --> 00:02:49.010 So what can we do with this formula? 00:02:49.010 --> 00:02:54.300 Well we can solve problems. So let's say 00:02:54.300 --> 00:02:55.490 that I have an object. 00:02:55.490 --> 00:02:58.350 So my object weighs-- not weighs. 00:02:58.350 --> 00:03:00.400 The mass of my object. 00:03:00.400 --> 00:03:03.190 And I'll differentiate between weight and mass in a second. 00:03:03.190 --> 00:03:07.110 Let's say the mass of some object is-- I don't know-- 50 00:03:07.110 --> 00:03:12.085 kilograms. That's how much a normal person might weigh or a 00:03:12.085 --> 00:03:13.480 light person. 00:03:13.480 --> 00:03:19.300 Mass weighs 50 kilograms. And let's say we're in an inertial 00:03:19.300 --> 00:03:20.100 frame of reference. 00:03:20.100 --> 00:03:23.390 We're in deep space, so we don't have all these other-- 00:03:23.390 --> 00:03:26.130 the force of wind and the force of gravity 00:03:26.130 --> 00:03:27.380 acting on us, et cetera. 00:03:31.600 --> 00:03:34.260 My force, let's say I apply it to the right. 00:03:34.260 --> 00:03:36.530 So we know that force is a vector. 00:03:36.530 --> 00:03:43.686 Let's say I apply a force of-- I don't know-- 100 Newtons. 00:03:43.686 --> 00:03:45.400 And let's say I apply it to the right. 00:03:49.600 --> 00:03:56.820 So this is the object, 50 kilograms. And I'm applying a 00:03:56.820 --> 00:04:02.220 force to the right of 100 Newtons. 00:04:02.220 --> 00:04:05.060 So what's going to happen to this object? 00:04:05.060 --> 00:04:07.410 Well, let's use the formula. 00:04:07.410 --> 00:04:12.640 Force is equal to mass times acceleration. 00:04:12.640 --> 00:04:14.790 The force is 100 Newtons. 00:04:14.790 --> 00:04:18.730 100 Newtons is equal to the mass. 00:04:18.730 --> 00:04:24.260 The mass is 50 kilograms. 50 kilograms times the 00:04:24.260 --> 00:04:26.700 acceleration. 00:04:26.700 --> 00:04:33.830 So we can divide both sides by 50 and you get 100 Newtons 00:04:33.830 --> 00:04:40.580 over 50 kilograms is equal to the acceleration. 00:04:40.580 --> 00:04:42.930 And it's 100 Newtons to the right. 00:04:42.930 --> 00:04:43.950 I'll use this little arrow here. 00:04:43.950 --> 00:04:46.160 That's not a traditional convention, but that's how we 00:04:46.160 --> 00:04:47.220 know it's to the right. 00:04:47.220 --> 00:04:48.770 So it's 100 divided by 50. 00:04:48.770 --> 00:04:51.150 So it's 2. 00:04:51.150 --> 00:04:56.010 We get this weird units here, Newtons per kilogram is equal 00:04:56.010 --> 00:04:58.250 to the acceleration to the right. 00:04:58.250 --> 00:04:59.590 This is also going to be to the right because the 00:04:59.590 --> 00:05:02.380 direction of the force is going to be the same as the 00:05:02.380 --> 00:05:04.540 direction of the acceleration. 00:05:04.540 --> 00:05:06.400 So what is this, 2 Newtons per kilogram? 00:05:06.400 --> 00:05:09.210 Well, if you remember-- well you could just guess that the 00:05:09.210 --> 00:05:10.730 unit of acceleration is meters per second squared. 00:05:10.730 --> 00:05:12.590 But let's show that this simplifies to that. 00:05:12.590 --> 00:05:15.060 So we said earlier that-- let me just switch colors. 00:05:15.060 --> 00:05:22.270 That a Newton is kilogram meter per second squared. 00:05:22.270 --> 00:05:27.470 And we're taking this Newton over this kilogram over 00:05:27.470 --> 00:05:29.140 kilogram, right? 00:05:29.140 --> 00:05:31.680 So that will cancel out with that and you get meters per 00:05:31.680 --> 00:05:32.850 second squared. 00:05:32.850 --> 00:05:35.900 And you wouldn't have to do this on a test. Essentially, 00:05:35.900 --> 00:05:37.690 if you did everything right, you would know that the unit 00:05:37.690 --> 00:05:40.000 acceleration is meters per second squared. 00:05:40.000 --> 00:05:42.170 So you would have the acceleration-- I'm just 00:05:42.170 --> 00:05:46.620 switching the two sides-- is equal to 2 00:05:46.620 --> 00:05:50.000 meters per second squared. 00:05:50.000 --> 00:05:51.250 And it'll be to the right. 00:05:54.600 --> 00:05:55.630 So that's useful. 00:05:55.630 --> 00:06:00.720 We just figured out based on how hard I push something, how 00:06:00.720 --> 00:06:03.760 fast it's going to accelerate while I push it. 00:06:03.760 --> 00:06:04.990 And you could use the same formula to 00:06:04.990 --> 00:06:05.850 figure out other things. 00:06:05.850 --> 00:06:11.820 Let's say I know that an object is accelerating-- let's 00:06:11.820 --> 00:06:16.430 say my acceleration is 3 meters per second 00:06:16.430 --> 00:06:18.600 squared to the right. 00:06:18.600 --> 00:06:20.910 Let's say to the left, just to switch things. 00:06:20.910 --> 00:06:26.880 And let's say that I know the force being applied on it is-- 00:06:26.880 --> 00:06:31.950 I don't know-- 30 Newtons to the left. 00:06:31.950 --> 00:06:33.520 And I want to figure out the mass. 00:06:33.520 --> 00:06:34.520 Well you use the same thing. 00:06:34.520 --> 00:06:39.740 You say force, 30 Newtons to the left is equal to mass 00:06:39.740 --> 00:06:41.440 times acceleration. 00:06:41.440 --> 00:06:45.530 Times 3 meters per second squared to the left. 00:06:45.530 --> 00:06:49.660 Divide both sides by the 3 meters per second and you get 00:06:49.660 --> 00:06:54.370 30 Newtons over 3 meters per second squared is 00:06:54.370 --> 00:06:56.030 equal to the mass. 00:06:56.030 --> 00:06:59.520 30 divided by 3 is 10. 00:06:59.520 --> 00:07:02.050 You can figure out that Newtons is kilogram meters per 00:07:02.050 --> 00:07:02.750 second squared. 00:07:02.750 --> 00:07:05.370 So you're just left with 10 kilograms is 00:07:05.370 --> 00:07:07.580 equal to the mass. 00:07:07.580 --> 00:07:10.590 It's very important that if you see a problem where the 00:07:10.590 --> 00:07:14.060 answer's given in-- I don't know-- kilometers per second 00:07:14.060 --> 00:07:17.410 squared or you know, instead of giving it in kilograms it's 00:07:17.410 --> 00:07:20.110 giving it in grams or decagrams, you should convert 00:07:20.110 --> 00:07:24.360 back to kilograms or meters just so you make sure you're 00:07:24.360 --> 00:07:25.210 using the right units. 00:07:25.210 --> 00:07:27.180 And that tends to be frankly, I think, the 00:07:27.180 --> 00:07:28.120 hardest thing for people. 00:07:28.120 --> 00:07:32.750 And we'll do all of that when we tackle harder problems. 00:07:32.750 --> 00:07:35.020 I think now is a good time back to actually differentiate 00:07:35.020 --> 00:07:37.220 between mass and weight. 00:07:37.220 --> 00:07:39.340 And you've probably thought the two were interchangeable, 00:07:39.340 --> 00:07:40.710 but they're not. 00:07:40.710 --> 00:07:46.100 Mass is how much of an object there is. 00:07:46.100 --> 00:07:48.440 You can almost view it as how much of the stuff there is or 00:07:48.440 --> 00:07:50.300 you can almost it view it-- how many atoms there are. 00:07:50.300 --> 00:07:51.435 But even atoms have mass. 00:07:51.435 --> 00:07:54.050 So just how much stuff there is. 00:07:54.050 --> 00:07:57.530 And another way to view mass is, how much does the object 00:07:57.530 --> 00:07:58.880 resist change? 00:07:58.880 --> 00:08:02.250 And that actually falls out of F equals ma. 00:08:02.250 --> 00:08:07.260 Because if our mass is bigger, it's going to take a lot more 00:08:07.260 --> 00:08:09.220 force to make it accelerate a certain amount. 00:08:09.220 --> 00:08:11.760 If the mass is smaller it'll take less force. 00:08:11.760 --> 00:08:15.340 So mass can be viewed as how much stuff there is, of an 00:08:15.340 --> 00:08:16.400 object there is. 00:08:16.400 --> 00:08:20.440 Or you can view it as how hard is it to change what that 00:08:20.440 --> 00:08:21.040 object is doing. 00:08:21.040 --> 00:08:24.480 If it's stationary, how hard is it to accelerate it? 00:08:24.480 --> 00:08:27.190 If it's moving, how hard is it to maybe stop it? 00:08:27.190 --> 00:08:28.930 Which would essentially be decelerating. 00:08:28.930 --> 00:08:31.400 How hard is it to accelerate an object? 00:08:31.400 --> 00:08:35.600 Weight is actually how much is-- what is the force of 00:08:35.600 --> 00:08:37.650 earth upon an object? 00:08:37.650 --> 00:08:39.840 So you're weight would actually change if you go from 00:08:39.840 --> 00:08:42.600 one planet to another because the force of gravity changes. 00:08:42.600 --> 00:08:46.760 So your weight is 1/6 on the moon as it is on earth because 00:08:46.760 --> 00:08:48.310 the pull of gravity is 1/6. 00:08:48.310 --> 00:08:49.450 But your mass doesn't change. 00:08:49.450 --> 00:08:51.810 There's still the same amount of Sal on earth as 00:08:51.810 --> 00:08:54.400 there is on the moon. 00:08:54.400 --> 00:08:57.550 So your weight really-- when you ask someone in Europe and 00:08:57.550 --> 00:08:59.950 they say hey, you know, I weigh 50 kilograms. You should 00:08:59.950 --> 00:09:03.190 say, no, you don't weigh 50 kilograms. You weigh whatever 00:09:03.190 --> 00:09:04.870 50 times 9.8 is. 00:09:04.870 --> 00:09:08.890 That's like 400 something-- you weigh 00:09:08.890 --> 00:09:11.860 490 Newtons or something. 00:09:11.860 --> 00:09:14.450 This is mass. 00:09:14.450 --> 00:09:17.400 And it's interesting because in the English system, and all 00:09:17.400 --> 00:09:18.860 of us Americans, we use the English system. 00:09:18.860 --> 00:09:21.790 When we say that we weigh 10 pounds, we're actually using 00:09:21.790 --> 00:09:25.680 the correct terminology because pounds 00:09:25.680 --> 00:09:28.730 are a unit of force. 00:09:28.730 --> 00:09:32.390 We're saying, if I weigh-- and I do weigh about 150 pounds. 00:09:32.390 --> 00:09:33.940 That means the earth is this pulling on me with 00:09:33.940 --> 00:09:36.280 150 pounds of force. 00:09:36.280 --> 00:09:39.770 And actually, turns out that my mass is measured in the 00:09:39.770 --> 00:09:41.880 unit called a slug, which we might discuss later. 00:09:41.880 --> 00:09:43.485 Actually, we'll do some problems where we do it in the 00:09:43.485 --> 00:09:45.620 metric system and the English system. 00:09:45.620 --> 00:09:47.930 And I'll see you in the next presentation.

Newton's Third Law of Motion

https://www.youtube.com/watch?v=NfuKfbpkIrQ

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https://www.youtube.com/api/timedtext?v=NfuKfbpkIrQ&ei=YmeUZeeKMIn6vdIP67GowAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=DA204DDFE72FBFB630FD09419662B3B9728368EA.953E5123FD9A8E37E67A473CE6E9476326F47E8B&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.860 --> 00:00:02.710 Now we're ready for Newton's third law. 00:00:02.710 --> 00:00:05.040 And Newton's third law, in some ways, I think is the most 00:00:05.040 --> 00:00:09.230 fun because it's-- at least it was to me, the least intuitive 00:00:09.230 --> 00:00:10.020 of all the laws. 00:00:10.020 --> 00:00:12.570 But once you really kind of understand it a lot of things 00:00:12.570 --> 00:00:13.290 start to make sense. 00:00:13.290 --> 00:00:16.970 So Newton's third law essentially says-- actually, I 00:00:16.970 --> 00:00:19.100 can tell you kind of what you might have heard. 00:00:19.100 --> 00:00:21.160 A lot of people say every action has an equal and 00:00:21.160 --> 00:00:22.690 opposite reaction. 00:00:22.690 --> 00:00:26.080 Another way of thinking about it is, if there's an object 00:00:26.080 --> 00:00:28.460 and it exerts a force on another object. 00:00:28.460 --> 00:00:34.150 So let's say I have a-- oh, let me think of something. 00:00:34.150 --> 00:00:41.460 Let's say that I have a fist. So let me draw the fist. So I 00:00:41.460 --> 00:00:45.660 have a fist and it is punching someone's face. 00:00:45.660 --> 00:00:48.110 So this is a face, and they're not happy. 00:00:51.030 --> 00:00:56.830 And let's say this fist is punching the face with a force 00:00:56.830 --> 00:01:00.840 of-- I don't know-- 10 Newtons. 00:01:00.840 --> 00:01:01.700 Let's not make it so violent. 00:01:01.700 --> 00:01:04.860 Let's say that this is a hand massaging the face. 00:01:04.860 --> 00:01:09.940 It's pressing upon the face with a force of 10 Newtons. 00:01:09.940 --> 00:01:14.210 So what Newton's third law tells us is that the face is 00:01:14.210 --> 00:01:17.380 also, I guess, we could say punching the hand. 00:01:17.380 --> 00:01:19.280 Or-- well no, we're not using the violent. 00:01:19.280 --> 00:01:23.280 The face is also pressing upon the hand with a force of 10 00:01:23.280 --> 00:01:28.280 Newtons in the opposite direction. 00:01:28.280 --> 00:01:30.360 I guess you would say along the same line. 00:01:30.360 --> 00:01:32.080 So does that make any sense? 00:01:32.080 --> 00:01:36.540 Because it seems like the hand is doing something to the face 00:01:36.540 --> 00:01:37.830 and not the other way around. 00:01:37.830 --> 00:01:42.420 But if you think about it, when you press on someone's 00:01:42.420 --> 00:01:46.330 face, their face might kind of press in a little bit, but you 00:01:46.330 --> 00:01:50.530 also feel something on your fist or on your hand. 00:01:50.530 --> 00:01:53.190 I mean think of it, maybe a better example would be 00:01:53.190 --> 00:01:57.950 instead of someone's face, imagine if it was a tree that 00:01:57.950 --> 00:02:01.700 you've decided to massage or, I guess, punch. 00:02:01.700 --> 00:02:05.510 So here is the tree. 00:02:05.510 --> 00:02:06.790 That is the tree. 00:02:06.790 --> 00:02:08.880 And the same thing would happen. 00:02:08.880 --> 00:02:11.970 If you were to press upon the tree with-- if you were to 00:02:11.970 --> 00:02:15.350 punch the tree essentially, the tree is essentially 00:02:15.350 --> 00:02:17.540 punching back with the exact same force. 00:02:17.540 --> 00:02:20.650 And here it makes sense because your hand will hurt. 00:02:20.650 --> 00:02:24.020 And maybe in this case, the face will hurt because the 00:02:24.020 --> 00:02:26.900 face kind of gives way while your fist doesn't. 00:02:26.900 --> 00:02:29.750 But here the tree's not giving way and your fist will. 00:02:29.750 --> 00:02:35.170 Another way to think about it is if-- well, and this is 00:02:35.170 --> 00:02:36.840 probably the least intuitive. 00:02:36.840 --> 00:02:42.380 If I have the earth and here am I standing 00:02:42.380 --> 00:02:43.630 on top of the earth. 00:02:45.940 --> 00:02:48.330 So we already figured out that the pull of the earth or the 00:02:48.330 --> 00:02:51.940 force of gravity upon me, it's pulling down 00:02:51.940 --> 00:02:54.820 upon me at 150 pounds. 00:02:54.820 --> 00:02:55.580 That's the force. 00:02:55.580 --> 00:02:56.710 And you know, we could say the Newton. 00:02:56.710 --> 00:02:59.120 But pounds is a unit of force, it's weight. 00:02:59.120 --> 00:03:02.710 But also, Newton's third law tells us that I'm actually, at 00:03:02.710 --> 00:03:06.870 the same moment, pulling on the earth with a 00:03:06.870 --> 00:03:10.620 force of 150 pounds. 00:03:10.620 --> 00:03:15.890 And this might not make a lot of sense to you, but you can 00:03:15.890 --> 00:03:17.210 think about it this way. 00:03:17.210 --> 00:03:20.360 When I'm stepping on-- let's say I'm stepping on a soft 00:03:20.360 --> 00:03:23.010 surface, like sand or something. 00:03:23.010 --> 00:03:26.440 My feet will compress a little bit, but so does the sand. 00:03:26.440 --> 00:03:29.460 Well, depending on which one's softer. 00:03:29.460 --> 00:03:32.360 And another way to think about it also is, if me and the 00:03:32.360 --> 00:03:35.440 earth are both in deep space. 00:03:35.440 --> 00:03:38.990 And I am, you could say, falling towards the earth 00:03:38.990 --> 00:03:40.780 because the earth is pulling on me. 00:03:40.780 --> 00:03:43.140 How do we know that the earth isn't falling towards me? 00:03:43.140 --> 00:03:44.850 I mean it's kind of arbitrary. 00:03:44.850 --> 00:03:45.960 There's no frame of reference. 00:03:45.960 --> 00:03:47.100 We're both in deep space. 00:03:47.100 --> 00:03:48.360 There's nothing else to look at. 00:03:48.360 --> 00:03:51.070 We're essentially, falling towards each other. 00:03:51.070 --> 00:03:53.200 I'm not necessarily falling to the earth, the earth's not 00:03:53.200 --> 00:03:54.580 necessarily falling to me, we're just falling towards 00:03:54.580 --> 00:03:55.310 each other. 00:03:55.310 --> 00:03:57.740 And that's another way of thinking about this. 00:03:57.740 --> 00:03:59.870 So you could think about every example where a force applies 00:03:59.870 --> 00:04:00.500 to something. 00:04:00.500 --> 00:04:03.390 And if you really think about it, the force is going the 00:04:03.390 --> 00:04:04.150 other way as well. 00:04:04.150 --> 00:04:09.600 For example, if I were to take I bat to this tree and swing 00:04:09.600 --> 00:04:10.700 on it really hard. 00:04:10.700 --> 00:04:13.490 So I were to swing on this tree really hard, I have a 00:04:13.490 --> 00:04:16.040 good chance of breaking that bat. 00:04:16.040 --> 00:04:18.839 Even though you would have thought, hey, that bat is 00:04:18.839 --> 00:04:20.690 applying the force to the tree. 00:04:20.690 --> 00:04:22.240 But why is the bat breaking? 00:04:22.240 --> 00:04:25.700 Because the tree is applying an equal force to the bat. 00:04:25.700 --> 00:04:27.780 And actually, if I did it perfectly, if I had like-- 00:04:27.780 --> 00:04:33.540 let's say I had two bats or two swords. 00:04:33.540 --> 00:04:37.080 For some reason I think I'm going a little too violent 00:04:37.080 --> 00:04:37.700 with these example. 00:04:37.700 --> 00:04:41.250 But I guess you know we're talking about forces. 00:04:41.250 --> 00:04:44.990 So maybe violence is justified here. 00:04:44.990 --> 00:04:46.810 But let's say I have two swords that are completely 00:04:46.810 --> 00:04:48.060 identical hitting each other. 00:04:51.260 --> 00:04:53.540 And let's say I keep increasing the force at which 00:04:53.540 --> 00:04:56.110 they're kind of going in opposite directions. 00:04:56.110 --> 00:04:58.600 At some point, they're going to break. 00:04:58.600 --> 00:05:01.640 If I just keep increasing the force on-- you know, one guy 00:05:01.640 --> 00:05:03.490 is swinging in one direction, one guy is swinging in the 00:05:03.490 --> 00:05:04.590 exact opposite direction. 00:05:04.590 --> 00:05:08.180 And the force just keeps increasing, at some point, 00:05:08.180 --> 00:05:09.720 they're going to break. 00:05:09.720 --> 00:05:14.480 And you could say that well-- this guy says well, I was the 00:05:14.480 --> 00:05:17.650 only guy swinging because this guy was stationery. 00:05:17.650 --> 00:05:18.880 And the other guy will say, well, I was really guy 00:05:18.880 --> 00:05:21.530 swinging because this guy was stationary. 00:05:21.530 --> 00:05:23.140 Not one of them is going to break, they're both going to 00:05:23.140 --> 00:05:28.230 break because even though this mauve sword was pushing on 00:05:28.230 --> 00:05:32.020 this blue sword with some force, the blue sword was 00:05:32.020 --> 00:05:34.850 essentially pushing back with the exact same force. 00:05:34.850 --> 00:05:36.780 So these are completely identical swords. 00:05:36.780 --> 00:05:38.320 At some point, they're going to break. 00:05:38.320 --> 00:05:39.780 Another way we could think about it, one of the swords 00:05:39.780 --> 00:05:41.410 could have just been held. 00:05:41.410 --> 00:05:42.390 You know, stationary. 00:05:42.390 --> 00:05:44.620 It could've been held stationary by somebody and 00:05:44.620 --> 00:05:47.530 this other sword that-- if you were to press down on it, kept 00:05:47.530 --> 00:05:51.040 increasing the force with which you press, at some point 00:05:51.040 --> 00:05:52.780 they both would break. 00:05:52.780 --> 00:05:53.780 Because they're identical. 00:05:53.780 --> 00:05:55.570 If one was harder than the other, than that one would 00:05:55.570 --> 00:05:56.920 stick around. 00:05:56.920 --> 00:05:58.130 Hopefully that gives you a little intuition. 00:05:58.130 --> 00:06:01.940 I mean we could do a bunch of more examples. 00:06:01.940 --> 00:06:02.690 I'm trying to think. 00:06:02.690 --> 00:06:05.460 Oh, let me think of another one. 00:06:05.460 --> 00:06:06.250 A less violent one. 00:06:06.250 --> 00:06:09.770 Let's say we're in deep space again and I have a-- I don't 00:06:09.770 --> 00:06:11.330 know-- I have a basketball. 00:06:11.330 --> 00:06:15.100 Let me do it in orange. 00:06:15.100 --> 00:06:19.260 I have a basketball and it weighs 1 kilogram. 00:06:21.930 --> 00:06:32.360 And let's say that I weigh 50 kilograms. So let's say I 00:06:32.360 --> 00:06:38.120 push-- so my hand pushes on this ball with a force of-- I 00:06:38.120 --> 00:06:40.200 don't know-- let's say I push on that ball with a force of 00:06:40.200 --> 00:06:41.582 10 Newtons. 00:06:41.582 --> 00:06:44.930 10 Newtons to the right. 00:06:44.930 --> 00:06:48.220 What Newton's third law tells me is that essentially, that 00:06:48.220 --> 00:06:51.530 basketball is going to push on my hand with an equal and 00:06:51.530 --> 00:06:52.240 opposite force. 00:06:52.240 --> 00:06:55.830 So it's going to push on me with a force of 10 Newtons. 00:06:55.830 --> 00:06:57.200 So what's going to happen? 00:06:57.200 --> 00:06:58.700 So we're touching. 00:06:58.700 --> 00:07:00.850 I'm pushing on 10 Newtons on the basketball and we're in 00:07:00.850 --> 00:07:01.570 deep space. 00:07:01.570 --> 00:07:04.310 There's no gravity from random stars, et cetera. 00:07:04.310 --> 00:07:06.200 And then the basketball's going to push on me with the 00:07:06.200 --> 00:07:09.040 force of 10 Newtons simultaneously. 00:07:09.040 --> 00:07:12.290 We know F equals ma. 00:07:12.290 --> 00:07:17.090 So the basketball, so 10 Newtons is equal to 1 kilogram 00:07:17.090 --> 00:07:18.030 times acceleration. 00:07:18.030 --> 00:07:21.700 So acceleration is going to be 10 meters per second squared 00:07:21.700 --> 00:07:22.460 to the right. 00:07:22.460 --> 00:07:24.580 So as long as we're touching, the basketball's going to 00:07:24.580 --> 00:07:29.620 accelerate at 10 meters per second squared to the right. 00:07:29.620 --> 00:07:33.410 And simultaneously, I'm going to accelerate at a certain 00:07:33.410 --> 00:07:34.460 acceleration to the left. 00:07:34.460 --> 00:07:36.880 And what's that going to be? 00:07:36.880 --> 00:07:41.710 50 kilograms. We know that the force to the left is also 00:07:41.710 --> 00:07:42.700 going to be 10 Newtons. 00:07:42.700 --> 00:07:46.470 That equals 50 kilograms times acceleration. 00:07:46.470 --> 00:07:52.530 So here, the acceleration is going to be 1/5 meters per 00:07:52.530 --> 00:07:54.700 second squared. 00:07:54.700 --> 00:07:57.470 So we're both in deep space floating around and I push on 00:07:57.470 --> 00:08:01.710 this 1 kilogram basketball with a force of 10 Newtons. 00:08:01.710 --> 00:08:05.350 As long as I'm pushing on it, it's going to accelerate at 10 00:08:05.350 --> 00:08:07.950 meters per second squared. 00:08:07.950 --> 00:08:11.950 But simultaneously, while I'm pushing on it, it's exerting 00:08:11.950 --> 00:08:14.630 an equal and opposite force on me of 10 Newtons. 00:08:14.630 --> 00:08:18.980 So I'm going to actually move back a little bit at a slower 00:08:18.980 --> 00:08:19.500 acceleration. 00:08:19.500 --> 00:08:21.690 That's just because I have more mass. 00:08:21.690 --> 00:08:24.090 At 1/5 meters per second squared. 00:08:24.090 --> 00:08:26.190 Another example you could think of is if 00:08:26.190 --> 00:08:27.520 someone shoots a gun. 00:08:27.520 --> 00:08:29.100 There's that-- I forgot the term because 00:08:29.100 --> 00:08:29.950 I don't shoot guns. 00:08:29.950 --> 00:08:34.929 But your shoulder jerks back as I've seen in the movies 00:08:34.929 --> 00:08:36.500 when a bullet is shot. 00:08:36.500 --> 00:08:39.090 That's because the gun is exerting a force on that 00:08:39.090 --> 00:08:42.429 bullet and the bullet is exerting an equal and opposite 00:08:42.429 --> 00:08:45.360 force on the gun, which kind of pushes 00:08:45.360 --> 00:08:46.450 back on your shoulder. 00:08:46.450 --> 00:08:49.940 And the reason why the bullet just goes a lot, lot faster 00:08:49.940 --> 00:08:53.060 forward than you and the gun go backwards is because your 00:08:53.060 --> 00:08:57.770 mass is much, much, much larger than the bullet. 00:08:57.770 --> 00:08:59.450 Hopefully that gives you a little bit of intuition on 00:08:59.450 --> 00:09:00.580 Newton's third law. 00:09:00.580 --> 00:09:02.150 And this is kind of non-intuitive. 00:09:02.150 --> 00:09:05.320 So look around you in the world, look at all the forces 00:09:05.320 --> 00:09:07.860 that are being applied, and I want you to think about when 00:09:07.860 --> 00:09:11.050 one force is being applied in one direction, why does it 00:09:11.050 --> 00:09:13.785 make sense that another force, an equal and opposite force, 00:09:13.785 --> 00:09:16.590 is being applied in the exact opposite direction? 00:09:16.590 --> 00:09:19.000 I'll see you all in the next video.

Newton's First Law of Motion

https://www.youtube.com/watch?v=D9y0RlF_DqA

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https://www.youtube.com/api/timedtext?v=D9y0RlF_DqA&ei=YmeUZf2sL-qjhcIP84-nkA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=77A82EB19077328AD55E67163050402B46BD6CA3.D19FCB66BA0CD155C3B6EF651DC94E49C8229911&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.680 --> 00:00:01.730 Good morning. 00:00:01.730 --> 00:00:02.900 It's February already. 00:00:02.900 --> 00:00:05.760 I'm back from my hiatus. 00:00:05.760 --> 00:00:08.020 I was so burned out doing all those SAT problems. But now 00:00:08.020 --> 00:00:10.840 I'm ready and I will start doing some physics. 00:00:10.840 --> 00:00:12.950 So we had done a bunch of projectile motion, what 00:00:12.950 --> 00:00:14.310 happens you throw something in the air or 00:00:14.310 --> 00:00:15.270 drop it from a cliff. 00:00:15.270 --> 00:00:17.990 But now I want to introduce you to is how do you actually 00:00:17.990 --> 00:00:20.080 affect the acceleration of an object? 00:00:20.080 --> 00:00:23.560 And to do that I'm going to introduce you to Newton's 00:00:23.560 --> 00:00:24.810 three laws. 00:00:27.520 --> 00:00:30.660 To some degree what we were doing before was derivative of 00:00:30.660 --> 00:00:31.550 what I'm going to do now. 00:00:31.550 --> 00:00:34.720 But this is kind of the backbone of classical physics. 00:00:34.720 --> 00:00:36.510 So Newton's three laws. 00:00:36.510 --> 00:00:38.950 And you've probably heard of these before. 00:00:38.950 --> 00:00:41.820 Newtow's three laws. 00:00:41.820 --> 00:00:44.270 Sometimes they're called Newton's Laws of Motion. 00:00:44.270 --> 00:00:46.160 I've actually looked this up on the web just to make sure 00:00:46.160 --> 00:00:49.520 and see if there's any correct way of writing it, but every 00:00:49.520 --> 00:00:51.470 website seems to have a different 00:00:51.470 --> 00:00:53.430 paraphrase of the laws. 00:00:53.430 --> 00:00:55.480 But hopefully, I can give you an intuitive sense 00:00:55.480 --> 00:00:56.310 of what they are. 00:00:56.310 --> 00:01:08.010 So the first law is an object at rest. An object at rest 00:01:08.010 --> 00:01:21.250 tends to stay at rest. And an object in motion 00:01:21.250 --> 00:01:22.430 tends to stay in motion. 00:01:22.430 --> 00:01:25.520 This is what I learned when I was a kid and now when I look 00:01:25.520 --> 00:01:27.110 at Wikipedia and things, there are some paraphrases. 00:01:27.110 --> 00:01:28.720 And we'll go over those paraphrases because I think 00:01:28.720 --> 00:01:29.740 they're instructive. 00:01:29.740 --> 00:01:31.910 Stay in motion. 00:01:31.910 --> 00:01:33.690 And you might say, Sal, this is obvious. 00:01:33.690 --> 00:01:35.300 Why does Newton get so much credit 00:01:35.300 --> 00:01:36.130 for stating the obvious? 00:01:36.130 --> 00:01:40.020 Obviously, if I look at my sofa for example, it is an 00:01:40.020 --> 00:01:44.030 object at rest and if I keep staring at it, it tends to 00:01:44.030 --> 00:01:49.590 stay at rest. Likewise, when I look at a car crossing an 00:01:49.590 --> 00:01:52.170 intersection-- that's not a red light, that's crossing an 00:01:52.170 --> 00:01:54.520 intersection, it's an object in motion. 00:01:54.520 --> 00:01:59.040 And then, I don't know-- 10 seconds later, it's still 00:01:59.040 --> 00:02:01.480 staying in motion and of course, it will stay in motion 00:02:01.480 --> 00:02:03.120 unless you press the brakes or whatever. 00:02:03.120 --> 00:02:05.620 So you might say, well Sal, this is the most 00:02:05.620 --> 00:02:06.670 obvious thing ever. 00:02:06.670 --> 00:02:09.380 This doesn't even need to be written down. 00:02:09.380 --> 00:02:12.840 But let's say you were Newton and you came to me-- it was in 00:02:12.840 --> 00:02:14.060 the 17th century. 00:02:14.060 --> 00:02:15.790 And you said, Sal, I have these new laws. 00:02:15.790 --> 00:02:18.630 And the first is an object at rest tends to stay at rest, 00:02:18.630 --> 00:02:20.860 and an object in motion tends to stay in motion. 00:02:20.860 --> 00:02:23.290 And I would say Newton, I can already disprove your law. 00:02:23.290 --> 00:02:27.140 Let's say I have an apple and I'm holding it up at let's say 00:02:27.140 --> 00:02:31.270 my-- I'm holding it up with my arm, so it's roughly my 00:02:31.270 --> 00:02:32.720 shoulder level. 00:02:32.720 --> 00:02:33.800 So I'm holding an apple. 00:02:33.800 --> 00:02:35.690 This is an apple. 00:02:35.690 --> 00:02:37.530 Looks like a heart, but it's an apple. 00:02:37.530 --> 00:02:41.610 So I'm holding it with my hand, I'm drawing my hand. 00:02:41.610 --> 00:02:43.170 I don't know if that makes sense to you, but I'm holding 00:02:43.170 --> 00:02:44.010 it with my hand. 00:02:44.010 --> 00:02:46.760 And what happens when I let go of that apple? 00:02:46.760 --> 00:02:49.890 So while I'm holding it with my hand it's an object at 00:02:49.890 --> 00:02:51.180 rest, right? 00:02:51.180 --> 00:02:52.490 But then when I let go, what happens? 00:02:52.490 --> 00:02:53.740 It falls. 00:02:53.740 --> 00:02:54.840 Falls to the ground. 00:02:54.840 --> 00:02:57.990 So I'll say, Newton, I just disproved your first law. 00:02:57.990 --> 00:03:00.170 Because this was an object at rest. And I did nothing to it. 00:03:00.170 --> 00:03:00.980 I just let go. 00:03:00.980 --> 00:03:03.680 I didn't apply, I didn't push it, I didn't pull it. 00:03:03.680 --> 00:03:04.400 I didn't throw it. 00:03:04.400 --> 00:03:05.480 I didn't do anything. 00:03:05.480 --> 00:03:09.610 And when I let go it just fell to the ground. 00:03:09.610 --> 00:03:12.100 It started moving without me doing anything, even though it 00:03:12.100 --> 00:03:13.410 was an object at rest. 00:03:13.410 --> 00:03:15.610 And then Newton will say, oh, well that's because there's a 00:03:15.610 --> 00:03:16.660 thing called gravity. 00:03:16.660 --> 00:03:17.670 And it's the force of gravity. 00:03:17.670 --> 00:03:20.690 And I would say, Newton, you need to start to learn to not 00:03:20.690 --> 00:03:22.480 make up things. 00:03:22.480 --> 00:03:26.350 Just because you're law doesn't make sense, you don't 00:03:26.350 --> 00:03:28.960 need to make up artificial forces in the universe. 00:03:28.960 --> 00:03:31.410 But anyway, he would end up being right. 00:03:31.410 --> 00:03:34.700 And the way to think about this, if I did this exact same 00:03:34.700 --> 00:03:41.090 experiment while I was in space and let's just say-- I 00:03:41.090 --> 00:03:43.260 was going to say orbit because it would look like that, but 00:03:43.260 --> 00:03:47.190 even orbit is kind of a-- you're still kind of falling 00:03:47.190 --> 00:03:49.000 towards the earth, it's just you're moving-- well, I won't 00:03:49.000 --> 00:03:49.520 go into that. 00:03:49.520 --> 00:03:51.060 I'll go into orbit at another time. 00:03:51.060 --> 00:03:53.850 But let's say we were just in deep space and me and the 00:03:53.850 --> 00:03:57.870 apple were just floating around in space. 00:03:57.870 --> 00:03:59.100 Maybe we're stationary. 00:03:59.100 --> 00:03:59.690 It's hard to say. 00:03:59.690 --> 00:04:01.620 We're floating with respect to what? 00:04:01.620 --> 00:04:05.530 And then, if we're in space and I let go of this apple, 00:04:05.530 --> 00:04:07.090 what happens to the apple? 00:04:07.090 --> 00:04:07.590 Nothing. 00:04:07.590 --> 00:04:08.690 It's not going to fall anywhere. 00:04:08.690 --> 00:04:09.950 It's not going to move. 00:04:09.950 --> 00:04:12.260 And so whenever you think about Newton's laws-- and 00:04:12.260 --> 00:04:13.440 that's why this is so amazing. 00:04:13.440 --> 00:04:15.330 He didn't know about space. 00:04:15.330 --> 00:04:18.570 He's living in this planet that everything tends to fall 00:04:18.570 --> 00:04:20.850 and things start moving for no reason because of whatever, 00:04:20.850 --> 00:04:23.470 gravity, and the wind and whatever else. 00:04:23.470 --> 00:04:26.580 And he actually theorized that there could be a place where 00:04:26.580 --> 00:04:30.030 there's no forces acting on objects where if I were to let 00:04:30.030 --> 00:04:33.260 go of this apple, it would just stay where it is. 00:04:33.260 --> 00:04:35.060 And similarly, the object in motion 00:04:35.060 --> 00:04:35.970 tends to stay in motion. 00:04:35.970 --> 00:04:37.820 And there again I would've told Newton, well, that 00:04:37.820 --> 00:04:38.850 doesn't make sense. 00:04:38.850 --> 00:04:41.920 If I were to-- I don't know. 00:04:41.920 --> 00:04:47.470 If I were to push a-- well, I don't know if they had bowling 00:04:47.470 --> 00:04:48.200 balls back then. 00:04:48.200 --> 00:04:51.730 But if I were to roll a bowling ball down a-- well 00:04:51.730 --> 00:04:55.870 let's say up a hill-- At some point that bowling ball's 00:04:55.870 --> 00:04:57.120 going to slow down. 00:04:59.250 --> 00:05:00.960 If I rolled it up a hill, at some point it's just 00:05:00.960 --> 00:05:01.590 going to slow down. 00:05:01.590 --> 00:05:03.570 And maybe if I got it right it would just stop at the top if 00:05:03.570 --> 00:05:05.500 I did it perfectly. 00:05:05.500 --> 00:05:07.170 And I could say, look, this was an object in motion. 00:05:07.170 --> 00:05:09.750 At some point it stops or it actually turns back around. 00:05:09.750 --> 00:05:12.990 Or even if I were to roll it this way, at some point it's 00:05:12.990 --> 00:05:13.860 just going to stop. 00:05:13.860 --> 00:05:17.610 Right The bowling ball's going to stop. 00:05:17.610 --> 00:05:21.080 If I were to push something as hard as I could, maybe it 00:05:21.080 --> 00:05:23.900 travels for a couple of feet, but then it's going to stop. 00:05:23.900 --> 00:05:25.730 And he'll say, oh, well you know, there's these forces 00:05:25.730 --> 00:05:27.230 that you're not realizing there's a force. 00:05:27.230 --> 00:05:31.100 There's the wind resistance in the bowling ball example. 00:05:31.100 --> 00:05:34.250 There's the force of friction in the example where I just 00:05:34.250 --> 00:05:34.540 pushed something. 00:05:34.540 --> 00:05:36.370 And I would've said, well Newton, you're just making up 00:05:36.370 --> 00:05:37.340 these forces again. 00:05:37.340 --> 00:05:38.690 And this is why this is so not intuitive. 00:05:38.690 --> 00:05:42.400 Because he had to essentially realize that there were all of 00:05:42.400 --> 00:05:45.280 these forces acting on something when to someone at 00:05:45.280 --> 00:05:47.110 that time, you wouldn't have realized that and you wouldn't 00:05:47.110 --> 00:05:49.420 have been able to even conceive that there's a place 00:05:49.420 --> 00:05:52.830 called space, for example, where these 00:05:52.830 --> 00:05:54.100 things wouldn't happen. 00:05:54.100 --> 00:05:57.820 If I push something in space, it will keep going. 00:05:57.820 --> 00:06:00.900 It would be an object in motion and it will keep that 00:06:00.900 --> 00:06:03.800 velocity until some other force acts on it. 00:06:03.800 --> 00:06:05.860 So it wasn't that intuitive. 00:06:05.860 --> 00:06:13.050 And so a more modern way to write this is to say that 00:06:13.050 --> 00:06:16.440 there is a frame of reference, there exists a frame of 00:06:16.440 --> 00:06:18.460 reference-- and I'll explain what a frame of reference is. 00:06:18.460 --> 00:06:22.810 But there exists a frame of reference where this is true. 00:06:22.810 --> 00:06:25.530 That could be the new way of saying Newton's 00:06:25.530 --> 00:06:27.160 first law of motion. 00:06:27.160 --> 00:06:29.850 So what's a frame of reference? 00:06:29.850 --> 00:06:32.430 So everything in physics-- if I'm moving, 00:06:32.430 --> 00:06:33.710 moving relative to what? 00:06:33.710 --> 00:06:35.290 Moving relative to the observer? 00:06:35.290 --> 00:06:36.480 Moving relative to the earth? 00:06:36.480 --> 00:06:37.490 You don't know. 00:06:37.490 --> 00:06:40.700 So a frame of reference is what is the observer doing? 00:06:40.700 --> 00:06:45.870 So example: when I'm in space and I let go of the apple, me 00:06:45.870 --> 00:06:48.810 and the apple are kind of in this-- I am observing the 00:06:48.810 --> 00:06:52.420 apple from what I call an inertial frame of reference. 00:06:52.420 --> 00:06:53.850 So this is a frame of reference actually where 00:06:53.850 --> 00:06:55.380 Newton's laws hold. 00:06:55.380 --> 00:07:00.210 If I take the apple on earth and I let go and it drops, the 00:07:00.210 --> 00:07:02.450 reason why this first law didn't hold is because I'm not 00:07:02.450 --> 00:07:04.450 really in an inertial frame of reference. 00:07:04.450 --> 00:07:08.190 Because me and the apple are both constantly being pulled 00:07:08.190 --> 00:07:10.590 on by this force called gravity. 00:07:10.590 --> 00:07:13.610 So although it looks like nothing's going on, me and the 00:07:13.610 --> 00:07:17.260 apple are in the same-- nothing's really acting on us. 00:07:17.260 --> 00:07:17.830 There is. 00:07:17.830 --> 00:07:19.380 There's this force of acceleration. 00:07:19.380 --> 00:07:21.930 Similarly, if I'm in a car and that car is 00:07:21.930 --> 00:07:22.880 accelerating, right? 00:07:22.880 --> 00:07:28.085 So let's say the car-- looks more like a pickup truck, so 00:07:28.085 --> 00:07:29.335 I'll go with the pickup truck. 00:07:33.430 --> 00:07:36.425 Let's say you have a pair of dice hanging from your rear 00:07:36.425 --> 00:07:37.510 view mirror. 00:07:37.510 --> 00:07:38.950 This is the dice right here. 00:07:38.950 --> 00:07:41.890 What happens when the car accelerates? 00:07:41.890 --> 00:07:44.150 Well the dice move back, right? 00:07:44.150 --> 00:07:48.810 And so when you're sitting in the truck itself, it looks 00:07:48.810 --> 00:07:50.670 like the dice are just moving back. 00:07:50.670 --> 00:07:53.110 No one's really doing anything to it. 00:07:53.110 --> 00:07:55.950 Let's say the car had no windows and you would just all 00:07:55.950 --> 00:07:58.573 of a sudden mysteriously feel-- well, you'd feel a 00:07:58.573 --> 00:08:00.600 little squeezing on your chest too, but you would also just 00:08:00.600 --> 00:08:02.020 see these dice move back. 00:08:02.020 --> 00:08:03.300 And you'd say, hey. 00:08:03.300 --> 00:08:05.040 Newton's first law doesn't hold. 00:08:05.040 --> 00:08:07.100 And what I would say is well that's because you're in a 00:08:07.100 --> 00:08:08.980 non-inertial frame of reference. 00:08:08.980 --> 00:08:11.170 To someone outside of the truck, they would see, oh 00:08:11.170 --> 00:08:13.660 well, the truck is moving, the truck is actually accelerating 00:08:13.660 --> 00:08:16.150 and that's why the dice move back. 00:08:16.150 --> 00:08:20.210 So in I guess you could say the horizontal dimension, and 00:08:20.210 --> 00:08:21.980 I'm probably just confusing you, but I want to give you a 00:08:21.980 --> 00:08:25.660 really intuitive feel about why this isn't so intuitive. 00:08:25.660 --> 00:08:28.940 In the horizontal direction, because there are no forces of 00:08:28.940 --> 00:08:32.760 gravity or whatever acting in this direction, and if I'm 00:08:32.760 --> 00:08:35.880 outside of the truck, I could then-- I would be in an 00:08:35.880 --> 00:08:37.900 inertial frame of reference in at least 00:08:37.900 --> 00:08:38.860 the horizontal dimension. 00:08:38.860 --> 00:08:41.159 I mean we always have gravity pulling down on us. 00:08:41.159 --> 00:08:43.870 But from the outside of the truck, I could observe that 00:08:43.870 --> 00:08:48.080 oh, you know, Newton's law holds because the whole frame 00:08:48.080 --> 00:08:51.340 of reference, this truck is actually being accelerated. 00:08:51.340 --> 00:08:54.440 So me being outside of that, I would be in an inertial frame 00:08:54.440 --> 00:08:55.870 of reference. 00:08:55.870 --> 00:08:58.570 Hopefully I haven't confused you too much. 00:08:58.570 --> 00:09:00.710 The way to think about it is that an inertial frame of 00:09:00.710 --> 00:09:03.860 reference is just a frame of reference where there's no 00:09:03.860 --> 00:09:07.070 outside forces acting on the whole frame of reference. 00:09:07.070 --> 00:09:10.870 And a frame of reference is just what is 00:09:10.870 --> 00:09:11.680 the observer doing? 00:09:11.680 --> 00:09:12.450 What am I doing? 00:09:12.450 --> 00:09:13.900 Am I moving with the object? 00:09:13.900 --> 00:09:15.760 Am I being accelerated with the object? 00:09:15.760 --> 00:09:19.840 Or, are neither me nor the object being acted upon? 00:09:19.840 --> 00:09:22.245 And that's the way to think about it. 00:09:22.245 --> 00:09:23.270 Oh, I already ran out of time. 00:09:23.270 --> 00:09:24.540 I only got one law done. 00:09:24.540 --> 00:09:26.570 So I'll see you in the next video.

Ferris Wheel Trig Problem (part 2)

https://www.youtube.com/watch?v=_Kw4hLGMkm4

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https://www.youtube.com/api/timedtext?v=_Kw4hLGMkm4&ei=ZWeUZduSDoa3mLAP-4O_IA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249813&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=43CE56AA0E2925B64A3057D35A0332CE2F9CD817.151F2AAA4E6415BDE32D811CBBC9E7B16F71FE2D&key=yt8&lang=en&name=English&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.660 --> 00:00:03.410 In the last part of the problem we figured out that the 00:00:03.410 --> 00:00:06.420 function of the height of the ferris wheel, the people at the 00:00:06.420 --> 00:00:10.140 ferris wheel at any time is a function of t. h of t is equal 00:00:10.140 --> 00:00:18.590 to 9 minus 8 cosine of 18t where t is in seconds. 00:00:18.590 --> 00:00:21.730 Now the second part of this problem they want us to graph h 00:00:21.730 --> 00:00:28.530 as a function of t between 0 is less than or equal to t, is 00:00:28.530 --> 00:00:30.800 less than or equal to 30. 00:00:30.800 --> 00:00:34.500 So let me draw axes. 00:00:34.500 --> 00:00:41.690 So let's say that that's my h-axis. 00:00:41.690 --> 00:00:43.660 Let's say that this is my t-axis. 00:00:53.280 --> 00:00:56.845 So this is t equals 0, and this is t is equal to 30 seconds. 00:01:00.740 --> 00:01:03.660 I get confused when I see this 18 here or whatever. 00:01:03.660 --> 00:01:06.040 So what I'm going to do first of all is I'm going to graph a 00:01:06.040 --> 00:01:08.230 different function, slightly different function, then I'll 00:01:08.230 --> 00:01:09.870 translate it to this function. 00:01:09.870 --> 00:01:15.820 I'm going to graph h of theta is equal to 9 00:01:15.820 --> 00:01:23.430 minus 8 cosine of theta. 00:01:23.430 --> 00:01:25.010 I think you'll see where I'm going with this 00:01:25.010 --> 00:01:26.320 when I'm all done. 00:01:26.320 --> 00:01:28.600 So let's try to graph h of theta is equal to 9 00:01:28.600 --> 00:01:31.260 minus 8 cosine of theta. 00:01:31.260 --> 00:01:33.810 So when t is equal to 30 seconds, what 00:01:33.810 --> 00:01:35.090 is theta equal to? 00:01:35.090 --> 00:01:38.800 So 30 times 18, that's 540. 00:01:38.800 --> 00:01:40.780 So this is 540 degrees. 00:01:40.780 --> 00:01:41.370 Same thing. 00:01:41.370 --> 00:01:44.340 I'll write the thetas in red above the t-axis. 00:01:44.340 --> 00:01:47.310 This is 540 degrees, so that's like two times 00:01:47.310 --> 00:01:47.940 around the circle. 00:01:47.940 --> 00:01:51.470 So that's 540 degrees, then this is going to 00:01:51.470 --> 00:01:53.890 be roughly 270 degrees. 00:01:58.400 --> 00:02:01.450 So, 90 degrees will be about 1/3 of this. 00:02:01.450 --> 00:02:04.830 That would be 90 degrees, that would be 180, so that would be 00:02:04.830 --> 00:02:11.680 90 degrees, that would be 180 degrees, this would be 360 00:02:11.680 --> 00:02:19.910 degrees, and this would be 360 plus 90 so this will 00:02:19.910 --> 00:02:22.140 be 450 degrees. 00:02:22.140 --> 00:02:24.390 If you wanted to figure out the corresponding time, 00:02:24.390 --> 00:02:27.680 you just take this degree and divide by 18. 00:02:27.680 --> 00:02:31.300 So it's 90 divided by 18 is what? 00:02:31.300 --> 00:02:34.175 It's five, right? 00:02:34.175 --> 00:02:39.360 So if I were to write here, this is at 5 seconds, this is 00:02:39.360 --> 00:02:45.400 at 10 seconds, this is 15 seconds, this is 20 seconds, 00:02:45.400 --> 00:02:51.910 this is -- sorry, this is 25 seconds, this is 30 seconds. 00:02:51.910 --> 00:02:56.140 Actually, a simple thing we could do is let's just figure 00:02:56.140 --> 00:02:59.640 out what the value of the function is at these points. 00:02:59.640 --> 00:03:02.690 Because these are pretty easy degrees to figure out 00:03:02.690 --> 00:03:04.810 what the cosine value is. 00:03:04.810 --> 00:03:09.160 So let's figure out -- let me draw a table. 00:03:09.160 --> 00:03:11.740 Tables are always good and I'll do it in yellow. 00:03:11.740 --> 00:03:17.520 So I'll draw a t theta and h. 00:03:20.720 --> 00:03:23.160 This might be kind of an unconventional way of doing 00:03:23.160 --> 00:03:26.170 things, but I have a simple mind so this is actually 00:03:26.170 --> 00:03:27.680 how I like to do it. 00:03:27.680 --> 00:03:34.460 So I like to think of theta as 0, 90, 180, 00:03:34.460 --> 00:03:42.810 270, 360, 450 and 540. 00:03:42.810 --> 00:03:47.160 And t, the corresponding time of those, as 0, 00:03:47.160 --> 00:03:54.170 5, 10, 15, 20, 25, 30. 00:03:54.170 --> 00:03:55.850 It's not rocket science here. 00:03:55.850 --> 00:04:00.470 When t equals 15 seconds, 15 times 18, we're trying 00:04:00.470 --> 00:04:04.120 to find the cosine of 270 degrees, right? 00:04:04.120 --> 00:04:06.080 15 times 18 is 270 degrees. 00:04:06.080 --> 00:04:07.740 I'm just doing this because I don't have a calculator and 00:04:07.740 --> 00:04:10.480 this will help me pick good points. 00:04:10.480 --> 00:04:13.850 So when t is equal to 0, what is height? 00:04:13.850 --> 00:04:17.750 Or t is equal to 0, theta is equal to 0, so cosine of theta 00:04:17.750 --> 00:04:19.800 is -- cosine of 0 is 1. 00:04:19.800 --> 00:04:23.090 So 9 minus 8 is 1. 00:04:23.090 --> 00:04:24.610 I'm going to do h in a different color. 00:04:24.610 --> 00:04:26.540 So this is 1. 00:04:26.540 --> 00:04:28.430 Cosine of 90 degrees? 00:04:28.430 --> 00:04:30.180 Cosine of 90 degrees is 0. 00:04:30.180 --> 00:04:34.170 So 9 minus 0 is 9. 00:04:34.170 --> 00:04:35.885 Cosine of 180 degrees? 00:04:35.885 --> 00:04:37.960 So we're going all the way around the unit circle. 00:04:37.960 --> 00:04:40.310 Cosine of 180 degrees is minus 1. 00:04:40.310 --> 00:04:47.770 So minus 1 times minus 8 is plus 8, so 9 plus 8, that's 15. 00:04:47.770 --> 00:04:50.500 Cosine of 270 degrees are pointing straight down, so the 00:04:50.500 --> 00:04:51.670 x-coord is going to be 0. 00:04:51.670 --> 00:04:55.540 So once again, we're at 9 again. 00:04:55.540 --> 00:04:56.930 9 minus 0. 00:04:56.930 --> 00:04:57.900 360 degrees. 00:04:57.900 --> 00:04:59.900 Cosine of 360 degrees is the same thing as 00:04:59.900 --> 00:05:02.390 cosine of 0, right? 00:05:02.390 --> 00:05:07.080 So once again, I mean we've gone around the circle once. 00:05:07.080 --> 00:05:10.480 So it's going to be the same as 0, so it's going to be 1. 00:05:10.480 --> 00:05:12.420 And 450 is going to be the same thing as 90. 00:05:12.420 --> 00:05:16.250 So it's going to be 9 and then 15 degrees. 00:05:16.250 --> 00:05:19.690 So let's plot these points. 00:05:19.690 --> 00:05:22.980 Actually, let me just draw 15 up here. 00:05:22.980 --> 00:05:24.820 So what are the points that keep showing up? 00:05:24.820 --> 00:05:33.310 So this is 1, that's 1, that's 1, and then we have 9. 00:05:33.310 --> 00:05:44.685 1, there's 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. 00:05:48.320 --> 00:05:50.970 Fair enough. 00:05:50.970 --> 00:05:54.380 So let me draw some guidelines just to help us. 00:05:57.760 --> 00:06:02.270 Actually, let me do them kind of hard to see, because I 00:06:02.270 --> 00:06:07.400 don't want to draw too much attention to the guidelines. 00:06:07.400 --> 00:06:10.100 I could do one guideline there. 00:06:12.970 --> 00:06:16.075 Then I'll do a bottom guideline right there. 00:06:20.040 --> 00:06:21.690 Then the 9 keeps showing up. 00:06:25.020 --> 00:06:26.260 Oh, you know what, I can't add. 00:06:26.260 --> 00:06:28.060 What's 9 plus 8? 00:06:28.060 --> 00:06:30.330 It's not 15, it's 17. 00:06:30.330 --> 00:06:37.580 Sorry, clearly, I need to practice my addition. 00:06:37.580 --> 00:06:39.970 So this is 9 plus 8, this is 17. 00:06:39.970 --> 00:06:41.890 And I realized that because I was like, well 9 should be in 00:06:41.890 --> 00:06:43.820 the middle, so this is actually 17. 00:06:43.820 --> 00:06:45.820 Ignore my little marks here. 00:06:45.820 --> 00:06:48.520 That's 17, this is 1. 00:06:48.520 --> 00:06:49.470 Ignore the marks. 00:06:49.470 --> 00:06:54.130 9 would be right in the middle between 1 and 17. 00:06:54.130 --> 00:06:58.290 So let me draw kind of mediant point right there. 00:06:58.290 --> 00:06:59.820 So this is 9. 00:06:59.820 --> 00:07:03.150 Sorry I can't add properly. 00:07:03.150 --> 00:07:06.230 Then let's draw the graph or at least plot the 00:07:06.230 --> 00:07:07.820 points on the graph. 00:07:07.820 --> 00:07:15.240 So, t equals 0 where h equals 1, so that's this point. 00:07:15.240 --> 00:07:18.020 That's right here. 00:07:18.020 --> 00:07:23.540 When t equals 5, h is equal to 9, right here. 00:07:23.540 --> 00:07:26.350 When t is equal to 10, h is equal to 17. 00:07:30.380 --> 00:07:34.280 When t is equal to 15, h is 9 again. 00:07:34.280 --> 00:07:37.510 So it's right here. 00:07:37.510 --> 00:07:40.040 At 20 we're back at 1. 00:07:40.040 --> 00:07:41.590 I think you see the pattern. 00:07:41.590 --> 00:07:44.570 At 25 we're back at 9. 00:07:44.570 --> 00:07:49.500 And then at 30 we're back at 17, not 15, because now I 00:07:49.500 --> 00:07:51.330 have corrected my mistake. 00:07:51.330 --> 00:07:53.270 And this is going to be sined graph, it's going to look 00:07:53.270 --> 00:07:56.360 something like this. 00:07:56.360 --> 00:07:59.080 Let me do it in a vibrant color so I can overwrite everything 00:07:59.080 --> 00:08:01.130 and it's going to look something like this. 00:08:01.130 --> 00:08:05.466 Go oops, and then up and them down here. 00:08:11.180 --> 00:08:19.100 Curve up, come back down, curve up and then come back down. 00:08:19.100 --> 00:08:20.060 Like that. 00:08:20.060 --> 00:08:21.490 So that's our graph. 00:08:21.490 --> 00:08:23.850 I think in the problem they tell us to approximate. 00:08:23.850 --> 00:08:30.910 Actually, let me open up my cousin's problem -- my other 00:08:30.910 --> 00:08:35.120 account has timed-out on me while I recorded this. 00:08:35.120 --> 00:08:40.080 They wanted to approximate when t equals 4 what the height is. 00:08:40.080 --> 00:08:43.030 So when t equals 4 the height is like right 00:08:43.030 --> 00:08:44.440 around there, right? 00:08:44.440 --> 00:08:46.650 So the height is a little bit less than 9. 00:08:46.650 --> 00:08:50.500 And I don't know, 7 or 8 meters in the air. 00:08:50.500 --> 00:08:53.770 And when time is equal to 10 -- well, time equal 10, we figured 00:08:53.770 --> 00:08:57.930 out exactly, we know that there are 17 meters in the air. 00:08:57.930 --> 00:09:00.510 So I know this was kind of a little messy and graphing 00:09:00.510 --> 00:09:03.770 trick functions tend to be, but hopefully you found 00:09:03.770 --> 00:09:06.160 this vaguely useful. 00:09:06.160 --> 00:09:07.620 Have fun.

Ferris Wheel Trig Problem

https://www.youtube.com/watch?v=clXSqjs1wgQ

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https://www.youtube.com/api/timedtext?v=clXSqjs1wgQ&ei=YmeUZaz7L_mdvdIPopOJmA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=979B6710DCF4ABB377E6325A5E738E3A1996932C.515B8EACB692FA68F4166C5AE80C9FBAA5E98246&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.800 --> 00:00:01.890 All right, I have a problem here. 00:00:01.890 --> 00:00:04.840 Jacob and Emily ride a ferris wheel at a carnival 00:00:04.840 --> 00:00:06.290 in Billings, Montana. 00:00:06.290 --> 00:00:09.280 The wheel has a 16-meter diameter. 00:00:09.280 --> 00:00:10.440 So let me draw the wheel. 00:00:10.440 --> 00:00:13.110 It has a 16-meter diameter. 00:00:13.110 --> 00:00:15.225 So let me draw it big. 00:00:15.225 --> 00:00:16.886 Give me a lot of space. 00:00:16.886 --> 00:00:20.910 So it has a 16-meter diameter, so what's its 00:00:20.910 --> 00:00:21.710 radius going to be? 00:00:21.710 --> 00:00:24.570 Its radius is going to be half of that, right? 00:00:24.570 --> 00:00:33.280 So if I were to draw its radius, just draw it like that. 00:00:36.370 --> 00:00:39.780 It's a 16-meter diameter, so it's radius is going to be 8 00:00:39.780 --> 00:00:44.390 meters with its lowest point above the ground-- oh, 00:00:44.390 --> 00:00:46.540 with its lowest point 1 meter above the ground. 00:00:46.540 --> 00:00:49.690 So its lowest point is right here. 00:00:49.690 --> 00:00:53.580 This is its lowest point, and that is 1 meter 00:00:53.580 --> 00:00:54.270 above the ground. 00:00:54.270 --> 00:00:57.920 So this distance right here is 1 meter. 00:00:57.920 --> 00:00:58.570 Fair enough. 00:01:01.180 --> 00:01:05.335 Assume that Jacob and Emily's height h above the ground is a 00:01:05.335 --> 00:01:10.100 sinusoidal function of time where t equals 0 represents the 00:01:10.100 --> 00:01:11.700 lowest point on the wheel. 00:01:11.700 --> 00:01:15.720 So this is at point t equals 0 right here. t equals 0 is the 00:01:15.720 --> 00:01:16.860 lowest point of the wheel. 00:01:16.860 --> 00:01:18.440 Write an equation for h. 00:01:18.440 --> 00:01:21.060 Oh, I think I forgot, so let me reread it. 00:01:21.060 --> 00:01:23.490 Jacob and Emily ride a ferris wheel at a carnival 00:01:23.490 --> 00:01:24.170 in Billings, Montana. 00:01:24.170 --> 00:01:26.030 The wheel has a 16-meter diameter-- we did 00:01:26.030 --> 00:01:29.450 that-- and turns at 3 revolutions per minute. 00:01:29.450 --> 00:01:34.910 So it turns at 3 revolutions per minute with its lowest 00:01:34.910 --> 00:01:36.570 point 1 meter above the ground. 00:01:36.570 --> 00:01:39.790 Assume that Jacob and Emily's height h above the ground is 00:01:39.790 --> 00:01:43.250 a sinusoidal function of time, where t equals 0. 00:01:43.250 --> 00:01:46.160 So we need to write a function of h, their distance above the 00:01:46.160 --> 00:01:49.200 ground, as a function of time, and they're saying that 00:01:49.200 --> 00:01:51.280 time is given in seconds. 00:01:51.280 --> 00:01:53.860 So, first of all, they're telling us 3 revolutions 00:01:53.860 --> 00:01:55.760 every minute, right? 00:01:55.760 --> 00:02:03.950 So that's 3 revolutions per 60 seconds, and that's the 00:02:03.950 --> 00:02:08.500 same thing as 1 revolution per 20 seconds, right? 00:02:08.500 --> 00:02:13.630 I just divide both sides of the per by 3 for 20 seconds. 00:02:13.630 --> 00:02:15.710 And one revolution is how many degrees? 00:02:15.710 --> 00:02:17.840 One revolution is 360 degrees. 00:02:17.840 --> 00:02:27.420 So it's 360 degrees per 20 seconds. 00:02:27.420 --> 00:02:30.280 And if you're going 360 degrees per 20 seconds, let's divide-- 00:02:30.280 --> 00:02:32.980 you know, the per you can just kind of use as an equal 00:02:32.980 --> 00:02:35.270 sign of the equation. 00:02:35.270 --> 00:02:37.180 360 degrees for 20 seconds. 00:02:37.180 --> 00:02:38.270 That means you're going to go what? 00:02:38.270 --> 00:02:40.740 18 degrees, Just divide both sides by 20. 00:02:40.740 --> 00:02:46.895 18 degrees per second. 00:02:49.430 --> 00:02:51.450 And we could have done it with a numerator and a denominator. 00:02:51.450 --> 00:02:54.260 3 revs per-- you know, you could have said 3 00:02:54.260 --> 00:02:57.940 revs over 60 seconds. 00:02:57.940 --> 00:02:59.140 That's actually how I should have done it. 00:02:59.140 --> 00:03:05.430 3 revs over 60 seconds is equal to 1 rev over 20 seconds, which 00:03:05.430 --> 00:03:11.600 is equal to 360 degrees over 20 seconds, which is equal to 18 00:03:11.600 --> 00:03:15.790 degrees per second, right? 00:03:15.790 --> 00:03:18.350 So we're going to travel 18 degrees per second. 00:03:18.350 --> 00:03:23.510 So the total number of degrees we've traveled in t seconds is 00:03:23.510 --> 00:03:29.890 going to be-- so see, if we say the angle, that's the angle 00:03:29.890 --> 00:03:31.180 from our starting point. 00:03:31.180 --> 00:03:37.270 So let's say we've traveled t seconds, and we're right there. 00:03:37.270 --> 00:03:42.460 What is-- let's drop a little altitude right here. 00:03:42.460 --> 00:03:46.270 What is this angle going to be, where this angle is right here? 00:03:46.270 --> 00:03:48.070 What is this angle going to be? 00:03:48.070 --> 00:03:51.000 How many degrees have we traveled? 00:03:51.000 --> 00:03:54.180 Well, we say we traveled 18 degrees per second, so if we 00:03:54.180 --> 00:04:03.090 travel t seconds, this is going to be 18t degrees, right? 00:04:03.090 --> 00:04:05.920 All right, so let's see if we can figure out how their height 00:04:05.920 --> 00:04:09.600 as a function of this-- well, as a function of t or as 00:04:09.600 --> 00:04:12.170 a function of this angle right here. 00:04:12.170 --> 00:04:16.270 So what is this height right here? 00:04:16.270 --> 00:04:17.220 Up here? 00:04:17.220 --> 00:04:20.420 It's 1 meter plus the radius because this distance 00:04:20.420 --> 00:04:21.710 right here is 8. 00:04:21.710 --> 00:04:25.000 So we could say that this is-- this point right here is h is 00:04:25.000 --> 00:04:27.420 equal to 9 at this point, right? 00:04:27.420 --> 00:04:29.650 We could almost view that as the h axis. 00:04:29.650 --> 00:04:30.990 So that's h is equal to 9. 00:04:30.990 --> 00:04:34.020 So at this point, how high are they? 00:04:34.020 --> 00:04:37.430 If this is h-- so right now, let me draw a little 00:04:37.430 --> 00:04:41.810 drop and go flat here. 00:04:41.810 --> 00:04:47.510 So their height above the ground is this distance h, 00:04:47.510 --> 00:04:53.280 which is the same thing as this distance h. 00:04:53.280 --> 00:04:55.050 So what is that distance? 00:04:55.050 --> 00:04:58.100 Well, it's going to be-- well, if this distance is h, what is 00:04:58.100 --> 00:05:00.590 this distance going to be? 00:05:00.590 --> 00:05:03.540 This distance is going to be 9 minus h. 00:05:03.540 --> 00:05:04.060 How do I know this? 00:05:04.060 --> 00:05:06.170 This whole distance is 9. 00:05:06.170 --> 00:05:09.770 This distance is h, so-- let me do it in a better color-- so 00:05:09.770 --> 00:05:15.690 that this distance right here is 9 minus h. 00:05:15.690 --> 00:05:17.410 So let's see what we can do. 00:05:17.410 --> 00:05:17.860 What do we know? 00:05:17.860 --> 00:05:19.550 We know this distance. 00:05:19.550 --> 00:05:23.300 We know this angle is 18t degrees. 00:05:23.300 --> 00:05:24.720 And do we know this side? 00:05:24.720 --> 00:05:25.010 Sure. 00:05:25.010 --> 00:05:25.730 That's the radius. 00:05:25.730 --> 00:05:26.490 That's 8. 00:05:26.490 --> 00:05:28.610 8 meters. 00:05:28.610 --> 00:05:32.780 9 minus h meters, 8 meters, and 18 degrees. 00:05:32.780 --> 00:05:36.260 And what are these sides relative to this angle? 00:05:36.260 --> 00:05:42.800 Well, if we were to draw a triangle here relative to this 00:05:42.800 --> 00:05:47.400 angle right here, the 9 minus h is adjacent, and the 8 meters 00:05:47.400 --> 00:05:51.430 is, of course, the hypotenuse, right? 00:05:51.430 --> 00:05:55.780 So what trig function deals with adjacent and hypotenuse. 00:05:55.780 --> 00:05:57.500 SOHCAHTOA. 00:05:57.500 --> 00:06:01.120 CAH, cosine is adjacent over hypotenuse. 00:06:01.120 --> 00:06:05.900 So we could say the cosine at 18 degrees, the cosine of 18t 00:06:05.900 --> 00:06:11.445 degrees, is equal to its adjacent side. 00:06:11.445 --> 00:06:14.462 The adjacent side is 9 minus h. 00:06:14.462 --> 00:06:21.670 It's equal to 9 minus h over the hypotenuse, over 8. 00:06:21.670 --> 00:06:24.720 And now we can solve for h, and we'll have h 00:06:24.720 --> 00:06:25.910 as a function of t. 00:06:25.910 --> 00:06:27.810 So we multiply both sides by 8. 00:06:27.810 --> 00:06:37.090 You get 8 cosine of 18t is equal to 9 minus h. 00:06:37.090 --> 00:06:39.870 Maybe we could subtract 9 from both sides. 00:06:39.870 --> 00:06:48.190 So we get minus 9 plus 8 cosine of 18t is equal to minus h, and 00:06:48.190 --> 00:06:52.340 then multiply both sides by negative 1, and then you get 00:06:52.340 --> 00:07:00.130 9-- positive 9, right-- minus 8 cosine of 18t is equals 00:07:00.130 --> 00:07:10.500 to h, or h is equal to 9 minus 8 cosine of 18t. 00:07:10.500 --> 00:07:11.330 So there we have it. 00:07:11.330 --> 00:07:16.230 We have expressed h as a function of t. 00:07:16.230 --> 00:07:18.050 And in the next video, I'm actually going to 00:07:18.050 --> 00:07:18.835 graph this function. 00:07:18.835 --> 00:07:20.420 See you soon.

Navigation Word Problem

https://www.youtube.com/watch?v=XTWZ_M8d-4g

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https://www.youtube.com/api/timedtext?v=XTWZ_M8d-4g&ei=YmeUZfqLM5Cip-oPvZec0AY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=97336058A1D7C2A8B09388DCB2A57CC62303AECF.545DD4218FEEDCB57983696F1743FDBF56751E77&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:02.710 I have a problem here from my cousin. 00:00:02.710 --> 00:00:03.410 Let's see what it says. 00:00:03.410 --> 00:00:07.450 It says Milwaukee, Wisconsin is directly west of Grand 00:00:07.450 --> 00:00:10.630 Haven, Michigan, on opposite sides of Lake Michigan. 00:00:10.630 --> 00:00:12.020 So let me draw that. 00:00:12.020 --> 00:00:19.535 So if we say that this right here is Milwaukee. 00:00:23.570 --> 00:00:25.760 It's due west of Grand Haven, Michigan. 00:00:25.760 --> 00:00:31.110 So if I were to draw-- let me draw a horizontal line. 00:00:31.110 --> 00:00:34.145 We're going over water so let me do it in blue. 00:00:34.145 --> 00:00:37.710 It's due west of Grand Haven, Michigan. 00:00:37.710 --> 00:00:40.640 You know, you go straight east. 00:00:40.640 --> 00:00:44.240 If I were to go straight east, I would get to 00:00:44.240 --> 00:00:45.870 Grand Haven, Michigan. 00:00:45.870 --> 00:00:48.430 Which is right here. 00:00:48.430 --> 00:00:50.540 Let me just label that, g. 00:00:50.540 --> 00:00:51.350 All right. 00:00:51.350 --> 00:00:54.840 On a foggy night, a law enforcement boat leaves 00:00:54.840 --> 00:00:58.190 from Milwaukee on a course of 105 degrees. 00:01:00.910 --> 00:01:04.570 The hardest part of these problems, in my opinion, is 00:01:04.570 --> 00:01:05.970 really just trying to figure out the convention 00:01:05.970 --> 00:01:06.300 they're using. 00:01:06.300 --> 00:01:08.560 When they say a course of 105 degrees, what does that mean? 00:01:08.560 --> 00:01:09.860 What direction is it? 00:01:09.860 --> 00:01:13.330 And I checked with my cousin, and her book says that in, I 00:01:13.330 --> 00:01:17.710 guess in the boating world, the course is how many 00:01:17.710 --> 00:01:19.800 degrees clockwise you're going, of due north. 00:01:19.800 --> 00:01:22.230 So due north is 0 degrees. 00:01:22.230 --> 00:01:23.370 So 105 degrees. 00:01:23.370 --> 00:01:25.590 So due north would be 0 degrees. 00:01:25.590 --> 00:01:29.310 Due north would be-- that's 0 degrees. 00:01:29.310 --> 00:01:33.100 So he's going 105 degrees clockwise of that. 00:01:33.100 --> 00:01:37.090 So if we 105 degrees-- so he's going 105 degrees. 00:01:37.090 --> 00:01:39.060 So that's like 105 degrees. 00:01:39.060 --> 00:01:41.700 Something like that. 00:01:41.700 --> 00:01:43.090 He's going 105 degrees. 00:01:43.090 --> 00:01:45.860 And I'll do him in magenta. 00:01:45.860 --> 00:01:47.160 His course is 105 degrees. 00:01:47.160 --> 00:01:53.100 So that's 105 degrees clockwise of due north. 00:01:53.100 --> 00:01:56.120 And what is that-- oh, it's like my screen just backed up. 00:01:56.120 --> 00:02:00.030 So what is that angle, in kind of what we're familiar with? 00:02:00.030 --> 00:02:01.970 Well, this would be 90 degrees right here. 00:02:01.970 --> 00:02:02.520 Going here. 00:02:02.520 --> 00:02:04.220 And then he goes 15 more degrees. 00:02:04.220 --> 00:02:05.770 So in kind of unit circle terms, this would be 00:02:05.770 --> 00:02:07.040 negative 15 degrees. 00:02:07.040 --> 00:02:09.540 Or if we wanted to figure out the angle of this vertex right 00:02:09.540 --> 00:02:12.710 here, that would be 90, and then we'd go another 15. 00:02:12.710 --> 00:02:17.530 So this angle right here is going to be what? 00:02:17.530 --> 00:02:20.080 It's going to be 15 degrees. 00:02:20.080 --> 00:02:21.530 Because he said his course is 105. 00:02:21.530 --> 00:02:23.700 So 90 plus 15 is 105. 00:02:23.700 --> 00:02:28.610 So how much south he's going of, kind of, straight 00:02:28.610 --> 00:02:31.590 west-east, is 15 degrees. 00:02:31.590 --> 00:02:33.990 This whole thing is 105. 00:02:33.990 --> 00:02:35.880 OK, let me keep reading the problem. 00:02:35.880 --> 00:02:38.660 He leaves from Milwaukee at a course of 105 degrees at the 00:02:38.660 --> 00:02:41.910 same time that a small smuggling craft steers a 00:02:41.910 --> 00:02:45.820 course of 195 degrees from Grand Haven. 00:02:45.820 --> 00:02:47.710 195 degrees. 00:02:47.710 --> 00:02:50.510 So once again, due north is 0 degrees. 00:02:50.510 --> 00:02:52.560 And this guy's going 195. 00:02:52.560 --> 00:02:54.940 So we figure-- this is just a convention. 00:02:54.940 --> 00:02:57.910 You figure out-- well, he's going 195 degrees 00:02:57.910 --> 00:02:59.260 clockwise of due north. 00:02:59.260 --> 00:03:02.350 So 195 degrees is going to be-- let's see, it's going to be 00:03:02.350 --> 00:03:04.790 a 180 degrees and then some. 00:03:04.790 --> 00:03:08.070 It's going to be like that. 00:03:08.070 --> 00:03:10.350 So his course is going to look something like this. 00:03:12.920 --> 00:03:16.090 His course is going to look something like that. 00:03:16.090 --> 00:03:17.770 And let's see if we can figure out what this 00:03:17.770 --> 00:03:18.870 angle right here is. 00:03:18.870 --> 00:03:21.180 Because we, as you can kind of see where this is going, we're 00:03:21.180 --> 00:03:23.820 trying to figure out probably the size of this triangle, 00:03:23.820 --> 00:03:24.330 if I had to guess. 00:03:24.330 --> 00:03:26.330 I haven't even read the whole problem yet. 00:03:26.330 --> 00:03:27.490 Let's see. 00:03:27.490 --> 00:03:29.150 So he's going 195 degrees. 00:03:29.150 --> 00:03:32.690 So if we were to drop, well, like here. 00:03:32.690 --> 00:03:36.450 This right here is a 180 degrees, to go clockwise from 00:03:36.450 --> 00:03:39.530 straight up to straight down. 00:03:39.530 --> 00:03:40.950 This is 180. 00:03:40.950 --> 00:03:42.480 And so he went 195. 00:03:42.480 --> 00:03:45.515 So this is going to be-- that's going to be 15 degrees. 00:03:48.090 --> 00:03:51.750 And if this angle is 15 degrees, what is this 00:03:51.750 --> 00:03:53.960 angle going to be? 00:03:53.960 --> 00:03:57.050 Well, this entire angle is 90 degrees, right? 00:03:57.050 --> 00:03:58.790 It's kind of the third quadrant when we're thinking 00:03:58.790 --> 00:04:00.330 in unit circle terms. 00:04:00.330 --> 00:04:03.810 So this angle right here is going to be 90 00:04:03.810 --> 00:04:06.100 minus this 15 degrees. 00:04:06.100 --> 00:04:08.200 So what's 90 minus 15? 00:04:08.200 --> 00:04:09.930 It's 75, right? 00:04:09.930 --> 00:04:10.885 75 degrees. 00:04:13.880 --> 00:04:15.890 And if we wanted to convert his, kind of, course angles 00:04:15.890 --> 00:04:18.220 into unit circles-- you know, with unit circles, you start 00:04:18.220 --> 00:04:19.980 here and you go all the way around this way. 00:04:19.980 --> 00:04:22.720 So I think you would get something like 255 degrees. 00:04:22.720 --> 00:04:23.020 But anyway. 00:04:23.020 --> 00:04:25.200 So we figured out that this angle is 15 degrees, this 00:04:25.200 --> 00:04:27.100 angle is 75 degrees. 00:04:27.100 --> 00:04:29.380 What's this angle going to be? 00:04:29.380 --> 00:04:33.260 This angle is going to be-- these all have to add 00:04:33.260 --> 00:04:34.290 up to 180, right? 00:04:34.290 --> 00:04:40.255 So this is going to be 180 minus 15 minus 75. 00:04:40.255 --> 00:04:41.250 And what's that? 00:04:41.250 --> 00:04:43.550 That's 180 minus 90. 00:04:43.550 --> 00:04:45.210 So 180 minus 90 is 90 degrees! 00:04:45.210 --> 00:04:46.600 So this angle here is 90 degrees. 00:04:46.600 --> 00:04:48.060 It's a right angle. 00:04:48.060 --> 00:04:49.770 It's a right angle. 00:04:49.770 --> 00:04:50.610 Interesting. 00:04:50.610 --> 00:04:52.210 OK, so what do they tell us? 00:04:52.210 --> 00:04:57.660 They tell us the law enforcement boat 00:04:57.660 --> 00:04:59.580 averages 23 knots. 00:04:59.580 --> 00:05:05.710 So he's traveling in this direction at 23 knots. 00:05:05.710 --> 00:05:08.790 All right, that's a little bit faster than 23 miles per hour. 00:05:08.790 --> 00:05:11.790 And collides with the smuggling craft. 00:05:11.790 --> 00:05:14.850 What was the smuggling boat's average speed? 00:05:14.850 --> 00:05:16.940 So they both leave their respective sites at the 00:05:16.940 --> 00:05:19.210 same time, and they both collide, right? 00:05:19.210 --> 00:05:22.420 So the time they traveled is the same. 00:05:22.420 --> 00:05:24.200 Let's call that time, t. 00:05:24.200 --> 00:05:24.640 Right? 00:05:24.640 --> 00:05:25.060 I don't know. 00:05:25.060 --> 00:05:26.890 They both left at the same time and it took some 00:05:26.890 --> 00:05:28.780 time for them to collide. 00:05:28.780 --> 00:05:31.230 So let's say that the time between when they left and 00:05:31.230 --> 00:05:32.340 the time they collided is t. 00:05:32.340 --> 00:05:35.350 So how far did the patrol boat travel? 00:05:35.350 --> 00:05:40.590 Well, he traveled at a speed of 23 knots, and it took him time, 00:05:40.590 --> 00:05:42.410 t, to get to the collision. 00:05:42.410 --> 00:05:44.105 So the distance he traveled is 23t. 00:05:47.220 --> 00:05:49.570 Speed times time is equal to distance. 00:05:49.570 --> 00:05:52.900 So the length of this side is 23t. 00:05:52.900 --> 00:05:55.260 Similarly, this guy, we don't know his speed. 00:05:55.260 --> 00:05:56.650 Let's call it, I don't know. 00:05:56.650 --> 00:05:58.240 Let's call it x. 00:05:58.240 --> 00:05:59.390 His speed is x. 00:05:59.390 --> 00:06:03.160 But the distance he travels is x times t. 00:06:03.160 --> 00:06:04.920 x times t, right? 00:06:04.920 --> 00:06:07.110 So that's the length of this side. 00:06:07.110 --> 00:06:10.850 So let's see if we can figure out what x is. 00:06:10.850 --> 00:06:13.340 So what do we know? 00:06:13.340 --> 00:06:14.900 We know a lot about this. 00:06:14.900 --> 00:06:17.510 We know this is a right triangle, et cetera. 00:06:17.510 --> 00:06:19.225 We know this angle. 00:06:19.225 --> 00:06:23.190 So if we wanted to solve for xt and use this 23t 00:06:23.190 --> 00:06:24.130 information, let's see. 00:06:24.130 --> 00:06:26.650 We know-- look at this angle. 00:06:26.650 --> 00:06:30.230 If we use the 75 degrees, we know the opposite angle. 00:06:30.230 --> 00:06:31.950 The opposite side, sorry. 00:06:31.950 --> 00:06:33.120 Which is 23t. 00:06:33.120 --> 00:06:36.600 And we know the adjacent side, which is xt. 00:06:36.600 --> 00:06:38.120 So let me write SOHCAHTOA here. 00:06:42.480 --> 00:06:46.900 So what deals with opposite and adjacent? 00:06:46.900 --> 00:06:48.480 Well, that's tangent, right? 00:06:48.480 --> 00:06:49.500 TOA. 00:06:49.500 --> 00:06:59.130 So if we say the tan of 75 degrees is going to be equal to 00:06:59.130 --> 00:07:05.340 the opposite side-- 23t-- over the adjacent side-- that's this 00:07:05.340 --> 00:07:09.170 side, opposite over adjacent-- xt. 00:07:09.170 --> 00:07:11.270 Well, the t's cancel out, right? 00:07:11.270 --> 00:07:12.425 The t's cancel out. 00:07:12.425 --> 00:07:14.460 And let's see if we can solve for x. 00:07:14.460 --> 00:07:16.320 Multiply x times both sides. 00:07:16.320 --> 00:07:22.450 You get x tangent of 75 is equal to 23. 00:07:22.450 --> 00:07:26.540 And then divide both sides by the tan of 75 and you get x is 00:07:26.540 --> 00:07:33.610 equal to 23 divided by the tangent of 75 degrees. 00:07:33.610 --> 00:07:34.840 And so that's our answer. 00:07:34.840 --> 00:07:38.480 And if I had to-- well, actually, I have-- let's see. 00:07:38.480 --> 00:07:39.950 Tangent of 75 degrees. 00:07:39.950 --> 00:07:41.350 I don't have a calculator in front of me. 00:07:41.350 --> 00:07:43.285 You could calculate it. 00:07:43.285 --> 00:07:47.240 It's actually going to be a pretty high number. 00:07:47.240 --> 00:07:48.200 So you could try to fit, you know. 00:07:48.200 --> 00:07:50.760 If you have a calculator, just type in 75 degrees. 00:07:50.760 --> 00:07:53.230 Take the tangent of it, and perform this calculation. 00:07:53.230 --> 00:07:56.790 But we've essentially solved this problem. 00:07:56.790 --> 00:07:59.110 I'll see you in the next video.

Proof: Law of sines

https://www.youtube.com/watch?v=APNkWrD-U1k

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https://www.youtube.com/api/timedtext?v=APNkWrD-U1k&ei=YmeUZYHzMLWkmLAPx52gkAs&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D3CAA9109E153BBAF8B721C580C759313012460B.1CA7F208E977F2BF52D166FC0DD824C6F3DF8DB5&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.870 --> 00:00:04.180 I will now do a proof of the law of sines. 00:00:04.180 --> 00:00:08.920 So, let's see, let me draw an arbitrary triangle. 00:00:08.920 --> 00:00:11.380 That's one side right there. 00:00:11.380 --> 00:00:13.510 And then I've got another side here. 00:00:13.510 --> 00:00:15.320 I'll try to make it look a little strange so you realize 00:00:15.320 --> 00:00:19.410 it can apply to any triangle. 00:00:19.410 --> 00:00:24.920 And let's say we know the following information. 00:00:24.920 --> 00:00:28.530 We know this angle -- well, actually, I'm not going to say 00:00:28.530 --> 00:00:30.940 what we know or don't know, but the law of sines is just a 00:00:30.940 --> 00:00:32.980 relationship between different angles and different sides. 00:00:32.980 --> 00:00:36.500 Let's say that this angle right here is alpha. 00:00:36.500 --> 00:00:38.470 This side here is A. 00:00:38.470 --> 00:00:40.490 The length here is A. 00:00:40.490 --> 00:00:44.720 Let's say that this side here is beta, and that 00:00:44.720 --> 00:00:47.270 the length here is B. 00:00:47.270 --> 00:00:50.185 Beta is just B with a long end there. 00:00:50.185 --> 00:00:53.700 So let's see if we can find a relationship that connects A 00:00:53.700 --> 00:00:56.120 and B, and alpha and beta. 00:00:56.120 --> 00:00:58.300 So what can we do? 00:00:58.300 --> 00:01:00.130 And hopefully that relationship we find will 00:01:00.130 --> 00:01:01.610 be the law of sines. 00:01:01.610 --> 00:01:04.390 Otherwise, I would have to rename this video. 00:01:04.390 --> 00:01:06.980 So let me draw an altitude here. 00:01:06.980 --> 00:01:10.612 I think that's the proper term. 00:01:10.612 --> 00:01:15.230 If I just draw a line from this side coming straight down, and 00:01:15.230 --> 00:01:18.350 it's going to be perpendicular to this bottom side, which I 00:01:18.350 --> 00:01:21.130 haven't labeled, but I'll probably, if I have to label 00:01:21.130 --> 00:01:23.630 it, probably label it C, because that's A and B. 00:01:23.630 --> 00:01:25.780 And this is going to be a 90 degree angle. 00:01:29.870 --> 00:01:31.280 I don't know the length of that. 00:01:31.280 --> 00:01:32.480 I don't know anything about it. 00:01:32.480 --> 00:01:38.510 All I know is I went from this vertex and I dropped a line 00:01:38.510 --> 00:01:40.550 that's perpendicular to this other side. 00:01:40.550 --> 00:01:42.300 So what can we do with this line? 00:01:42.300 --> 00:01:45.330 Well let me just say that it has length x. 00:01:45.330 --> 00:01:49.320 The length of this line is x. 00:01:49.320 --> 00:01:52.950 Can we find a relationship between A, the length of 00:01:52.950 --> 00:01:55.620 this line x, and beta? 00:01:55.620 --> 00:01:56.570 Well, sure. 00:01:56.570 --> 00:01:57.670 Let's see. 00:01:57.670 --> 00:02:02.790 Let me find an appropriate color. 00:02:02.790 --> 00:02:03.160 OK. 00:02:03.160 --> 00:02:05.290 That's, I think, a good color. 00:02:05.290 --> 00:02:07.380 So what's the relationship? 00:02:07.380 --> 00:02:10.730 If we look at this angle right here, beta, x is opposite to it 00:02:10.730 --> 00:02:13.140 and A is the hypotenuse, if we look at this right triangle 00:02:13.140 --> 00:02:14.430 right here, right? 00:02:14.430 --> 00:02:18.180 So what deals with opposite and hypotenuse? 00:02:18.180 --> 00:02:20.100 Whenever we do trigonometry, we should always just right soh 00:02:20.100 --> 00:02:21.180 cah toa at the top of the page. 00:02:21.180 --> 00:02:22.010 Soh cah toa. 00:02:22.010 --> 00:02:24.190 So what deals with opposite of hypotenuse? 00:02:24.190 --> 00:02:24.960 Sine, right? 00:02:24.960 --> 00:02:26.820 Soh, and you should probably guess that, because I'm 00:02:26.820 --> 00:02:28.770 proving the law of sines. 00:02:28.770 --> 00:02:36.490 So the sine of beta is equal to the opposite 00:02:36.490 --> 00:02:37.370 over the hypotenuse. 00:02:37.370 --> 00:02:42.130 It's equal to this opposite, which is x, over the 00:02:42.130 --> 00:02:46.440 hypotenuse, which is A, in this case. 00:02:46.440 --> 00:02:48.500 And if we wanted to solve for x, and I'll just do that, 00:02:48.500 --> 00:02:51.160 because it'll be convenient later, we can multiply both 00:02:51.160 --> 00:02:56.550 sides of this equation by A and you get A sine of 00:02:56.550 --> 00:02:59.240 beta is equal to x. 00:02:59.240 --> 00:03:00.350 Fair enough. 00:03:00.350 --> 00:03:02.100 That got us someplace. 00:03:02.100 --> 00:03:03.470 Well, let's see if we can find a relationship 00:03:03.470 --> 00:03:06.680 between alpha, B, and x. 00:03:06.680 --> 00:03:09.170 Well, similarly, if we look at this right triangle, because 00:03:09.170 --> 00:03:13.730 this is also a right triangle, of course, x here, relative to 00:03:13.730 --> 00:03:16.330 alpha, is also the opposite side, and B now is 00:03:16.330 --> 00:03:16.945 the hypotenuse. 00:03:16.945 --> 00:03:22.560 So we can also write that sine of alpha -- let me do it in a 00:03:22.560 --> 00:03:30.270 different color -- is equal to opposite over hypotenuse. 00:03:32.840 --> 00:03:36.995 The opposite is x and the hypotenuse is B. 00:03:40.530 --> 00:03:42.900 And let's solve for x again, just to do it. 00:03:42.900 --> 00:03:47.910 Multiply both sides by B and you get B sine of 00:03:47.910 --> 00:03:50.280 alpha is equal to x. 00:03:50.280 --> 00:03:51.130 So now what do we have? 00:03:51.130 --> 00:03:55.080 We have two different ways that we solved for this thing that I 00:03:55.080 --> 00:03:56.890 dropped down from this side, this x, right? 00:03:56.890 --> 00:03:59.770 We have A sine of beta is equal to x. 00:03:59.770 --> 00:04:03.180 And then B sine of alpha is equal to x. 00:04:03.180 --> 00:04:05.120 Well, if they're both equal to x, then they're both 00:04:05.120 --> 00:04:06.390 equal to each other. 00:04:06.390 --> 00:04:08.060 So let me write that down. 00:04:08.060 --> 00:04:13.060 Let me write that down in a soothing color. 00:04:13.060 --> 00:04:21.520 So we know that A sine of beta is equal to x, which is also 00:04:21.520 --> 00:04:24.900 equal to B sine of beta -- sorry, B sine of alpha. 00:04:28.160 --> 00:04:32.280 If we divide both sides of this equation by A, what do we get? 00:04:32.280 --> 00:04:36.600 We get sine of beta, right, because the A on this side 00:04:36.600 --> 00:04:42.360 cancels out, is equal to B sine of alpha over A. 00:04:42.360 --> 00:04:47.790 And if we divide both sides of this equation by B, we get 00:04:47.790 --> 00:05:04.920 sine of beta over B is equal to sine of alpha over A. 00:05:04.920 --> 00:05:07.880 So this is the law of sines. 00:05:07.880 --> 00:05:12.280 The ratio between the sine of beta and its opposite side -- 00:05:12.280 --> 00:05:15.550 and it's the side that it corresponds to, this B -- is 00:05:15.550 --> 00:05:20.670 equal to the ratio of the sine of alpha and its opposite side. 00:05:20.670 --> 00:05:23.200 And a lot of times in the books, let's say, if this angle 00:05:23.200 --> 00:05:27.430 was theta, and this was C, then they would also write that's 00:05:27.430 --> 00:05:32.560 also equal to the sine of theta over C. 00:05:32.560 --> 00:05:37.130 And the proof of adding this here is identical. 00:05:37.130 --> 00:05:39.380 We've picked B arbitrarily, B as a side, we could have done 00:05:39.380 --> 00:05:43.830 the exact same thing with theta and C, but instead of dropping 00:05:43.830 --> 00:05:46.350 the altitude here, we would have had to drop one of 00:05:46.350 --> 00:05:47.160 the other altitudes. 00:05:47.160 --> 00:05:49.800 And I think you could figure out that part. 00:05:49.800 --> 00:05:51.530 But the important thing is we have this ratio. 00:05:51.530 --> 00:05:53.800 And of course, you could have written it -- since it's a 00:05:53.800 --> 00:05:55.920 ratio, you could flip both sides of the ratio -- you could 00:05:55.920 --> 00:06:02.730 write it B over the sine of B is equal to A over 00:06:02.730 --> 00:06:04.480 the sine of alpha. 00:06:04.480 --> 00:06:09.740 And this is useful, because if you know one side and its 00:06:09.740 --> 00:06:12.640 corresponding angle, the angle opposite it that kind of opens 00:06:12.640 --> 00:06:15.680 up into that side, and say you know the other side, then you 00:06:15.680 --> 00:06:19.640 could figure out the angle that opens up into it. 00:06:19.640 --> 00:06:21.370 If you know three of these things, you can figure 00:06:21.370 --> 00:06:21.960 out the fourth. 00:06:21.960 --> 00:06:26.100 And that's what's useful about the law of sines. 00:06:26.100 --> 00:06:30.490 So maybe now I will do a few law of sines word problems. 00:06:30.490 --> 00:06:32.450 I'll see you in the next video.

SAT Prep: Test 8 Section 8 Part 2

https://www.youtube.com/watch?v=nwW5IMuRCaM

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https://www.youtube.com/api/timedtext?v=nwW5IMuRCaM&ei=YmeUZfeVM5u9mLAP6-Gn4A0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=0C854995F58F58C169623394E8B116AD83F885AD.67528A5603F2A7FC8682BD69DBBCD430DB919CC7&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:04.090 We're on problem 8. 00:00:04.090 --> 00:00:08.070 How many more degrees of arc are there in 1/4 of a circle 00:00:08.070 --> 00:00:10.020 than 1/5 of a circle? 00:00:10.020 --> 00:00:10.730 All right. 00:00:10.730 --> 00:00:14.400 So 1/4 of a circle has how many degrees? 00:00:14.400 --> 00:00:18.240 So it's 1/4 times the degrees in a circle, so that's 360 00:00:18.240 --> 00:00:19.460 degrees which is equal to what? 00:00:19.460 --> 00:00:21.860 What's 360 divided by 4? 00:00:21.860 --> 00:00:23.510 That's 90 degrees. 00:00:23.510 --> 00:00:23.840 Right? 00:00:23.840 --> 00:00:25.580 1/4 times 360 is 90 degrees. 00:00:25.580 --> 00:00:31.200 So that's 1/4 of a circle. 00:00:31.200 --> 00:00:35.090 1/5 fifth of a circle times 360 it equals what? 00:00:35.090 --> 00:00:39.850 360 divided by 5 which is equal to 72, right? 00:00:39.850 --> 00:00:40.490 35 right. 00:00:40.490 --> 00:00:42.090 72. 00:00:42.090 --> 00:00:43.870 So that equals 72 degrees. 00:00:43.870 --> 00:00:46.470 That's in 1/5 of a circle. 00:00:46.470 --> 00:00:48.560 And they're asking, how many more degrees of an arc are 00:00:48.560 --> 00:00:50.890 there in 1/4 of a circle than 1/5? 00:00:50.890 --> 00:00:53.760 So what's 90 minus 72? 00:00:53.760 --> 00:00:56.020 That's 18 degrees. 00:00:56.020 --> 00:00:59.120 That's answer B. 00:00:59.120 --> 00:01:02.100 Next problem. 00:01:02.100 --> 00:01:05.220 Problem 9. 00:01:05.220 --> 00:01:08.140 Let me draw the axes. 00:01:12.410 --> 00:01:18.640 And then they have a curve that looks something like this 00:01:18.640 --> 00:01:22.080 which goes to the origin and looks like it's symmetric. 00:01:22.080 --> 00:01:27.040 It goes like this and it keeps going. 00:01:27.040 --> 00:01:33.715 And they tell us this is the point minus 6 common 0. 00:01:33.715 --> 00:01:37.580 And they tell us-- well do they tell us? 00:01:37.580 --> 00:01:40.710 They have this point and they say that this is 1. 00:01:43.370 --> 00:01:46.920 This is the point 7 comma 6 up here. 00:01:46.920 --> 00:01:49.480 7 comma 6. 00:01:49.480 --> 00:01:52.360 And they're saying that this is 6 comma 0. 00:01:52.360 --> 00:01:54.560 Although they don't draw a little point there, which 00:01:54.560 --> 00:01:56.560 makes me a little suspicious. 00:01:56.560 --> 00:01:58.550 Maybe they just forgot. 00:01:58.550 --> 00:02:01.060 Based on the graph of the function f above, what are the 00:02:01.060 --> 00:02:04.430 values of x for which f of x is negative? 00:02:04.430 --> 00:02:06.930 So when is f of x negative? 00:02:06.930 --> 00:02:08.850 Well, it's negative in this range. 00:02:08.850 --> 00:02:12.280 From here to here. 00:02:12.280 --> 00:02:14.100 That's when it's negative. 00:02:14.100 --> 00:02:15.860 So what values of x are its negative? 00:02:15.860 --> 00:02:19.470 Well, at 0 it's 0, so 0 doesn't count. 00:02:19.470 --> 00:02:23.300 So x has to be greater than 0. 00:02:23.300 --> 00:02:25.510 Because we can't count 0, because at 0 the function is 00:02:25.510 --> 00:02:28.040 actually 0 and we want to know negative and 0 isn't a 00:02:28.040 --> 00:02:29.550 negative number. 00:02:29.550 --> 00:02:31.450 And what does x have to be less than? 00:02:31.450 --> 00:02:33.240 Well this is point 6. 00:02:33.240 --> 00:02:36.020 So x has to be less than 6. 00:02:36.020 --> 00:02:37.640 So that is choice B. 00:02:37.640 --> 00:02:38.380 Not too hard, huh? 00:02:38.380 --> 00:02:41.010 You just had to say, when does it dip below the x-axis? 00:02:41.010 --> 00:02:44.370 Well when x is between 0 and 6. 00:02:44.370 --> 00:02:45.620 Next problem. 00:02:48.540 --> 00:02:50.340 Let me switch colors. 00:02:50.340 --> 00:02:53.230 In the figure above-- oh what do I have to draw-- the figure 00:02:53.230 --> 00:02:55.910 above shows the dimensions of a pedestal constructed of 4 00:02:55.910 --> 00:02:56.610 layers of marble. 00:02:56.610 --> 00:02:59.440 Each layer is a rectangular solid that is 1 foot high and 00:02:59.440 --> 00:03:00.530 has a square base. 00:03:00.530 --> 00:03:02.910 How many cubic feet of marble-- yes I have to draw 00:03:02.910 --> 00:03:09.616 this-- so at the top I have cube that looks like that. 00:03:09.616 --> 00:03:18.630 It's 1 by 1 by 1, then they have another marble-- that 00:03:18.630 --> 00:03:22.330 looks like this. 00:03:22.330 --> 00:03:26.280 It's 1 high and then, of course, it's 2 wide. 00:03:26.280 --> 00:03:28.700 Right, so let's just start doing the volumes immediately. 00:03:28.700 --> 00:03:30.230 The volume of the top one is 1. 00:03:30.230 --> 00:03:32.230 The volume of the second one is what? 00:03:32.230 --> 00:03:33.730 It's 2 by 2 by 1. 00:03:33.730 --> 00:03:37.360 It's 2 times 2 times 1 which equals 4. 00:03:37.360 --> 00:03:40.704 The volume of the next one is what? 00:03:40.704 --> 00:03:42.670 It goes out like this. 00:03:47.460 --> 00:03:50.570 It's 3 by 3 by 1, right? 00:03:50.570 --> 00:03:53.710 So what is its volume? 00:03:53.710 --> 00:03:56.982 3 by 3 by 1. 00:03:56.982 --> 00:03:58.870 So it equals 9. 00:03:58.870 --> 00:04:00.880 I think you start seeing the pattern here. 00:04:00.880 --> 00:04:02.820 And then finally we have the fourth one. 00:04:02.820 --> 00:04:06.390 The fourth one looks like that. 00:04:06.390 --> 00:04:09.180 I'm not going to draw it fully, you get the point. 00:04:09.180 --> 00:04:15.250 It's going to be 4 by 4 by 1. 00:04:15.250 --> 00:04:16.970 Which is equal to 16. 00:04:16.970 --> 00:04:21.480 So the volume of all of them combined is 1 plus 4 plus 9 00:04:21.480 --> 00:04:26.980 plus 16 that's 5 plus 9 plus 16. 00:04:26.980 --> 00:04:28.440 14 plus 16. 00:04:28.440 --> 00:04:32.790 That equals 30 and that's choice C. 00:04:32.790 --> 00:04:35.290 Next problem. 00:04:35.290 --> 00:04:41.890 Problem 11. 00:04:41.890 --> 00:04:48.780 If x and y are positive integers and 4 to the 2x is 00:04:48.780 --> 00:04:52.600 equal to 2y, what is x in terms of y? 00:04:52.600 --> 00:04:55.150 Whenever you see something like this-- when you see a 4 00:04:55.150 --> 00:04:57.550 and a 2-- just convert them all to the same base. 00:04:57.550 --> 00:05:02.070 So how do you write 4 as 2 to some power is 4, right? 00:05:02.070 --> 00:05:07.700 How do you write 4 as an exponential expression with 2? 00:05:07.700 --> 00:05:09.240 My brain is malfunctioning. 00:05:09.240 --> 00:05:11.870 Well that's the same thing is 2 squared, right? 00:05:11.870 --> 00:05:16.695 So that's 2 squared times 2 to the x is equal to 2 the y. 00:05:16.695 --> 00:05:19.280 And we add exponents when we multiply two numbers 00:05:19.280 --> 00:05:19.965 with the same base. 00:05:19.965 --> 00:05:24.690 So that's 2 to the 2 plus x is equal to y-- sorry is equal to 00:05:24.690 --> 00:05:26.350 2 to the y. 00:05:26.350 --> 00:05:29.780 So 2 plus x must equal y. 00:05:29.780 --> 00:05:32.230 And they do this all the time on the SAT so you should 00:05:32.230 --> 00:05:35.190 really-- these are easy problems if you just remember 00:05:35.190 --> 00:05:37.210 to do this and remember exponent rules. 00:05:37.210 --> 00:05:38.810 And they want x in terms of y. 00:05:38.810 --> 00:05:40.230 So subtract 2 from both sides. 00:05:40.230 --> 00:05:43.250 You get x is equal to y minus 2. 00:05:43.250 --> 00:05:46.570 And that is choice A. 00:05:46.570 --> 00:05:47.400 Next problem. 00:05:47.400 --> 00:05:52.650 Problem 12. 00:05:52.650 --> 00:05:55.960 If the degree measures of the angles of a triangle are the 00:05:55.960 --> 00:06:02.840 ratio of 2 to 3 to 4, how many degrees does the measure of 00:06:02.840 --> 00:06:07.760 the largest angle exceed the measure of the smallest angle? 00:06:07.760 --> 00:06:12.090 So let's say that the smallest angle is-- I don't know-- 00:06:12.090 --> 00:06:15.920 let's say the smallest angle is 2x. 00:06:15.920 --> 00:06:18.130 Then the middle angle is going to be 3x. 00:06:18.130 --> 00:06:20.870 And then the largest angle is going to be 4x. 00:06:20.870 --> 00:06:23.380 If I were to draw a triangle it would look like this. 00:06:23.380 --> 00:06:27.180 2x, 3x, 4x. 00:06:27.180 --> 00:06:27.625 4x, not 45. 00:06:27.625 --> 00:06:29.310 4x. 00:06:29.310 --> 00:06:30.780 And they all have to add up to 180 because they're the angles 00:06:30.780 --> 00:06:32.050 of a triangle. 00:06:32.050 --> 00:06:37.270 So 2x plus 3x x plus 4x is going to equal 180. 00:06:37.270 --> 00:06:37.870 And what is this? 00:06:37.870 --> 00:06:41.200 This is 2 plus 3 which is 5, plus 4 is 9. 00:06:41.200 --> 00:06:42.940 So 9x is equal to 180. 00:06:42.940 --> 00:06:45.990 x is equal to 20. 00:06:45.990 --> 00:06:50.640 So the smallest angle is going to be 40 degrees and the 00:06:50.640 --> 00:06:53.680 largest angle is going to be 80 degrees, right? 00:06:53.680 --> 00:06:54.770 4 times 20. 00:06:54.770 --> 00:06:57.240 And they want to know, by how many degrees does the measure 00:06:57.240 --> 00:06:59.500 of the largest angle exceed the measure 00:06:59.500 --> 00:07:00.340 the smallest angle? 00:07:00.340 --> 00:07:02.180 So it's 80 minus 40. 00:07:02.180 --> 00:07:05.360 The largest minus the smallest. 80 minus 40 is equal 00:07:05.360 --> 00:07:10.130 to 40 and that is choice C. 00:07:10.130 --> 00:07:11.380 Next problem. 00:07:14.974 --> 00:07:16.910 Switch colors. 00:07:16.910 --> 00:07:18.920 Problem 13. 00:07:18.920 --> 00:07:23.150 The rate for a telephone call between city A and city B is 00:07:23.150 --> 00:07:33.400 $0.50 for the first minute and $0.30 for each minute or 00:07:33.400 --> 00:07:34.860 portion thereof. 00:07:34.860 --> 00:07:41.310 So $0.50 for the first minute and then every extra minute it 00:07:41.310 --> 00:07:42.210 charges $0.30. 00:07:42.210 --> 00:07:44.920 So $0.30 for every extra minute. 00:07:44.920 --> 00:07:47.630 So if n is the number of minutes-- the first minute is 00:07:47.630 --> 00:07:50.160 there, so it's every minute above the first minute-- so 00:07:50.160 --> 00:07:51.370 that's n minus 1, right? 00:07:51.370 --> 00:07:54.590 If you do it for 2 minutes you get charged $0.50, and then 2 00:07:54.590 --> 00:07:58.130 minus 1 is 1, so then plus $0.30. 00:07:58.130 --> 00:08:01.150 And this is for n minutes-- I'm assuming n is minutes. 00:08:01.150 --> 00:08:02.860 Which of the following functions describes the cost, 00:08:02.860 --> 00:08:05.160 in dollars, of a phone call between the two cities that 00:08:05.160 --> 00:08:07.780 lasts for n minutes, if n is a positive integer? 00:08:07.780 --> 00:08:09.460 All right, well. 00:08:09.460 --> 00:08:10.600 What did I write here? 00:08:10.600 --> 00:08:11.950 What choice is that? 00:08:11.950 --> 00:08:13.980 That is choice D. 00:08:13.980 --> 00:08:16.070 And once again, how did I think about this? 00:08:16.070 --> 00:08:18.890 If I say the cost of n minutes-- it's a function of n 00:08:18.890 --> 00:08:21.570 minutes-- it equals, well the first minute 00:08:21.570 --> 00:08:23.980 is going to be $0.50. 00:08:23.980 --> 00:08:26.680 And then for every minute after the first minute-- so 00:08:26.680 --> 00:08:29.210 every minute after the first minute can be represented by n 00:08:29.210 --> 00:08:30.370 minus 1, right? 00:08:30.370 --> 00:08:34.299 The second minute-- so 2 minus 1-- I get charged 1 minute 00:08:34.299 --> 00:08:37.770 after the first minute, if I speak for 2 minutes. 00:08:37.770 --> 00:08:39.720 For every minute after the first minute, I'm going to get 00:08:39.720 --> 00:08:40.840 charged $0.30. 00:08:40.840 --> 00:08:42.750 So that's how we did it. 00:08:42.750 --> 00:08:44.169 Problem 14. 00:08:44.169 --> 00:08:46.750 I don't know if I have-- actually let me just do this 00:08:46.750 --> 00:08:48.840 next video because I have to do just three more problems so 00:08:48.840 --> 00:08:50.740 I might as well just do a video for it. 00:08:50.740 --> 00:08:52.130 I'll see you.

SAT Prep: Test 8 Section 8 Part 1

https://www.youtube.com/watch?v=q2TqEsD1t4U

vtt

https://www.youtube.com/api/timedtext?v=q2TqEsD1t4U&ei=YmeUZeb6MtfLp-oPkZiW2AY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=6AED09BA462111F7C3D6482F2121145521C9C4E6.E6E0DE678F5A6737FD3197A10C4DCCD7393EA2F4&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.800 --> 00:00:03.310 We're on the last section of the last test. So this is the 00:00:03.310 --> 00:00:04.730 home stretch. 00:00:04.730 --> 00:00:06.670 Problem 1. 00:00:06.670 --> 00:00:10.040 A restaurant menu lists 8 dinners and 3 desserts. 00:00:10.040 --> 00:00:12.220 How many different dinner-dessert combinations 00:00:12.220 --> 00:00:13.400 are possible from this menu? 00:00:13.400 --> 00:00:14.940 So you have 8 dinners. 00:00:14.940 --> 00:00:18.070 And for each of them you can have 3 different desserts. 00:00:18.070 --> 00:00:18.350 Right? 00:00:18.350 --> 00:00:20.500 So that's 8 times 3 combinations. 00:00:20.500 --> 00:00:21.920 And that's 24. 00:00:21.920 --> 00:00:23.350 And you could list them all out, right? 00:00:23.350 --> 00:00:26.420 For dinner 1 you could have 1, 2, 3, desserts. 00:00:26.420 --> 00:00:28.200 For dinner 2 you can have 1, 2, 3, desserts. 00:00:28.200 --> 00:00:31.740 So you would have 3 for each of the 8, 3 possible desserts. 00:00:31.740 --> 00:00:33.830 So that's how you get to 24. 00:00:33.830 --> 00:00:36.030 Problem 2. 00:00:36.030 --> 00:00:47.410 The sum of 3x and 5 is equal to the product of x times 1/3. 00:00:47.410 --> 00:00:50.630 So I just read it and while I read it I wrote it out. 00:00:50.630 --> 00:00:55.960 The sum of 3x and 5 is equal to the product of x and 1/3. 00:00:55.960 --> 00:00:57.090 Which of the following equations gives the 00:00:57.090 --> 00:00:59.830 relationship stated in the problem above? 00:00:59.830 --> 00:01:03.690 Well, exactly as I wrote it, that's choice E. 00:01:03.690 --> 00:01:06.560 x times 1/3, that's the same thing as 1/3 x. 00:01:06.560 --> 00:01:08.010 So that's 1/3 x. 00:01:08.010 --> 00:01:09.840 So that's choice E. 00:01:09.840 --> 00:01:11.090 Problem 3. 00:01:14.190 --> 00:01:16.590 Let me see if I can do it right here. 00:01:16.590 --> 00:01:20.410 A clerk accidentally threw a valuable document into one of 00:01:20.410 --> 00:01:21.365 90 trash cans. 00:01:21.365 --> 00:01:23.150 All right, 1 of 90 trash cans. 00:01:23.150 --> 00:01:25.070 Let's say there are 90 trash cans. 00:01:25.070 --> 00:01:28.330 It is equally likely that the document is in any one of 00:01:28.330 --> 00:01:29.950 these 90 trash cans. 00:01:29.950 --> 00:01:34.820 If exactly 15 of these 90 trash cans are blue, what is 00:01:34.820 --> 00:01:36.890 the probability that the document will be in 00:01:36.890 --> 00:01:38.890 a blue trash can? 00:01:38.890 --> 00:01:43.030 Well, 15 of these 90 are blue. 00:01:43.030 --> 00:01:45.800 So the probability it's going to be in one of those 15 is 00:01:45.800 --> 00:01:49.520 going to be is essentially 15/90. 00:01:49.520 --> 00:01:50.050 How do I know that? 00:01:50.050 --> 00:01:54.690 Well there's 100% probability it's in 1 of the 90, right? 00:01:54.690 --> 00:01:57.090 And so if I want to know, what is the probability it's in one 00:01:57.090 --> 00:01:58.430 of the blue, I take the fraction of the 00:01:58.430 --> 00:01:59.640 blue over the whole. 00:01:59.640 --> 00:02:01.650 And that's the probability that it's in one of the blue 00:02:01.650 --> 00:02:02.160 trash cans. 00:02:02.160 --> 00:02:06.360 And if I divide the top and the bottom-- let's see 15 goes 00:02:06.360 --> 00:02:08.695 into 90-- 15 goes into 30 2 times, so it 00:02:08.695 --> 00:02:09.600 goes into 90 6 times. 00:02:09.600 --> 00:02:11.610 So this is 1/6. 00:02:11.610 --> 00:02:13.663 Just divide the numerator and the denominator by 15. 00:02:13.663 --> 00:02:17.750 And that's choice C. 00:02:17.750 --> 00:02:20.300 Next problem. 00:02:20.300 --> 00:02:22.380 Problem 4. 00:02:22.380 --> 00:02:28.040 How many different integer pairs satisfy the equation, x 00:02:28.040 --> 00:02:30.480 over y is equal to 1/2? 00:02:30.480 --> 00:02:33.680 So another way of writing this, you could say that x is 00:02:33.680 --> 00:02:35.290 equal to 1/2 y. 00:02:35.290 --> 00:02:38.870 I just multiply both sides by y. 00:02:38.870 --> 00:02:40.470 Or you can multiply both sides by 2. 00:02:40.470 --> 00:02:42.470 And you say 2x is equal to y. 00:02:42.470 --> 00:02:44.350 So how many integer pairs satisfy this? 00:02:44.350 --> 00:02:48.200 Well, any integer I've put in here, I double 00:02:48.200 --> 00:02:48.980 it and I get a y. 00:02:48.980 --> 00:02:49.940 So what of the pairs? 00:02:49.940 --> 00:02:56.720 1, 2, 2, 4, 3, 6, 4, 8, 5, 10. 00:02:56.720 --> 00:02:57.840 They're infinite. 00:02:57.840 --> 00:02:58.720 I can keep going. 00:02:58.720 --> 00:03:00.150 So the answer is more than 4. 00:03:00.150 --> 00:03:01.690 I mean, I listed more than 4 just in the 00:03:01.690 --> 00:03:02.900 span of about 5 seconds. 00:03:02.900 --> 00:03:04.320 So the answer is definitely E. 00:03:04.320 --> 00:03:06.020 There are infinite integer pairs that 00:03:06.020 --> 00:03:08.230 satisfy that equation. 00:03:08.230 --> 00:03:10.790 Next problem. 00:03:10.790 --> 00:03:13.840 Problem 5. 00:03:13.840 --> 00:03:15.890 Let me see what they're asking first. According to the graph 00:03:15.890 --> 00:03:20.170 above, during which of the following 2 month periods did 00:03:20.170 --> 00:03:22.590 Ellen's bookstore sell the least number of books? 00:03:22.590 --> 00:03:23.580 So a 2 month period. 00:03:23.580 --> 00:03:24.930 Interesting. 00:03:24.930 --> 00:03:27.720 So let's draw the graph. 00:03:27.720 --> 00:03:35.810 And so this is June, July, August, 00:03:35.810 --> 00:03:39.033 September, October, November. 00:03:39.033 --> 00:03:46.650 And this is 10, 20, 30, 40 and I'm going to try to draw as 00:03:46.650 --> 00:03:49.750 best as they drew, as best as I can. 00:03:49.750 --> 00:03:52.770 So in June, it looks like they sold a little under, I don't 00:03:52.770 --> 00:03:54.480 know, 10 books. 00:03:54.480 --> 00:03:56.915 June looks something like that. 00:03:56.915 --> 00:03:59.000 In July they sold a ton. 00:03:59.000 --> 00:04:01.270 They sold 40 books, it looks like. 00:04:01.270 --> 00:04:04.391 July looks something like that. 00:04:04.391 --> 00:04:07.685 In August they sold exactly 10 books, it looks like. 00:04:07.685 --> 00:04:09.420 I'll just eyeball it. 00:04:09.420 --> 00:04:13.096 August. September looks like 30, or a little over 30. 00:04:13.096 --> 00:04:16.740 September is like there. 00:04:16.740 --> 00:04:24.790 October is like 25. 00:04:24.790 --> 00:04:30.624 And then November is 30. 00:04:30.624 --> 00:04:32.750 November looks about 30. 00:04:32.750 --> 00:04:36.070 So they say, during which of the following 2 month periods 00:04:36.070 --> 00:04:38.160 did Ellen's bookstore sell the least number of books? 00:04:38.160 --> 00:04:39.670 So let's look at the choices. 00:04:39.670 --> 00:04:44.280 Choice A is June and July. 00:04:44.280 --> 00:04:45.530 Whoops. 00:04:47.150 --> 00:04:48.660 So how many books did she sell in June and July? 00:04:48.660 --> 00:04:49.840 We can eyeball it. 00:04:49.840 --> 00:04:53.260 Let's say that this is 7 and this is 40. 00:04:53.260 --> 00:04:55.870 So this is going to be 47. 00:04:55.870 --> 00:04:58.890 July and August is this period. 00:05:01.770 --> 00:05:03.780 So I can already tell you that July and August is going to be 00:05:03.780 --> 00:05:04.740 higher than June and July. 00:05:04.740 --> 00:05:05.920 Why? 00:05:05.920 --> 00:05:08.430 Because August is higher than June. 00:05:08.430 --> 00:05:09.720 August looks like 10. 00:05:09.720 --> 00:05:10.720 This looks like 40. 00:05:10.720 --> 00:05:14.480 So this is going to be 50. 00:05:14.480 --> 00:05:20.320 And then if we look at August and September and if we say 00:05:20.320 --> 00:05:23.290 that this is 10 and that looks like about-- I 00:05:23.290 --> 00:05:27.580 don't know-- 30, 33. 00:05:27.580 --> 00:05:29.900 So what's 10 plus 33? 00:05:29.900 --> 00:05:31.160 43. 00:05:31.160 --> 00:05:32.850 So that's our winner so far. 00:05:32.850 --> 00:05:34.755 And then they're asking September and October. 00:05:34.755 --> 00:05:35.860 Let me switch colors. 00:05:35.860 --> 00:05:37.782 September and October would be that. 00:05:37.782 --> 00:05:43.655 If that's 33, October looks like about 25. 00:05:43.655 --> 00:05:44.800 So there's a little bit of approximation here. 00:05:44.800 --> 00:05:46.410 So what's 33 plus 25? 00:05:46.410 --> 00:05:49.160 That's 58 and that's actually the most so far. 00:05:49.160 --> 00:05:53.520 And then finally October and November. 00:05:53.520 --> 00:05:55.495 November looks like 30. 00:05:55.495 --> 00:05:57.240 So what's 30 plus 25. 00:05:57.240 --> 00:05:58.450 It's 55. 00:05:58.450 --> 00:06:02.200 So the smallest is definitely August and September. 00:06:02.200 --> 00:06:03.450 Next problem. 00:06:05.470 --> 00:06:07.940 Maybe I'll stick with this white. 00:06:07.940 --> 00:06:09.550 Problem 6. 00:06:09.550 --> 00:06:12.896 OK, they drew us a line. 00:06:12.896 --> 00:06:20.730 And they say that this is point A, this is point B and 00:06:20.730 --> 00:06:22.420 this is point C. 00:06:22.420 --> 00:06:27.750 In the figure above, AC equals 24. 00:06:27.750 --> 00:06:32.000 And they tell us that AB is equal to BC. 00:06:32.000 --> 00:06:35.700 So this side, this length has to equal that length and 00:06:35.700 --> 00:06:37.390 combined they add up to 24. 00:06:37.390 --> 00:06:39.760 So we know that's going to be 12 and that's 00:06:39.760 --> 00:06:41.180 also going to be 12. 00:06:41.180 --> 00:06:44.140 Because they have to add up to 24 and they're equal. 00:06:44.140 --> 00:06:49.320 Point D, not shown, is on the line between A and B-- so 00:06:49.320 --> 00:06:55.470 point D is someplace here-- such that AD is equal to DB. 00:06:55.470 --> 00:06:59.165 So if we put D here, I'll do it in a different color. 00:06:59.165 --> 00:07:05.210 If this is D, they want us-- so 12 is this whole length and 00:07:05.210 --> 00:07:07.930 they're saying AD is equal to DB. 00:07:07.930 --> 00:07:09.780 So what are each of these smaller lengths 00:07:09.780 --> 00:07:10.540 going to be, equal? 00:07:10.540 --> 00:07:13.220 This is equal to this, so this is going to have to be 6 and 00:07:13.220 --> 00:07:15.370 this is going to have to be 6, right? 00:07:15.370 --> 00:07:17.970 Because the two sides are equal and the two lengths are 00:07:17.970 --> 00:07:18.780 equal and they add up to 12. 00:07:18.780 --> 00:07:19.750 Now what are they asking? 00:07:19.750 --> 00:07:20.980 What does DC equal? 00:07:20.980 --> 00:07:24.090 So what is this distance? 00:07:24.090 --> 00:07:25.420 Just to here. 00:07:25.420 --> 00:07:26.100 From there to there. 00:07:26.100 --> 00:07:27.480 OK. 00:07:27.480 --> 00:07:31.600 So we'd have to go 6 to get to B. 00:07:31.600 --> 00:07:35.410 And then we'd have to go 12 more to get to c. 00:07:35.410 --> 00:07:37.880 6 plus 12 and that gets us 18. 00:07:37.880 --> 00:07:40.000 So this distance is 18. 00:07:40.000 --> 00:07:40.880 Not too bad. 00:07:40.880 --> 00:07:44.320 This problem is that's answer D. 00:07:44.320 --> 00:07:45.570 Problem 7. 00:07:49.100 --> 00:07:55.610 If n is a positive integer then 6 times 10 to the 00:07:55.610 --> 00:08:05.030 negative n, plus 1 times 10 to the negative 00:08:05.030 --> 00:08:07.820 n must equal what? 00:08:07.820 --> 00:08:08.740 So let's simplify that. 00:08:08.740 --> 00:08:13.210 This is essentially-- I mean they wrote it a little 00:08:13.210 --> 00:08:16.250 complicated-- but we have six 10s to the negative n here. 00:08:16.250 --> 00:08:18.260 And then we have 1 more, right? 00:08:18.260 --> 00:08:19.870 I mean 6 times 10 to the negative n. 00:08:19.870 --> 00:08:22.690 You could rewrite this as 10 to the negative n plus 10 to 00:08:22.690 --> 00:08:26.040 the negative n plus 10 to the negative n plus 10 to the 00:08:26.040 --> 00:08:27.595 negative n-- not that I'd recommend you to do this on 00:08:27.595 --> 00:08:30.540 the exam-- 10 to the negative n plus 10 to the negative n. 00:08:30.540 --> 00:08:30.970 How many have I drawn? 00:08:30.970 --> 00:08:33.130 1, 2, 3, 4, 5, 6. 00:08:33.130 --> 00:08:33.360 Right. 00:08:33.360 --> 00:08:37.090 That's 6 times 10 to the negative n. 00:08:37.090 --> 00:08:40.980 And to that we add 1 more 10 to the negative n. 00:08:40.980 --> 00:08:43.010 So how many 10 to the negative ns do we have? 00:08:43.010 --> 00:08:44.440 We have 7 now. 00:08:44.440 --> 00:08:45.100 Right? 00:08:45.100 --> 00:08:49.470 So it's 7 times 10 to the negative n. 00:08:49.470 --> 00:08:50.890 And that is not a choice. 00:08:50.890 --> 00:08:52.860 So what's another way of writing 10 to the negative n? 00:08:52.860 --> 00:08:54.870 Well, that's the same thing as 1 over 10 to the n. 00:08:54.870 --> 00:08:58.500 So that's 7 times 1 over 10 to the n. 00:08:58.500 --> 00:09:02.600 Which is, of course, equal to 7 over 10 to the n. 00:09:02.600 --> 00:09:03.850 And that is choice B. 00:09:06.480 --> 00:09:07.840 Next problem. 00:09:07.840 --> 00:09:10.170 Actually, I'll do it in the next video because I have less 00:09:10.170 --> 00:09:11.040 than a minute left. 00:09:11.040 --> 00:09:12.450 I'll see--

SAT Prep: Test 8 Section 8 Part 3

https://www.youtube.com/watch?v=Ghy5jasP6Ek

vtt

https://www.youtube.com/api/timedtext?v=Ghy5jasP6Ek&ei=ZWeUZeahJcmup-oPqYipwAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249813&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E57BE98970526CE1C133B00D262E0C1810A56A89.E771E787231CD7929CEA6D75E26E2B43463C5EA4&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:02.830 This might be the last video if I can squeeze in three 00:00:02.830 --> 00:00:04.676 problems. So problem 14. 00:00:04.676 --> 00:00:07.650 The last video at least for SAT prep. 00:00:07.650 --> 00:00:08.915 Hopefully I will keep doing videos. 00:00:13.490 --> 00:00:15.630 OK, it looks something like this. 00:00:15.630 --> 00:00:17.145 And then it comes back down like this. 00:00:17.145 --> 00:00:18.640 And what are they telling us? 00:00:18.640 --> 00:00:25.500 They're saying that this is line l, line m. 00:00:25.500 --> 00:00:27.110 This is x. 00:00:27.110 --> 00:00:30.820 This is z degrees, this is y degrees. 00:00:30.820 --> 00:00:32.100 Looks like we're going to have to play the angle game. 00:00:32.100 --> 00:00:37.760 In the figure above, line l is parallel to line m. 00:00:37.760 --> 00:00:41.620 What does z equal in terms of x and y? 00:00:41.620 --> 00:00:41.930 OK. 00:00:41.930 --> 00:00:44.270 So these two lines are parallel. 00:00:44.270 --> 00:00:46.510 If this is y degrees, what other 00:00:46.510 --> 00:00:47.810 angles are also y degrees? 00:00:47.810 --> 00:00:49.840 Well, what's the corresponding angle? 00:00:49.840 --> 00:00:50.630 Right here. 00:00:50.630 --> 00:00:52.540 Well, this is also going to be y degrees, right? 00:00:52.540 --> 00:00:53.530 These lines are parallel. 00:00:53.530 --> 00:00:54.790 This is a transversal. 00:00:54.790 --> 00:00:56.030 These are corresponding angles. 00:00:56.030 --> 00:00:56.840 This is going to be y. 00:00:56.840 --> 00:00:57.740 And it makes sense too. 00:00:57.740 --> 00:01:00.230 I mean, if you tilted this angle, you would visually see 00:01:00.230 --> 00:01:01.770 that these angles would be the same. 00:01:01.770 --> 00:01:03.850 If this is y, what is this angle? 00:01:03.850 --> 00:01:06.870 Well, they're opposite, so this is also going to be y. 00:01:06.870 --> 00:01:11.460 And then we have z plus x plus y has to equal 180. 00:01:11.460 --> 00:01:13.150 Because they're all in the same triangle. 00:01:13.150 --> 00:01:16.860 z plus x plus y is equal to 180. 00:01:16.860 --> 00:01:18.105 We want to solve for z. 00:01:18.105 --> 00:01:21.320 So subtract x and y from both sides, and you get z is equal 00:01:21.320 --> 00:01:25.000 to 180 minus x minus y. 00:01:25.000 --> 00:01:26.380 And that is choice E. 00:01:29.170 --> 00:01:30.420 Next problem. 00:01:33.150 --> 00:01:36.650 Problem 15. 00:01:36.650 --> 00:01:49.180 If n over n minus 1 times 1/n times n over n plus 00:01:49.180 --> 00:01:52.420 1 is equal to 5/k. 00:01:52.420 --> 00:01:55.290 For positive integers n and k, what is the value of k? 00:01:55.290 --> 00:01:57.050 So these are positive integers. 00:01:57.050 --> 00:02:00.140 So before multiplying all of this out, we can simplify a 00:02:00.140 --> 00:02:00.570 little bit. 00:02:00.570 --> 00:02:03.620 This n can cancel out with this n. 00:02:03.620 --> 00:02:05.660 And now let's see if we can. 00:02:05.660 --> 00:02:08.020 What does the top-- what does this left-hand side become? 00:02:08.020 --> 00:02:10.169 And the numerator always left-- this is just a 1 now. 00:02:10.169 --> 00:02:11.180 This is a 1. 00:02:11.180 --> 00:02:13.030 So we're left with 1 times 1 times n. 00:02:13.030 --> 00:02:19.190 So that's n over n minus 1, times 1-- I can ignore that 00:02:19.190 --> 00:02:20.440 1-- times n plus 1. 00:02:23.110 --> 00:02:28.570 Is equal to 5 over k. 00:02:28.570 --> 00:02:29.540 So what are they asking? 00:02:29.540 --> 00:02:33.010 Well, what is the value of k? 00:02:33.010 --> 00:02:37.330 Well, we can say that n is equal to 5. 00:02:37.330 --> 00:02:39.520 Or let's assume that n is equal to 5. 00:02:39.520 --> 00:02:40.950 Because we don't know definitely that 00:02:40.950 --> 00:02:41.710 n is equal to 5. 00:02:41.710 --> 00:02:44.860 It could be some multiple-- I'll show you. 00:02:44.860 --> 00:02:48.260 Let's assume that n is equal to 5. 00:02:48.260 --> 00:02:51.560 If n is equal to 5, then what is k? 00:02:51.560 --> 00:02:55.000 Well, then k would be this denominator. 00:02:55.000 --> 00:02:55.420 Right? 00:02:55.420 --> 00:02:57.710 If n is 5, then k is this. 00:02:57.710 --> 00:03:04.700 Then k would be-- so this could be 5 over 5 minus 1 00:03:04.700 --> 00:03:06.900 times 5 plus 1. 00:03:06.900 --> 00:03:11.720 And that equals 5 over 4 times 6, which is 00:03:11.720 --> 00:03:15.160 equal to 5 over 24. 00:03:15.160 --> 00:03:18.150 So this could be 5 over 24. 00:03:18.150 --> 00:03:21.240 And they're all positive integers so k is 24. 00:03:21.240 --> 00:03:22.120 k is 24. 00:03:22.120 --> 00:03:23.710 And that's choice C. 00:03:23.710 --> 00:03:25.900 So the trick here is really once again-- simplify a little 00:03:25.900 --> 00:03:27.440 bit, multiply it out and then pattern matching. 00:03:27.440 --> 00:03:30.146 Let me just set n is equal to 5. 00:03:30.146 --> 00:03:32.490 If n is equal to 5, then what is k? 00:03:32.490 --> 00:03:34.980 It's 5 minus 1 times 5 plus 1. 00:03:34.980 --> 00:03:36.850 It's just pattern matching. 00:03:36.850 --> 00:03:39.290 Next problem. 00:03:39.290 --> 00:03:41.010 Problem 6. 00:03:41.010 --> 00:03:43.530 I will do it in magenta because this is the last 00:03:43.530 --> 00:03:45.270 problem in the book. 00:03:45.270 --> 00:03:49.170 To celebrate a colleague's graduation the m coworkers in 00:03:49.170 --> 00:03:52.530 an office agreed to contribute equally to a catered lunch 00:03:52.530 --> 00:03:54.810 that cost a total of y dollars. 00:03:54.810 --> 00:04:00.070 So there's m workers, and the total price is y 00:04:00.070 --> 00:04:02.410 dollars for the lunch. 00:04:02.410 --> 00:04:05.770 If p of the workers fail to contribute, which of the 00:04:05.770 --> 00:04:08.260 following represents the additional amount in dollars 00:04:08.260 --> 00:04:10.490 that each of the remaining coworkers must contribute to 00:04:10.490 --> 00:04:12.050 pay for the lunch? 00:04:12.050 --> 00:04:14.620 The additional amount in dollars. 00:04:14.620 --> 00:04:18.670 So if everyone paid, how much would we have to pay? 00:04:18.670 --> 00:04:20.850 Well, the total lunch is y, right? 00:04:20.850 --> 00:04:24.730 So if everyone was a good coworker, we would each have 00:04:24.730 --> 00:04:28.820 to pay y divided by the number of coworkers. 00:04:28.820 --> 00:04:30.680 This is the ideal situation. 00:04:30.680 --> 00:04:33.340 But we know some of the coworkers didn't pay. 00:04:33.340 --> 00:04:34.200 p didn't pay. 00:04:34.200 --> 00:04:36.500 So how many are we going to have to divvy it up by now? 00:04:36.500 --> 00:04:40.260 So then that means only m minus p paid. 00:04:40.260 --> 00:04:43.120 These are the deadbeats that did not pay for the lunch. 00:04:43.120 --> 00:04:45.520 So only the m minus p paid. 00:04:45.520 --> 00:04:49.320 So now we have to actually divide the y dollars between a 00:04:49.320 --> 00:04:51.610 smaller group of people who actually paid. 00:04:51.610 --> 00:04:53.140 And the smaller group of people who actually 00:04:53.140 --> 00:04:55.420 paid is m minus p. 00:04:55.420 --> 00:04:57.220 So if you wanted to figure out-- and this is going to be 00:04:57.220 --> 00:04:57.930 a larger number. 00:04:57.930 --> 00:04:58.250 Why? 00:04:58.250 --> 00:04:59.750 Because its denominator is smaller. 00:04:59.750 --> 00:05:01.140 When the denominator is smaller and you have the same 00:05:01.140 --> 00:05:03.180 numerator, there's going to be a larger number. 00:05:03.180 --> 00:05:07.120 So if you want to know what is the additional amount you have 00:05:07.120 --> 00:05:11.240 to pay-- well, this is how much we're having to pay, 00:05:11.240 --> 00:05:13.080 which is a larger amount than how much we would have paid if 00:05:13.080 --> 00:05:14.400 everyone paid. 00:05:14.400 --> 00:05:15.750 So how much are we paying extra? 00:05:15.750 --> 00:05:18.000 Well, we subtract this from this. 00:05:18.000 --> 00:05:20.110 This is how much we end up paying. 00:05:20.110 --> 00:05:23.200 And we subtract how much we should have paid. 00:05:23.200 --> 00:05:24.560 And we'll get the additional amount. 00:05:24.560 --> 00:05:26.410 Let me draw a line so we don't get confused. 00:05:29.960 --> 00:05:32.390 That doesn't look like one of the choices, so let's actually 00:05:32.390 --> 00:05:33.400 get a common denominator. 00:05:33.400 --> 00:05:38.580 Common denominator would be m times m minus p. 00:05:38.580 --> 00:05:40.720 I just multiply the denominators. 00:05:40.720 --> 00:05:42.140 So y over m minus p. 00:05:42.140 --> 00:05:47.110 That's the same thing as m y over m times m minus p. 00:05:47.110 --> 00:05:50.040 I just multiply the numerator and the denominator by m. 00:05:50.040 --> 00:05:55.600 And this is minus-- m minus p times y. 00:06:01.610 --> 00:06:03.100 For this one I just multiplied the numerator and the 00:06:03.100 --> 00:06:05.250 denominator by m minus p. 00:06:05.250 --> 00:06:07.066 I just found a common denominator and added the 00:06:07.066 --> 00:06:09.090 fractions, or subtracted the fractions. 00:06:09.090 --> 00:06:12.240 And so, the denominator stays. 00:06:12.240 --> 00:06:14.810 m times m minus p. 00:06:14.810 --> 00:06:16.970 Let me see if I can simplify the numerator. 00:06:16.970 --> 00:06:25.250 That becomes m y minus m y plus py. 00:06:25.250 --> 00:06:27.270 Minus times a minus here is a plus. 00:06:27.270 --> 00:06:28.280 Plus py. 00:06:28.280 --> 00:06:29.600 These cancel out. 00:06:29.600 --> 00:06:35.800 So you get py over m times m minus p. 00:06:35.800 --> 00:06:41.680 And that is choice E. 00:06:41.680 --> 00:06:44.470 py over m times m minus p. 00:06:44.470 --> 00:06:49.240 And we have now done something like, what, 8 tests. 00:06:49.240 --> 00:06:54.170 8 tests and 54 problems per test. So that's 54 times 8. 00:06:54.170 --> 00:06:55.940 54 times 8. 00:06:55.940 --> 00:06:56.300 That's, what? 00:06:56.300 --> 00:07:00.180 432 SAT problems. And I think you're ready to go take the 00:07:00.180 --> 00:07:03.200 SAT and get a perfect score. 00:07:03.200 --> 00:07:03.760 And let me know if you do. 00:07:03.760 --> 00:07:05.100 That would be very exciting. 00:07:05.100 --> 00:07:05.870 All right. 00:07:05.870 --> 00:07:08.070 I'll see you in, I guess, other videos 00:07:08.070 --> 00:07:09.240 that are not SAT related. 00:07:09.240 --> 00:07:10.820 Or maybe when a new book comes out I'll 00:07:10.820 --> 00:07:12.530 have to do this again.

SAT Prep: Test 8 Section 5 Part 1

https://www.youtube.com/watch?v=yKiU7kzjTQw

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en

WEBVTT Kind: captions Language: en 00:00:00.600 --> 00:00:04.190 We're on test 8, section 5, page 855. 00:00:04.190 --> 00:00:05.830 Problem 1. 00:00:05.830 --> 00:00:15.720 If x over x minus 2 is equal to 39 over 37, 00:00:15.720 --> 00:00:17.520 then what does x equal? 00:00:17.520 --> 00:00:19.760 And here you could try to solve this situation, but 00:00:19.760 --> 00:00:21.130 there's just a pattern here that you could actually 00:00:21.130 --> 00:00:22.370 identify immediately. 00:00:22.370 --> 00:00:26.400 If x is 39, then what is x minus 2? 00:00:26.400 --> 00:00:28.080 x minus 2's going to be 37. 00:00:28.080 --> 00:00:30.110 So you could immediately just try out these numbers. 00:00:30.110 --> 00:00:31.920 That just kind of struck you because 37 is 00:00:31.920 --> 00:00:33.495 two less than 39. 00:00:33.495 --> 00:00:35.750 x minus 2 is 2 less than x. 00:00:35.750 --> 00:00:37.450 So x must be 39. 00:00:37.450 --> 00:00:39.450 You could do that problem really fast. 00:00:39.450 --> 00:00:41.420 If you don't understand what I'm saying you could kind of 00:00:41.420 --> 00:00:43.800 solve it with the traditional method, cross multiply. 00:00:43.800 --> 00:00:50.190 You could say 37 times x, 37x is equal to 39 00:00:50.190 --> 00:00:51.810 times x minus 2. 00:00:55.190 --> 00:01:04.379 So you get 37x is equal to 39x minus, what is that, minus 78. 00:01:04.379 --> 00:01:09.510 And then you could subtract 39x from both sides and you 00:01:09.510 --> 00:01:12.990 get minus 2x is equal to minus 78. 00:01:12.990 --> 00:01:16.400 Divide both sides by 2 and you get x is equal to 39. 00:01:16.400 --> 00:01:19.310 But that would have taken a lot of valuable time. 00:01:19.310 --> 00:01:22.040 The first problem with any section on the SAT, if you 00:01:22.040 --> 00:01:23.900 can't do it in 10 seconds you might be doing 00:01:23.900 --> 00:01:25.560 it the wrong way. 00:01:25.560 --> 00:01:27.690 It's normally a very, very quick problem. 00:01:27.690 --> 00:01:31.610 And that's why this probably was not the correct 00:01:31.610 --> 00:01:32.250 way of doing it. 00:01:32.250 --> 00:01:36.080 You could have just said well if x is 39, x minus 2 is 37. 00:01:36.080 --> 00:01:39.180 Next problem. 00:01:39.180 --> 00:01:41.910 Problem 2. 00:01:41.910 --> 00:01:46.440 Students in advanced biology class-- so they have boys, 00:01:46.440 --> 00:01:47.990 girls, total. 00:01:56.050 --> 00:02:00.570 And then on this side, this juniors, 00:02:00.570 --> 00:02:03.100 seniors, and then total. 00:02:03.100 --> 00:02:16.730 And then they write-- this is k, n, m, r, s, t, w, x, z. 00:02:16.730 --> 00:02:19.070 In the table above, each letter represents the number 00:02:19.070 --> 00:02:20.210 of students in that category. 00:02:20.210 --> 00:02:23.360 k would be the number of junior boys in the advanced 00:02:23.360 --> 00:02:24.830 biology class. 00:02:24.830 --> 00:02:29.610 Which of the following must be equal to z? 00:02:29.610 --> 00:02:31.970 So z is the total number of kids, right? 00:02:31.970 --> 00:02:34.710 So there's a couple of things that could be equal to z. 00:02:34.710 --> 00:02:38.160 It could be w plus x. 00:02:38.160 --> 00:02:39.810 That's not a choice. 00:02:39.810 --> 00:02:43.280 It could also be m plus t. 00:02:43.280 --> 00:02:45.130 But that's not a choice either, right, m 00:02:45.130 --> 00:02:46.890 plus t isn't a choice. 00:02:46.890 --> 00:02:49.990 And the other way is you could just add up all of the kids 00:02:49.990 --> 00:02:53.120 that are in each of the categories. 00:02:53.120 --> 00:02:55.450 So z is the total of all the kids. 00:02:55.450 --> 00:02:57.930 So if you say the number of junior boys plus senior boys 00:02:57.930 --> 00:03:03.320 plus junior girls plus senior girls, so that's k plus 00:03:03.320 --> 00:03:05.740 n plus r plus s. 00:03:05.740 --> 00:03:09.750 That's all the students in the class, right? 00:03:09.750 --> 00:03:11.390 And that must equal z. 00:03:11.390 --> 00:03:14.610 And that is choice E. 00:03:14.610 --> 00:03:15.860 Next problem. 00:03:21.710 --> 00:03:24.450 They have a line here like that. 00:03:30.800 --> 00:03:32.170 And then what do they tell us? 00:03:32.170 --> 00:03:38.350 They tell us that this right here is 25 degrees. 00:03:38.350 --> 00:03:46.880 This is a, b, c, this is 60 degrees, and this is x. 00:03:46.880 --> 00:03:48.990 In the triangle ABC above, what is the value of x. 00:03:48.990 --> 00:03:50.510 This is just the angle game. 00:03:50.510 --> 00:03:53.280 So the first thing we want to figure out is this angle. 00:03:53.280 --> 00:03:55.600 And we know that this angle plus 60 is going to be 180, 00:03:55.600 --> 00:03:56.750 right, because it's supplementary. 00:03:56.750 --> 00:04:00.980 They kind of combined to form 180 degrees, or they kind of 00:04:00.980 --> 00:04:03.220 created a line to go halfway around the circle. 00:04:03.220 --> 00:04:05.790 So this has to be 120. 00:04:05.790 --> 00:04:08.100 Because this plus 60 is 180. 00:04:08.100 --> 00:04:12.340 And then we see that 25 plus 120 plus x have to be 180 00:04:12.340 --> 00:04:14.280 because they're all in the same triangle. 00:04:14.280 --> 00:04:21.230 So 25 plus 120 plus x is going to be equal to 180 as well. 00:04:21.230 --> 00:04:26.250 So 145 plus x is equal to 180. 00:04:26.250 --> 00:04:28.130 Subtract 145 from both sides. 00:04:28.130 --> 00:04:30.620 You get x is equal to 35. 00:04:30.620 --> 00:04:32.080 80 minus 45. 00:04:32.080 --> 00:04:32.690 Right. 00:04:32.690 --> 00:04:35.660 35 and that's choice C. 00:04:35.660 --> 00:04:36.910 Next problem. 00:04:38.890 --> 00:04:41.040 Problem 4. 00:04:41.040 --> 00:04:42.920 The Martin's refrigerator is broken and will 00:04:42.920 --> 00:04:46.570 cost $300 to fix. 00:04:46.570 --> 00:04:50.827 And new energy efficient refrigerator costing $900, so 00:04:50.827 --> 00:04:54.640 let's say this other refrigerator costs $900. 00:04:54.640 --> 00:04:57.850 It'll save 15 dollars a month on the electric bill. 00:05:00.540 --> 00:05:03.410 If they buy the new refrigerator in x months, the 00:05:03.410 --> 00:05:06.460 Martin's will have saved an amount equal to the difference 00:05:06.460 --> 00:05:09.900 between the cost of the new refrigerator and the cost of 00:05:09.900 --> 00:05:10.800 the old one. 00:05:10.800 --> 00:05:13.990 What is the value of x? 00:05:13.990 --> 00:05:19.180 So they're saying that in x months this new refrigerator-- 00:05:19.180 --> 00:05:21.760 they're saying if in x months the Martin's would have saved 00:05:21.760 --> 00:05:25.580 an amount equal to the difference between the cost of 00:05:25.580 --> 00:05:28.950 the new refrigerator and the cost of fixing the old one. 00:05:28.950 --> 00:05:30.910 So what's the cost of the new refrigerator? 00:05:30.910 --> 00:05:32.970 It's $900. 00:05:32.970 --> 00:05:35.370 And what's the cost of fixing the old one? 00:05:35.370 --> 00:05:36.970 It's $300. 00:05:36.970 --> 00:05:39.170 So that's the difference between the cost of fixing the 00:05:39.170 --> 00:05:41.110 new refrigerator and the cost of fixing the old one. 00:05:41.110 --> 00:05:43.220 And they say that that's how much they will 00:05:43.220 --> 00:05:44.970 have saved in x months. 00:05:44.970 --> 00:05:46.700 So they're going to save that in x months. 00:05:46.700 --> 00:05:48.910 They save $15 a month, right? 00:05:48.910 --> 00:05:54.250 So $15 times x months has to be equal to the difference in 00:05:54.250 --> 00:05:56.850 price, essentially, of fixing or buying. 00:05:56.850 --> 00:05:59.120 And this is a times, not a minus. 00:05:59.120 --> 00:06:04.330 So you get $600 is equal to 15x. 00:06:04.330 --> 00:06:08.240 So x is equal to 600 over 15. 00:06:08.240 --> 00:06:13.450 So x is equal to 40-- 15 goes in 60 four times, it goes into 00:06:13.450 --> 00:06:14.920 600 40 times. 00:06:14.920 --> 00:06:16.090 So that's choice D. 00:06:16.090 --> 00:06:19.140 It takes 40 months for them to break even. 00:06:19.140 --> 00:06:21.900 Next problem. 00:06:21.900 --> 00:06:25.040 Problem 5. 00:06:25.040 --> 00:06:29.690 The perimeter of an equilateral triangle ABC is 3 00:06:29.690 --> 00:06:32.860 times the perimeter of an equilateral triangle DEF. 00:06:32.860 --> 00:06:39.040 Let me draw them real fast. This is one of them, and the 00:06:39.040 --> 00:06:40.290 other one actually is smaller. 00:06:43.860 --> 00:06:54.350 If I called this one ABC, this one is DEF. 00:06:54.350 --> 00:06:56.690 They say the perimeter of this one is 3 times this one-- and 00:06:56.690 --> 00:07:00.170 they're both equilateral, right, so this is x, x, x. 00:07:00.170 --> 00:07:02.930 And so we don't know what these sides are. 00:07:02.930 --> 00:07:06.660 But the perimeter of this one will be 3x-- oh, 00:07:06.660 --> 00:07:07.680 then they tell us. 00:07:07.680 --> 00:07:09.880 If the perimeter of DEF is 10, what is the length 00:07:09.880 --> 00:07:11.810 of one side of ABC? 00:07:11.810 --> 00:07:17.640 So perimeter of triangle DEF is equal to 10. 00:07:17.640 --> 00:07:20.320 And they tell us that the perimeter of 00:07:20.320 --> 00:07:22.650 ABC is 3 times this. 00:07:22.650 --> 00:07:27.150 So that means that the perimeter of triangle ABC is 00:07:27.150 --> 00:07:31.550 going to be 3 times this, so it's 30, right? 00:07:31.550 --> 00:07:34.080 So the perimeter of this triangle is 30 and it's three 00:07:34.080 --> 00:07:36.860 equal sides, so each side has to be 10. 00:07:36.860 --> 00:07:38.620 30 divided by 3. 00:07:38.620 --> 00:07:42.290 And that's choice B. 00:07:42.290 --> 00:07:43.540 Next problem. 00:07:48.720 --> 00:07:49.995 Problem 6. 00:07:49.995 --> 00:07:56.210 A machine mints coins at the rate of one coin per second. 00:07:56.210 --> 00:08:02.970 If it does this for 10 hours each day, approximately how 00:08:02.970 --> 00:08:09.600 many days will it take the machine to mint 360,000 coins? 00:08:09.600 --> 00:08:10.860 How many days? 00:08:10.860 --> 00:08:13.880 So let's say how much does it produce in one day? 00:08:13.880 --> 00:08:20.560 So let's see, in one day is equal to-- so it'll be 00:08:20.560 --> 00:08:28.640 producing for 10 hours per day times how 00:08:28.640 --> 00:08:31.620 many seconds per hour? 00:08:31.620 --> 00:08:36.610 Times 3,600 seconds per hour. 00:08:36.610 --> 00:08:37.850 And how did I get 3,600? 00:08:37.850 --> 00:08:41.340 60 minutes per hour times 60 seconds per minute, right, 00:08:41.340 --> 00:08:46.140 that gives you 3,600 times one coin per second. 00:08:46.140 --> 00:08:47.360 And the units actually cancel out. 00:08:47.360 --> 00:08:50.800 The hours cancel out with hours, and then you get 00:08:50.800 --> 00:08:52.400 seconds cancel out with seconds, and you're left with 00:08:52.400 --> 00:08:53.860 coins per day. 00:08:53.860 --> 00:09:00.330 So that equals 10 times 3,600 is 36, 1, 2, add another 0, 00:09:00.330 --> 00:09:05.490 3,600 coins per day. 00:09:05.490 --> 00:09:08.690 So the amount of coins I produce in d days is going to 00:09:08.690 --> 00:09:15.680 be 3,600 times d, times the number of days. 00:09:15.680 --> 00:09:20.470 And that we're saying has to be equal to 360,000. 00:09:20.470 --> 00:09:28.600 Well immediately let's divide both sides-- d will be 360,000 00:09:28.600 --> 00:09:32.150 divided by 3,600. 00:09:32.150 --> 00:09:34.150 So that's cancels out with that. 00:09:34.150 --> 00:09:38.420 You have 360 divided by 36, well that's just equal to 10. 00:09:38.420 --> 00:09:41.600 So it will take 10 days to produce 360,000 coins, and 00:09:41.600 --> 00:09:42.930 that's choice A. 00:09:42.930 --> 00:09:44.780 I'll see you in the next video.

SAT Prep: Test 8 Section 5 Part 2

https://www.youtube.com/watch?v=4-JYxNfqp8g

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en

WEBVTT Kind: captions Language: en 00:00:00.670 --> 00:00:03.700 We're on problem number 7. 00:00:03.700 --> 00:00:08.189 If the average of x and 3x is 12, what is the value of x? 00:00:08.189 --> 00:00:12.370 So the average, so x plus 3x-- and I'm averaging two numbers 00:00:12.370 --> 00:00:15.270 so divide by 2-- that's going to be equal to 12. 00:00:15.270 --> 00:00:16.180 So we just solve for this. 00:00:16.180 --> 00:00:19.800 Multiply both sides of the equation by 2 and you get 2 00:00:19.800 --> 00:00:22.090 times, times 2, this cancels with this. 00:00:22.090 --> 00:00:26.020 You get x plus 3x is equal to 24. 00:00:26.020 --> 00:00:26.890 And what's x plus 3x. 00:00:26.890 --> 00:00:28.670 That's 4x, right? 00:00:28.670 --> 00:00:30.645 4x is equal to 24. 00:00:30.645 --> 00:00:35.720 x is equal to 6, and that's choice C. 00:00:35.720 --> 00:00:36.970 Next problem. 00:00:40.080 --> 00:00:42.970 Problem 8. 00:00:42.970 --> 00:00:46.700 At Maple Creek High School, some members of the chess club 00:00:46.700 --> 00:00:50.320 are also on the swim team, and no members of the swim team 00:00:50.320 --> 00:00:52.130 are tenth graders. 00:00:52.130 --> 00:00:54.830 Which of the following must be true. 00:00:54.830 --> 00:00:57.220 This seems like it'll call for a Venn diagram. 00:00:57.220 --> 00:01:02.680 So let's say that that represents the chess club. 00:01:02.680 --> 00:01:04.790 And they say some members of the chess club 00:01:04.790 --> 00:01:06.210 are on the swim team. 00:01:06.210 --> 00:01:08.930 So some members are on the swim team. 00:01:08.930 --> 00:01:12.620 Maybe I should put the swim team in like blue. 00:01:12.620 --> 00:01:16.100 So let's say the swim team. 00:01:16.100 --> 00:01:17.350 That's the swim team. 00:01:19.740 --> 00:01:21.510 And these are the members, right, that are 00:01:21.510 --> 00:01:23.980 in both right here. 00:01:23.980 --> 00:01:27.200 But then it tells us no members of the swim team are 00:01:27.200 --> 00:01:30.170 tenth graders. 00:01:30.170 --> 00:01:32.970 So if I draw another circle for the tenth graders, it 00:01:32.970 --> 00:01:35.560 can't intersect with the swim team, but it could intersect 00:01:35.560 --> 00:01:36.080 with the chess team. 00:01:36.080 --> 00:01:36.540 I don't know. 00:01:36.540 --> 00:01:38.510 I mean it could be like that. 00:01:38.510 --> 00:01:41.240 That could be tenth graders. 00:01:41.240 --> 00:01:42.830 It could be like that. 00:01:42.830 --> 00:01:45.930 Or it could be out here some place. 00:01:45.930 --> 00:01:47.150 But we don't know. 00:01:47.150 --> 00:01:49.880 There could be chess and tenth graders, just not the same 00:01:49.880 --> 00:01:52.440 people who are on the swim team. 00:01:52.440 --> 00:01:53.690 So let's see. 00:01:55.820 --> 00:01:57.180 So which of the following must be true? 00:01:57.180 --> 00:01:58.960 No members of the chess club are tenth graders. 00:01:58.960 --> 00:01:59.370 No. 00:01:59.370 --> 00:02:01.800 This is a situation where you could have some members of the 00:02:01.800 --> 00:02:03.500 chess club who aren't on the swim team who 00:02:03.500 --> 00:02:05.420 could be tenth graders. 00:02:05.420 --> 00:02:09.039 B, some members of the chess club are tenth graders. 00:02:09.039 --> 00:02:11.840 Well some members could be, but we don't know for sure. 00:02:11.840 --> 00:02:12.880 This could be tenth grade. 00:02:12.880 --> 00:02:13.690 We don't know. 00:02:13.690 --> 00:02:16.530 This could be the tenth grade kind of set or this could be 00:02:16.530 --> 00:02:16.970 the tenth grade. 00:02:16.970 --> 00:02:19.590 There might be no tenth graders in either the chess 00:02:19.590 --> 00:02:20.310 team or the swim team. 00:02:20.310 --> 00:02:21.890 We don't know for sure. 00:02:21.890 --> 00:02:25.940 And then choice C, some members of the chess club are 00:02:25.940 --> 00:02:27.590 not tenth graders. 00:02:27.590 --> 00:02:28.890 This we know for sure. 00:02:28.890 --> 00:02:30.240 How do we know it for sure? 00:02:30.240 --> 00:02:37.030 Because these kids who are on both, they're in the chess 00:02:37.030 --> 00:02:39.210 club, but they're also on the swim team. 00:02:39.210 --> 00:02:41.960 The fact that they're in swim team, we know that they can't 00:02:41.960 --> 00:02:43.550 be tenth graders. 00:02:43.550 --> 00:02:46.670 So this is some members of the chess club-- this little 00:02:46.670 --> 00:02:50.330 intersection here-- that are not tenth graders. 00:02:50.330 --> 00:02:52.440 So choice C is the correct choice. 00:02:55.280 --> 00:02:56.530 Next problem. 00:03:01.830 --> 00:03:08.380 If 3x plus n is equal to x plus 1, what is 00:03:08.380 --> 00:03:09.540 n in terms of x? 00:03:09.540 --> 00:03:11.555 So we essentially just solve for n. 00:03:11.555 --> 00:03:13.790 Let's subtract 3x from both sides. 00:03:13.790 --> 00:03:17.100 You get n is equal to-- what's x minus 3x? 00:03:17.100 --> 00:03:18.580 It's minus 2x. 00:03:18.580 --> 00:03:21.000 And n plus 1. 00:03:21.000 --> 00:03:21.710 And we're done. 00:03:21.710 --> 00:03:23.870 And that choice isn't there, but if you just switch these 00:03:23.870 --> 00:03:27.000 two terms you just get that equals 1 minus 2x and 00:03:27.000 --> 00:03:28.900 that's choice D. 00:03:28.900 --> 00:03:31.850 Pretty quick problem, especially for one that's the 00:03:31.850 --> 00:03:32.450 ninth problem. 00:03:32.450 --> 00:03:34.110 They normally get a little harder by this point. 00:03:34.110 --> 00:03:36.630 Problem 10. 00:03:36.630 --> 00:03:41.290 If k is a positive integer, let k be defined as a set of 00:03:41.290 --> 00:03:42.470 all multiples of k. 00:03:42.470 --> 00:03:46.480 So k with a square around it is equal to the set of 00:03:46.480 --> 00:03:52.490 multiples of k. 00:03:52.490 --> 00:03:56.900 All of the numbers in which of the following sets are also in 00:03:56.900 --> 00:03:59.770 all three of the set-- OK. 00:03:59.770 --> 00:04:04.690 All of the numbers in which of the following sets are also in 00:04:04.690 --> 00:04:10.900 all three of the sets of 2, 3 and 5? 00:04:10.900 --> 00:04:22.330 So the what they're saying is 2, 3, 5, this donates all the 00:04:22.330 --> 00:04:23.580 multiples of 2. 00:04:26.730 --> 00:04:30.450 This is all multiples of 3. 00:04:30.450 --> 00:04:37.340 This is all multiples of 5. 00:04:37.340 --> 00:04:41.800 So what they're essentially saying is let's find a number 00:04:41.800 --> 00:04:45.970 where all of its multiples, all of this number's multiples 00:04:45.970 --> 00:04:49.740 are also going to be multiples of each of these. 00:04:49.740 --> 00:04:53.290 So it has to be a multiple-- so every number that-- 00:04:53.290 --> 00:04:55.920 whatever this mystery number is, let's call it x-- every 00:04:55.920 --> 00:05:00.790 multiple of x has to be a multiple of 2, 3 and 5. 00:05:00.790 --> 00:05:05.950 Well the simple way is if x is a multiple of 2, 3 and 5, then 00:05:05.950 --> 00:05:07.340 every multiple of x is going to be a 00:05:07.340 --> 00:05:08.970 multiple of 2, 3 and 5. 00:05:08.970 --> 00:05:11.160 So what's 2 times 3 times 5? 00:05:11.160 --> 00:05:13.680 It's 2 times 3 times 5. 00:05:13.680 --> 00:05:16.090 That's 6 times 5, that's 30. 00:05:16.090 --> 00:05:20.160 So 30 is a multiple of all of them, so any multiple of 30 00:05:20.160 --> 00:05:21.990 will be a multiple of all of these. 00:05:21.990 --> 00:05:25.050 When we look at the choices we don't see 30. 00:05:25.050 --> 00:05:27.360 But do we see any other number that is a 00:05:27.360 --> 00:05:30.010 multiple of 2, 3 and 5? 00:05:30.010 --> 00:05:32.160 Well sure, 60 is, right? 00:05:32.160 --> 00:05:33.620 We just multiply by 2 again. 00:05:33.620 --> 00:05:36.100 But 60 is still a multiple of 2, 3 and 5. 00:05:36.100 --> 00:05:38.530 If you were to do 2, 4, 6, 8 all the way you'd get 60, if 00:05:38.530 --> 00:05:41.260 you go 3, 9, 12, 15 all the way, you'd get to 60. 00:05:41.260 --> 00:05:44.580 You go 5, 10, 15, 20, 25, you'd get to 60. 00:05:44.580 --> 00:05:46.630 So 60 is a multiple of all of them. 00:05:46.630 --> 00:05:50.600 So what we're saying is-- so what's the set of all the 00:05:50.600 --> 00:05:51.410 multiples of 60? 00:05:51.410 --> 00:05:58.260 It's 60, 120, 180, 240, et cetera, right? 00:05:58.260 --> 00:06:02.510 And all of these numbers are in each of these sets. 00:06:02.510 --> 00:06:06.020 Because all of these numbers are multiples of 2, 3 and 5. 00:06:06.020 --> 00:06:07.190 So our answer is 60. 00:06:07.190 --> 00:06:09.320 If you look at the other choices, some of them are 00:06:09.320 --> 00:06:11.740 divisible by 5, some are divisible by 2 or 3, 00:06:11.740 --> 00:06:13.100 some are 3 and 5. 00:06:13.100 --> 00:06:17.870 But none of them are divisible by 2, 3 and 5, only 60 is. 00:06:17.870 --> 00:06:19.120 Next problem. 00:06:21.760 --> 00:06:23.890 That problem was a little hard to read initially though. 00:06:23.890 --> 00:06:25.140 That's how they confuse you. 00:06:31.210 --> 00:06:33.383 So we're going to go from A to D-- I should have drawn all 00:06:33.383 --> 00:06:38.570 the lines first. Let me draw the lines first. It's like a 00:06:38.570 --> 00:06:40.741 hexagon kind of. 00:06:40.741 --> 00:06:43.900 The top, the outside of the hexagon there. 00:06:47.100 --> 00:06:53.330 A, B, C, D, E, F. 00:06:53.330 --> 00:06:54.740 And then this is the origin. 00:06:54.740 --> 00:06:59.640 And the figure above, AD is equal to BE. 00:06:59.640 --> 00:07:00.020 Oh, no, no. 00:07:00.020 --> 00:07:00.810 They don't tell us that. 00:07:00.810 --> 00:07:01.820 I'm hallucinating. 00:07:01.820 --> 00:07:05.720 In the figure above AD, BE, and CF intersect at 0.0. 00:07:05.720 --> 00:07:07.880 The intersect's here at the origin. 00:07:07.880 --> 00:07:12.710 If the measure of AOB, the measure of that, is 80 00:07:12.710 --> 00:07:22.520 degrees, and CF bisects BOD, so it 00:07:22.520 --> 00:07:26.110 bisects this larger angle. 00:07:26.110 --> 00:07:28.880 CF bisect BOD, that angle. 00:07:28.880 --> 00:07:32.230 So that tells us that this angle has to be 00:07:32.230 --> 00:07:34.100 equal to this angle. 00:07:34.100 --> 00:07:35.520 That's the definition of bisecting an angle. 00:07:35.520 --> 00:07:37.310 You're splitting this larger angle in half. 00:07:37.310 --> 00:07:41.110 So these angles have to be equal to each other. 00:07:41.110 --> 00:07:43.270 So what is the measure of EOF? 00:07:47.600 --> 00:07:51.770 So we want to figure out this angle. 00:07:51.770 --> 00:07:54.285 Well this angle is opposite to this angle, so they're going 00:07:54.285 --> 00:07:54.840 to be equal. 00:07:54.840 --> 00:07:56.970 So if we can figure out this angle we're done. 00:07:56.970 --> 00:07:59.370 So let's call this angle x. 00:07:59.370 --> 00:08:03.770 If that angle's x this angle is also x. 00:08:03.770 --> 00:08:05.900 This x, this x, and this 80 degrees, they're all 00:08:05.900 --> 00:08:10.440 supplementary because they all go halfway around the circle. 00:08:10.440 --> 00:08:16.140 So x plus x plus 80 is going to be equal to 180 degrees. 00:08:16.140 --> 00:08:20.090 2x plus 80 is equal to 180. 00:08:20.090 --> 00:08:24.590 2x is equal to 100, x is equal to 50. 00:08:24.590 --> 00:08:29.390 And as we said before, x is equal to 50, the angle EOF, 00:08:29.390 --> 00:08:31.750 which you're trying to figure out, is opposite to it so it's 00:08:31.750 --> 00:08:32.850 going to be equal. 00:08:32.850 --> 00:08:34.909 So this is also going to be 50 degrees. 00:08:34.909 --> 00:08:38.610 And that's choice B. 00:08:38.610 --> 00:08:40.559 Next problem. 00:08:40.559 --> 00:08:43.590 I don't know if I have time for this, but I'll try. 00:08:43.590 --> 00:08:45.780 Problem 12. 00:08:45.780 --> 00:08:47.310 k is a positive integer. 00:08:47.310 --> 00:08:50.890 What is the least value of k for which the 00:08:50.890 --> 00:08:53.310 square root of-- OK. 00:08:53.310 --> 00:08:59.390 So what is the least value of k for which 5k 00:08:59.390 --> 00:09:02.390 over 3 is an integer. 00:09:02.390 --> 00:09:04.600 So this has to be a whole number, right? 00:09:04.600 --> 00:09:07.720 So essentially if we want to find the least value of k, we 00:09:07.720 --> 00:09:09.700 essentially want to say, well what's the least integer that 00:09:09.700 --> 00:09:11.910 this could be? 00:09:11.910 --> 00:09:15.240 And they're telling us that k is a positive integer. 00:09:15.240 --> 00:09:19.330 So first of all, in order for the square root to be an 00:09:19.330 --> 00:09:24.230 integer, this whole thing has to be an integer, right? 00:09:24.230 --> 00:09:27.710 So let's see, k has to be a multiple of 3. 00:09:27.710 --> 00:09:31.525 In order for this expression to be an integer, k has to be 00:09:31.525 --> 00:09:32.670 a multiple of 3. 00:09:32.670 --> 00:09:37.500 If k is 3, we get square root of 15 over 3-- well that 00:09:37.500 --> 00:09:40.140 doesn't work. 00:09:40.140 --> 00:09:44.080 If k is 3 we just get 5 in there. 00:09:44.080 --> 00:09:45.850 Actually, let me continue this into the next problem because 00:09:45.850 --> 00:09:46.730 I don't want to rush this. 00:09:46.730 --> 00:09:48.470 I'll see you in the next video.

SAT Prep: Test 8 Section 5 Part 3

https://www.youtube.com/watch?v=lPLm4pM4KXU

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en

WEBVTT Kind: captions Language: en 00:00:00.220 --> 00:00:00.960 Welcome back. 00:00:00.960 --> 00:00:04.540 We're on problem number 12. 00:00:04.540 --> 00:00:07.400 If k is a positive integer, what is the least value of k 00:00:07.400 --> 00:00:14.330 for which the square root of 5k/3 is an integer? 00:00:14.330 --> 00:00:16.250 So this whole thing has to be an integer. 00:00:16.250 --> 00:00:19.230 If the whole thing is going to be an integer, the thing 00:00:19.230 --> 00:00:22.680 inside of the square root has to be an integer as well. 00:00:22.680 --> 00:00:24.270 The square root of a fraction is never 00:00:24.270 --> 00:00:25.620 going to be an integer. 00:00:25.620 --> 00:00:28.400 So 5k/3 is going to be an integer. 00:00:28.400 --> 00:00:30.290 And we could also immediately say that k has to be a 00:00:30.290 --> 00:00:30.960 multiple of 3. 00:00:30.960 --> 00:00:32.479 That's the only way this is going to be an integer. 00:00:32.479 --> 00:00:34.410 But let's think about what type of a perfect 00:00:34.410 --> 00:00:36.030 square this has to be. 00:00:36.030 --> 00:00:39.650 This perfect square, it has to be divisible by 5. 00:00:39.650 --> 00:00:41.720 How do I know that? 00:00:41.720 --> 00:00:44.360 Well, let's just rewrite it as 5 times k/3. 00:00:46.900 --> 00:00:51.220 It has to be a multiple of 5, and this has to be an integer, 00:00:51.220 --> 00:00:54.470 which we just said. k has to be divisible by 3. 00:00:54.470 --> 00:00:55.820 The only way that this whole thing is going to be an 00:00:55.820 --> 00:00:58.540 integer is if k/3 is an integer. 00:00:58.540 --> 00:01:01.670 So what is the smallest perfect square that is 00:01:01.670 --> 00:01:02.920 divisible by 5? 00:01:05.260 --> 00:01:08.650 Because we want k to be the smallest possible number. 00:01:08.650 --> 00:01:10.700 The smallest possible integer. 00:01:10.700 --> 00:01:14.820 So the smallest perfect square that is a multiple of 5 is, of 00:01:14.820 --> 00:01:17.060 course, 5 squared, which is 25. 00:01:17.060 --> 00:01:19.750 So let's set this equal to 25. 00:01:19.750 --> 00:01:25.150 So 5k/3 is going to be equal to 25. 00:01:25.150 --> 00:01:27.680 Let's see, we can multiply both sides by 3. 00:01:27.680 --> 00:01:30.590 You get 5k is equal to 75. 00:01:30.590 --> 00:01:32.090 Divide both sides by 5. 00:01:32.090 --> 00:01:34.550 You get k is equal to 15. 00:01:34.550 --> 00:01:35.600 And you can try it out. 00:01:35.600 --> 00:01:39.110 Put 15 in here, this whole thing becomes 25. 00:01:39.110 --> 00:01:40.780 And then the square root becomes 5. 00:01:40.780 --> 00:01:41.980 And they're all integers. 00:01:41.980 --> 00:01:43.230 Next problem. 00:01:46.050 --> 00:01:49.100 OK, they drew some boxes or some shapes. 00:01:49.100 --> 00:01:51.550 This one's 1. 00:01:51.550 --> 00:01:53.170 And this one looks something like this. 00:01:56.264 --> 00:01:58.410 That's 1, 1. 00:01:58.410 --> 00:02:00.020 And then this one looks something like this. 00:02:03.490 --> 00:02:06.340 This is 2, 1. 00:02:06.340 --> 00:02:09.190 The figures above represent three pieces of cardboard. 00:02:09.190 --> 00:02:11.820 All angles of the cardboard pieces are right angles. 00:02:11.820 --> 00:02:13.070 Fair enough. 00:02:13.070 --> 00:02:15.910 All short sides have length 1, and all long 00:02:15.910 --> 00:02:18.090 sides have length 2. 00:02:18.090 --> 00:02:20.900 Which of the following figures could have been made from only 00:02:20.900 --> 00:02:23.920 the three pieces of cardboard without overlapping them. 00:02:23.920 --> 00:02:26.200 So let's draw these choices. 00:02:26.200 --> 00:02:30.150 So choice one looks like this. 00:02:30.150 --> 00:02:33.990 It comes down like this. 00:02:33.990 --> 00:02:34.950 This is 2. 00:02:34.950 --> 00:02:36.610 This comes down like this. 00:02:36.610 --> 00:02:39.750 This goes like this. 00:02:39.750 --> 00:02:41.360 And this length is 3. 00:02:41.360 --> 00:02:44.020 So maybe I'll try to fit the biggest piece in and see what 00:02:44.020 --> 00:02:44.860 I have left over. 00:02:44.860 --> 00:02:46.240 So the only place where that big piece can 00:02:46.240 --> 00:02:47.490 fit in is right here. 00:02:51.330 --> 00:02:53.420 That's the only place the big piece can fit in. 00:02:53.420 --> 00:02:55.050 And then I could fit in the small piece. 00:02:55.050 --> 00:02:55.710 I could put it here. 00:02:55.710 --> 00:02:57.020 But then have no space. 00:02:57.020 --> 00:02:59.430 That or that is not enough space for this piece. 00:02:59.430 --> 00:03:01.800 So one does not work. 00:03:01.800 --> 00:03:03.050 Choice two. 00:03:07.170 --> 00:03:18.142 So it has length 4 and it goes 2, 1, like that. 00:03:18.142 --> 00:03:21.520 And then it goes 2, then it comes back. 00:03:21.520 --> 00:03:24.730 So this is 2, 2, this is 1. 00:03:24.730 --> 00:03:25.960 So where can I fit the big piece? 00:03:25.960 --> 00:03:28.585 Let's try that first. Well, the really only place where it 00:03:28.585 --> 00:03:30.450 will fit, because it's 3 wide, is right here. 00:03:30.450 --> 00:03:32.520 This is the only place where the big piece will fit. 00:03:36.200 --> 00:03:38.970 And then where can I fit the small piece? 00:03:38.970 --> 00:03:42.410 Well, this looks like a pretty obvious place for the small 00:03:42.410 --> 00:03:44.416 piece, because, obviously, the medium piece isn't going to be 00:03:44.416 --> 00:03:45.250 fit right there. 00:03:45.250 --> 00:03:46.550 And can I fit the medium piece? 00:03:46.550 --> 00:03:48.770 Can this piece fit here? 00:03:48.770 --> 00:03:50.640 Well, sure, I just have to rotate it. 00:03:50.640 --> 00:03:51.460 And how do I know that? 00:03:51.460 --> 00:03:53.970 Well, let me show, if I can visualize it. 00:03:53.970 --> 00:04:00.010 If I rotate it so that this side becomes this side, and 00:04:00.010 --> 00:04:06.530 that this side, right here, becomes this side. 00:04:06.530 --> 00:04:07.910 And then I could keep going. 00:04:07.910 --> 00:04:11.830 I could say this side is this side. 00:04:11.830 --> 00:04:13.660 This is actually strangely fun. 00:04:13.660 --> 00:04:16.102 This side is this side. 00:04:16.102 --> 00:04:18.269 And I think you get the picture. 00:04:18.269 --> 00:04:20.769 This side is this side. 00:04:20.769 --> 00:04:21.380 So I think you get the idea. 00:04:21.380 --> 00:04:22.050 I just rotated it. 00:04:22.050 --> 00:04:23.290 So this one works. 00:04:23.290 --> 00:04:24.780 Choice two works. 00:04:24.780 --> 00:04:26.810 Let's look at choice three. 00:04:26.810 --> 00:04:30.740 Choice three looks like this. 00:04:30.740 --> 00:04:35.080 It's also 4 high, go out 1, down step-- this looks like a 00:04:35.080 --> 00:04:37.940 step-- goes down 2. 00:04:37.940 --> 00:04:38.925 Then it comes out like that. 00:04:38.925 --> 00:04:40.670 It comes like that. 00:04:40.670 --> 00:04:44.580 And this is 4, 1, 1, 2, 1. 00:04:44.580 --> 00:04:45.885 So where can we fit the big piece? 00:04:45.885 --> 00:04:47.730 The big piece is 3 wide. 00:04:47.730 --> 00:04:49.970 So this is the only part that's 3 wide, but there's no 00:04:49.970 --> 00:04:51.790 place to stick the bottom part of the big piece. 00:04:51.790 --> 00:04:55.670 The big piece could fit like this, but there's no place to 00:04:55.670 --> 00:04:56.610 put this part. 00:04:56.610 --> 00:04:58.410 So the big piece, I can't figure out a way to make the 00:04:58.410 --> 00:04:58.930 big piece fit. 00:04:58.930 --> 00:05:00.650 And I don't think it can, because this is the only part 00:05:00.650 --> 00:05:03.260 that's 3 wide, because it has to be 3 wide. 00:05:03.260 --> 00:05:05.600 So choice three is also not the answer. 00:05:05.600 --> 00:05:09.970 So the answer is two only, and that's choice C. 00:05:09.970 --> 00:05:11.220 Next problem. 00:05:14.850 --> 00:05:17.230 Problem 14. 00:05:17.230 --> 00:05:20.770 How many integers greater than 20 and less than 30 are each 00:05:20.770 --> 00:05:24.860 the product of exactly two different numbers, both of 00:05:24.860 --> 00:05:28.200 which are prime? 00:05:28.200 --> 00:05:31.020 Exactly two different numbers, both of which are-- so let's 00:05:31.020 --> 00:05:32.380 just list out all the numbers. 00:05:32.380 --> 00:05:34.500 Between 20 and 30, not including those two, because 00:05:34.500 --> 00:05:35.810 greater than 20, less than 30. 00:05:35.810 --> 00:05:48.110 So 21, 22, 23, 24, 25, 26, 27, 28, 29. 00:05:48.110 --> 00:05:50.770 So should we count 1 and the number as factors? 00:05:50.770 --> 00:05:52.830 Well, no, because 1 is not a prime number. 00:05:52.830 --> 00:05:56.710 Prime numbers are all the numbers greater than or equal 00:05:56.710 --> 00:06:01.330 to 2 that have 1 and itself as the only factors. 00:06:01.330 --> 00:06:04.510 So let's see how many factors other than 1 and itself are 00:06:04.510 --> 00:06:06.510 both different and both prime. 00:06:06.510 --> 00:06:09.970 So this has 3 and 7, and that's all I 00:06:09.970 --> 00:06:10.610 can get out of it. 00:06:10.610 --> 00:06:13.300 And these are both prime, so this works. 00:06:13.300 --> 00:06:16.330 This has 2 times 11. 00:06:16.330 --> 00:06:17.280 These are both prime. 00:06:17.280 --> 00:06:19.810 I can't think of any other two numbers that divide into 22. 00:06:19.810 --> 00:06:21.880 So that one works. 00:06:21.880 --> 00:06:24.660 23 only has 1 times 23. 00:06:24.660 --> 00:06:25.930 1 isn't prime. 00:06:25.930 --> 00:06:28.140 I mean, all of these also have the kind of identity factors, 00:06:28.140 --> 00:06:29.550 so it's not 23. 00:06:29.550 --> 00:06:33.110 24 has 2 times 12, and 6 times 4, and none 00:06:33.110 --> 00:06:33.920 of those are prime. 00:06:33.920 --> 00:06:35.690 So 24 doesn't work. 00:06:35.690 --> 00:06:37.920 25 has 5 times 5. 00:06:37.920 --> 00:06:40.500 Both are prime, but they're not different numbers, so that 00:06:40.500 --> 00:06:41.840 doesn't work. 00:06:41.840 --> 00:06:45.120 2 times 13 goes to 26 and that's the only factor, so 00:06:45.120 --> 00:06:47.420 that works. 00:06:47.420 --> 00:06:50.970 27 has 3 times 9, but 9 isn't prime, it's a composite 00:06:50.970 --> 00:06:53.520 number, divisible by 3, so that doesn't work. 00:06:53.520 --> 00:06:57.440 28 has 2 times 14, and 7 times 2 and-- so it has multiple 00:06:57.440 --> 00:07:00.940 factors and 14 isn't prime, so it doesn't work. 00:07:00.940 --> 00:07:03.990 29 is prime, so it doesn't have any other different 00:07:03.990 --> 00:07:06.265 numbers, two different numbers that divide 00:07:06.265 --> 00:07:06.850 into it that are prime. 00:07:06.850 --> 00:07:07.730 It only has 1 times 29. 00:07:07.730 --> 00:07:10.760 It doesn't have anything else, and 1 isn't prime. 00:07:10.760 --> 00:07:13.290 So there's 1, 2, 3 choices. 00:07:13.290 --> 00:07:15.330 So that's D. 00:07:15.330 --> 00:07:18.360 I actually think this was a little badly worded because, I 00:07:18.360 --> 00:07:25.360 don't know, you could argue that the number itself is a 00:07:25.360 --> 00:07:29.890 factor-- or I guess the product of exactly two numbers 00:07:29.890 --> 00:07:31.810 and 1 can't count. 00:07:31.810 --> 00:07:34.220 It has to be the product of two numbers, and so you can't 00:07:34.220 --> 00:07:36.740 count 1 and itself because 1 is not a prime number. 00:07:36.740 --> 00:07:38.510 Both of the numbers that make up the 00:07:38.510 --> 00:07:40.620 product have to be prime. 00:07:40.620 --> 00:07:46.940 Problem number 15. 00:07:46.940 --> 00:07:48.190 So they drew this triangle. 00:07:50.960 --> 00:07:54.580 They tell us that this is 7 minus x. 00:07:54.580 --> 00:07:57.580 This is 7 plus x. 00:07:57.580 --> 00:07:58.480 This is 10. 00:07:58.480 --> 00:07:59.840 This is 90 degrees. 00:07:59.840 --> 00:08:01.750 And they say, in the figure above is a right triangle. 00:08:01.750 --> 00:08:05.490 What is the value of 49 plus x squared? 00:08:05.490 --> 00:08:06.680 So let's do the Pythagorean theoreum. 00:08:06.680 --> 00:08:09.610 This squared plus this squared has to equal that squared. 00:08:09.610 --> 00:08:16.120 So 7 minus x, squared plus 7 plus x, squared is 00:08:16.120 --> 00:08:18.870 going to equal 100. 00:08:18.870 --> 00:08:23.170 This equals 49 minus 2x, plus x squared. 00:08:23.170 --> 00:08:28.950 This is equal to plus 49 plus 2x, plus x squared, and that 00:08:28.950 --> 00:08:30.300 equals 100. 00:08:30.300 --> 00:08:35.260 This minus 2x and this plus 2x are going to cancel out. 00:08:35.260 --> 00:08:36.330 And now this is what? 00:08:36.330 --> 00:08:39.750 49 plus 49, that's the same thing as 2 times 49. 00:08:39.750 --> 00:08:40.970 And then you have x squared plus x 00:08:40.970 --> 00:08:43.260 squared, plus 2x squared. 00:08:43.260 --> 00:08:44.950 I could have multiplied this out, but you'll see why I 00:08:44.950 --> 00:08:47.550 didn't, because I want to preserve that 49 in there. 00:08:47.550 --> 00:08:50.780 And that equals, so this is 100, this equals 100. 00:08:50.780 --> 00:08:53.950 Factor out the 2. 00:08:53.950 --> 00:09:00.430 You get 2 times 49, plus x squared is equal to 100. 00:09:00.430 --> 00:09:02.050 Preserve the 49. 00:09:02.050 --> 00:09:03.450 And then divide both sides by 2. 00:09:03.450 --> 00:09:07.790 You get 49 plus x squared is equal to 50. 00:09:07.790 --> 00:09:08.780 And we are done. 00:09:08.780 --> 00:09:09.630 That's the answer. 00:09:09.630 --> 00:09:11.350 That's choice A. 00:09:11.350 --> 00:09:13.150 See you in the next video.

SAT Prep: Test 8 Section 5 Part 4

https://www.youtube.com/watch?v=Q5z0MSvTtRg

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WEBVTT Kind: captions Language: en 00:00:00.650 --> 00:00:04.540 We're on problem 16. 00:00:04.540 --> 00:00:07.225 They draw the axes. 00:00:07.225 --> 00:00:10.880 x-axis, that's the y-axis. 00:00:10.880 --> 00:00:12.250 I wish there was the parabola drawing tool 00:00:12.250 --> 00:00:14.430 here, but there isn't. 00:00:14.430 --> 00:00:16.240 There's the parabola, it looks something like this. 00:00:19.750 --> 00:00:23.840 They tell us that this is the point 1. 00:00:23.840 --> 00:00:24.450 Fair enough. 00:00:24.450 --> 00:00:27.460 This is, of course, the y-axis, this is the x-axis. 00:00:27.460 --> 00:00:29.870 The figure above show that the graph of a quadratic function 00:00:29.870 --> 00:00:33.000 h who's maximum value is h of 2. 00:00:33.000 --> 00:00:36.560 So this is its maximum value right here. 00:00:36.560 --> 00:00:39.760 When x is equal to 2, its maximum is h of 2. 00:00:43.350 --> 00:00:51.510 If h of a is equal to 0-- so where does this 00:00:51.510 --> 00:00:52.400 function equal 0? 00:00:52.400 --> 00:00:54.630 It equals 0 two places. 00:00:54.630 --> 00:00:55.540 Here and here, right? 00:00:55.540 --> 00:00:57.950 Its y value is 0 in those two places. 00:00:57.950 --> 00:01:00.610 So this could be a or that could be a, we don't know. 00:01:00.610 --> 00:01:02.150 Which of the following could be a? 00:01:02.150 --> 00:01:04.050 So we're saying this could be a or this could 00:01:04.050 --> 00:01:06.420 be a, we don't know. 00:01:06.420 --> 00:01:08.780 And we know that this is 2, so a is either going to be 00:01:08.780 --> 00:01:13.260 greater than 2 or less than 0. 00:01:13.260 --> 00:01:17.340 Or actually, we can actually figure out what a is, right? 00:01:17.340 --> 00:01:21.620 Because if this is the maximum point, it's going to be 00:01:21.620 --> 00:01:25.070 symmetric around the point 2. 00:01:25.070 --> 00:01:28.990 So the distance from 2 to a on this side has to be the same 00:01:28.990 --> 00:01:30.810 as the distance from 2 to a on this one-- Actually that 00:01:30.810 --> 00:01:31.740 doesn't help. 00:01:31.740 --> 00:01:32.960 So let's look at the choices. 00:01:32.960 --> 00:01:36.040 It could be negative 1-- that looks fair because 00:01:36.040 --> 00:01:38.060 it's less than 0. 00:01:38.060 --> 00:01:40.290 a can't be 0-- 0 is this point. 00:01:40.290 --> 00:01:42.990 Let me pick a color. 00:01:42.990 --> 00:01:43.840 That's 0. 00:01:43.840 --> 00:01:46.330 That's definitely not where you inter-- they draw it so 00:01:46.330 --> 00:01:47.710 that you can't be there. 00:01:47.710 --> 00:01:51.120 It can't be 2 because that's actually where the maximum 00:01:51.120 --> 00:01:54.800 point is, so a definitely isn't 2. 00:01:54.800 --> 00:01:56.730 Can a be 3? 00:01:56.730 --> 00:01:59.280 And this is where the symmetry comes into play. 00:01:59.280 --> 00:02:02.220 We know that it does not intersect the x-axis for at 00:02:02.220 --> 00:02:06.850 least-- on the left-hand side you go 1, 2, and then some 00:02:06.850 --> 00:02:09.250 distance which we don't really know, but you're going more 00:02:09.250 --> 00:02:10.930 than 2 on this side. 00:02:10.930 --> 00:02:12.060 So you're also going to have to go more 00:02:12.060 --> 00:02:12.760 than two on this side. 00:02:12.760 --> 00:02:16.140 You're going to have to go 1, 2, and then some distance. 00:02:16.140 --> 00:02:18.860 So this is 2, 3, 4. 00:02:18.860 --> 00:02:24.940 So you know that a has to be less than 0, or a has to be 00:02:24.940 --> 00:02:26.430 greater than 4. 00:02:26.430 --> 00:02:31.550 And if you look at all of the choices, the only-- negative 1 00:02:31.550 --> 00:02:32.500 is less than 0. 00:02:32.500 --> 00:02:34.880 None of the other ones are greater than 4. 00:02:34.880 --> 00:02:36.020 And how did I know greater than 4? 00:02:36.020 --> 00:02:37.440 Because it's symmetric. 00:02:37.440 --> 00:02:42.190 If this is the maximum point at 2, and one of the places 00:02:42.190 --> 00:02:45.660 where I intersect the x-axis is more than 2 to the left, 00:02:45.660 --> 00:02:48.050 the other place where I intersect the x-axis has to be 00:02:48.050 --> 00:02:49.380 more than 2 to the right. 00:02:49.380 --> 00:02:50.860 So it has to be greater than 4. 00:02:50.860 --> 00:02:53.850 This has to be less than 0, this has to be greater than 4. 00:02:53.850 --> 00:02:55.480 And that's the answer. 00:02:55.480 --> 00:02:56.120 Negative 1. 00:02:56.120 --> 00:02:57.770 Next problem. 00:02:57.770 --> 00:02:59.750 And you really didn't have to-- once you felt comfortable 00:02:59.750 --> 00:03:01.330 with negative 1 you could have just said well that's the 00:03:01.330 --> 00:03:03.970 answer because it can very easily have intersected the 00:03:03.970 --> 00:03:06.400 x-axis at x equals negative 1. 00:03:06.400 --> 00:03:08.490 Problem 17-- and that was the first choice, which tends to 00:03:08.490 --> 00:03:08.900 be convenient. 00:03:08.900 --> 00:03:11.330 Let me get a brighter color. 00:03:11.330 --> 00:03:11.850 Problem 17. 00:03:11.850 --> 00:03:19.870 If k and h are constants and x squared plus kx plus 7 is 00:03:19.870 --> 00:03:28.100 equivalent to, equals, x plus 1 times x plus h, what is the 00:03:28.100 --> 00:03:30.500 value of k? 00:03:30.500 --> 00:03:33.330 Well at first sight it doesn't seem like we could solve this, 00:03:33.330 --> 00:03:35.950 but maybe we can, let's try. 00:03:35.950 --> 00:03:45.470 x squared plus kx plus 7 is equal to x squared plus x 00:03:45.470 --> 00:03:52.800 plus hx plus h. 00:03:52.800 --> 00:03:55.100 And so this equals x squared. 00:03:55.100 --> 00:03:58.350 What's 1x-- right, x is the same thing as 1x-- 00:03:58.350 --> 00:04:00.420 what's 1x plus xh? 00:04:00.420 --> 00:04:08.810 Well that's 1 plus hx plus h, right, I just combined these 00:04:08.810 --> 00:04:10.610 two terms, added their coefficients. 00:04:10.610 --> 00:04:15.780 And we say x squared plus kx plus-- oh, well actually, 00:04:15.780 --> 00:04:19.209 maybe we can solve this because we can just match up 00:04:19.209 --> 00:04:26.840 terms. We say this term matches up to this term. 00:04:26.840 --> 00:04:29.130 This term matches up to that term. 00:04:29.130 --> 00:04:32.590 So we know, h is equal to 7. 00:04:32.590 --> 00:04:35.790 And if h is equal to 7, what is this term equal to? 00:04:35.790 --> 00:04:40.860 That term is going to be equal to 7 plus 1 is 8. 00:04:40.860 --> 00:04:43.570 And this term, what does this term match up to? 00:04:43.570 --> 00:04:44.270 It matches up to this. 00:04:44.270 --> 00:04:45.970 We're just kind of matching coefficients. 00:04:45.970 --> 00:04:48.870 If this is 8 than k is equal to 8. 00:04:48.870 --> 00:04:49.650 And that's choice D. 00:04:49.650 --> 00:04:51.430 At first I was like how am I going to solve this? 00:04:51.430 --> 00:04:53.710 That's why you should always just move forward and see if 00:04:53.710 --> 00:04:55.340 you see any patterns. 00:04:55.340 --> 00:04:56.590 Next problem. 00:05:00.490 --> 00:05:22.530 Problem 18. 00:05:22.530 --> 00:05:29.590 y, x, and then they draw a line here, 00:05:29.590 --> 00:05:30.840 something like that. 00:05:33.610 --> 00:05:45.440 You go from A to C-- oh, the line keeps going, and then 00:05:45.440 --> 00:05:50.010 there's the point up here, which is the point 00:05:50.010 --> 00:05:52.510 here, this is 4,10. 00:05:52.510 --> 00:05:56.740 So this is 10 and this is 4. 00:05:56.740 --> 00:06:00.080 This is A, B, C. 00:06:00.080 --> 00:06:02.700 In the figure above if the legs of triangle ABC are 00:06:02.700 --> 00:06:05.540 parallel to the axes-- this is parallel to that, this is 00:06:05.540 --> 00:06:08.620 parallel to that, so this is going to be a right angle. 00:06:08.620 --> 00:06:10.950 Which of the following could be the lengths of 00:06:10.950 --> 00:06:14.230 the sides of ABC. 00:06:14.230 --> 00:06:15.110 Interesting. 00:06:15.110 --> 00:06:18.200 So this is really just a slope problem because what they want 00:06:18.200 --> 00:06:20.990 you do is figure out the ratio of this side to this side. 00:06:20.990 --> 00:06:24.270 So what is the slope of this line? 00:06:24.270 --> 00:06:33.470 Well it rose 10-- change in y over change in x is equal to 00:06:33.470 --> 00:06:37.520 10 minus 0, which is 10 over 4 minus 0, which is 4. 00:06:37.520 --> 00:06:42.300 That equals 5 over 2. 00:06:42.300 --> 00:06:44.640 So for every one it moved to the right, it's going 00:06:44.640 --> 00:06:46.030 to go 5 over 2 up. 00:06:46.030 --> 00:06:49.220 So if this is x, this is going to be 5 over 2x. 00:06:52.120 --> 00:06:54.510 So first we should look for-- the two shorter sides are 00:06:54.510 --> 00:06:56.000 going to have this ratio. 00:06:56.000 --> 00:06:58.030 1 to 5 over 2. 00:06:58.030 --> 00:07:02.000 So if you look at the first choice, choice A, 2 and 5, if 00:07:02.000 --> 00:07:08.760 this is 2, if x is equal to 2, then what's 2 times 5 or 2? 00:07:08.760 --> 00:07:11.210 Well this will be 5, right? 00:07:11.210 --> 00:07:12.370 That'll be 5. 00:07:12.370 --> 00:07:14.240 So that satisfies that ratio. 00:07:14.240 --> 00:07:18.300 And you could even figure out rise is 5 when run is 2, which 00:07:18.300 --> 00:07:19.870 is the slope of 5 over 2. 00:07:19.870 --> 00:07:22.400 Now you just have to confirm that they gave the right 00:07:22.400 --> 00:07:23.960 length for this longer side. 00:07:23.960 --> 00:07:25.830 So we just us Pythagorean Theorem. 00:07:25.830 --> 00:07:28.050 The length of that side's going to be the square root of 00:07:28.050 --> 00:07:29.490 this squared plus this squared. 00:07:29.490 --> 00:07:32.980 The square root of 4 plus 25, which equals the 00:07:32.980 --> 00:07:35.310 square root of 29. 00:07:35.310 --> 00:07:38.370 So the sides could be 2, 5 squared to 29. 00:07:38.370 --> 00:07:41.270 And that is choice A. 00:07:41.270 --> 00:07:44.405 Choice B has 2 and 5, but then it gives 7, assuming that you 00:07:44.405 --> 00:07:46.360 didn't know how to do the Pythagorean Theorem. 00:07:46.360 --> 00:07:47.610 Next problem. 00:07:50.000 --> 00:07:52.760 Problem 19. 00:07:52.760 --> 00:08:00.140 Let f be defined by f of x is equal to 2x minus 1. 00:08:00.140 --> 00:08:08.020 If 1/2 times f of the square root of t is equal to 4, what 00:08:08.020 --> 00:08:10.200 is the value of t? 00:08:10.200 --> 00:08:12.490 So let's just evaluate this expression. 00:08:12.490 --> 00:08:14.620 So it's 1/2 times what? 00:08:14.620 --> 00:08:17.265 f of the square root of t-- so everywhere we see x we stick 00:08:17.265 --> 00:08:18.870 in a square root of t. 00:08:18.870 --> 00:08:22.570 So 1/2-- that 1/2 is just this 1/2. 00:08:22.570 --> 00:08:24.960 f of the square root of t is 2 times the square 00:08:24.960 --> 00:08:27.570 root of t minus 1. 00:08:27.570 --> 00:08:30.040 I just replaced x with the square root of t. 00:08:30.040 --> 00:08:31.930 And we know that this equals 4. 00:08:31.930 --> 00:08:35.280 Let's multiply both sides by 1/2 and you get-- or multiply 00:08:35.280 --> 00:08:37.510 both sides by 2-- the 1/2 gets canceled out here. 00:08:37.510 --> 00:08:41.429 So you get 2 square root of t minus 1 is equal to 8. 00:08:41.429 --> 00:08:45.480 And you get 2 square root of 5, adding 1 to both sides is 00:08:45.480 --> 00:08:47.420 equal to 9. 00:08:47.420 --> 00:08:50.010 Let me continue it here. 00:08:50.010 --> 00:08:52.960 Divide both sides by 2 you get the square root of t is equal 00:08:52.960 --> 00:08:55.140 to 9 over 2. 00:08:55.140 --> 00:08:57.130 And now we can square both sides of this. 00:08:57.130 --> 00:09:00.220 So you get t is equal to-- what's 9 over 2 squared? 00:09:00.220 --> 00:09:03.910 It's 81 over 4, and that is choice E. 00:09:07.130 --> 00:09:08.610 And then let's see if I have time. 00:09:08.610 --> 00:09:09.520 I have 50 seconds. 00:09:09.520 --> 00:09:14.040 Let me see if I can squeeze in video 20 in here. 00:09:14.040 --> 00:09:16.450 If k is a positive integer, which of the following must 00:09:16.450 --> 00:09:19.330 represent an even integer that is twice the 00:09:19.330 --> 00:09:21.260 value of an odd integer. 00:09:24.580 --> 00:09:26.840 So k is any positive integer. 00:09:26.840 --> 00:09:29.060 So I want an even number that is twice the 00:09:29.060 --> 00:09:30.930 value of an odd integer. 00:09:30.930 --> 00:09:34.920 So an odd integer-- if this is any integer, an odd integer 00:09:34.920 --> 00:09:38.080 can be represented this-- 2 times k plus 1. 00:09:38.080 --> 00:09:39.970 This is definitely going to be odd integer. 00:09:39.970 --> 00:09:42.580 You can try it out with k is 1, 2 or 3. 00:09:42.580 --> 00:09:44.840 So if I want double this, I just multiply it by 2, 00:09:44.840 --> 00:09:50.110 I get 4k plus 2. 00:09:50.110 --> 00:09:51.590 And that is choice E. 00:09:51.590 --> 00:09:53.330 And you could try it out with some numbers. 00:09:53.330 --> 00:09:55.380 You just have to take my word that that has to be an odd 00:09:55.380 --> 00:09:57.550 integer and then you double it. 00:09:57.550 --> 00:09:59.350 I'll see you in the next section.

SAT Prep: Test 8 Section 2 Part 1

https://www.youtube.com/watch?v=FyoZaqF2dsY

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https://www.youtube.com/api/timedtext?v=FyoZaqF2dsY&ei=YmeUZer3MJHwvdIPleqT6AU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=55E287B012DCED531F97ABA708345BA1768A380D.3860003EC151329B2FB5AABE0308F8FF9F6BA80E&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.760 --> 00:00:01.450 Welcome back. 00:00:01.450 --> 00:00:04.590 We're now in the final test, test eight. 00:00:04.590 --> 00:00:08.660 And we'll start in Section 2, problem one. 00:00:08.660 --> 00:00:11.790 The total cost of three equally priced mechanical 00:00:11.790 --> 00:00:13.770 pencils is $4.50. 00:00:13.770 --> 00:00:17.210 If the cost per pencil is increased by $0.50, how much 00:00:17.210 --> 00:00:19.260 will five of these pencils cost? 00:00:19.260 --> 00:00:24.600 So three pencils, three times the cost of the pencil is 00:00:24.600 --> 00:00:28.290 equal to $4.50. 00:00:28.290 --> 00:00:32.150 So pencil is equal to $4.50 divided by 3. 00:00:32.150 --> 00:00:34.810 Just divide both sides of that equation by 3 00:00:34.810 --> 00:00:37.710 and that equals $1.50. 00:00:37.710 --> 00:00:41.340 Then they say the cost per pencil is increased by $0.50. 00:00:41.340 --> 00:00:46.050 So we're going to go from $1.50 plus $0.50, so that'll 00:00:46.050 --> 00:00:50.880 get us to $2,00 per pencil. 00:00:50.880 --> 00:00:53.990 How much will five of these pencils cost at the new rate? 00:00:53.990 --> 00:00:58.636 So it's $2.00 per pencil, times 5, and that's $10.00. 00:00:58.636 --> 00:01:01.980 And that's choice E. 00:01:01.980 --> 00:01:03.230 Problem two. 00:01:07.140 --> 00:01:09.140 OK, the table above represents a relationship 00:01:09.140 --> 00:01:10.510 between x and y. 00:01:10.510 --> 00:01:17.555 So x can be-- x and y, so it's 1, 2, 3, 4. 00:01:20.280 --> 00:01:28.870 And then y, when x is 1, is 3, 7, 11, 15. 00:01:28.870 --> 00:01:31.510 Which of the following linear equations describes the 00:01:31.510 --> 00:01:32.725 relationship? 00:01:32.725 --> 00:01:36.530 So let's see if we can think of something here on our own. 00:01:39.710 --> 00:01:44.510 Let's see, this looks like-- well, I don't know how you 00:01:44.510 --> 00:01:47.410 would get to 4 times-- let's see the choices, actually, see 00:01:47.410 --> 00:01:49.230 which ones work. 00:01:49.230 --> 00:01:54.660 So A is y is equal to x plus 1. 00:01:54.660 --> 00:01:56.160 Try this out, x plus 1. 00:01:56.160 --> 00:01:58.640 1 plus 1 is 2, so that's not right. 00:01:58.640 --> 00:02:00.180 They've got 3 here. 00:02:00.180 --> 00:02:01.670 So that fails. 00:02:01.670 --> 00:02:05.900 B, y is equal to x plus 4. 00:02:05.900 --> 00:02:10.020 When x is 1-- if it was x plus 4, then this would be 5. 00:02:10.020 --> 00:02:12.750 But this is 3, so that's not right. 00:02:12.750 --> 00:02:16.120 Choice C, y is equal to 3x. 00:02:16.120 --> 00:02:20.410 That works here, right? x is 1, y is 3 times 1. 00:02:20.410 --> 00:02:23.250 But it fails here, because when x is 2, then 00:02:23.250 --> 00:02:24.190 this should be 6. 00:02:24.190 --> 00:02:26.340 But it's 7, so this isn't right. 00:02:26.340 --> 00:02:28.040 These are all wrong. 00:02:28.040 --> 00:02:31.440 Choice D, y is equal to 4x. 00:02:31.440 --> 00:02:32.710 Well, that fails on this first one. 00:02:32.710 --> 00:02:36.320 When x is 1, if it was 4x, this would be 4. 00:02:36.320 --> 00:02:38.110 But it's not, so that fails. 00:02:38.110 --> 00:02:39.630 Choice E is probably going to be our answer. 00:02:39.630 --> 00:02:40.720 Let's try. 00:02:40.720 --> 00:02:43.620 y is equal to 4x minus 1. 00:02:43.620 --> 00:02:48.360 So when x is 1, 4 times 1 is 4 minus 1 is 3. 00:02:48.360 --> 00:02:51.950 4 times 2 is 8 minus 1 is 7. 00:02:51.950 --> 00:02:55.920 4 times 3 is 12, minus 1 is 11. 00:02:55.920 --> 00:02:58.395 4 times 4 is 16, minus 1 is 15. 00:02:58.395 --> 00:03:00.370 So it works. 00:03:00.370 --> 00:03:06.830 Next problem, soon. problem 3. 00:03:06.830 --> 00:03:08.250 OK, they drew two circles. 00:03:08.250 --> 00:03:10.710 They look tangent to each other. 00:03:10.710 --> 00:03:12.420 One circle looks like that. 00:03:12.420 --> 00:03:16.790 Let me see if my skills of drawing tangent circles, how 00:03:16.790 --> 00:03:18.350 good they are. 00:03:18.350 --> 00:03:21.606 You got one circle, and not bad, not bad if I 00:03:21.606 --> 00:03:23.180 have to say so myself. 00:03:23.180 --> 00:03:26.140 And then they draw a line that looks like from the center of 00:03:26.140 --> 00:03:27.140 one circle to the other. 00:03:27.140 --> 00:03:30.520 I'll do it in a different color. 00:03:30.520 --> 00:03:38.020 So we are going from here roughly to there. 00:03:38.020 --> 00:03:40.020 I know it doesn't look like center to center, but I think 00:03:40.020 --> 00:03:41.820 you get the idea. 00:03:41.820 --> 00:03:46.080 They're saying this point here is A. 00:03:46.080 --> 00:03:48.300 This point right here is B. 00:03:48.300 --> 00:03:51.040 And then this point right here is C. 00:03:51.040 --> 00:03:53.380 In the figure above, the two circles are tangent at the 00:03:53.380 --> 00:03:57.125 point B, and AC is equal to 6, so this whole distance is 00:03:57.125 --> 00:03:58.920 equal to 6, and they're tangent right here, so these 00:03:58.920 --> 00:04:01.130 circles just touch point B. 00:04:01.130 --> 00:04:04.210 If the circumference of circle with center A is twice the 00:04:04.210 --> 00:04:07.250 circumference of the circle with center C, what is the 00:04:07.250 --> 00:04:09.700 length of BC? 00:04:09.700 --> 00:04:12.260 So we want to know what BC is. 00:04:12.260 --> 00:04:16.200 So this circumference is twice the length of this 00:04:16.200 --> 00:04:17.529 circumference. 00:04:17.529 --> 00:04:19.260 So what's the formula for circumference? 00:04:19.260 --> 00:04:24.650 Circumference is equal to 2 pi r, right? 00:04:24.650 --> 00:04:27.660 So let's say that this is r. 00:04:27.660 --> 00:04:29.030 Let's call this distance r. 00:04:31.690 --> 00:04:35.580 What do we know about this distance? 00:04:35.580 --> 00:04:39.320 Let's call that x, I guess, right? 00:04:39.320 --> 00:04:42.180 So if this distance is r, the radius of the large circle is 00:04:42.180 --> 00:04:47.220 r, its circumference is going to be 2 pi r, right? 00:04:47.220 --> 00:04:48.640 So what's the circumference of the smaller circle? 00:04:48.640 --> 00:04:49.940 Well, it's 1/2 of that. 00:04:49.940 --> 00:04:57.450 So c/2 is equal to 1/2 of the big circle circumference is 00:04:57.450 --> 00:05:00.765 equal to the circumference of the smaller circle, which is 2 00:05:00.765 --> 00:05:02.320 pi x, right? 00:05:02.320 --> 00:05:03.375 Because the radius is x. 00:05:03.375 --> 00:05:05.650 But what's c/2? 00:05:05.650 --> 00:05:07.760 That's this divided by 2. 00:05:07.760 --> 00:05:11.050 So c/2 is the same thing as this divided 00:05:11.050 --> 00:05:13.020 by 2, which is what? 00:05:13.020 --> 00:05:16.480 The 2 would just disappear, so it equals pi r. 00:05:16.480 --> 00:05:23.080 You could take both pi's away, and you get 2x is equal to r, 00:05:23.080 --> 00:05:25.350 or x is equal to r/2. 00:05:25.350 --> 00:05:27.100 And you really didn't have to do that. 00:05:27.100 --> 00:05:29.920 You could just know that circumference is directly 00:05:29.920 --> 00:05:32.540 proportional to radius, unlike area, which is proportional to 00:05:32.540 --> 00:05:33.490 the square. 00:05:33.490 --> 00:05:35.750 So circumference is directly proportional to radius. 00:05:35.750 --> 00:05:37.900 So if this circle has twice the circumference as this 00:05:37.900 --> 00:05:40.860 circle, its radius is also going to be twice the radius 00:05:40.860 --> 00:05:42.230 of this circle. 00:05:42.230 --> 00:05:43.350 But that's interesting now. 00:05:43.350 --> 00:05:46.660 So now we know that if this is x, this distance here is x, BC 00:05:46.660 --> 00:05:50.460 is x, this distance here is going to be 2x, right? 00:05:50.460 --> 00:05:53.440 It's going to be 2 times the small radius. 00:05:53.440 --> 00:05:57.580 And we know that when you add all of them together, 2x plus 00:05:57.580 --> 00:06:02.200 x, you get the whole length from AC, A to C, and you so 00:06:02.200 --> 00:06:03.590 that equals y. 00:06:03.590 --> 00:06:06.990 So you get 3x is equal to 6 and then you get 00:06:06.990 --> 00:06:08.460 x is equal to 2. 00:06:08.460 --> 00:06:09.830 And that's our answer, because we wanted to 00:06:09.830 --> 00:06:11.190 know just what x is. 00:06:11.190 --> 00:06:13.610 And so the really fast way of doing this is, if you said, 00:06:13.610 --> 00:06:17.040 well, this is going to be twice-- let's define x as this 00:06:17.040 --> 00:06:18.480 small radius. 00:06:18.480 --> 00:06:21.140 And if this circle has twice the circumference, it's going 00:06:21.140 --> 00:06:22.000 to have twice the radius. 00:06:22.000 --> 00:06:23.490 So this would be 2x. 00:06:23.490 --> 00:06:25.530 And then you could just go straight to this step without 00:06:25.530 --> 00:06:27.620 having to do all of this. 00:06:27.620 --> 00:06:34.200 Next problem, problem 4. 00:06:34.200 --> 00:06:35.835 OK, they drew a bunch of points. 00:06:35.835 --> 00:06:38.730 Let's see, a lot of drawing for me. 00:06:38.730 --> 00:06:40.600 I'm a little tired. 00:06:40.600 --> 00:06:44.570 My goal is to finish these problems this weekend, and 00:06:44.570 --> 00:06:46.740 it's midnight and past my bedtime. 00:06:46.740 --> 00:06:51.010 But I'm hanging in there and trying to finish all of these. 00:06:51.010 --> 00:06:55.175 OK, so let's see, they have point C, which is up here. 00:06:55.175 --> 00:06:57.120 And I don't know if I have to be accurate with the 00:06:57.120 --> 00:06:59.110 coordinates yet. 00:06:59.110 --> 00:07:02.610 Point B is roughly here. 00:07:02.610 --> 00:07:08.090 Point A is here. 00:07:08.090 --> 00:07:13.542 So then you have point E that's right around here. 00:07:13.542 --> 00:07:16.950 And then you have point D, which is right there. 00:07:16.950 --> 00:07:18.310 What are they going to ask us? 00:07:18.310 --> 00:07:20.550 Which of the letter points in the figure above has 00:07:20.550 --> 00:07:26.480 coordinates x, y, such that the absolute value of x minus 00:07:26.480 --> 00:07:30.600 the absolute value of y is equal to 3? 00:07:30.600 --> 00:07:32.570 So we could just try out the points, but they're really 00:07:32.570 --> 00:07:37.250 just saying that no matter what the x and y-- they say it 00:07:37.250 --> 00:07:38.410 doesn't matter what quadrant it is. 00:07:38.410 --> 00:07:42.850 We want the difference between the x and the y to be 3. 00:07:42.850 --> 00:07:47.210 So the intuition here is that the x is going to be larger 00:07:47.210 --> 00:07:49.460 than the y, right? 00:07:49.460 --> 00:07:53.800 Or the distance that x is away from the origin is going to be 00:07:53.800 --> 00:07:57.030 larger by 3 than the distance that y is away from the 00:07:57.030 --> 00:07:57.720 origin, right? 00:07:57.720 --> 00:07:59.840 Because the absolute value of x, you kind of view it as the 00:07:59.840 --> 00:08:03.800 distance x is from the origin, and absolute value of y is the 00:08:03.800 --> 00:08:05.790 distance y is from the origin. 00:08:05.790 --> 00:08:08.980 So which ones have a larger distance x from the origin 00:08:08.980 --> 00:08:10.910 than a y from the origin? 00:08:10.910 --> 00:08:12.200 Well, it looks like D, right? 00:08:12.200 --> 00:08:13.920 D looks further in this direction. 00:08:13.920 --> 00:08:15.360 Let me switch colors. 00:08:15.360 --> 00:08:20.110 D is further in this direction than it is in this direction. 00:08:20.110 --> 00:08:24.370 And if we look at the other choices, B also is like that. 00:08:24.370 --> 00:08:25.660 And so let's look at the coordinates. 00:08:25.660 --> 00:08:26.970 Those are my prime coordinates. 00:08:26.970 --> 00:08:28.920 If I actually look at the graph, B looks like 00:08:28.920 --> 00:08:31.700 it's 1, 2, 3, 4. 00:08:31.700 --> 00:08:35.419 So this looks like minus 4 and minus 1. 00:08:35.419 --> 00:08:38.282 So B looks like minus 4, minus 1. 00:08:38.282 --> 00:08:40.049 And actually, that looks like our answer, right? 00:08:40.049 --> 00:08:42.049 Because what's the absolute value of 4? 00:08:42.049 --> 00:08:45.720 That's the absolute value of minus 4 minus the absolute 00:08:45.720 --> 00:08:47.440 value of minus 1. 00:08:47.440 --> 00:08:51.040 That equals 4 minus 1, which equals 3. 00:08:51.040 --> 00:08:54.170 So that is choice B. 00:08:54.170 --> 00:08:56.020 We didn't have to try D. 00:08:56.020 --> 00:08:57.870 Next problem. 00:08:57.870 --> 00:09:00.370 Actually, I'm coming up on nine minutes. 00:09:00.370 --> 00:09:02.450 So I'll do the next problem in the next video. 00:09:02.450 --> 00:09:03.870 I'll see

SAT Prep: Test 8 Section 2 Part 2

https://www.youtube.com/watch?v=lumGHA9JGNY

vtt

https://www.youtube.com/api/timedtext?v=lumGHA9JGNY&ei=YmeUZf7jMqaJp-oPh-OQiAY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D849A0642B0AC8FD80E614A6DF3F8B383A25C282.A1C7C2EF295B1957E76926E43A8FDA687B3F605F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:02.100 We're on problem number five. 00:00:02.100 --> 00:00:04.985 Let's see, we have a pie graph here of survey results. 00:00:10.206 --> 00:00:15.030 And let's see, that looks like almost a straight line. 00:00:15.030 --> 00:00:17.030 Dividing it in half there. 00:00:17.030 --> 00:00:21.690 And then this gets divided in half again. 00:00:21.690 --> 00:00:25.635 It looks like that, roughly. 00:00:25.635 --> 00:00:28.010 It looks like that. 00:00:28.010 --> 00:00:32.479 And then this is slightly bigger like that. 00:00:32.479 --> 00:00:37.960 All right, and this tells us x is less than 20 00:00:37.960 --> 00:00:40.090 for 30% of the time. 00:00:40.090 --> 00:00:41.740 I don't even know what these represent. 00:00:41.740 --> 00:00:48.430 x is greater than or equal to 60 25% of the time. 00:00:48.430 --> 00:00:55.670 x is between 40 and less than 60, this is 25% of the time. 00:00:55.670 --> 00:00:58.500 So it can equal 40, but it's less than 60. 00:00:58.500 --> 00:01:05.019 And then it's greater than or equal to 20 and less than 40 00:01:05.019 --> 00:01:06.475 20% of the time. 00:01:06.475 --> 00:01:07.835 That's the survey results. 00:01:07.835 --> 00:01:11.120 The chart above shows the results when 1,000 people were 00:01:11.120 --> 00:01:13.610 asked how old are you? 00:01:13.610 --> 00:01:16.210 The age they gave is represented by x. 00:01:16.210 --> 00:01:19.960 How many people said their age was less than 40? 00:01:19.960 --> 00:01:23.240 So x is less than 40. 00:01:23.240 --> 00:01:27.780 Well, this category right here is x is between 20 and 40, so 00:01:27.780 --> 00:01:37.070 it's this one plus all the people who said that they were 00:01:37.070 --> 00:01:38.940 younger than 20 because if you're less than 20, you're 00:01:38.940 --> 00:01:41.130 definitely less than 40 as well, so it's 00:01:41.130 --> 00:01:42.970 both of these combined. 00:01:42.970 --> 00:01:45.490 And this is 30%, this is 20%. 00:01:45.490 --> 00:01:51.620 So if you combine it, it's 30% plus 20% equals 50% of the 00:01:51.620 --> 00:01:53.105 entire population asked. 00:01:53.105 --> 00:01:58.760 And 1,000 people were asked, so 50% times 1,000, well, 00:01:58.760 --> 00:01:59.690 that's straightforward. 00:01:59.690 --> 00:02:01.960 That equals 500 people. 00:02:01.960 --> 00:02:04.490 And that's choice D. 00:02:04.490 --> 00:02:09.840 Next problem, problem 6. 00:02:09.840 --> 00:02:13.630 Which of the following could be the remainders when four 00:02:13.630 --> 00:02:20.320 consecutive positive integers are each divided by 3? 00:02:20.320 --> 00:02:21.550 Interesting. 00:02:21.550 --> 00:02:28.650 So let's say we have x plus 1, x plus 2, x plus 3, right? 00:02:28.650 --> 00:02:31.216 These are four consecutive positive integers. 00:02:35.980 --> 00:02:37.940 Well, let's assume that this first one is 00:02:37.940 --> 00:02:39.830 divisible by 3, right? 00:02:39.830 --> 00:02:42.850 Let's say that x divided by 3, there's no remainder, so the 00:02:42.850 --> 00:02:45.260 remainder is 0. 00:02:45.260 --> 00:02:50.780 So when this number's divisible by this number, x is 00:02:50.780 --> 00:02:53.450 divisible by 3, so now when you divide x plus 1 by 3, 00:02:53.450 --> 00:02:55.200 you're going to have 1 left over. 00:02:55.200 --> 00:02:57.660 Similarly, you're going to have 2 left over. 00:02:57.660 --> 00:03:00.810 Here, you're not going to 3 left over, right? 00:03:00.810 --> 00:03:06.440 Because if x is divisible by 3, then x plus 3 is also 00:03:06.440 --> 00:03:08.670 divisible by 3, right? 00:03:08.670 --> 00:03:10.150 You're going to have a cycle. 00:03:10.150 --> 00:03:12.700 It's going to go back to a remainder of zero. 00:03:12.700 --> 00:03:15.250 So this is a possible situation. 00:03:15.250 --> 00:03:19.360 And similarly, if x had a remainder of 1, then x plus 1 00:03:19.360 --> 00:03:20.880 would have a remainder of 2. 00:03:20.880 --> 00:03:24.790 Then x plus 2 would have a remainder of 0, and then this 00:03:24.790 --> 00:03:26.360 would have a remainder of 1 again. 00:03:26.360 --> 00:03:27.800 So I don't know, let's look at the choices. 00:03:27.800 --> 00:03:29.810 Are any of these out there? 00:03:29.810 --> 00:03:30.310 Well, sure. 00:03:30.310 --> 00:03:31.510 This is choice D, actually. 00:03:31.510 --> 00:03:37.440 The first thing we did, choice D was 0, 1, 2, 0, which is D. 00:03:37.440 --> 00:03:42.770 And the key here is realizing that you can't have a 00:03:42.770 --> 00:03:48.210 remainder of 3 or 4 when you're dividing by 3, right? 00:03:48.210 --> 00:03:51.170 Your remainder can only be 0, 1 or 2. 00:03:51.170 --> 00:03:54.780 So with that realization alone, you can cancel out all 00:03:54.780 --> 00:03:56.460 the choices that have a 3 or a 4 in it. 00:03:56.460 --> 00:03:58.150 You can't have a remainder of 3 or 4 if 00:03:58.150 --> 00:03:59.510 you're dividing by 3. 00:03:59.510 --> 00:04:01.250 And you can try that out, right? 00:04:01.250 --> 00:04:02.810 Because if you have a remainder by 3, that means you 00:04:02.810 --> 00:04:05.310 could divide one more 3 into the number and have a 00:04:05.310 --> 00:04:06.500 remainder of 0. 00:04:06.500 --> 00:04:11.670 So you could cancel out A, B, C and E. 00:04:11.670 --> 00:04:12.900 You can actually cancel out everything. 00:04:12.900 --> 00:04:15.430 So you would have just had to have that one realization, and 00:04:15.430 --> 00:04:18.079 you would have said the choice is D because you can't have a 00:04:18.079 --> 00:04:21.970 remainder of 3 or 4 if you're dividing by 3. 00:04:21.970 --> 00:04:28.430 Next problem, problem 7. 00:04:31.090 --> 00:04:36.710 If y is inversely proportional to x, so that means that y is 00:04:36.710 --> 00:04:39.620 proportional to the inverse of x, so it's equal to some 00:04:39.620 --> 00:04:40.910 constant times 1/x. 00:04:40.910 --> 00:04:43.390 Because that's what inversely proportional means. 00:04:43.390 --> 00:04:47.030 If we said proportional, it'd be y equals k times x, but 00:04:47.030 --> 00:04:49.050 it's inversely proportional. 00:04:49.050 --> 00:04:56.120 And they tell us that y is 15 when x is 5. 00:04:56.120 --> 00:05:00.320 So y is 15 is equal to k times 1/5, right? 00:05:00.320 --> 00:05:03.300 y is 15 when x is equal to 5. 00:05:03.300 --> 00:05:06.810 Let's see, let's multiply both sides of this by 5. 00:05:06.810 --> 00:05:14.290 So you get 5 times 15 is 75 is equal to k, right? 00:05:14.290 --> 00:05:17.890 So y is equal to 75/x. 00:05:17.890 --> 00:05:19.900 I just rewrote this. 00:05:19.900 --> 00:05:22.910 What is the value of y when x is 25? 00:05:22.910 --> 00:05:25.210 So y is equal to 75. 00:05:25.210 --> 00:05:27.430 x is now 25. 00:05:27.430 --> 00:05:28.780 What's 75 divided by 25? 00:05:28.780 --> 00:05:30.150 Well, it's 3. 00:05:30.150 --> 00:05:33.680 And that's choice C. 00:05:33.680 --> 00:05:34.930 Problem 8. 00:05:40.620 --> 00:05:50.950 If 2x plus z is equal to 2y, and 2x-- and then they also 00:05:50.950 --> 00:05:57.880 tell us that 2x plus 2y plus z is equal to 20. 00:05:57.880 --> 00:06:00.005 What is the value of y? 00:06:02.620 --> 00:06:05.170 y is equal to what? 00:06:05.170 --> 00:06:06.820 So there's something interesting here. 00:06:06.820 --> 00:06:14.330 we can rewrite this second equation as subtract 2y from 00:06:14.330 --> 00:06:18.430 both sides of this equation right here, and you get 2x 00:06:18.430 --> 00:06:24.560 plus z is equal to 20 minus y, right? 00:06:24.560 --> 00:06:27.880 So essentially, you have a 2x plus z here, and you have a 2x 00:06:27.880 --> 00:06:29.540 plus z here. 00:06:29.540 --> 00:06:31.920 Let's make a new variable. 00:06:31.920 --> 00:06:34.120 You don't have to do this step, but I think this'll 00:06:34.120 --> 00:06:35.550 simplify things. 00:06:35.550 --> 00:06:39.880 Let's call the variable Q. 00:06:39.880 --> 00:06:43.280 Let's say Q is equal to 2x plus z. 00:06:43.280 --> 00:06:44.360 That's where they're trying to confuse you. 00:06:44.360 --> 00:06:46.230 They're giving you two equations with three unknowns. 00:06:46.230 --> 00:06:48.520 And you're like, how can I solve for one of them? 00:06:48.520 --> 00:06:52.610 Well, what's interesting is they have-- you're solving for 00:06:52.610 --> 00:06:55.240 one of them, and then the relationship has a 2x plus z 00:06:55.240 --> 00:06:56.480 in both equations. 00:06:56.480 --> 00:06:59.830 So if you say Q is equal to 2x plus z, everything starts to 00:06:59.830 --> 00:07:03.990 make sense because then this top equation will become Q is 00:07:03.990 --> 00:07:05.360 equal to 2y. 00:07:05.360 --> 00:07:07.780 And what will this bottom equation be? 00:07:07.780 --> 00:07:12.560 That would be Q is equal to 20 minus y, right? 00:07:12.560 --> 00:07:15.040 And now you could set these equal to each other. 00:07:15.040 --> 00:07:19.670 2y is equal to 20 minus y. 00:07:19.670 --> 00:07:25.960 Add y to both sides, you get 3y is equal to 20. 00:07:25.960 --> 00:07:27.210 Am I doing that right? 00:07:27.210 --> 00:07:30.740 3y is equal to 20? 00:07:30.740 --> 00:07:34.005 You add y to both sides. y is equal to 20/3. 00:07:34.005 --> 00:07:35.690 And they don't have that choice, so I 00:07:35.690 --> 00:07:38.810 must have made a mistake. 00:07:38.810 --> 00:07:39.915 Let me redo the problem. 00:07:39.915 --> 00:07:41.260 I must've made a mistake. 00:07:41.260 --> 00:07:42.680 Problem 8. 00:07:42.680 --> 00:07:47.600 So they're telling us 2x plus z is equal to 2y, and then 00:07:47.600 --> 00:07:56.140 they tell us 2x plus 2y plus z is equal to 20. 00:07:56.140 --> 00:07:57.290 What is the value of y? 00:07:57.290 --> 00:08:02.510 OK, the second equation can be written as 2x plus z is equal 00:08:02.510 --> 00:08:06.010 to 20 minus 2y. 00:08:06.010 --> 00:08:10.470 And this top equation is still 2x plus z is equal to 2y. 00:08:10.470 --> 00:08:11.510 Oh, I see. 00:08:11.510 --> 00:08:13.870 I had dropped a y someplace. 00:08:13.870 --> 00:08:16.450 So this must equal this, because they both 00:08:16.450 --> 00:08:17.530 equal 2x plus z. 00:08:17.530 --> 00:08:19.630 I don't even have to do all that substitution Q. 00:08:19.630 --> 00:08:22.830 I think that you can see that 2x plus z equals this, 2x plus 00:08:22.830 --> 00:08:26.130 z equals this, so this must equal this. 00:08:26.130 --> 00:08:30.030 So 2y is equal to 20 minus 2y. 00:08:30.030 --> 00:08:35.000 Add 2y to both sides, you get 4y is equal to 20, 00:08:35.000 --> 00:08:36.820 y is equal to 5. 00:08:36.820 --> 00:08:40.100 And that is choice A. 00:08:40.100 --> 00:08:41.500 I'll see you in the next video. 00:08:55.690 --> 00:08:56.090 Oh, whoops! 00:08:56.090 --> 00:08:57.350 I didn't delete. 00:08:57.350 --> 00:08:58.600 Sorry.

SAT Prep: Test 8 Section 2 Part 3

https://www.youtube.com/watch?v=hbsWxarO5d4

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https://www.youtube.com/api/timedtext?v=hbsWxarO5d4&ei=YmeUZb7gMPO3hcIP6OaY0Ag&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E8F33E4DCE01A408565C3C32E647B696783EA75E.C16D543FD55C38CBACF7805F88137C6022111185&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.250 --> 00:00:05.460 OK, we are in problem number 9. 00:00:09.070 --> 00:00:14.490 If 2 times x minus 3 is equal to 7, what is the value of x? 00:00:14.490 --> 00:00:15.550 Well, this is straightforward. 00:00:15.550 --> 00:00:20.660 Just solve for x problem, divide both sides by 2. 00:00:20.660 --> 00:00:23.100 Or actually, yeah, we could do it either way. 00:00:23.100 --> 00:00:26.550 We could say this is 2x minus 6 is equal to 7. 00:00:26.550 --> 00:00:30.420 Add 6 to both sides, you get 2x is equal to 13. 00:00:30.420 --> 00:00:34.740 Divide both sides by 2. x is equal to 13/2, which is the 00:00:34.740 --> 00:00:36.710 same thing as 6 and 1/2, which is the same 00:00:36.710 --> 00:00:39.510 thing as 6.5, right? 00:00:39.510 --> 00:00:41.210 That's 12, yep. 00:00:41.210 --> 00:00:48.820 Problem 10. 00:00:48.820 --> 00:00:55.470 Point P lies on the line with the equation y minus 4 is 00:00:55.470 --> 00:01:00.060 equal to 3 times x minus 2. 00:01:00.060 --> 00:01:08.910 If the x-coordinate of P is 4-- OK, so P is the 0.4, what 00:01:08.910 --> 00:01:10.510 is the y-coordinate? 00:01:10.510 --> 00:01:13.820 So we don't know what the y-coordinate is, so we 00:01:13.820 --> 00:01:17.525 literally just have to solve-- we have to just input 4 for x 00:01:17.525 --> 00:01:19.290 and then solve for y. 00:01:19.290 --> 00:01:25.690 So you get y minus 4 is equal to 3 times 4 minus 2. 00:01:25.690 --> 00:01:30.980 y minus 4 is equal to 3 times 4 minus 2 is 2. 00:01:30.980 --> 00:01:33.740 y minus 4 is equal to 6. 00:01:33.740 --> 00:01:34.805 y is equal to 10. 00:01:34.805 --> 00:01:36.930 And that's our answer. 00:01:36.930 --> 00:01:38.840 The point is 4, 10, but they just wanted to know the 00:01:38.840 --> 00:01:41.490 y-coordinate, and that's 10. 00:01:41.490 --> 00:01:46.970 Problem 11. 00:01:46.970 --> 00:01:52.620 Car A traveled 60 miles and averaged 20 miles per gallon 00:01:52.620 --> 00:01:53.380 of gasoline. 00:01:53.380 --> 00:02:03.910 So car A traveled 60 miles and got 20 miles per gallon, mpg, 00:02:03.910 --> 00:02:05.660 miles per gallon. 00:02:05.660 --> 00:02:13.450 Car B traveled 15 miles for each gallon. 00:02:13.450 --> 00:02:16.810 OK, so it got 15 miles per gallon, right? 00:02:16.810 --> 00:02:21.310 It traveled 15 miles for each gallon of gasoline it used. 00:02:21.310 --> 00:02:25.040 How many miles had car B traveled when it had used the 00:02:25.040 --> 00:02:30.490 same amount of gasoline that car A used to travel 60 miles? 00:02:30.490 --> 00:02:34.340 So they're saying we need to figure out how many gallons A 00:02:34.340 --> 00:02:36.890 used, and then how far can B get with that 00:02:36.890 --> 00:02:37.990 same number of gallons? 00:02:37.990 --> 00:02:39.790 That's essentially what they're asking. 00:02:39.790 --> 00:02:42.240 So how many gallons did A use to go 60 miles? 00:02:42.240 --> 00:02:45.860 Well, we've got 20 miles per gallon and went 60 miles. 00:02:45.860 --> 00:02:53.750 So car A could do this on 3 gallons, right? 00:02:53.750 --> 00:03:03.120 The way you can view this, gallons times miles per gallon 00:03:03.120 --> 00:03:05.490 is equal to miles, right? 00:03:05.490 --> 00:03:09.920 And so gallons, we'll say g times 20 miles per gallon is 00:03:09.920 --> 00:03:13.170 equal to 60 miles. 00:03:13.170 --> 00:03:16.540 So gallons, divide both sides by 20, is equal to 30. 00:03:16.540 --> 00:03:18.330 That's times right there. 00:03:18.330 --> 00:03:19.790 I'm sorry, it's equal to 3. 00:03:19.790 --> 00:03:21.460 60 divided by 20 is 3. 00:03:21.460 --> 00:03:25.000 So car A used 3 gallons to go 60 miles. 00:03:25.000 --> 00:03:39.800 And so car B can go 15 miles per gallon times 3 gallons 00:03:39.800 --> 00:03:46.090 equals 15 times 3 is 45 gallons. 00:03:46.090 --> 00:03:47.340 Next problem. 00:03:54.020 --> 00:03:56.500 OK, see if I can draw that. 00:03:56.500 --> 00:03:59.680 It's a straight line here. 00:03:59.680 --> 00:04:05.025 Straight line, point A comes up to this point to B. 00:04:05.025 --> 00:04:07.850 It comes down a little bit. 00:04:07.850 --> 00:04:09.660 And then point C comes down here. 00:04:12.390 --> 00:04:16.190 OK, and they're telling us that this is 65 degrees. 00:04:16.190 --> 00:04:19.180 This is 100 degrees. 00:04:19.180 --> 00:04:21.940 This is 120. 00:04:21.940 --> 00:04:25.430 And this, right here, is x degrees. 00:04:25.430 --> 00:04:28.700 In the figure above, points A, D, and E lie on the same line. 00:04:28.700 --> 00:04:30.190 This is A. 00:04:30.190 --> 00:04:32.050 This is D. 00:04:32.050 --> 00:04:33.220 This is E. 00:04:33.220 --> 00:04:34.330 They lie on the same line. 00:04:34.330 --> 00:04:37.690 What is the value of x? 00:04:37.690 --> 00:04:39.700 So we really just need to figure out the value of this, 00:04:39.700 --> 00:04:42.160 and then we can say x is supplementary to that. 00:04:42.160 --> 00:04:45.410 So let's think about how we can think about what the value 00:04:45.410 --> 00:04:48.240 of this angle is. 00:04:48.240 --> 00:04:54.150 So the way I think about it is draw a line here, and you 00:04:54.150 --> 00:04:57.500 could automatically know what all the angles in a 00:04:57.500 --> 00:04:58.770 quadrilateral are. 00:04:58.770 --> 00:05:00.710 But I'll prove it to you that all the angles in a 00:05:00.710 --> 00:05:03.110 quadrilateral are going to add up to 360. 00:05:03.110 --> 00:05:04.590 How do I know that? 00:05:04.590 --> 00:05:11.010 Because think of it this way: this angle plus this angle 00:05:11.010 --> 00:05:14.630 plus this angle is going to be 180. 00:05:14.630 --> 00:05:18.280 And then this angle plus this angle plus this angle 00:05:18.280 --> 00:05:20.200 is going to be 180. 00:05:20.200 --> 00:05:22.730 And if you add all of those together, you're essentially 00:05:22.730 --> 00:05:26.570 adding all of the angles of the quadrilateral-- it's 00:05:26.570 --> 00:05:30.190 late-- because this and this is equal to the angle of the 00:05:30.190 --> 00:05:30.900 whole quadrilateral. 00:05:30.900 --> 00:05:34.390 So the angles in the entire quadrilateral are 360 degrees. 00:05:34.390 --> 00:05:36.570 And you can memorize that if you want. 00:05:36.570 --> 00:05:39.190 So if we know that all of these angles add up to 360 00:05:39.190 --> 00:05:42.060 degrees, what is this big angle right here? 00:05:42.060 --> 00:05:50.150 Well, we know that 65 plus 100 plus 120 plus-- let's call 00:05:50.150 --> 00:05:54.010 this angle y, and y is this whole thing-- plus y is equal 00:05:54.010 --> 00:05:55.980 to 360 degrees. 00:05:55.980 --> 00:05:56.660 And this is what? 00:05:56.660 --> 00:06:05.220 This is 220 plus 65, 285 plus y is equal to 360. 00:06:05.220 --> 00:06:13.300 So y is equal to 360 minus 285, and that is what? 00:06:13.300 --> 00:06:15.620 That is 75 degrees, right? 00:06:15.620 --> 00:06:17.890 Because 70 would be 350, right? 00:06:17.890 --> 00:06:21.100 So y is 75 degrees. 00:06:21.100 --> 00:06:25.220 So this whole angle right there is 75. 00:06:25.220 --> 00:06:29.860 And we know that x plus this big angle here, 75-- and 00:06:29.860 --> 00:06:30.840 ignore this green line. 00:06:30.840 --> 00:06:32.660 I'm looking about this big angle. 00:06:32.660 --> 00:06:35.470 x plus 75 is equal to 180. 00:06:35.470 --> 00:06:39.280 So x has to be equal to 105 degrees. 00:06:39.280 --> 00:06:41.820 Subtract 75 from both sides. 00:06:41.820 --> 00:06:50.700 Next question, problem 13. 00:06:50.700 --> 00:06:52.770 The first term of a sequence is 20 and the 00:06:52.770 --> 00:06:53.830 second term is 8. 00:06:53.830 --> 00:06:56.450 So it goes from 20, then it goes to 8. 00:06:56.450 --> 00:07:00.090 The third term and each term thereafter is the average of 00:07:00.090 --> 00:07:02.380 the two terms immediately preceding it. 00:07:02.380 --> 00:07:04.990 What is the value of the first term in the sequence that is 00:07:04.990 --> 00:07:06.490 not an integer? 00:07:06.490 --> 00:07:06.790 OK. 00:07:06.790 --> 00:07:08.700 So we just have to average these two. 00:07:08.700 --> 00:07:10.540 So this is going to be what? 00:07:10.540 --> 00:07:12.500 It's going to be 28 divided by 2. 00:07:12.500 --> 00:07:16.940 28/2, which equals 14, right? 00:07:16.940 --> 00:07:20.950 I just took 20 plus 8 divided by 2 is 14. 00:07:20.950 --> 00:07:23.910 And now, what's the average of 8 and 14? 00:07:23.910 --> 00:07:28.202 Well, 8 plus 14 is 22. 00:07:28.202 --> 00:07:31.840 So 22 divided by 2, which is equal to 11. 00:07:31.840 --> 00:07:33.260 Delete my work. 00:07:33.260 --> 00:07:36.750 Now, what's the average of 14 and 11? 00:07:36.750 --> 00:07:39.900 You add them, you get 25 divided by 00:07:39.900 --> 00:07:43.480 2, that equals 12.5. 00:07:43.480 --> 00:07:44.180 So here we go. 00:07:44.180 --> 00:07:47.150 This is the first number in the series that is not an 00:07:47.150 --> 00:07:48.500 integer, 12.5. 00:07:48.500 --> 00:07:50.310 And you could fill that in. 00:07:50.310 --> 00:07:52.970 Problem 14. 00:07:52.970 --> 00:07:56.050 If x is 1/5 of y-- so x is equal to 1/5y. 00:07:58.600 --> 00:08:00.970 y is 3/10 of z. 00:08:00.970 --> 00:08:07.870 y is equal to 3/10z and z is greater than 0, then x is what 00:08:07.870 --> 00:08:09.200 fraction of z? 00:08:09.200 --> 00:08:12.160 So you could literally just substitute here, right? 00:08:12.160 --> 00:08:14.720 You can just substitute this y. 00:08:14.720 --> 00:08:17.330 You could substitute 3/10z for y, right? 00:08:17.330 --> 00:08:20.510 Because we say y is equal to 3 times z, so x is equal 00:08:20.510 --> 00:08:23.600 to 1/5 times y. 00:08:23.600 --> 00:08:29.220 We're going to substitute for y, 3/10z. 00:08:29.220 --> 00:08:31.670 So x is equal to-- what is this? 00:08:31.670 --> 00:08:33.475 3/50z. 00:08:33.475 --> 00:08:35.830 So x is 3/50 of z. 00:08:35.830 --> 00:08:40.370 So you would fill in 3/50 in your answer box. 00:08:40.370 --> 00:08:42.580 OK, Next problem. 00:08:42.580 --> 00:08:44.895 Well, I don't know if I have time to do it in this video. 00:08:44.895 --> 00:08:46.145 Well, let me see. 00:08:50.410 --> 00:08:51.380 No, I don't want to rush it. 00:08:51.380 --> 00:08:52.930 I'll do it in the next video. 00:08:52.930 --> 00:08:54.180 See you soon.

SAT Prep: Test 8 Section 2 Part 4

https://www.youtube.com/watch?v=423zK3ev1vM

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https://www.youtube.com/api/timedtext?v=423zK3ev1vM&ei=ZWeUZfCeJKy0p-oP7c64-AQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249813&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=24EF19DB8828565F86C0102F4B616CB93DB193AF.A949DFB6D1C78FCE7513B4A7484A9DE2A76D0914&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.860 --> 00:00:03.646 Problem 15. 00:00:03.646 --> 00:00:06.501 Let's see, they have a square. 00:00:06.501 --> 00:00:07.560 It looks like a square. 00:00:07.560 --> 00:00:08.940 I don't know if it's a square just yet. 00:00:08.940 --> 00:00:11.540 And they have a triangle that comes off of 00:00:11.540 --> 00:00:12.790 the square like this. 00:00:15.220 --> 00:00:17.300 Ah, close enough. 00:00:17.300 --> 00:00:18.420 And what do they tell us? 00:00:18.420 --> 00:00:20.790 They tell us that this is 90 degrees. 00:00:20.790 --> 00:00:22.585 This is 60 degrees. 00:00:22.585 --> 00:00:30.445 And this is A, B, C, D, and E. 00:00:30.445 --> 00:00:31.870 And what other thing? 00:00:31.870 --> 00:00:38.640 In the figure above, E, B, C, D is a square, OK, and A, E is 00:00:38.640 --> 00:00:40.210 equal to 8. 00:00:40.210 --> 00:00:45.030 What is the area of E, B, C, D? 00:00:45.030 --> 00:00:46.760 So we just have to figure out one side of the square. 00:00:46.760 --> 00:00:48.040 And it's a square, so all the sides are equal 00:00:48.040 --> 00:00:49.130 and then we're done. 00:00:49.130 --> 00:00:53.080 And this is actually just a 30-60-90 triangle problem. 00:00:53.080 --> 00:00:53.600 How do I know it? 00:00:53.600 --> 00:00:55.700 Because this angle is 60, this is 90, so this 00:00:55.700 --> 00:00:56.590 one has to be 30. 00:00:56.590 --> 00:00:58.450 Or I could add it up to 180. 00:00:58.450 --> 00:01:00.950 And if you don't even remember what a 30-60-90 triangle is, 00:01:00.950 --> 00:01:03.000 where they add up, you could go back to page 838, 00:01:03.000 --> 00:01:03.670 and they tell you. 00:01:03.670 --> 00:01:06.020 If you look in the middle of that page, they have special 00:01:06.020 --> 00:01:11.920 right triangles, and they tell you that if the hypotenuse is 00:01:11.920 --> 00:01:20.040 2x that the side opposite the 30 degree side is x, and then 00:01:20.040 --> 00:01:24.060 the side opposite the 60 degree side is x square root 00:01:24.060 --> 00:01:26.430 of 3, right? 00:01:26.430 --> 00:01:29.670 So if x is equal to 8, what's x squared of 3? 00:01:29.670 --> 00:01:32.500 x squared of 3 is going to be equal to 8 square roots of 3. 00:01:32.500 --> 00:01:35.680 And, of course, this side will be 2 times 8, which is 16. 00:01:35.680 --> 00:01:37.760 I always think of the 30 degree side as half of the 00:01:37.760 --> 00:01:40.680 hypotenuse, and then the 60-degree side as square root 00:01:40.680 --> 00:01:42.430 of 3 times the 30-degree side. 00:01:42.430 --> 00:01:43.420 That's how I think about it. 00:01:43.420 --> 00:01:46.410 But you can just look at what they do on page 00:01:46.410 --> 00:01:48.140 838 and you'll know. 00:01:48.140 --> 00:01:51.280 If this is 2x, the side opposite the 30-degree side is 00:01:51.280 --> 00:01:55.186 x, And the side opposite the 60-degree side is x times the 00:01:55.186 --> 00:01:57.153 square root of 3, so this has to be 8 times the 00:01:57.153 --> 00:01:58.660 square root of 3. 00:01:58.660 --> 00:02:00.790 If this side is 8 square root times the square root of 3, 00:02:00.790 --> 00:02:01.990 then so is this side. 00:02:01.990 --> 00:02:05.150 And so to figure out the area of the square, you just say 8 00:02:05.150 --> 00:02:08.170 times the square root of 3 times 8 times the 00:02:08.170 --> 00:02:08.860 square root of 3. 00:02:08.860 --> 00:02:09.930 And what does that equal? 00:02:09.930 --> 00:02:13.450 That's 8 times 8, that's 64, times the 00:02:13.450 --> 00:02:15.560 square root of 3 squared. 00:02:15.560 --> 00:02:17.540 And what's the square root of three squared? 00:02:17.540 --> 00:02:19.170 That's just 3. 00:02:19.170 --> 00:02:22.520 So it would become 64 times 3, and that's what? 00:02:22.520 --> 00:02:27.870 180 plus 12, that's equal to 192. 00:02:27.870 --> 00:02:29.120 Next problem. 00:02:32.450 --> 00:02:35.660 It's really good to become comfortable with 00:02:35.660 --> 00:02:38.140 the 30-60-90 triangles. 00:02:38.140 --> 00:02:41.800 You'll probably get one or two extra problems on the SAT if 00:02:41.800 --> 00:02:43.550 you get really proficient at that. 00:02:43.550 --> 00:02:47.750 In a mixture of peanuts and cashews, the ratio by weight 00:02:47.750 --> 00:02:49.640 of peanuts to cashews is 5:2. 00:02:49.640 --> 00:02:54.620 Peanuts to cashews, and that's really the weight is 5:2. 00:02:54.620 --> 00:02:58.380 How many pounds of cashews will there be in four pounds 00:02:58.380 --> 00:02:59.670 of this mixture? 00:02:59.670 --> 00:03:04.600 How many pounds of cashews there'll be in 4 pounds of 00:03:04.600 --> 00:03:05.700 this mixture? 00:03:05.700 --> 00:03:09.300 So we know that cashews plus peanuts is 00:03:09.300 --> 00:03:11.590 going to be 4 pounds. 00:03:11.590 --> 00:03:15.890 And we also know-- I could multiply both sides of this 00:03:15.890 --> 00:03:17.660 equation by C, this top one. 00:03:17.660 --> 00:03:22.400 We also know that peanuts are going to be 5/2 the number of 00:03:22.400 --> 00:03:24.160 cashews, right? 00:03:24.160 --> 00:03:27.290 Because the weight of peanuts to cashews is 5:2. 00:03:27.290 --> 00:03:29.440 So peanuts is going to be 5/2 of cashews. 00:03:29.440 --> 00:03:31.890 And the cashews plus the peanuts is equal to 4. 00:03:31.890 --> 00:03:36.850 So why don't we just substitute this in for this? 00:03:36.850 --> 00:03:44.550 So we get cashews plus 5/2 of cashews is equal to 4. 00:03:44.550 --> 00:03:45.830 And what is this equal to? 00:03:45.830 --> 00:03:49.850 This is equal to dot common denominator 2. 00:03:49.850 --> 00:03:51.400 This is like one cashew, right? 00:03:51.400 --> 00:03:57.080 So that's 2/2 plus 5/2 cashews is equal to 4. 00:03:57.080 --> 00:04:02.120 So going up here, that's 7/2 times cashews is 00:04:02.120 --> 00:04:03.850 equal to the 4. 00:04:03.850 --> 00:04:06.800 Let's multiply both sides of this times 2. 00:04:06.800 --> 00:04:11.650 So then you get 7 cashews is equal to 8. 00:04:11.650 --> 00:04:18.329 Divide both sides by 7, you get cashews is equal to 8/7 00:04:18.329 --> 00:04:22.770 pounds, right? 00:04:22.770 --> 00:04:27.380 8/7 pound, so 1 and 1/7 pound is going to be cashews. 00:04:27.380 --> 00:04:29.930 And this stuff here, I could just multiply both sides by 00:04:29.930 --> 00:04:34.590 2/7, times 2/7. 00:04:34.590 --> 00:04:36.040 This would have canceled out, and I would have gotten the 00:04:36.040 --> 00:04:37.000 same thing, 8/7. 00:04:37.000 --> 00:04:41.110 So there are 8/7 pounds of cashews here. 00:04:41.110 --> 00:04:42.360 Next problem. 00:04:45.040 --> 00:04:48.860 Kind of an odd number, but I think on these free answer, 00:04:48.860 --> 00:04:52.480 they sometimes give you problems that might have not 00:04:52.480 --> 00:04:58.810 the most clean numbers just to trip you up. 00:04:58.810 --> 00:05:01.240 OK, so that's the y-axis. 00:05:01.240 --> 00:05:03.650 That's my x-axis. 00:05:03.650 --> 00:05:06.680 After this problem and the next, I will go to bed. 00:05:06.680 --> 00:05:10.420 So, let's see, that's the y-axis. 00:05:10.420 --> 00:05:13.326 And then they drew a line here. 00:05:13.326 --> 00:05:15.510 It looks something like that. 00:05:15.510 --> 00:05:16.760 They call that line l. 00:05:19.580 --> 00:05:22.185 They draw another line like this. 00:05:27.070 --> 00:05:31.770 Let's see, this is A, this is B, and they say that this 00:05:31.770 --> 00:05:34.610 point right here is the point 8, 3. 00:05:34.610 --> 00:05:35.580 OK. 00:05:35.580 --> 00:05:39.980 line m, not shown, passes through the origin, that's o, 00:05:39.980 --> 00:05:43.370 and intersects A, B between A and B. 00:05:43.370 --> 00:05:46.320 What is one possible value of the slope of line m? 00:05:46.320 --> 00:05:49.440 So line m intersects at the origin and intersects this 00:05:49.440 --> 00:05:50.110 line someplace. 00:05:50.110 --> 00:05:53.980 It's going to look something like this. 00:05:53.980 --> 00:05:57.630 So the whole issue here is that this line is going to 00:05:57.630 --> 00:06:01.450 have a lower slope than this line, than line l. 00:06:01.450 --> 00:06:02.860 So what is the slope of line l? 00:06:02.860 --> 00:06:05.210 Well, it goes through the point 8, 3, and it also goes 00:06:05.210 --> 00:06:09.330 through the point 0, 0. 00:06:09.330 --> 00:06:11.040 So what's the slope of line l? 00:06:11.040 --> 00:06:12.780 Change in y over change in x. 00:06:15.340 --> 00:06:25.570 3 minus 0 over 8 minus 0 is equal to 3/8, right? 00:06:25.570 --> 00:06:28.780 So this slope is 3/8. 00:06:28.780 --> 00:06:31.950 So in order for this slope-- it has to be less than this. 00:06:31.950 --> 00:06:33.750 And it's also going to be greater than 1, right? 00:06:33.750 --> 00:06:35.300 Because it's still in the first quadrant, so it's still 00:06:35.300 --> 00:06:36.580 going to be in this range. 00:06:36.580 --> 00:06:39.250 So what's a slope that's less than 3/8 and greater than 1? 00:06:39.250 --> 00:06:40.960 Well, I don't know. 00:06:40.960 --> 00:06:43.240 2/8, which equals 1/4? 00:06:43.240 --> 00:06:44.190 That would work. 00:06:44.190 --> 00:06:46.690 That's less than 3/8 and greater than 1. 00:06:46.690 --> 00:06:49.460 I mean, you could say 1/8. 00:06:49.460 --> 00:06:50.010 That would work. 00:06:50.010 --> 00:06:51.100 1/16 would work. 00:06:51.100 --> 00:06:52.410 1/1,000 would work. 00:06:52.410 --> 00:06:56.120 Any of those, that's all possible values for the slope 00:06:56.120 --> 00:06:58.770 of m because they just say it intersects A, B 00:06:58.770 --> 00:06:59.610 between A and B. 00:06:59.610 --> 00:07:01.000 It doesn't say where. 00:07:01.000 --> 00:07:04.685 So any line, you know, it could look like this. 00:07:04.685 --> 00:07:07.090 The line could look really small, so that would be like a 00:07:07.090 --> 00:07:11.100 slope of 1/1,000, but it would still work. 00:07:11.100 --> 00:07:19.430 Next problem, problem 18. 00:07:19.430 --> 00:07:20.990 OK, so they gave us a table. 00:07:25.030 --> 00:07:34.808 So they say year, number of students. 00:07:34.808 --> 00:07:43.410 And then they go 92, 93, 94, 95, 96. 00:07:43.410 --> 00:07:58.680 In 92, there was x, then 1,552, 1,238 1,459, and 1,351 00:07:58.680 --> 00:08:01.320 The table above shows student enrollment in Weston High 00:08:01.320 --> 00:08:03.780 School from 1992 to 1996. 00:08:03.780 --> 00:08:07.920 If the median enrollment for the five years was 1,351-- so 00:08:07.920 --> 00:08:12.490 this is the median; this is the middle number-- and no two 00:08:12.490 --> 00:08:14.140 years have the same enrollment, what is the 00:08:14.140 --> 00:08:16.480 greatest possible value for x? 00:08:16.480 --> 00:08:20.270 So 1,351 has to be the middle number. 00:08:20.270 --> 00:08:22.440 Let me make another list. 1,351 has to 00:08:22.440 --> 00:08:24.150 be the middle number. 00:08:24.150 --> 00:08:25.890 And let's see, what are the numbers that we know are 00:08:25.890 --> 00:08:26.910 definitely greater? 00:08:26.910 --> 00:08:35.169 We have 1,451, and then we have 1,552. 00:08:35.169 --> 00:08:37.220 And the numbers that are less, we know-- 00:08:37.220 --> 00:08:38.630 so this is the median. 00:08:38.630 --> 00:08:43.440 We know that 1,238 is less. 00:08:43.440 --> 00:08:49.610 And we know that if 1,351 is the median, then x is also 00:08:49.610 --> 00:08:51.850 going to have to be less than-- there has to be two 00:08:51.850 --> 00:08:57.470 numbers less than 1,351, so x has to be less than 1,351, 00:08:57.470 --> 00:08:59.290 although x could be here as well. 00:08:59.290 --> 00:09:00.880 It could be larger than 1,238. 00:09:00.880 --> 00:09:10.110 So we know that x is less than 1,351. 00:09:10.110 --> 00:09:12.750 And it could have been equal to 1,351, but they tell us 00:09:12.750 --> 00:09:15.010 that no two years have the same enrollment. 00:09:15.010 --> 00:09:17.770 So what is the largest possible value for x? 00:09:17.770 --> 00:09:21.100 Well, it's 1,350, right? 00:09:21.100 --> 00:09:23.040 Because you know that's the largest number that's less 00:09:23.040 --> 00:09:24.180 than 1,351. 00:09:24.180 --> 00:09:26.410 And then this would be 1,350 here. 00:09:26.410 --> 00:09:28.505 And then 1,351 would still be the median. 00:09:28.505 --> 00:09:29.900 And we are done! 00:09:29.900 --> 00:09:31.890 I'll see you in the next section.

SAT Prep: Test 7 Section 8 Part 1

https://www.youtube.com/watch?v=GZOp27tWARg

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en

WEBVTT Kind: captions Language: en 00:00:00.720 --> 00:00:03.590 We are in Section 8 on the seventh test. We're almost 00:00:03.590 --> 00:00:04.960 done with this book, so let's start. 00:00:04.960 --> 00:00:06.210 Problem number one. 00:00:08.760 --> 00:00:19.520 If 6,700 is equal to 100 times 6k plus 7, then k equals what? 00:00:19.520 --> 00:00:22.060 So let's divide both sides of this by 100. 00:00:22.060 --> 00:00:25.060 So you get 100 and you get these two zeroes, so you get 00:00:25.060 --> 00:00:28.310 67 is equal to 6k plus 7. 00:00:28.310 --> 00:00:30.520 I just divided both sides by 100. 00:00:30.520 --> 00:00:31.980 Subtract 7 from both sides. 00:00:31.980 --> 00:00:34.490 You get 60 is equal to 6k. 00:00:34.490 --> 00:00:38.310 Divide both sides by 6, you get 10 is equal to k. 00:00:38.310 --> 00:00:40.740 That's choice C. 00:00:40.740 --> 00:00:43.040 Problem 2. 00:00:43.040 --> 00:00:49.850 If 3 more than n is a negative number-- so n plus 3 is a 00:00:49.850 --> 00:00:52.270 negative number-- that's what that says, right? 00:00:52.270 --> 00:00:57.210 And 5 more than n is a positive number, so n plus 5 00:00:57.210 --> 00:01:01.320 is greater than 0, which of the following could be the 00:01:01.320 --> 00:01:02.770 value of n? 00:01:02.770 --> 00:01:07.810 So n plus 3 is negative, n plus 5 is greater, so n plus 3 00:01:07.810 --> 00:01:12.710 has to be less than negative 3, right? 00:01:12.710 --> 00:01:15.500 This means that n is less than negative 3. 00:01:15.500 --> 00:01:16.180 How did I get that? 00:01:16.180 --> 00:01:19.970 I subtracted 3 from both sides of this equation, right? 00:01:19.970 --> 00:01:23.520 If I subtract 5 from both sides of this equation, I 00:01:23.520 --> 00:01:28.570 would get that n is greater than negative 5. 00:01:28.570 --> 00:01:30.460 So you have to be greater than negative 5 and less than 00:01:30.460 --> 00:01:31.780 negative 3. 00:01:31.780 --> 00:01:34.240 What's the only number that I can think of that's that? 00:01:34.240 --> 00:01:37.140 Well, there's a lot of numbers, but a simple one is 00:01:37.140 --> 00:01:39.230 minus 4, right? 00:01:39.230 --> 00:01:42.140 Minus 4 is less than negative 3, right? 00:01:42.140 --> 00:01:46.540 If I draw a number line, this is minus 3, minus 4, minus 5. 00:01:46.540 --> 00:01:49.280 So minus 4 is less than negative 3 and it's greater 00:01:49.280 --> 00:01:50.100 than minus 5. 00:01:50.100 --> 00:01:51.120 So that's the answer. 00:01:51.120 --> 00:01:53.310 Choice B. 00:01:53.310 --> 00:01:55.100 And you can try it out with minus 4. 00:01:55.100 --> 00:01:57.110 Minus 4 plus 3 is minus 1. 00:01:57.110 --> 00:01:58.520 So that works. 00:01:58.520 --> 00:02:00.830 Minus 4 plus 5 is plus 1. 00:02:00.830 --> 00:02:03.270 So that works. 00:02:03.270 --> 00:02:04.520 Problem three. 00:02:08.960 --> 00:02:14.090 OK in the figure above-- OK let me see if I can draw this. 00:02:14.090 --> 00:02:16.640 I'll draw it big because it looks complicated. 00:02:16.640 --> 00:02:26.800 A line like that, a line like that, and one like that. 00:02:26.800 --> 00:02:41.180 They tell us this is x, this is y, and then they tell us in 00:02:41.180 --> 00:02:44.020 the figure above, if x is equal to 70, so this is equal 00:02:44.020 --> 00:02:49.710 to 70, and y is equal to 40, so this is equal to 40, and 00:02:49.710 --> 00:02:53.030 the dotted lines bisect the angles with measure x and y 00:02:53.030 --> 00:02:55.180 degrees, what is the value of z? 00:02:55.180 --> 00:02:57.370 OK, so the dotted lines, instead of drawing dotted 00:02:57.370 --> 00:03:00.950 lines, I'm going to draw green lines, so this green line will 00:03:00.950 --> 00:03:05.450 bisect this angle, so it goes right in between. 00:03:05.450 --> 00:03:08.600 So if it bisects the angle, what is the measure of this 00:03:08.600 --> 00:03:11.350 angle right here? 00:03:11.350 --> 00:03:13.630 Well, it's going to be half of this angle that it bisected. 00:03:13.630 --> 00:03:16.240 So this whole thing is 70, so this is 00:03:16.240 --> 00:03:18.860 going to be 35 degrees. 00:03:18.860 --> 00:03:21.990 Similarly, this other green line is going to bisect this 00:03:21.990 --> 00:03:25.960 y-- oh, I thought I was using the line tool. 00:03:25.960 --> 00:03:30.490 This other green line is going to bisect y right here, right? 00:03:30.490 --> 00:03:31.380 So it bisects it. 00:03:31.380 --> 00:03:35.100 So if it bisects it, what's this angle going to be? 00:03:35.100 --> 00:03:37.280 Well, it's going to be half of angle y, and they tell us 00:03:37.280 --> 00:03:38.240 angle y is 40. 00:03:38.240 --> 00:03:38.810 I wrote that. 00:03:38.810 --> 00:03:39.800 I just wrote over it. 00:03:39.800 --> 00:03:41.910 So this has to be 20 degrees, right? 00:03:41.910 --> 00:03:44.600 Because it's half of the full y. 00:03:44.600 --> 00:03:45.770 And they say, what is the values of z? 00:03:45.770 --> 00:03:48.840 Well, z is just this whole thing. 00:03:48.840 --> 00:03:52.890 So it's this angle plus this angle, so it's 55 degrees. 00:03:52.890 --> 00:03:54.140 And that's choice E. 00:03:56.840 --> 00:03:58.090 Problem four. 00:04:02.800 --> 00:04:04.960 A piece of fruit is to be chosen at random 00:04:04.960 --> 00:04:06.700 from a basket a fruit. 00:04:06.700 --> 00:04:09.560 The probability that the piece of fruit chosen will be an 00:04:09.560 --> 00:04:11.610 apple is 2/5. 00:04:11.610 --> 00:04:19.589 So the probability of an apple is equal to 2/5. 00:04:19.589 --> 00:04:23.620 Which of the following could not be the number of pieces of 00:04:23.620 --> 00:04:25.760 fruit in the basket? 00:04:25.760 --> 00:04:27.350 Could not be. 00:04:27.350 --> 00:04:35.280 So the secret here is you take 2/5 and whatever 2/5 is times 00:04:35.280 --> 00:04:37.610 the number of fruit, right? 00:04:37.610 --> 00:04:43.410 2/5 times the number of fruit has to equal 00:04:43.410 --> 00:04:44.660 the number of apples. 00:04:47.970 --> 00:04:53.430 And implicitly, there's not going to be a fractional 00:04:53.430 --> 00:04:54.980 number of apples in the basket. 00:04:54.980 --> 00:04:56.840 You can't pick up a piece of an apple. 00:04:56.840 --> 00:04:59.450 So this is going to have to be an integer, the number of 00:04:59.450 --> 00:05:01.140 apples, right? 00:05:01.140 --> 00:05:07.020 So if 2/5 times some number is going to be an integer, then 00:05:07.020 --> 00:05:10.020 this number has to be divisible by 5. 00:05:10.020 --> 00:05:13.140 It makes sense, you know? 00:05:13.140 --> 00:05:14.810 If there are 10 fruit, then this is going 00:05:14.810 --> 00:05:15.640 to be equal to 4. 00:05:15.640 --> 00:05:18.040 If there are 20 fruit, then this is going to be what? 00:05:18.040 --> 00:05:19.740 What's 2/5 of 20? 00:05:19.740 --> 00:05:21.350 It's going to be 8. 00:05:21.350 --> 00:05:25.605 But if there were 6 fruit, or if we thought the number of 00:05:25.605 --> 00:05:28.860 fruit was 6, 2/5 times 6 is 12/5. 00:05:28.860 --> 00:05:32.650 That means that there are 2 and 2/5 apples. 00:05:32.650 --> 00:05:33.650 That's not right. 00:05:33.650 --> 00:05:35.740 So what we have to find is the number of fruit, and it has to 00:05:35.740 --> 00:05:37.822 be some number divisible by 5. 00:05:37.822 --> 00:05:40.140 And if you look at the choices, they give you 20, 00:05:40.140 --> 00:05:41.200 that's divisible by 5. 00:05:41.200 --> 00:05:43.720 35 is divisible by 5. 00:05:43.720 --> 00:05:45.840 Choice C, 52. 00:05:45.840 --> 00:05:48.020 That's not divisible by 5. 00:05:48.020 --> 00:05:50.100 And then the other two choices, 70 and 80 are 00:05:50.100 --> 00:05:51.010 divisible by 5. 00:05:51.010 --> 00:05:53.650 So only C is not divisible by 5. 00:05:53.650 --> 00:05:54.920 So that's our choice. 00:05:54.920 --> 00:05:57.180 That could not be the number of fruit in the basket. 00:05:57.180 --> 00:05:58.260 And you could figure it out. 00:05:58.260 --> 00:06:01.770 If there were 52 fruit, and 2/5 were apples, you'd get a 00:06:01.770 --> 00:06:03.020 fractional number of apples. 00:06:03.020 --> 00:06:04.740 And that's not cool. 00:06:04.740 --> 00:06:05.990 Next problem. 00:06:10.080 --> 00:06:12.590 Problem five. 00:06:12.590 --> 00:06:14.960 A square and an equilateral triangle have equal 00:06:14.960 --> 00:06:16.130 perimeters. 00:06:16.130 --> 00:06:18.740 If the square has sides of length 3, what is the length 00:06:18.740 --> 00:06:21.190 of one side of the triangle? 00:06:21.190 --> 00:06:29.340 So a square and a triangle have equal perimeters. 00:06:29.340 --> 00:06:32.240 And the square has-- what does it say? 00:06:32.240 --> 00:06:34.050 It has sides of length 3. 00:06:34.050 --> 00:06:36.740 So 3, 3, 3, 3. 00:06:36.740 --> 00:06:38.300 So what's this perimeter? 00:06:38.300 --> 00:06:40.020 It's 3 plus 3 plus 3 plus 3. 00:06:40.020 --> 00:06:41.010 That's 12. 00:06:41.010 --> 00:06:42.500 Perimeter is 12. 00:06:42.500 --> 00:06:44.420 Well, this triangle is also going to have a perimeter of 00:06:44.420 --> 00:06:50.050 12, but it's going to have it amongst 3 equal sides, right? 00:06:50.050 --> 00:06:52.140 Because there's only 3 sides of a triangle. 00:06:52.140 --> 00:06:55.315 So it's going to have 12 divided by 3, so it's going to 00:06:55.315 --> 00:06:58.640 have to be 4, 4, and 4, so one side of the 00:06:58.640 --> 00:07:00.750 triangle has length 4. 00:07:00.750 --> 00:07:02.100 That's choice C. 00:07:05.400 --> 00:07:07.134 Problem six. 00:07:07.134 --> 00:07:10.930 We'll do it here. 00:07:10.930 --> 00:07:18.540 If x is equal to negative 1 and k is greater than 0, which 00:07:18.540 --> 00:07:22.110 of the following has the greatest value? 00:07:22.110 --> 00:07:29.080 OK, so choice A is 2 times k times x. 00:07:29.080 --> 00:07:32.550 So k is positive, x is minus 1, so this is going to 00:07:32.550 --> 00:07:34.230 be less than 0. 00:07:34.230 --> 00:07:36.890 So this is going to be a negative number, right? 00:07:36.890 --> 00:07:39.860 And so we know that's probably not going to be the answer. 00:07:39.860 --> 00:07:42.740 So if we look at all of the choices, the only positive 00:07:42.740 --> 00:07:45.650 answers, the only positive choices, are going to be the 00:07:45.650 --> 00:07:50.280 ones where I raise x to an even exponent, right? 00:07:50.280 --> 00:07:53.840 Because if I raised x to an odd exponent, like let's see, 00:07:53.840 --> 00:07:59.195 choice C, choice C is 6k x cubed. 00:07:59.195 --> 00:08:02.600 And that's equal to 6k times minus 1 cubed, right? 00:08:02.600 --> 00:08:03.980 Because x is minus 1. 00:08:03.980 --> 00:08:06.530 So that's going to be minus 6k. 00:08:06.530 --> 00:08:08.220 And we know k is a positive number. 00:08:08.220 --> 00:08:12.440 I mean, really, you could just ignore the k's, really. 00:08:12.440 --> 00:08:13.580 k is just a positive number. 00:08:13.580 --> 00:08:14.700 It's just kind of a scaling factor. 00:08:14.700 --> 00:08:16.050 And they all have k in them. 00:08:16.050 --> 00:08:17.660 So you just have to worry about the coefficient in the x 00:08:17.660 --> 00:08:21.850 terms. So one thing we can-- if x has an odd exponent on 00:08:21.850 --> 00:08:24.440 it, it's going to be a negative number, so that's not 00:08:24.440 --> 00:08:25.990 going to be the greatest value. 00:08:25.990 --> 00:08:28.560 So there are only two that have x with a-- so 00:08:28.560 --> 00:08:31.260 you have 4k x squared. 00:08:31.260 --> 00:08:32.690 That's choice B. 00:08:32.690 --> 00:08:40.900 And then you have choice D, which is 8k x to the fourth. 00:08:40.900 --> 00:08:44.230 Well, in both of these cases, x squared is equal to 1. 00:08:44.230 --> 00:08:45.950 Negative 1 squared is 1. 00:08:45.950 --> 00:08:49.170 Negative 1 to the fourth is also equal to 1, right? 00:08:49.170 --> 00:08:51.020 And if k is positive, what's bigger? 00:08:51.020 --> 00:08:52.710 8k or 4k? 00:08:52.710 --> 00:08:54.930 Well, 8k is going to be bigger. 00:08:54.930 --> 00:08:56.420 As I said, they all have k in them so you could kind of 00:08:56.420 --> 00:08:58.310 ignore k because it's a positive number. 00:08:58.310 --> 00:09:01.000 So the answer is choice D. 00:09:01.000 --> 00:09:02.860 I'll see you in the next video.

SAT Prep: Test 7 Section 8 Part 4

https://www.youtube.com/watch?v=RHfnCQCqohk

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WEBVTT Kind: captions Language: en 00:00:00.750 --> 00:00:01.640 In the home stretch. 00:00:01.640 --> 00:00:05.910 Problem 14. 00:00:05.910 --> 00:00:13.200 If n and p are integers greater than 1-- I didn't 00:00:13.200 --> 00:00:17.420 write that they're integers-- and if p is a factor of both n 00:00:17.420 --> 00:00:33.130 plus 3 and n plus 10, what is the value of p? 00:00:33.130 --> 00:00:36.610 So n and p are integers greater than 1 and p is a 00:00:36.610 --> 00:00:42.400 factor of both n plus 3 and n plus 10. 00:00:42.400 --> 00:00:43.860 This is interesting. 00:00:43.860 --> 00:00:52.890 So that means that n plus 3 is equal to some number times p, 00:00:52.890 --> 00:00:55.710 where k is just some random number. 00:00:55.710 --> 00:01:00.560 And we also know that n plus 10 is also equal to some-- 00:01:00.560 --> 00:01:05.700 maybe, probably, definitely some other number times p. 00:01:05.700 --> 00:01:09.720 And so we could subtract 3 from both 00:01:09.720 --> 00:01:10.730 sides in this equation. 00:01:10.730 --> 00:01:15.330 We'd get n is equal to k times p, some number 00:01:15.330 --> 00:01:17.330 times p minus 3. 00:01:17.330 --> 00:01:19.300 And we also could do it here. 00:01:19.300 --> 00:01:20.640 We can subtract 10 from both sides. 00:01:20.640 --> 00:01:27.520 We say n is equal to some other number times p minus 10. 00:01:27.520 --> 00:01:28.660 And I don't know where this is going. 00:01:28.660 --> 00:01:30.550 I'm just really playing around with this. 00:01:30.550 --> 00:01:31.630 So let's see. 00:01:31.630 --> 00:01:34.130 So both of these things are equal to n. 00:01:34.130 --> 00:01:35.560 What are we trying to solve for? 00:01:35.560 --> 00:01:37.460 We're trying to solve for p. 00:01:37.460 --> 00:01:40.560 So some number times p minus 3 is equal to some other number 00:01:40.560 --> 00:01:42.480 times p minus 10. 00:01:42.480 --> 00:01:44.360 Let me see where that gets me. 00:01:44.360 --> 00:01:52.300 So kp minus 3 is equal to m times p minus 10. 00:01:52.300 --> 00:01:57.870 Let's add 10 to both sides of this equation. 00:01:57.870 --> 00:02:05.340 You'll get kp plus 7 is equal to m times p. 00:02:05.340 --> 00:02:07.460 Now, this is interesting. 00:02:07.460 --> 00:02:12.460 So if I multiply some integer times p, right? 00:02:12.460 --> 00:02:15.590 So this is some multiple of p, right? 00:02:15.590 --> 00:02:17.860 This is actually n plus 3. 00:02:17.860 --> 00:02:22.910 But if I have some multiple of p right here and I add 7 to 00:02:22.910 --> 00:02:26.610 it, I get another multiple of p. 00:02:26.610 --> 00:02:31.210 So that means that 7 has to be divisible by p. 00:02:31.210 --> 00:02:38.060 There are only two numbers that 7 is divisible by. 00:02:38.060 --> 00:02:38.885 1 and 7. 00:02:38.885 --> 00:02:42.210 And it's not going to be 1, because it tells us that n and 00:02:42.210 --> 00:02:44.030 p are both greater than 1, right? 00:02:44.030 --> 00:02:47.120 So p has to be 7. 00:02:47.120 --> 00:02:51.670 p is equal to 7, which is choice B. 00:02:51.670 --> 00:02:59.960 And so the way I think about it is just when you add 10 to 00:02:59.960 --> 00:03:01.590 a number-- think of it this way. 00:03:01.590 --> 00:03:04.090 Ignore this whole n plus 3. 00:03:04.090 --> 00:03:08.390 Let's just say n plus 3, let's just say that we'll 00:03:08.390 --> 00:03:11.030 call this Q, right? 00:03:11.030 --> 00:03:15.020 We're saying p is a factor of Q, right? 00:03:15.020 --> 00:03:19.110 And then if Q is n plus 3, then this would be Q plus 7. 00:03:19.110 --> 00:03:21.590 I hope I'm not confusing you. 00:03:21.590 --> 00:03:24.540 So this is divisible by p. 00:03:24.540 --> 00:03:28.040 And then when you add 7 to it, it's also divisible by p. 00:03:28.040 --> 00:03:33.490 So p has to be 7, because when I add 7 to-- think of it this 00:03:33.490 --> 00:03:43.020 way: if n plus 3 was-- let's say this was 21. 00:03:43.020 --> 00:03:48.000 Not n, n plus 3 was 21, then n would be what, 18? 00:03:48.000 --> 00:03:56.220 n is 18, n plus 3 is 21, and n plus 10 would be 28. 00:03:56.220 --> 00:03:58.330 And these are both divisible by 7. 00:03:58.330 --> 00:03:59.340 So that's another way you could do it. 00:03:59.340 --> 00:04:01.350 You could actually just try out the numbers. 00:04:01.350 --> 00:04:02.500 But hopefully, that'll give you the intuition. 00:04:02.500 --> 00:04:06.590 But the big clue is some multiple times p plus 7 is 00:04:06.590 --> 00:04:08.800 equal to another multiple times p. 00:04:08.800 --> 00:04:13.140 So we know that p has to be 7 because p has to be divisible 00:04:13.140 --> 00:04:14.830 into 7 and greater than 1. 00:04:14.830 --> 00:04:17.570 And there's only one number that's greater than 1 and that 00:04:17.570 --> 00:04:18.510 is a factor of 7. 00:04:18.510 --> 00:04:20.360 And that's B. 00:04:20.360 --> 00:04:21.950 Or that's 7, sorry. 00:04:21.950 --> 00:04:25.160 Next problem. 00:04:25.160 --> 00:04:26.950 That was tricky, I think. 00:04:26.950 --> 00:04:29.860 Problem 15. 00:04:29.860 --> 00:04:32.680 In the queue shown above, points A, B and C are 00:04:32.680 --> 00:04:34.840 midpoints of the three edges. 00:04:34.840 --> 00:04:39.110 Which of the following angles has the least measure? 00:04:39.110 --> 00:04:40.980 Oh man, this is going to be a lot of drawing. 00:04:40.980 --> 00:04:42.670 This might take the whole time, but I'll 00:04:42.670 --> 00:04:45.430 try my best. OK. 00:04:45.430 --> 00:04:46.760 B, C and E. 00:04:46.760 --> 00:04:54.080 I'm going to draw it big just so-- actually, I'm 00:04:54.080 --> 00:04:55.330 drawing it too big. 00:04:59.930 --> 00:05:01.540 I'll do it the best I can. 00:05:17.130 --> 00:05:17.590 OK. 00:05:17.590 --> 00:05:19.315 And then they have these dotted lines in the back. 00:05:30.440 --> 00:05:32.570 OK, now they have-- let me draw this 00:05:32.570 --> 00:05:33.510 in a different color. 00:05:33.510 --> 00:05:34.760 This is point x. 00:05:37.800 --> 00:05:40.620 This is point y. 00:05:40.620 --> 00:05:41.970 They draw a bunch of points. 00:05:41.970 --> 00:05:43.850 B, C, and E are the midpoints. 00:05:43.850 --> 00:05:47.170 So this is point B. 00:05:47.170 --> 00:05:51.100 This is point A. 00:05:51.100 --> 00:05:54.430 This is point C. 00:05:54.430 --> 00:05:57.700 Point D. 00:05:57.700 --> 00:06:00.720 Point E. 00:06:00.720 --> 00:06:04.020 And everything they drew essentially starts at x, goes 00:06:04.020 --> 00:06:07.785 to one of these points, and then goes back to y, right? 00:06:07.785 --> 00:06:12.590 It goes from x to B to y, x to A to y. 00:06:12.590 --> 00:06:14.520 All the choices go to each of these points and 00:06:14.520 --> 00:06:15.290 then go back to y. 00:06:15.290 --> 00:06:17.510 And what they want to know is, which angle 00:06:17.510 --> 00:06:21.740 has the least measure? 00:06:21.740 --> 00:06:24.130 So the way we could think about this is, all of these 00:06:24.130 --> 00:06:31.136 angles-- we could just draw this part-- sorry, if this was 00:06:31.136 --> 00:06:33.680 x, this is one of the messier things I've ever drawn, this 00:06:33.680 --> 00:06:36.020 is y and then the other angles, you know it's going to 00:06:36.020 --> 00:06:37.955 go to some point and then come back. 00:06:37.955 --> 00:06:40.800 And what we want to know is, when do we 00:06:40.800 --> 00:06:44.540 get the least angle? 00:06:44.540 --> 00:06:47.540 Think of it this way, the longer this base angle is, the 00:06:47.540 --> 00:06:50.000 longer this is, the smaller the angle. 00:06:50.000 --> 00:06:51.730 Because this is the angle we're measuring. 00:06:51.730 --> 00:06:54.950 So we want to make this angle, this 00:06:54.950 --> 00:06:56.235 length as long as possible. 00:06:56.235 --> 00:06:59.900 The length from whatever letter we are to y, right? 00:06:59.900 --> 00:07:02.150 So B to y is very short, so this angle 00:07:02.150 --> 00:07:02.980 would be pretty big. 00:07:02.980 --> 00:07:05.420 A to y is even bigger. 00:07:05.420 --> 00:07:07.195 C to y is even bigger than that. 00:07:07.195 --> 00:07:10.530 D to y is actually the biggest. This is the biggest 00:07:10.530 --> 00:07:12.900 distance, right? 00:07:12.900 --> 00:07:16.746 So this will be the smallest angle. 00:07:16.746 --> 00:07:23.282 So that's ydx or xdy, so that's choice D. 00:07:23.282 --> 00:07:26.750 And remember, the underlying intuition is, if I were to-- 00:07:26.750 --> 00:07:30.500 let me do the colors so you know what I'm saying-- this 00:07:30.500 --> 00:07:34.440 line, if I were to draw that, that would be here. 00:07:34.440 --> 00:07:36.520 I'm saying that corresponds to that right there. 00:07:36.520 --> 00:07:37.155 That's x. 00:07:37.155 --> 00:07:38.130 That's y. 00:07:38.130 --> 00:07:40.610 And then we're taking a point from x to one of 00:07:40.610 --> 00:07:41.980 these points, right? 00:07:41.980 --> 00:07:47.770 xby, xay, xcy, and whatever point it is, the way we get 00:07:47.770 --> 00:07:49.460 the smallest angle is if we make this 00:07:49.460 --> 00:07:53.330 line as long as possible. 00:07:53.330 --> 00:07:54.830 And you can visualize that. 00:07:54.830 --> 00:07:57.090 Take this point further and further out and this angle 00:07:57.090 --> 00:08:00.040 gets smaller and smaller and smaller. 00:08:00.040 --> 00:08:04.130 So if you look at it that way, what is going to be the 00:08:04.130 --> 00:08:07.110 longest distance y to which of these points? 00:08:07.110 --> 00:08:10.480 y to C is the same as y to E. 00:08:10.480 --> 00:08:11.905 y to B is very short. 00:08:11.905 --> 00:08:12.990 y to A id a little longer. 00:08:12.990 --> 00:08:14.260 y to C is a little longer. 00:08:14.260 --> 00:08:16.320 y to D is the longest distance. 00:08:16.320 --> 00:08:20.365 So the least angle will be, if this point is D, xdy. 00:08:20.365 --> 00:08:23.428 And that's choice D. 00:08:23.428 --> 00:08:26.510 Let me see if I can squeeze problem 16 in here. 00:08:26.510 --> 00:08:28.360 Otherwise, I'll do it in the next video. 00:08:28.360 --> 00:08:29.930 Problem 16. 00:08:29.930 --> 00:08:41.130 if xy is equal to 7, x minus y is equal to 5 then x squared y 00:08:41.130 --> 00:08:46.370 minus xy squared is equal to what? 00:08:46.370 --> 00:08:48.430 All right, so we know what xy is and we know 00:08:48.430 --> 00:08:49.410 what x minus y is. 00:08:49.410 --> 00:08:51.220 So let's watch this. 00:08:51.220 --> 00:08:52.360 Factor an x out of here. 00:08:52.360 --> 00:08:56.830 You get the same thing as x times xy, minus-- let's factor 00:08:56.830 --> 00:08:57.890 a y out of here. 00:08:57.890 --> 00:09:02.410 This is the same thing as y times xy, right? 00:09:02.410 --> 00:09:04.430 All I did is I factored a y out. 00:09:04.430 --> 00:09:06.790 And now let's factor the x minus y out. 00:09:06.790 --> 00:09:11.680 This is the same thing as xy times x minus y, right? 00:09:11.680 --> 00:09:13.500 I'm just factoring the xy out. 00:09:13.500 --> 00:09:14.460 You could do it in reverse. 00:09:14.460 --> 00:09:17.840 You can distribute the xy and you'd get this up here. 00:09:17.840 --> 00:09:19.070 And this equals what? 00:09:19.070 --> 00:09:21.730 xy is 7. 00:09:21.730 --> 00:09:25.190 And x minus y, it gives us right there, is 5. 00:09:25.190 --> 00:09:28.420 7 times 5 is 35. 00:09:28.420 --> 00:09:29.885 That's choice D. 00:09:29.885 --> 00:09:32.600 And we are done with Test 7. 00:09:32.600 --> 00:09:35.260 I will see you in Test 8.

SAT Prep: Test 7 Section 8 Part 2

https://www.youtube.com/watch?v=Z7fM5fu7LAs

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en

WEBVTT Kind: captions Language: en 00:00:00.550 --> 00:00:04.200 I'm on problem number seven. 00:00:04.200 --> 00:00:07.250 Josephine's daily exercise routine consists of swimming, 00:00:07.250 --> 00:00:08.720 cycling, and running in that order. 00:00:08.720 --> 00:00:14.660 So swimming, cycling, and running in that order. 00:00:14.660 --> 00:00:23.340 She runs faster than she swims, and cycles faster than 00:00:23.340 --> 00:00:24.990 she runs, like most people. 00:00:27.640 --> 00:00:29.900 If she does not rest between the activities, which of the 00:00:29.900 --> 00:00:32.580 following could be the graph of the distance she covers 00:00:32.580 --> 00:00:35.650 during the entire time of her exercise routine? 00:00:35.650 --> 00:00:37.460 So I'm not going to even look at the choices. 00:00:37.460 --> 00:00:39.060 And I'll just give you an intuition and then you can 00:00:39.060 --> 00:00:42.150 look for the choices that match it. 00:00:42.150 --> 00:00:52.030 So if I were to graph distance versus time, the slope of my 00:00:52.030 --> 00:00:53.980 line is going to be my speed, right? 00:00:53.980 --> 00:00:54.830 Because what is speed? 00:00:54.830 --> 00:00:56.570 It's distance per time. 00:00:56.570 --> 00:00:58.550 And that's the same thing as slope. 00:00:58.550 --> 00:01:00.320 It's rise over run. 00:01:00.320 --> 00:01:02.700 So in this context, that would be the slope of the line. 00:01:02.700 --> 00:01:05.180 So what you want is a situation where she swims 00:01:05.180 --> 00:01:08.460 first-- which is her slowest-- then she cycles-- which is her 00:01:08.460 --> 00:01:11.640 fastest-- then she runs-- which is kind of in between. 00:01:11.640 --> 00:01:14.020 So what you're going to have is a low slope, then a very 00:01:14.020 --> 00:01:16.680 high slope, and then you want kind of a medium slope. 00:01:16.680 --> 00:01:18.550 So it's going to look something like this. 00:01:18.550 --> 00:01:21.990 Because she swims first, so that's the slowest exercise, 00:01:21.990 --> 00:01:24.050 so maybe it'll be like that. 00:01:24.050 --> 00:01:26.693 Then she cycles, which is her fastest exercise, so maybe it 00:01:26.693 --> 00:01:28.180 will look something like that. 00:01:28.180 --> 00:01:30.250 And then she runs, which is kind of an in-between 00:01:30.250 --> 00:01:33.390 exercise, so maybe it looks something like this. 00:01:33.390 --> 00:01:36.460 So which of the choices looks like that? 00:01:36.460 --> 00:01:39.630 Where you start relatively slow, you peak out pretty 00:01:39.630 --> 00:01:43.090 fast, and then you do something that looks a little 00:01:43.090 --> 00:01:45.300 bit faster-- that has a higher slope than this. 00:01:45.300 --> 00:01:47.730 And if I look at all of the choices, it 00:01:47.730 --> 00:01:51.040 looks like choice E. 00:01:51.040 --> 00:01:55.230 Choice E is actually pretty close to what I drew. 00:01:55.230 --> 00:01:56.880 If you look at the other choices, choice A she goes 00:01:56.880 --> 00:02:00.960 slow, then medium, then fast. So that's like if she cycled 00:02:00.960 --> 00:02:02.630 last, right? 00:02:02.630 --> 00:02:04.920 In choice B she does her slow thing last. That looks like 00:02:04.920 --> 00:02:07.690 she's swimming last. Choice C looks like she's swimming last 00:02:07.690 --> 00:02:10.780 as well, because that's the slowest. And then choice D 00:02:10.780 --> 00:02:13.620 looks like she cycles, then she swims, then she runs. 00:02:13.620 --> 00:02:15.390 So it's definitely choice E. 00:02:15.390 --> 00:02:18.150 You go slow, fast, and then medium. 00:02:18.150 --> 00:02:19.400 Next problem. 00:02:25.900 --> 00:02:28.510 Problem eight. 00:02:28.510 --> 00:02:34.000 In the xy-coordinate systems, square root of 6-- sorry, the 00:02:34.000 --> 00:02:38.600 point is square root of 6 comma k-- is one of the points 00:02:38.600 --> 00:02:40.075 of intersection of the graphs. 00:02:40.075 --> 00:02:41.820 So these are the graphs. 00:02:41.820 --> 00:02:49.560 y is equal to x squared minus 7, and y is equal to minus x 00:02:49.560 --> 00:02:54.890 squared plus j. 00:02:54.890 --> 00:02:55.640 Fair enough. 00:02:55.640 --> 00:02:56.860 Where j is a constant. 00:02:56.860 --> 00:02:58.930 What is the value of j? 00:02:58.930 --> 00:03:01.490 So we know the x-coordinate where it intersects, right? 00:03:01.490 --> 00:03:04.200 We know it intersects at square root of 6. 00:03:04.200 --> 00:03:07.590 So intersection means that the y values are the same for a 00:03:07.590 --> 00:03:08.670 given x value. 00:03:08.670 --> 00:03:10.210 That's what intersects. 00:03:10.210 --> 00:03:12.810 So we can set these two equations equal to each other. 00:03:12.810 --> 00:03:16.080 So we could say, x squared-- we could set the y values 00:03:16.080 --> 00:03:19.640 equal to each other-- x squared minus 7 is equal to 00:03:19.640 --> 00:03:24.860 this: minus x squared plus j. 00:03:24.860 --> 00:03:29.940 Now we could keep things in terms of x or we could-- well 00:03:29.940 --> 00:03:31.690 let's just algebraically play with this a little bit. 00:03:31.690 --> 00:03:35.300 And I hope you understand that this is what we would do to 00:03:35.300 --> 00:03:37.700 solve for the intersection of these two graphs, if we wanted 00:03:37.700 --> 00:03:39.495 to solve for the x value. 00:03:39.495 --> 00:03:40.960 And we could play with this a little bit. 00:03:40.960 --> 00:03:43.010 We could add x squared to both sides. 00:03:43.010 --> 00:03:49.530 You get 2x squared minus 7 is equal to j. 00:03:49.530 --> 00:03:50.340 And what's x? 00:03:50.340 --> 00:03:51.640 We know where these intersect. 00:03:51.640 --> 00:03:53.570 We know one of the points where they intersect is 00:03:53.570 --> 00:03:54.930 square root of 6. 00:03:54.930 --> 00:04:00.150 So let's input-- let's replace x with square root of 6. 00:04:00.150 --> 00:04:06.610 So 2 times square root of 6 squared minus 7 is equal to j. 00:04:06.610 --> 00:04:08.185 What's the square root of 6 squared? 00:04:08.185 --> 00:04:10.660 It's just 6, right? 00:04:10.660 --> 00:04:15.540 So it's 2 times 6 minus 7 is equal to j. 00:04:15.540 --> 00:04:19.079 So 12 minus 7 is equal to j. 00:04:19.079 --> 00:04:20.810 5 is equal to j. 00:04:20.810 --> 00:04:22.250 And that is choice A. 00:04:24.960 --> 00:04:26.210 Problem nine. 00:04:31.920 --> 00:04:38.850 If the absolute value of 2 minus x is less than 3, which 00:04:38.850 --> 00:04:41.270 of the following is a possible value of x? 00:04:41.270 --> 00:04:51.030 So this means that 2 minus x is less than 3. 00:04:51.030 --> 00:04:51.840 Right? 00:04:51.840 --> 00:04:53.950 So I guess the way you can think about is that the 00:04:53.950 --> 00:04:56.070 difference between 2 and x is going to be 00:04:56.070 --> 00:04:58.750 less than 3, right? 00:04:58.750 --> 00:05:04.030 The other way you could view it is that the negative of 00:05:04.030 --> 00:05:10.920 this-- x minus 2-- is going to be less than 3. 00:05:10.920 --> 00:05:11.520 Right? 00:05:11.520 --> 00:05:15.140 We could have also said 2 minus x is greater than 00:05:15.140 --> 00:05:16.140 negative 3. 00:05:16.140 --> 00:05:17.470 That's the other thing we could have written. 00:05:17.470 --> 00:05:21.080 And what I did is I just-- this is either going to be a 00:05:21.080 --> 00:05:22.970 positive or a negative, right? 00:05:22.970 --> 00:05:26.350 So if it's a positive, we just say 2 minus x is less than 3. 00:05:26.350 --> 00:05:30.030 If x is going to be greater than 2, then we could say that 00:05:30.030 --> 00:05:31.930 x minus 2 is less than 3. 00:05:31.930 --> 00:05:33.390 Because then it would be a negative number and then it 00:05:33.390 --> 00:05:34.636 would become positive. 00:05:34.636 --> 00:05:36.420 Hope I didn't confuse you. 00:05:36.420 --> 00:05:40.220 So if this is true you would get-- let's see-- you would 00:05:40.220 --> 00:05:44.840 get, if I were to add x to both sides I'd get 2 is less 00:05:44.840 --> 00:05:47.050 than 3 plus x. 00:05:47.050 --> 00:05:49.020 Subtract 3 from both sides. 00:05:49.020 --> 00:05:52.040 You get negative 1 is less than x. 00:05:52.040 --> 00:05:53.360 This one, add 2 to both sides. 00:05:53.360 --> 00:05:57.110 You get x is less than 5. 00:05:57.110 --> 00:06:02.670 So x is greater than negative 1 and it is less than 5. 00:06:02.670 --> 00:06:11.840 So if you look at all of the choices-- it has to be less 00:06:11.840 --> 00:06:18.750 than 5 and greater than negative 1-- so out of all of 00:06:18.750 --> 00:06:22.170 those choices, the only choice that is less than 5 is choice 00:06:22.170 --> 00:06:25.340 A, which is 4. 00:06:25.340 --> 00:06:26.410 And you could look at it. 00:06:26.410 --> 00:06:27.990 You could actually just try out the numbers. 00:06:27.990 --> 00:06:30.770 And frankly, even if you got confused by absolute value, 00:06:30.770 --> 00:06:33.250 you could have just tried each of the choices. 00:06:33.250 --> 00:06:34.890 2 minus 4 is minus 2. 00:06:34.890 --> 00:06:39.280 The absolute value of negative 2 is just 2, which 00:06:39.280 --> 00:06:40.070 is less than 3. 00:06:40.070 --> 00:06:41.150 So that's right. 00:06:41.150 --> 00:06:42.970 So you could have actually just tried out the choices if 00:06:42.970 --> 00:06:43.620 this confused you. 00:06:43.620 --> 00:06:46.160 And that actually might have been faster because in the 00:06:46.160 --> 00:06:48.240 worst case, you'd have to try out five choices and each of 00:06:48.240 --> 00:06:51.770 them would have taken a few seconds to evaluate. 00:06:51.770 --> 00:06:53.700 But sometimes it's good to do it mathematically. 00:06:53.700 --> 00:06:57.840 But I wouldn't wrong you for-- I wouldn't blame you for just 00:06:57.840 --> 00:07:01.040 trying things out as well. 00:07:01.040 --> 00:07:01.840 Ten. 00:07:01.840 --> 00:07:03.560 OK. 00:07:03.560 --> 00:07:08.410 If all interior angles of the polygon are congruent, then x 00:07:08.410 --> 00:07:11.450 is equal to-- OK so this is interesting. 00:07:11.450 --> 00:07:13.120 All of the angles of this polygon are congruent. 00:07:13.120 --> 00:07:13.985 So it's a pentagon. 00:07:13.985 --> 00:07:24.790 So it's 1, 2-- like that-- 3, 4, and 5. 00:07:24.790 --> 00:07:27.260 I didn't draw really that congruent. 00:07:27.260 --> 00:07:29.395 And actually this line keeps going like this. 00:07:29.395 --> 00:07:31.490 And the important thing to realize with any of these 00:07:31.490 --> 00:07:37.460 triangles is, if I were to draw-- there's a formula for 00:07:37.460 --> 00:07:42.750 the angles of a polygon, but I always-- so the way to think 00:07:42.750 --> 00:07:49.920 about it is that these are all going to be equal angles. 00:07:49.920 --> 00:07:52.490 So think of it this way. 00:07:52.490 --> 00:07:55.840 These are five equal triangles. 00:07:55.840 --> 00:07:56.990 Let me think of it the best way. 00:07:56.990 --> 00:08:00.880 So they're telling us-- we want to figure out what x 00:08:00.880 --> 00:08:03.050 degrees is, right? 00:08:03.050 --> 00:08:04.720 Well this x degrees is going to be the 00:08:04.720 --> 00:08:06.000 same as this x degrees. 00:08:09.990 --> 00:08:12.080 And how do I know that?

SAT Prep: Test 7 Section 8 Part 3

https://www.youtube.com/watch?v=Fq1BirUwrLQ

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en

WEBVTT Kind: captions Language: en 00:00:00.720 --> 00:00:01.420 Welcome back. 00:00:01.420 --> 00:00:02.050 I'm sorry. 00:00:02.050 --> 00:00:05.470 I actually had to take a five-second pause because 00:00:05.470 --> 00:00:07.560 something came up, but let me continue. 00:00:07.560 --> 00:00:11.180 Problem number 10. 00:00:11.180 --> 00:00:13.530 So let's see, they say that this is x degrees. 00:00:13.530 --> 00:00:16.260 So there's a formula for the angles inside of a polygon, 00:00:16.260 --> 00:00:17.990 but what I'm going to do is derive it for you, because 00:00:17.990 --> 00:00:24.320 frankly, on an exam you can-- well, later in life, it's very 00:00:24.320 --> 00:00:25.570 easy to forget everything. 00:00:27.750 --> 00:00:32.560 So what I do is, we know that all of the angles are 00:00:32.560 --> 00:00:34.290 congruent, right? 00:00:34.290 --> 00:00:38.390 So if we know that this is-- let's call this angle y, then 00:00:38.390 --> 00:00:41.890 this angle is going to be y, this angle going to be y, that 00:00:41.890 --> 00:00:49.950 is y, that's y, that's y, that's y, that's y, that's y, 00:00:49.950 --> 00:00:50.950 that's y, right? 00:00:50.950 --> 00:00:53.030 All of these angles are the same. 00:00:53.030 --> 00:00:55.580 And what do we know about these angles here? 00:00:55.580 --> 00:01:00.930 Let's called these z, z, z, z, z. 00:01:00.930 --> 00:01:02.545 There are five of these z angles and what are they going 00:01:02.545 --> 00:01:04.470 to add up to be? 00:01:04.470 --> 00:01:06.350 Well, they go around in a circle, right? 00:01:06.350 --> 00:01:09.470 So these five z angles are going to have to 00:01:09.470 --> 00:01:11.316 add up to 360 degrees. 00:01:11.316 --> 00:01:15.660 So we could say 5z is equal to 360 degrees. 00:01:15.660 --> 00:01:25.370 z is equal to-- let's see, 5 goes into 360, 7, 35, 10, 72. 00:01:25.370 --> 00:01:27.960 So z is equal to 72 degrees. 00:01:27.960 --> 00:01:31.480 So each of these z's is 72 degrees. 00:01:31.480 --> 00:01:34.760 That angle of 72 degrees, what are the y's? 00:01:34.760 --> 00:01:38.560 Well, y plus y plus z has to be 180, right? 00:01:38.560 --> 00:01:46.530 So you have y plus y plus 72 is equal to 180 degrees. 00:01:46.530 --> 00:01:50.350 2y plus 72 is equal to 180. 00:01:50.350 --> 00:01:54.430 2y is equal to what? 00:01:54.430 --> 00:01:58.166 108. 00:01:58.166 --> 00:01:59.670 And actually we could solve for y. 00:01:59.670 --> 00:02:00.330 y is 54. 00:02:00.330 --> 00:02:02.580 But really we just want to know what 2y is 00:02:02.580 --> 00:02:03.430 because look at this. 00:02:03.430 --> 00:02:05.656 We're trying to solve for x. 00:02:05.656 --> 00:02:11.850 x plus 2y is 180, right? 00:02:11.850 --> 00:02:12.912 How did I get that? 00:02:12.912 --> 00:02:15.280 x is supplementary with y and y. 00:02:15.280 --> 00:02:17.710 This is 2y right here and altogether they are 00:02:17.710 --> 00:02:18.480 supplementary. 00:02:18.480 --> 00:02:21.650 So that's where I get x plus 2y is 180. 00:02:21.650 --> 00:02:22.650 We know what 2y is. 00:02:22.650 --> 00:02:24.236 It's 108. 00:02:24.236 --> 00:02:28.868 x plus 108 is equal to 180. 00:02:28.868 --> 00:02:32.940 x is equal to-- and actually, it's interesting. x is going 00:02:32.940 --> 00:02:34.800 to be equal to z, right? 00:02:34.800 --> 00:02:38.250 x is 72 degrees. 00:02:38.250 --> 00:02:42.160 And that is choice C. 00:02:42.160 --> 00:02:43.590 And that actually is interesting that x 00:02:43.590 --> 00:02:44.410 is the same as z. 00:02:44.410 --> 00:02:47.130 And actually, when you think about it, it makes sense 00:02:47.130 --> 00:02:50.460 because y plus y plus z is 180, and here, y plus y plus x 00:02:50.460 --> 00:02:51.570 has to be 180. 00:02:51.570 --> 00:02:53.320 So x is going to be equal to z. 00:02:53.320 --> 00:02:55.660 You say there are 360 degrees in the circle. 00:02:55.660 --> 00:02:57.760 Divide by the number of z's there are, one, two, three, 00:02:57.760 --> 00:03:00.050 four, five, you get 72. 00:03:00.050 --> 00:03:01.300 Next problem. 00:03:04.880 --> 00:03:09.360 Problem 11. 00:03:09.360 --> 00:03:14.090 The length of a drawing of a tool is 3/8 of the length of 00:03:14.090 --> 00:03:15.190 the actual tool. 00:03:15.190 --> 00:03:23.630 OK, if the length of the drawing of the tool is 6 00:03:23.630 --> 00:03:31.600 inches, what is the length in inches of the actual tool? 00:03:31.600 --> 00:03:36.860 OK, 6 inches is equal to 3/8 times the actual tool. 00:03:36.860 --> 00:03:39.360 Let's multiply both sides of this equation by the 00:03:39.360 --> 00:03:40.830 reciprocal of this right here. 00:03:40.830 --> 00:03:49.090 So let's say 8/3 times 6 is equal to 8/3 times 3/8a. 00:03:49.090 --> 00:03:50.190 This, of course, cancels out. 00:03:50.190 --> 00:03:52.420 That's why I multiplied by the reciprocal. 00:03:52.420 --> 00:03:56.190 And this becomes a 2, this becomes a 1. 00:03:56.190 --> 00:03:58.530 8 times 2 is 16. 00:03:58.530 --> 00:04:00.530 So the actual tool is 16 inches. 00:04:00.530 --> 00:04:03.290 And that's choice C. 00:04:03.290 --> 00:04:04.590 Next problem. 00:04:04.590 --> 00:04:05.555 Problem 12. 00:04:05.555 --> 00:04:07.020 I might have space to do it here. 00:04:10.110 --> 00:04:16.779 If x plus 3/2 is an integer-- so they're saying x plus 3 is 00:04:16.779 --> 00:04:18.360 divisible by 2, right? 00:04:18.360 --> 00:04:21.269 Because in order for this to be an integer-- and so what do 00:04:21.269 --> 00:04:23.010 we know about x? 00:04:23.010 --> 00:04:28.130 Well, x plus 3 has to be even number, right? 00:04:28.130 --> 00:04:29.830 Because it's divisible by 2. 00:04:29.830 --> 00:04:35.510 So that must mean that x has to be an odd number. 00:04:35.510 --> 00:04:36.300 How do I know that? 00:04:36.300 --> 00:04:38.270 So I know that x plus 3 is even. 00:04:45.300 --> 00:04:51.080 If x plus 3 is even, then I know that x is odd. 00:04:51.080 --> 00:04:53.060 Why is that? 00:04:53.060 --> 00:04:54.680 Because 3 is odd. 00:04:54.680 --> 00:04:57.480 The only way, if I had one number plus an odd and I get 00:04:57.480 --> 00:05:00.500 an even, this number has to be odd as well. 00:05:00.500 --> 00:05:02.670 2 odds added together equal an even, right? 00:05:02.670 --> 00:05:07.210 For example, 5 plus 3 is 8, or 11 plus 3 is 14. 00:05:07.210 --> 00:05:10.330 So an odd plus an odd is an even. 00:05:10.330 --> 00:05:12.850 So x has to be odd. 00:05:12.850 --> 00:05:17.100 So when we look at the choices, we have choice E. 00:05:17.100 --> 00:05:18.100 x is an odd integer. 00:05:18.100 --> 00:05:22.390 That's the only thing that we can really assume based on 00:05:22.390 --> 00:05:26.370 this, and that's because x plus 3 has to be even. 00:05:26.370 --> 00:05:30.230 And anything plus an odd and you get an even, that thing 00:05:30.230 --> 00:05:32.050 you added to the odd has to be odd as well. 00:05:32.050 --> 00:05:34.420 And you can try it out with numbers. 00:05:34.420 --> 00:05:35.670 Next problem. 00:05:39.650 --> 00:05:41.610 Problem 13. 00:05:41.610 --> 00:05:43.000 All right. 00:05:43.000 --> 00:05:46.230 In the x, y plane above, points q and s are the centers 00:05:46.230 --> 00:05:47.140 of the circle. 00:05:47.140 --> 00:05:49.190 Which are tangent to the x-axis? 00:05:49.190 --> 00:05:51.670 Let me see if I can draw this. 00:05:55.260 --> 00:05:56.530 So this is the y-axis. 00:05:56.530 --> 00:06:00.420 This is the x-axis. 00:06:00.420 --> 00:06:01.930 And now they draw a couple of circles. 00:06:01.930 --> 00:06:04.035 A small one and a big one. 00:06:04.035 --> 00:06:06.210 Let's see if I can draw it. 00:06:06.210 --> 00:06:08.800 The small circle looks something like that. 00:06:11.750 --> 00:06:15.390 And the big circle doesn't touch the small circle. 00:06:15.390 --> 00:06:25.000 So the big circle looks something like that. 00:06:25.000 --> 00:06:26.766 Fair enough. 00:06:26.766 --> 00:06:31.650 Let's see, this is point Q. 00:06:31.650 --> 00:06:34.786 This is point S and those are the centers. 00:06:34.786 --> 00:06:41.686 And they tell us that this point right here is 3 comma 6, 00:06:41.686 --> 00:06:47.305 and this point up here is 11 comma 10. 00:06:47.305 --> 00:06:48.400 And what else? 00:06:48.400 --> 00:06:50.060 This is point R. 00:06:50.060 --> 00:06:58.590 this is point P, QS, oh, and then one last thing, I'm 00:06:58.590 --> 00:07:02.270 drawing this line here and a line here. 00:07:05.630 --> 00:07:09.500 In both cases, they say this is a perpendicular. 00:07:09.500 --> 00:07:10.350 Let's do the problem. 00:07:10.350 --> 00:07:13.790 In the x, y plane above, points Q and S are the centers 00:07:13.790 --> 00:07:16.880 of the circle, which are tangent to the x-axis, right? 00:07:16.880 --> 00:07:19.260 They just barely touch the x-axis right here. 00:07:19.260 --> 00:07:20.620 That's the x-axis. 00:07:20.620 --> 00:07:24.870 What is the slope of line QS? 00:07:24.870 --> 00:07:26.490 So what is the slope of this line? 00:07:29.000 --> 00:07:29.510 Line QS? 00:07:29.510 --> 00:07:32.233 And it could go further out like that, go further in this 00:07:32.233 --> 00:07:34.670 direction like that. 00:07:34.670 --> 00:07:35.850 So we really just have to figure out the 00:07:35.850 --> 00:07:37.640 coordinates Q and S. 00:07:37.640 --> 00:07:42.190 So Q is the center of this, so it's tangent to the x-axis. 00:07:42.190 --> 00:07:44.070 This entire diameter has what? 00:07:44.070 --> 00:07:46.680 Has height what? 00:07:46.680 --> 00:07:48.100 Height 6. 00:07:48.100 --> 00:07:50.190 So this the point 3. 00:07:50.190 --> 00:07:54.330 This is 3, 6, so what's this Q going to be? 00:07:54.330 --> 00:07:58.060 The x is definitely going to still be 3 and the y is going 00:07:58.060 --> 00:08:00.030 to be halfway to this top point. 00:08:00.030 --> 00:08:02.230 Well, what's halfway to 6? 00:08:02.230 --> 00:08:04.200 3. 00:08:04.200 --> 00:08:09.950 Similarly, this is point 11 comma 10, so the x-coordinate 00:08:09.950 --> 00:08:15.130 is 11 and this y-coordinate is to 10. 00:08:15.130 --> 00:08:17.320 So what's the coordinate of S going to be? 00:08:17.320 --> 00:08:20.520 The x-coordinate is, of course, still going to be 11. 00:08:20.520 --> 00:08:23.550 And the y-coordinate is going to be halfway to 10. 00:08:23.550 --> 00:08:24.340 What's halfway to 10? 00:08:24.340 --> 00:08:26.360 It's 5. 00:08:26.360 --> 00:08:27.220 So what's the slope? 00:08:27.220 --> 00:08:31.040 Slope is change in y over change in x. 00:08:31.040 --> 00:08:32.640 So let's take the y's. 00:08:32.640 --> 00:08:41.520 It's 5 minus 3 over 11 minus 3. 00:08:41.520 --> 00:08:42.770 That equals 2/9. 00:08:46.830 --> 00:08:48.740 No, sorry, 2/8. 00:08:48.740 --> 00:08:49.880 I always mess up on that stuff. 00:08:49.880 --> 00:08:51.420 11 minus 3 is 8. 00:08:51.420 --> 00:08:54.800 So 2/8 and that equals 1/4. 00:08:54.800 --> 00:08:57.534 And that is choice D. 00:08:57.534 --> 00:09:00.320 I'll see you in the next video.

SAT Prep: Test 7 Section 5 Part 2

https://www.youtube.com/watch?v=va9qsz7Q6b0

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https://www.youtube.com/api/timedtext?v=va9qsz7Q6b0&ei=Z2eUZcroNuPDmLAP4vOH6Ao&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249815&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E8BED3A2153007DA6F6094450A6F45EB5633E579.5542B1EB941FD436E82FA0CAC0E1891D52ABD4D7&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.820 --> 00:00:04.022 We're in problem number seven. 00:00:04.022 --> 00:00:05.580 Let me switch colors. 00:00:05.580 --> 00:00:10.330 The average of the weights of 14 books is p pounds. 00:00:10.330 --> 00:00:12.080 In terms of p, what is the total weight of 00:00:12.080 --> 00:00:13.470 the books in pounds? 00:00:13.470 --> 00:00:16.620 So this comes up all the time in the SAT. 00:00:16.620 --> 00:00:21.010 The average of weights of 14 books is p pounds. 00:00:21.010 --> 00:00:26.970 So if you had weight 1, plus weight 2, plus-- if I were to 00:00:26.970 --> 00:00:30.300 add all 14 books, that's just the weights of all the 14 00:00:30.300 --> 00:00:37.210 books, and I were to divide by 14, that is equal to p. 00:00:37.210 --> 00:00:38.310 This is just the definition of average. 00:00:38.310 --> 00:00:40.690 You add up the weights of all the books, divide by the 00:00:40.690 --> 00:00:42.680 number of books, and you get the average. 00:00:42.680 --> 00:00:46.500 Well, if you multiply both sides by 14 you get w1-- you 00:00:46.500 --> 00:00:51.390 get the sum of all of the weights of the books is equal 00:00:51.390 --> 00:00:54.100 to what? p times 14. 00:00:54.100 --> 00:00:55.760 And what are they asking? 00:00:55.760 --> 00:00:58.080 In terms of p, what is the total weight 00:00:58.080 --> 00:00:58.970 of all of the books? 00:00:58.970 --> 00:01:01.260 Well, this is the total weight of all of the books, right? 00:01:01.260 --> 00:01:03.350 I just take the weight of the first one, plus the weight of 00:01:03.350 --> 00:01:04.980 the second one, and keep adding until I get to the 00:01:04.980 --> 00:01:08.380 weight of the 14th, and that is equal to p times 14. 00:01:08.380 --> 00:01:10.700 And that is choice E. 00:01:10.700 --> 00:01:13.000 14p. 00:01:13.000 --> 00:01:14.770 Choice E. 00:01:14.770 --> 00:01:15.860 Next problem. 00:01:15.860 --> 00:01:17.110 Problem eight. 00:01:23.036 --> 00:01:24.286 Draw the axes. 00:01:34.990 --> 00:01:37.820 This is x and y. 00:01:37.820 --> 00:01:41.190 And then they draw-- let me do it in a different color. 00:01:41.190 --> 00:01:47.460 They have point A, which is 2, negative 1. 00:01:47.460 --> 00:01:49.010 So it's 2 comma negative 1. 00:01:49.010 --> 00:01:51.970 So this is the point x is equal to 2. 00:01:51.970 --> 00:01:54.560 If we were to go down, this is y equal negative 1. 00:01:54.560 --> 00:02:01.330 And then they have point B, which is k,t. 00:02:01.330 --> 00:02:07.450 And then they have point C up here-- C is 2 comma 5. 00:02:07.450 --> 00:02:11.080 So it's 2 comma and this is 5. 00:02:11.080 --> 00:02:13.530 Point B is the midpoint of AC. 00:02:13.530 --> 00:02:16.830 So point B is this midpoint of AC. 00:02:16.830 --> 00:02:19.670 So this distance is the same as that distance. 00:02:19.670 --> 00:02:22.250 What is the value of t? 00:02:22.250 --> 00:02:23.190 So what's its y value? 00:02:23.190 --> 00:02:26.540 We know what k is, k's going to have to be 2. 00:02:26.540 --> 00:02:27.790 So what's the y value? 00:02:27.790 --> 00:02:29.780 So the y value's essentially just going to be the average 00:02:29.780 --> 00:02:31.980 of this y value and that y value, because it's right in 00:02:31.980 --> 00:02:33.380 the middle of the two. 00:02:33.380 --> 00:02:39.680 So it's going to be 5 plus minus 1 over 2, 00:02:39.680 --> 00:02:41.070 which is equal to what? 00:02:41.070 --> 00:02:44.060 That's equal to 4 over 2, which is equal to 2. 00:02:44.060 --> 00:02:47.090 So this point, right here, is going to be 2 comma 2. 00:02:47.090 --> 00:02:49.910 And all I did is I took the average of the y values. 00:02:49.910 --> 00:02:52.610 The other way you could have thought about it is, what is 00:02:52.610 --> 00:02:54.000 this total distance? 00:02:54.000 --> 00:02:55.600 What's the distance from 5 to negative 1? 00:02:55.600 --> 00:02:57.010 Well, that's 6. 00:02:57.010 --> 00:02:58.435 So this distance would be 3. 00:02:58.435 --> 00:03:00.330 And what's 5 minus 3? 00:03:00.330 --> 00:03:01.690 It's 2 as well. 00:03:01.690 --> 00:03:04.340 But the easiest way is just to average this y value and this 00:03:04.340 --> 00:03:07.330 y value, and you'll get the y value for the midpoint. 00:03:07.330 --> 00:03:10.920 And that is choice C. 00:03:10.920 --> 00:03:12.170 Next problem. 00:03:17.160 --> 00:03:19.100 Problem nine. 00:03:19.100 --> 00:03:22.130 So far I didn't have to draw anything, this is pretty good. 00:03:22.130 --> 00:03:33.130 If k times 2x plus 3, times x minus 1 is equal to 0, and 00:03:33.130 --> 00:03:35.960 they also tell us that x is greater than 1, what is the 00:03:35.960 --> 00:03:39.370 value of k? 00:03:39.370 --> 00:03:43.250 So let's see, if for this times this, times this to 00:03:43.250 --> 00:03:46.770 equal 0, at least one of them has to be equal to 0. 00:03:46.770 --> 00:03:50.530 Maybe more than one, maybe all of them. 00:03:50.530 --> 00:03:54.670 In order for this term to be 0, what does x have to equal? 00:03:54.670 --> 00:04:00.225 Well, for x to be 0 here-- for this term to be 0, sorry-- x 00:04:00.225 --> 00:04:01.470 would have to be equal to 1. 00:04:01.470 --> 00:04:01.980 How did I get that? 00:04:01.980 --> 00:04:03.580 I said x minus 1 is equal to 0. 00:04:03.580 --> 00:04:05.640 Add 1 to both sides, x is equal to 1. 00:04:05.640 --> 00:04:08.030 Well, they told us that x is greater than 1, so x 00:04:08.030 --> 00:04:09.490 cannot equal 1. 00:04:09.490 --> 00:04:11.460 So this cannot happen. 00:04:11.460 --> 00:04:13.910 That cannot happen, because x is greater than 1. 00:04:13.910 --> 00:04:16.050 So we know that this is not 0. 00:04:16.050 --> 00:04:18.850 This term is not-- this might not equal-- that's 00:04:18.850 --> 00:04:20.480 not equal to 0. 00:04:20.480 --> 00:04:24.830 What has to be a value for x for this term to be 0? 00:04:24.830 --> 00:04:27.630 2x plus 3 equals 0. 00:04:27.630 --> 00:04:29.350 Subtract 3 from both sides. 00:04:29.350 --> 00:04:30.960 2x is equal to minus 3. 00:04:30.960 --> 00:04:33.500 x is equal to minus 3/2. 00:04:33.500 --> 00:04:37.650 Once again, they tell us that x has to be greater than 1. 00:04:37.650 --> 00:04:40.950 If x is greater than 1, we know that x cannot be minus 00:04:40.950 --> 00:04:43.760 3/2, because obviously this is less than 1. 00:04:43.760 --> 00:04:47.750 So once again, we know that this term does not equal 0. 00:04:47.750 --> 00:04:51.650 So if this term doesn't equal 0, this term doesn't equal 0. 00:04:51.650 --> 00:04:54.770 But when I multiply all three of these terms I get 0. 00:04:54.770 --> 00:04:58.420 This k has to equal 0. 00:04:58.420 --> 00:05:00.360 This is non-zero, this is non-zero. 00:05:00.360 --> 00:05:03.310 If I'm getting 0 when I multiply them, k has to be 0, 00:05:03.310 --> 00:05:07.150 and that's choice B. 00:05:07.150 --> 00:05:09.540 Next problem. 00:05:09.540 --> 00:05:10.390 Problem number 10. 00:05:10.390 --> 00:05:13.505 Let's see if I have enough space to do it. 00:05:13.505 --> 00:05:14.755 I should just clear it every time. 00:05:17.510 --> 00:05:21.650 If all men in the Williams family are over 6 feet tall-- 00:05:21.650 --> 00:05:24.880 that's not true of my family, of the Khan family; maybe one 00:05:24.880 --> 00:05:26.840 day if we eat right-- which of the following 00:05:26.840 --> 00:05:28.060 statements must be true? 00:05:28.060 --> 00:05:30.570 If all of the men in the Williams' family are over 6 00:05:30.570 --> 00:05:31.920 feet tall, which of the following 00:05:31.920 --> 00:05:33.490 statements must be true? 00:05:33.490 --> 00:05:38.000 Choice A, no man under 6 foot tall is a member of the 00:05:38.000 --> 00:05:40.180 Williams family. 00:05:40.180 --> 00:05:41.650 Sure. 00:05:41.650 --> 00:05:43.610 That sounds good to me. 00:05:43.610 --> 00:05:47.700 Choice A, no man under 6 feet tall is a member of the 00:05:47.700 --> 00:05:48.850 Williams family. 00:05:48.850 --> 00:05:51.730 For example, I am 5 foot 9. 00:05:51.730 --> 00:05:54.210 If I was a member of the Williams family, then the 00:05:54.210 --> 00:05:57.030 first statement could not be have been said, that all men 00:05:57.030 --> 00:06:00.795 in the Williams family are over 6 feet tall. 00:06:00.795 --> 00:06:03.190 So that makes sense, all men in the Williams family are 00:06:03.190 --> 00:06:04.920 over 6 feet tall. 00:06:04.920 --> 00:06:07.750 So that means that no man under 6 foot tall is a member 00:06:07.750 --> 00:06:10.010 of the Williams family, because if there were then you 00:06:10.010 --> 00:06:11.930 couldn't have said that all men in the Williams family are 00:06:11.930 --> 00:06:13.810 over 6 feet tall. 00:06:13.810 --> 00:06:15.880 That probably required the most talking and the least 00:06:15.880 --> 00:06:17.680 writing on my behalf, but hopefully that's 00:06:17.680 --> 00:06:20.000 a convincing answer. 00:06:20.000 --> 00:06:20.720 Problem 11. 00:06:20.720 --> 00:06:23.240 Let me know if it wasn't. 00:06:23.240 --> 00:06:28.410 What is the radius of a circle that has circumference of pi. 00:06:28.410 --> 00:06:33.820 So we know that circumference is equal 2 pi r. 00:06:33.820 --> 00:06:36.440 And they're telling us that the circumference is pi. 00:06:36.440 --> 00:06:39.870 So pi is equal to 2 pi r. 00:06:39.870 --> 00:06:43.660 Divide both sides by 2 pi, you get pi over 2 00:06:43.660 --> 00:06:45.980 pi is equal to radius. 00:06:45.980 --> 00:06:47.510 Divide the numerator and the denominator by 00:06:47.510 --> 00:06:49.550 pi, you get 1, 1. 00:06:49.550 --> 00:06:55.380 So you get 1/2 is equal to the radius, and that is choice B. 00:06:55.380 --> 00:06:57.590 Pretty straightforward, eh? 00:06:57.590 --> 00:07:05.270 Problem 12. 00:07:05.270 --> 00:07:09.950 If y is directly proportional to x squared and y equals-- 00:07:09.950 --> 00:07:12.440 OK, so when I say directly proportional, that means y is 00:07:12.440 --> 00:07:15.975 equal to some constant, we don't know what it is, y is 00:07:15.975 --> 00:07:18.620 equal to some constant times x squared. 00:07:18.620 --> 00:07:20.150 That's what directly proportional means. 00:07:20.150 --> 00:07:22.540 It means it's some constant times x squared. 00:07:22.540 --> 00:07:25.740 y is directly proportional to x squared, and y is equal to 00:07:25.740 --> 00:07:27.690 1/8 when x is equal to 1/2. 00:07:27.690 --> 00:07:35.110 So y is equal to 1/8 is equal to k times when x is 1/2. 00:07:35.110 --> 00:07:37.910 So when x is 1/2, that's all we're saying. 00:07:37.910 --> 00:07:38.760 So what does this tell us? 00:07:38.760 --> 00:07:43.340 That means that 1/8 is equal to k times 1/4. 00:07:43.340 --> 00:07:49.590 Multiply both sides by 4, times 4, you get 4/8 is equal 00:07:49.590 --> 00:07:51.570 to k, because this cancels out. 00:07:51.570 --> 00:07:53.890 And 4/8 is the same thing as 1/2. 00:07:53.890 --> 00:07:59.880 So the relationship is, y is equal to 1/2 x squared, and 00:07:59.880 --> 00:08:01.720 now what are they asking us? 00:08:01.720 --> 00:08:06.200 What is the positive value of x when y is equal to 9/2? 00:08:06.200 --> 00:08:08.786 So they're saying when y equals 9/2, let's solve for x 00:08:08.786 --> 00:08:10.150 in the positive value. 00:08:10.150 --> 00:08:13.130 9/2 is equal to 1/2 x squared. 00:08:13.130 --> 00:08:16.870 Multiply both sides by 2, this 2 cancels here, 00:08:16.870 --> 00:08:18.006 this 2 cancels here. 00:08:18.006 --> 00:08:19.410 I was multiplying. 00:08:19.410 --> 00:08:22.090 So you get 9 is equal to x squared. 00:08:22.090 --> 00:08:23.670 So they want the positive value of x. 00:08:23.670 --> 00:08:27.320 So x is equal to plus or minus 3. 00:08:27.320 --> 00:08:30.100 And they want the positive value, so x is equal to 00:08:30.100 --> 00:08:31.880 positive 3. 00:08:31.880 --> 00:08:35.260 And that's choice D. 00:08:35.260 --> 00:08:36.510 Choice D. 00:08:39.710 --> 00:08:41.120 Let me do the next problem in the next video. 00:08:41.120 --> 00:08:42.789 I'll see you soon.

SAT Prep: Test 7 Section 5 Part 1

https://www.youtube.com/watch?v=R6byhBey7eY

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https://www.youtube.com/api/timedtext?v=R6byhBey7eY&ei=YmeUZdqDM7H7vdIPqtGM4AY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B0D9F05EC2701728B94F3F188A55443BFBD9E0CA.45DBBB82E40F66DAB772079FD5AEDC5BAA73D236&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.740 --> 00:00:03.370 Problem number 1. 00:00:03.370 --> 00:00:09.400 It gives a series-- 2, 6, 14, 30. 00:00:09.400 --> 00:00:12.770 And they say in the sequence above the first term is 2. 00:00:12.770 --> 00:00:15.540 Each number after the first if obtained by adding 1 to the 00:00:15.540 --> 00:00:18.150 preceding number and then doubling the result. 00:00:18.150 --> 00:00:20.840 So you add 1, so you get 3, and then you 00:00:20.840 --> 00:00:22.240 double it you get 6. 00:00:22.240 --> 00:00:25.230 You add 1 to 6 to 7, right, this is plus 1 and then you 00:00:25.230 --> 00:00:26.500 multiply it by 2. 00:00:26.500 --> 00:00:29.030 You get 7 times 2 is 14. 00:00:29.030 --> 00:00:32.770 14 plus 1 is 15 times 2 is 30. 00:00:32.770 --> 00:00:34.380 And they want the sixth term. 00:00:34.380 --> 00:00:42.730 So then 30 plus 1 is 31 times 2 is 62 and then you add 1. 00:00:42.730 --> 00:00:45.290 We add 1 first. So you get to 63. 00:00:45.290 --> 00:00:46.875 And then you multiply that times 2. 00:00:46.875 --> 00:00:49.140 That's what, 126. 00:00:49.140 --> 00:00:52.660 So that is choice E. 00:00:52.660 --> 00:00:53.910 Problem 2. 00:00:56.250 --> 00:01:06.540 If a times x plus y is equal to 45, and ax is equal to 15, 00:01:06.540 --> 00:01:09.290 what is ay equal? 00:01:09.290 --> 00:01:10.630 So let's just distribute this a here. 00:01:10.630 --> 00:01:15.640 You get ax plus ay is equal to 45. 00:01:15.640 --> 00:01:18.350 They tell us ax is equal to 15. 00:01:18.350 --> 00:01:20.730 So that is equal to 15-- ax is equal to 15. 00:01:20.730 --> 00:01:24.790 So you get 15 plus ay is equal to 45. 00:01:24.790 --> 00:01:30.590 Subtract 15 from both sides and you get ay is equal to 30. 00:01:30.590 --> 00:01:31.210 That's our answer. 00:01:31.210 --> 00:01:32.200 They want to know what ay is. 00:01:32.200 --> 00:01:34.820 You don't have to solve for a or y, just ay, so 00:01:34.820 --> 00:01:37.640 that's choice E. 00:01:37.640 --> 00:01:38.890 Next problem. 00:01:40.890 --> 00:01:43.460 Before drawing this let me read it so I can figure out 00:01:43.460 --> 00:01:44.560 what I have to draw. 00:01:44.560 --> 00:01:47.780 On the speedometer above what is the speed in miles per hour 00:01:47.780 --> 00:01:50.580 indicated by the needle position? 00:01:50.580 --> 00:01:54.010 So let me see if I can draw this. 00:01:54.010 --> 00:01:59.860 So they give the bottom one like that, that's given in 00:01:59.860 --> 00:02:02.580 feet per second. 00:02:02.580 --> 00:02:06.540 And then the next one up is this one, 00:02:06.540 --> 00:02:09.960 that's miles per hour. 00:02:09.960 --> 00:02:14.170 And then the top one is kilometers per hour. 00:02:14.170 --> 00:02:16.340 I'm going to write it on this side, although they write it 00:02:16.340 --> 00:02:18.640 up here-- kilometers per hour. 00:02:18.640 --> 00:02:24.800 And then they indicate-- and on the miles per hour thing-- 00:02:24.800 --> 00:02:27.700 I don't know what you actually call this measurement. 00:02:27.700 --> 00:02:32.290 So there's 30 and then there's one slash, two slash, three 00:02:32.290 --> 00:02:35.350 slash, and then they go to 60. 00:02:35.350 --> 00:02:39.720 And then the line that they draw is right here. 00:02:39.720 --> 00:02:41.010 It goes like that. 00:02:45.450 --> 00:02:48.550 And they say in the speedometer above what is the 00:02:48.550 --> 00:02:50.460 speed in miles per hour indicated 00:02:50.460 --> 00:02:51.510 by the needle position. 00:02:51.510 --> 00:02:55.290 And the reason why I didn't do much detail on this line or on 00:02:55.290 --> 00:02:58.090 this line is because this one's kilometers per hour and 00:02:58.090 --> 00:02:58.880 this is feet per second. 00:02:58.880 --> 00:03:00.890 So I don't care about those, I just care about the miles per 00:03:00.890 --> 00:03:03.970 hour, and that is this one. 00:03:03.970 --> 00:03:05.520 And let's see, they drew it here. 00:03:05.520 --> 00:03:07.690 So how big is each of these slashes? 00:03:07.690 --> 00:03:09.510 1, 2, 3, 4. 00:03:09.510 --> 00:03:16.460 So each of these go 1/4 of the distance between 30 and 60. 00:03:16.460 --> 00:03:18.200 Each of them goes 1/4 of the distance. 00:03:18.200 --> 00:03:21.200 So you could say, for example, this is halfway between 30 and 00:03:21.200 --> 00:03:24.540 60, so this is going to be 45. 00:03:24.540 --> 00:03:26.940 And so what's halfway between 30 and 45, which is 00:03:26.940 --> 00:03:28.490 where the thing is? 00:03:28.490 --> 00:03:31.920 Well 45 is 15 more than 30, so this is going to be 7 and 1/2 00:03:31.920 --> 00:03:33.050 more than 30. 00:03:33.050 --> 00:03:37.040 So this point right here is going to be 37.5. 00:03:37.040 --> 00:03:37.820 And how do I know that? 00:03:37.820 --> 00:03:42.060 If you just say, well if I just take 30 divided by 4-- 1, 00:03:42.060 --> 00:03:46.050 2, 3, 4-- each of these are going to be 7.5-- right, 30 00:03:46.050 --> 00:03:48.940 divided by 4 is equal to 7.5. 00:03:48.940 --> 00:03:50.520 Or you could say well this middle one's definitely going 00:03:50.520 --> 00:03:54.120 to be 45 because 45 is right in between 30 and 60. 00:03:54.120 --> 00:03:57.360 And then what's right in between 30 and 45? 00:03:57.360 --> 00:03:59.760 Well that's 37.5 again. 00:03:59.760 --> 00:04:02.320 So the speedometer's indicating 37.5 miles per 00:04:02.320 --> 00:04:05.160 hour, that's choice B. 00:04:05.160 --> 00:04:08.560 Next problem, problem 4. 00:04:08.560 --> 00:04:12.390 How many different positive three-digit integers can be 00:04:12.390 --> 00:04:17.910 formed if the three digits, 4, 5 and 6, must be used in each 00:04:17.910 --> 00:04:19.160 of the integers. 00:04:22.580 --> 00:04:25.220 So they're essentially saying how many combinations can you 00:04:25.220 --> 00:04:27.480 get of the number 456? 00:04:27.480 --> 00:04:31.310 So I could write out all those combinations if you like. 00:04:31.310 --> 00:04:35.870 I mean you could 456, 465-- right, those are all of the 00:04:35.870 --> 00:04:37.650 ones that if you have 4 in front. 00:04:37.650 --> 00:04:41.670 Then if you have 5 in front, you have 546 or 00:04:41.670 --> 00:04:44.180 you can have 564. 00:04:44.180 --> 00:04:45.210 I just switched them. 00:04:45.210 --> 00:04:50.920 And if you have 6 in front you could have 645 or 654. 00:04:50.920 --> 00:04:52.170 So those are 6. 00:04:52.170 --> 00:04:53.880 The other way I think about it is I have 00:04:53.880 --> 00:04:55.270 three numbers, right? 00:04:55.270 --> 00:04:57.860 I can put one of three in the first position. 00:04:57.860 --> 00:04:59.880 I can put one of two in the second position. 00:04:59.880 --> 00:05:03.250 And then I would have one left over in the last position. 00:05:03.250 --> 00:05:04.790 I'd have three choices to put in the first 00:05:04.790 --> 00:05:06.680 number, 4, 5 or 6. 00:05:06.680 --> 00:05:08.890 Then after I put one in that first position I have two 00:05:08.890 --> 00:05:12.300 left, so there's two more possibilities, and then I have 00:05:12.300 --> 00:05:14.290 one left over for the third position. 00:05:14.290 --> 00:05:15.410 So you could also think of it this way. 00:05:15.410 --> 00:05:16.520 You could just write them out-- 1, 00:05:16.520 --> 00:05:18.440 2, 3, 4, 5, 6 numbers. 00:05:18.440 --> 00:05:21.360 Or you could use that latter technique I just talked about 00:05:21.360 --> 00:05:23.180 and it would be 3 times 2 times 1, which is 00:05:23.180 --> 00:05:25.720 also equal to 6. 00:05:25.720 --> 00:05:26.970 Next problem. 00:05:29.660 --> 00:05:31.120 Let me draw what they have drawn. 00:05:36.400 --> 00:05:41.570 Well I'll try my best. This line goes like this. 00:05:48.160 --> 00:05:50.270 And then there's a dotted line. 00:05:50.270 --> 00:05:51.610 Let me just see if I can do that. 00:06:00.190 --> 00:06:02.490 The hard part of these problems is drawing them. 00:06:02.490 --> 00:06:04.380 Good thing you don't have to do that on the SAT. 00:06:04.380 --> 00:06:07.490 The three-dimensional figure represented above consists of 00:06:07.490 --> 00:06:09.520 rectangular and triangle faces. 00:06:09.520 --> 00:06:11.270 Fair enough, triangular faces. 00:06:11.270 --> 00:06:14.720 Each rectangular face has area r. 00:06:14.720 --> 00:06:17.430 So this is a rectangular face here, there's three 00:06:17.430 --> 00:06:19.110 rectangular faces. 00:06:19.110 --> 00:06:24.190 So area of the rectangle is equal to r. 00:06:24.190 --> 00:06:25.980 And each triangular face is area t. 00:06:25.980 --> 00:06:29.130 So area of each of the triangle is equal to t. 00:06:29.130 --> 00:06:31.350 What is a total surface area of the figure in 00:06:31.350 --> 00:06:32.410 terms of r and t? 00:06:32.410 --> 00:06:35.510 So how many rectangular surfaces are there? 00:06:35.510 --> 00:06:37.470 Well there's this one in back. 00:06:40.660 --> 00:06:42.350 It's kind of behind the thing. 00:06:42.350 --> 00:06:46.550 You have this base that I'll drawn in magenta. 00:06:46.550 --> 00:06:48.610 This is the base rectangular surface. 00:06:48.610 --> 00:06:51.510 And then you have this one that's closest to us, which I 00:06:51.510 --> 00:06:54.760 could-- this one-- that's the closest. So you have three 00:06:54.760 --> 00:06:56.560 rectangular surfaces. 00:06:56.560 --> 00:06:59.895 So it's 3 times r, because each rectangular surface has 00:06:59.895 --> 00:07:01.110 an area or r. 00:07:01.110 --> 00:07:02.230 And then how many of the triangular 00:07:02.230 --> 00:07:03.130 surfaces do you have? 00:07:03.130 --> 00:07:06.020 You have the one in the back, back here, and then you have 00:07:06.020 --> 00:07:08.210 the one of the front right here. 00:07:08.210 --> 00:07:10.220 So you have two triangular surfaces. 00:07:10.220 --> 00:07:15.010 So it's 3 times the rectangular areas plus 2 times 00:07:15.010 --> 00:07:17.770 the triangular areas because there's two triangular areas. 00:07:17.770 --> 00:07:21.980 So 3r plus 2t, and that's choice B. 00:07:21.980 --> 00:07:23.230 Next problem. 00:07:25.730 --> 00:07:28.150 Problem 6. 00:07:28.150 --> 00:07:35.100 If n is a positive integer and n plus 1 over 2 to the n is 00:07:35.100 --> 00:07:42.380 equal to 1/2, n equals what? 00:07:42.380 --> 00:07:44.530 So let's think about how we can do this. 00:07:44.530 --> 00:07:49.110 Let's multiply both sides of this equation by 2 to the n. 00:07:49.110 --> 00:07:51.130 So if you multiply both sides of the equation times 2 to the 00:07:51.130 --> 00:07:59.090 n, you get n plus 1 is equal to 1/2 times 2 to the n. 00:07:59.090 --> 00:08:01.940 Let me think about this a little bit. 00:08:01.940 --> 00:08:03.210 Let's see, 1/2 times 2 to the n. 00:08:03.210 --> 00:08:07.150 1/2 is the same thing as 2 to the negative 1. 00:08:07.150 --> 00:08:14.000 So this is n plus 1 is equal to-- let me think about this. 00:08:14.000 --> 00:08:21.920 n plus 1 is equal to 2 to the negative 1 times 2 to the n. 00:08:21.920 --> 00:08:25.680 So n plus 1 is equal to 2-- sorry. 00:08:25.680 --> 00:08:28.260 2 to the n minus 1. 00:08:28.260 --> 00:08:29.430 Huh. 00:08:29.430 --> 00:08:30.950 So that's where I can get it. 00:08:30.950 --> 00:08:33.090 Really at this point, the best thing I can think of is just 00:08:33.090 --> 00:08:35.600 trying out the choices and see which one works. 00:08:35.600 --> 00:08:40.080 If you look at choice A, 1-- actually, we could just do it 00:08:40.080 --> 00:08:41.679 from the original, that's faster. 00:08:41.679 --> 00:08:45.850 1 plus 1 over 2 to the n is equal to 2 over 2 to the 1, 00:08:45.850 --> 00:08:46.820 which is equal to 1. 00:08:46.820 --> 00:08:49.790 So it's not 1/2, so it's not a equals 1. 00:08:49.790 --> 00:08:51.970 Choice B is 2. 00:08:51.970 --> 00:08:57.950 So it would be 2 plus 1, which is 3 over 2 squared, which 00:08:57.950 --> 00:08:59.510 equals 3/4. 00:08:59.510 --> 00:09:00.700 So that's not the answer. 00:09:00.700 --> 00:09:04.230 If you look at choice C, choice C is 3. 00:09:04.230 --> 00:09:10.450 So 3 plus 1 is 4 over 2 to the third. 00:09:10.450 --> 00:09:13.670 That's 4 over 8 equals 1/2. 00:09:13.670 --> 00:09:15.260 So it's choice C. 00:09:15.260 --> 00:09:16.780 And I'm trying to figure out if there's a way that you 00:09:16.780 --> 00:09:18.900 could solve for that easier other than trying out the 00:09:18.900 --> 00:09:22.500 numbers, but for some reason it's not popping into my head. 00:09:22.500 --> 00:09:25.010 Let me know a message if you figure it out. 00:09:25.010 --> 00:09:27.170 But I also do these under time pressure. 00:09:27.170 --> 00:09:30.450 But I will see you in the next video.

SAT Prep: Test 7 Section 5 Part 3

https://www.youtube.com/watch?v=RpCkWhPqTQM

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en

WEBVTT Kind: captions Language: en 00:00:00.760 --> 00:00:03.830 We're on problem 13. 00:00:03.830 --> 00:00:11.520 And they say if 4x is equal to 6u, which is equal to 5v, 00:00:11.520 --> 00:00:17.670 which is equal to 7w, all of which is greater than 0, which 00:00:17.670 --> 00:00:20.210 of the following is true. 00:00:20.210 --> 00:00:24.243 So they're essentially wanting you to order w to-- they want 00:00:24.243 --> 00:00:27.670 you to order these terms in order from least to greatest. 00:00:27.670 --> 00:00:30.710 So one thing I'm going to do, let me order these terms, the 00:00:30.710 --> 00:00:32.900 coefficients from least to greatest. That's the same 00:00:32.900 --> 00:00:37.330 thing as 4x is equal to 5v, which is equal to 6u, which is 00:00:37.330 --> 00:00:38.840 equal to 7w. 00:00:38.840 --> 00:00:41.290 They're all greater than 0. 00:00:41.290 --> 00:00:45.290 So what's going to be greater, x or w? 00:00:45.290 --> 00:00:48.350 So if I have 4 of something and that equals 7 of this 00:00:48.350 --> 00:00:52.090 thing, this is going to be the smallest, right? 00:00:52.090 --> 00:00:53.340 Think of it this way. 00:00:55.590 --> 00:01:00.990 If x was 7, then w would be 4. 00:01:00.990 --> 00:01:01.330 Try that out. 00:01:01.330 --> 00:01:04.660 If x was 7 and this term would be 28, the w would be 4. 00:01:04.660 --> 00:01:08.920 So w is the smallest. And by that same principle, u is 00:01:08.920 --> 00:01:09.800 going to be next. 00:01:09.800 --> 00:01:13.780 So w is less than u, which is less than v, which 00:01:13.780 --> 00:01:14.990 is less than x. 00:01:14.990 --> 00:01:19.790 And it's just on the principle, if it takes me 7 of 00:01:19.790 --> 00:01:23.080 something to make up the 4 of something else, this thing has 00:01:23.080 --> 00:01:24.900 to be bigger because I need fewer of it to 00:01:24.900 --> 00:01:27.000 make 7 of this thing. 00:01:27.000 --> 00:01:30.430 And by the same logic u is less than v, right, because 00:01:30.430 --> 00:01:33.640 you need 6 of that thing to make one of v. 00:01:33.640 --> 00:01:39.550 I mean if you really wanted to kind of try it out you could 00:01:39.550 --> 00:01:44.100 divide all of these by 4 and you'd get x is equal to 5/4v, 00:01:44.100 --> 00:01:47.710 which is equal to 6/4u, which is equal to 7/4w. 00:01:47.710 --> 00:01:49.800 All of these terms are greater than 1. 00:01:49.800 --> 00:01:53.560 So x is more than all of these. 00:01:53.560 --> 00:01:55.800 And you could do it for all of these and you would see that 00:01:55.800 --> 00:01:56.380 same order. 00:01:56.380 --> 00:01:58.655 That whichever one has the highest coefficient is going 00:01:58.655 --> 00:02:01.320 to be the smallest number, and then the next smallest is 00:02:01.320 --> 00:02:03.700 going to be u, then v, then x. 00:02:03.700 --> 00:02:06.210 So let's see what choice that is. w is less than u, which is 00:02:06.210 --> 00:02:07.780 less than u, which is less than x. 00:02:07.780 --> 00:02:10.930 That's choice D. 00:02:10.930 --> 00:02:14.800 Problem 14. 00:02:14.800 --> 00:02:22.210 Let the function h be defined by h of t is equal to 2 times 00:02:22.210 --> 00:02:26.910 t cubed minus 3. 00:02:26.910 --> 00:02:33.280 When h of t is equal to minus 60, so h of t is equal to 00:02:33.280 --> 00:02:38.730 minus 60, what is 2 minus 3t? 00:02:38.730 --> 00:02:41.280 What is 2 minus 3t. 00:02:41.280 --> 00:02:45.260 So let's see what we can do here. 00:02:45.260 --> 00:02:47.452 2 minus 3t. 00:02:47.452 --> 00:02:53.710 So h of t is equal to minus 60, so minus 60 is equal to 2 00:02:53.710 --> 00:02:59.210 times t cubed minus 3. 00:02:59.210 --> 00:03:00.810 Fair enough. 00:03:00.810 --> 00:03:02.410 And then divide both sides by 2. 00:03:02.410 --> 00:03:04.460 I'm just trying to see where I can go with this. 00:03:04.460 --> 00:03:06.650 Divide both sides by 2. 00:03:06.650 --> 00:03:13.350 So you get minus 30 is equal to t cubed minus 3. 00:03:13.350 --> 00:03:20.050 Add 3 to both sides you get minus 27 is equal to t cubed. 00:03:20.050 --> 00:03:21.450 This is pretty straightforward. 00:03:21.450 --> 00:03:23.960 Something to the third power is equal to minus 27. 00:03:23.960 --> 00:03:25.920 Well what's the cubed root of 27? 00:03:25.920 --> 00:03:26.680 It's 3. 00:03:26.680 --> 00:03:29.830 So the cubed root of minus 27 is minus 3. 00:03:29.830 --> 00:03:32.360 So t is equal to minus 3. 00:03:32.360 --> 00:03:33.100 And you could try that out. 00:03:33.100 --> 00:03:35.430 What's minus 3 to the third power? 00:03:35.430 --> 00:03:39.140 Minus 3 times minus 3 is 9 times minus 3 is minus 27. 00:03:39.140 --> 00:03:44.340 So when h of t is minus 60, t is equal to minus 3. 00:03:44.340 --> 00:03:46.810 And so what is 2 minus 3t, which is what they asked us 00:03:46.810 --> 00:03:47.285 originally? 00:03:47.285 --> 00:03:51.670 It's going to be 2 minus 3 times minus 3. 00:03:51.670 --> 00:03:58.790 So it's 2 minus minus 9 or 2 plus 9, which is equal to 11. 00:03:58.790 --> 00:04:01.910 And that is choice B. 00:04:01.910 --> 00:04:03.160 Next problem. 00:04:07.410 --> 00:04:08.840 I'll switch colors. 00:04:08.840 --> 00:04:11.100 Where was I? 00:04:11.100 --> 00:04:13.770 My brain is fried from doing all of these SAT problems. But 00:04:13.770 --> 00:04:16.430 I figure I do it once an it'll be there forever for students 00:04:16.430 --> 00:04:19.589 for the rest of eternity to learn SAT problems and compete 00:04:19.589 --> 00:04:22.220 with my future children and they'll be [UNINTELLIGIBLE]. 00:04:22.220 --> 00:04:23.630 Problem 15. 00:04:23.630 --> 00:04:28.410 If x is divisible by 3 and y is divisible by 5, which of 00:04:28.410 --> 00:04:31.630 the following must be divisible by 15? 00:04:31.630 --> 00:04:37.040 So x divisible by 3, so x is a multiple of 3-- you can view 00:04:37.040 --> 00:04:38.130 it that way. 00:04:38.130 --> 00:04:42.130 y is the divisible by 5. 00:04:42.130 --> 00:04:45.970 Which of the following must also be divisible by 15? 00:04:45.970 --> 00:04:46.710 Do you know how I'm going to do it? 00:04:46.710 --> 00:04:50.400 I'm going to say look, you could write x is equal to 3k 00:04:50.400 --> 00:04:51.660 where k is some integer. 00:04:51.660 --> 00:04:56.120 We don't know what multiple x is of 3, but we know 3 times 00:04:56.120 --> 00:04:58.270 some integer, k, is equal to x. 00:04:58.270 --> 00:05:01.550 And similarly, you could say y is equal to 5, I don't know, 00:05:01.550 --> 00:05:04.690 5m where you could say some integer m times 00:05:04.690 --> 00:05:06.550 5 is equal to y. 00:05:06.550 --> 00:05:08.960 Because we know that 5 times something's equal to y. 00:05:08.960 --> 00:05:10.510 That's what divisible means. 00:05:10.510 --> 00:05:12.520 So let's look at the choices. 00:05:12.520 --> 00:05:13.890 1. 00:05:13.890 --> 00:05:15.090 They want to know which are the ones that are 00:05:15.090 --> 00:05:16.350 divisible by 15. 00:05:16.350 --> 00:05:19.170 So choice 1 is x times y. 00:05:19.170 --> 00:05:25.540 Well x times y is the same thing as 3k times 5m. 00:05:25.540 --> 00:05:29.530 And that's the same thing as 15km. 00:05:29.530 --> 00:05:34.430 So xy is the same thing as 15 times a product of integers, 00:05:34.430 --> 00:05:36.040 and the product of two integers is going to be an 00:05:36.040 --> 00:05:37.330 integer, right? 00:05:37.330 --> 00:05:39.560 So this is definitely divisible by 15. 00:05:39.560 --> 00:05:42.720 So choice 1 is divisible by 15. 00:05:42.720 --> 00:05:43.970 Problem 2. 00:05:47.750 --> 00:05:52.930 3x plus 5y. 00:05:52.930 --> 00:05:55.260 So we could do the same thing. 00:05:55.260 --> 00:06:03.610 x is 3k, so that's 3 times 3k plus 5y plus 5 times 5m. 00:06:07.350 --> 00:06:12.210 That equals 9k plus 25m. 00:06:12.210 --> 00:06:14.270 This isn't necessarily divisible by 15. 00:06:14.270 --> 00:06:16.810 I mean what if k and m are both 1? 00:06:16.810 --> 00:06:20.680 Then you get 34 and-- this is not divisible by 15. 00:06:20.680 --> 00:06:21.930 Choice 3. 00:06:24.730 --> 00:06:28.700 5x plus 3y. 00:06:28.700 --> 00:06:29.510 Doing the same thing. 00:06:29.510 --> 00:06:33.600 That equals 5 times 3k, right, this is x. 00:06:33.600 --> 00:06:38.360 5 times 3k plus 3 times 5m. 00:06:38.360 --> 00:06:44.480 And this equals 15k plus 15m. 00:06:44.480 --> 00:06:46.670 An you can factor out the 15. 00:06:46.670 --> 00:06:50.020 That's 15 times k plus m. 00:06:50.020 --> 00:06:54.270 And once again, so this thing is equal to 15 times k plus m. 00:06:54.270 --> 00:06:55.900 k plus m is going to be some integer. 00:06:55.900 --> 00:06:57.760 We said k and m are both integers. 00:06:57.760 --> 00:07:00.420 So this term right here is definitely divisible by 15. 00:07:00.420 --> 00:07:02.720 So one and three are our answer. 00:07:02.720 --> 00:07:05.940 And that is choice D. 00:07:05.940 --> 00:07:08.070 And you know, if you didn't want to do this fancy stuff 00:07:08.070 --> 00:07:09.720 where you say it's multiplying by some integer, 00:07:09.720 --> 00:07:11.160 et cetera, et cetera. 00:07:11.160 --> 00:07:14.090 Just x is divisible by 3, y is divisible by 5, well let's 00:07:14.090 --> 00:07:17.810 just say x is equal to 3 and y is equal to 5 and 00:07:17.810 --> 00:07:18.920 then try them out. 00:07:18.920 --> 00:07:22.090 Then xy is going to be 15, which is, of course, 00:07:22.090 --> 00:07:23.810 divisible by 15. 00:07:23.810 --> 00:07:26.670 This would be, let's see, 3 times 3 plus 5 times 5, this 00:07:26.670 --> 00:07:27.620 would be 34. 00:07:27.620 --> 00:07:29.380 Not divisible by 15. 00:07:29.380 --> 00:07:33.205 And then the final one, 3 times 5 is 15 plus 3 times 5 00:07:33.205 --> 00:07:35.970 is 15, this would be 30, which is divisible by 15. 00:07:35.970 --> 00:07:37.950 So that would have been the quick and dirty, not 00:07:37.950 --> 00:07:39.920 necessarily the most mathematically rigorous way of 00:07:39.920 --> 00:07:42.170 doing it, but it would have gotten you the right answer. 00:07:42.170 --> 00:07:45.400 And that really is what matters, I guess, on the SAT. 00:07:49.610 --> 00:07:51.455 That would have been the quick and dirty solution. 00:07:51.455 --> 00:07:55.900 Let me see if I can draw this thing. 00:07:55.900 --> 00:07:57.920 One line like that. 00:07:57.920 --> 00:08:00.270 One line like that. 00:08:00.270 --> 00:08:02.900 One line something like that. 00:08:02.900 --> 00:08:06.460 And what do they tell us about this? 00:08:06.460 --> 00:08:07.750 I'll use yellow. 00:08:07.750 --> 00:08:14.020 They tell us that this angle right here is 115 degrees. 00:08:14.020 --> 00:08:16.910 This is line l. 00:08:16.910 --> 00:08:20.290 This is z degrees. 00:08:20.290 --> 00:08:24.190 Line m, line n, and this is y degrees. 00:08:24.190 --> 00:08:26.730 Looks like we're going to have to play the angle game here. 00:08:26.730 --> 00:08:27.920 This is y degrees. 00:08:27.920 --> 00:08:30.370 In the figure above what is y plus z? 00:08:34.039 --> 00:08:36.370 So when I play the angle game I just try to figure out as 00:08:36.370 --> 00:08:37.750 many sides as I can. 00:08:37.750 --> 00:08:41.330 So if this is 115 what is this going to be? 00:08:41.330 --> 00:08:44.320 Well 115 plus this supplementary, right? 00:08:44.320 --> 00:08:47.440 So these are going to add up to be 180, so this is going to 00:08:47.440 --> 00:08:49.380 be 180 minus 115. 00:08:49.380 --> 00:08:51.070 What's 80 minus 15? 00:08:51.070 --> 00:08:54.910 It's 65, right, because 80 minus 10 is 70. 00:08:54.910 --> 00:08:56.010 So this is 65 degrees. 00:08:56.010 --> 00:08:57.720 And I just said that because this angle plus 00:08:57.720 --> 00:09:00.390 this has to be 180. 00:09:00.390 --> 00:09:03.370 Now what is this angle right here? 00:09:03.370 --> 00:09:05.760 This angle right here is going to be 180 minus y for the same 00:09:05.760 --> 00:09:08.650 reason because it's supplementary to y. 00:09:08.650 --> 00:09:15.330 This angle here, same reason, it's going to be 180 minus z. 00:09:15.330 --> 00:09:17.690 And we know that this angle plus this angle plus this 00:09:17.690 --> 00:09:19.680 angle has to add up to 180. 00:09:19.680 --> 00:09:29.060 So 65 plus 180 minus y plus 180 minus z is equal to 180. 00:09:29.060 --> 00:09:35.150 Let's see, so that's-- you could subtract 180 from both 00:09:35.150 --> 00:09:36.770 sides, you get a 0 here. 00:09:36.770 --> 00:09:38.740 What's 65 plus 180? 00:09:38.740 --> 00:09:47.160 65 plus 180 is 245 minus y minus z, is equal to 0. 00:09:47.160 --> 00:09:53.110 Add y and z to both sides, you get 245 is equal to y plus z. 00:09:53.110 --> 00:09:55.780 And that is choice E. 00:09:55.780 --> 00:09:57.600 See you in the next video.

SAT Prep: Test 7 Section 5 Part 4

https://www.youtube.com/watch?v=U2FLQYvioeY

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WEBVTT Kind: captions Language: en 00:00:00.780 --> 00:00:05.590 We're on problem 17. 00:00:05.590 --> 00:00:09.240 The sum of three consecutive odd integers is 111. 00:00:09.240 --> 00:00:10.920 That's what they tell us at the top. 00:00:10.920 --> 00:00:11.810 I won't write that. 00:00:11.810 --> 00:00:14.440 If n represents the least of the three integers, which of 00:00:14.440 --> 00:00:15.900 the following equations represents 00:00:15.900 --> 00:00:17.010 the statement above? 00:00:17.010 --> 00:00:19.610 So n is the least of the three integers, so 00:00:19.610 --> 00:00:21.040 let's say that's n. 00:00:21.040 --> 00:00:25.440 So it's the least of three consecutive odd integers. 00:00:25.440 --> 00:00:27.970 So what's the next odd integer going to be? 00:00:27.970 --> 00:00:30.080 It's not going to be n plus 1, because n plus 1 is going to 00:00:30.080 --> 00:00:31.110 be an even integer, right? 00:00:31.110 --> 00:00:34.430 If this was 3 then the next odd integer's going to be 5, 00:00:34.430 --> 00:00:36.870 which is going to be n plus 2. 00:00:36.870 --> 00:00:41.470 And then the next one above that is going to be n plus 4. 00:00:41.470 --> 00:00:42.240 And you can try it out. 00:00:42.240 --> 00:00:45.000 Let's say that n was 3, and this is 5, and this is 7. 00:00:45.000 --> 00:00:47.460 That's three consecutive odd integers. 00:00:47.460 --> 00:00:50.820 And they're telling us that the sum is 111. 00:00:50.820 --> 00:00:51.900 So the sum is 111. 00:00:51.900 --> 00:00:54.870 What's the sum of these three terms? 00:00:54.870 --> 00:00:56.400 Well, you add up the three n's. 00:00:56.400 --> 00:00:59.110 n plus n plus n is 3n. 00:00:59.110 --> 00:01:02.380 And then 2 plus 4, 3n plus 6. 00:01:02.380 --> 00:01:05.500 And that's going to be equal to 111. 00:01:05.500 --> 00:01:09.060 And that is choice D. 00:01:09.060 --> 00:01:10.640 Maybe I just confused you how I wrote this. 00:01:10.640 --> 00:01:13.880 All I did is I said that's the same thing as n plus n plus 2 00:01:13.880 --> 00:01:19.640 plus n plus 4, which is equal to 3n plus 6. 00:01:19.640 --> 00:01:20.890 Next problem. 00:01:30.300 --> 00:01:31.160 Problem 8. 00:01:31.160 --> 00:01:32.150 And they do this arc. 00:01:32.150 --> 00:01:33.720 It looks something like that. 00:01:33.720 --> 00:01:35.920 It actually looks better than what I just drew, 00:01:35.920 --> 00:01:39.190 but you get the point. 00:01:39.190 --> 00:01:42.030 So there's like a distance of 2, and then there's a distance 00:01:42.030 --> 00:01:43.700 of b, it looks like. 00:01:43.700 --> 00:01:53.800 So it's 2 then b then 2 then b then 2 then b again. 00:01:53.800 --> 00:01:55.630 The figure above shows part of a circle whose 00:01:55.630 --> 00:01:57.660 circumference is 45. 00:01:57.660 --> 00:02:01.260 So the circumference of the circle is 45. 00:02:01.260 --> 00:02:05.600 If arcs of length 2 and length b continue to alternate around 00:02:05.600 --> 00:02:08.509 the entire circle, so this is just part of the circle and 00:02:08.509 --> 00:02:10.889 they just keep going around the circle. 00:02:10.889 --> 00:02:12.470 They just keep going around the circle so that there are 00:02:12.470 --> 00:02:16.020 18 arcs of each length. 00:02:16.020 --> 00:02:19.990 What is the degree measure of each of the arcs of length b? 00:02:19.990 --> 00:02:23.690 So they say there's 18 arcs of each length. 00:02:23.690 --> 00:02:27.880 They say that there are 18 of these arcs of lengths 2, and 00:02:27.880 --> 00:02:31.690 there's 18 of these b segments because they're saying this 00:02:31.690 --> 00:02:33.620 is-- well, they drew this as part of the 00:02:33.620 --> 00:02:35.300 whole circle, right? 00:02:35.300 --> 00:02:40.050 So that means that if the entire circumference is 45, 00:02:40.050 --> 00:02:43.385 but in terms of 2's and b's, we know that there's 18 2's , 00:02:43.385 --> 00:02:45.660 so it's 18 times 2. 00:02:45.660 --> 00:02:49.990 And then we also know that there are 18 b's, plus 18b. 00:02:49.990 --> 00:02:51.940 This is another way of writing the circumference of the 00:02:51.940 --> 00:02:54.540 circle. because they say that the circumference is made of 00:02:54.540 --> 00:02:59.660 18 of these arcs and 18 of these b arcs here, so this is 00:02:59.660 --> 00:03:01.270 also equal to the circumference of the circle, 00:03:01.270 --> 00:03:04.890 which they told us is 45. 00:03:04.890 --> 00:03:07.500 Now let's see, this is 36. 00:03:07.500 --> 00:03:13.800 18 times 2 plus 18b is equal to 45. 00:03:13.800 --> 00:03:15.260 Subtract 36 from both sides. 00:03:15.260 --> 00:03:18.900 You get 18b is equal to 9. 00:03:18.900 --> 00:03:26.710 b is equal to 9/18, which is equal to 1/2. 00:03:26.710 --> 00:03:28.070 Now what are they asking? 00:03:28.070 --> 00:03:33.310 What is the degree measure of each of the arcs of length b? 00:03:33.310 --> 00:03:40.190 So we know that each b is 1/2. 00:03:40.190 --> 00:03:40.760 And actually, you know what? 00:03:40.760 --> 00:03:42.720 We could have gone to this step. 00:03:42.720 --> 00:03:47.180 We could have said that 18 b's-- well, no, actually, this 00:03:47.180 --> 00:03:51.680 is-- each b is of length 1/2, right? 00:03:51.680 --> 00:03:55.410 So what fraction is that of the entire circumference? 00:03:55.410 --> 00:03:57.120 This length right here is 1/2. 00:03:57.120 --> 00:03:58.370 That's what we just solved, right? 00:03:58.370 --> 00:04:02.840 You saying there's going to be 18 b's and 18 arcs of length 2 00:04:02.840 --> 00:04:04.120 and they all add up to 45. 00:04:04.120 --> 00:04:04.715 We solve for b. 00:04:04.715 --> 00:04:05.770 It's 1/2. 00:04:05.770 --> 00:04:07.910 We want to figure out what's the degree measure. 00:04:07.910 --> 00:04:12.600 So to figure out the degree measure, we say, well, 1/2 is 00:04:12.600 --> 00:04:18.230 to the circumference of the entire circle, which is 45, is 00:04:18.230 --> 00:04:22.000 equal to the degree measure over the total degrees in the 00:04:22.000 --> 00:04:25.460 entire circle, because the fraction in terms of the 00:04:25.460 --> 00:04:27.520 circumference is the same thing it's going to be in 00:04:27.520 --> 00:04:30.430 terms of degrees as a fraction of 360 degrees. 00:04:30.430 --> 00:04:30.680 Why? 00:04:30.680 --> 00:04:33.320 Because there's 360 degrees in the circle. 00:04:33.320 --> 00:04:35.010 So let's cross-multiply this. 00:04:35.010 --> 00:04:39.690 1/2 times 360 degrees, so you get 180 degrees-- that's 1/2 00:04:39.690 --> 00:04:44.020 times 360-- is equal to 45x. 00:04:44.020 --> 00:04:49.130 So x is equal to 180 divided by 45. 00:04:49.130 --> 00:04:51.780 And what's 180 divided by 45? 00:04:51.780 --> 00:04:55.740 Let's see, 40 goes into 160, it's 4 times, right? 00:04:55.740 --> 00:04:56.730 x is equal to 4. 00:04:56.730 --> 00:04:58.540 4 times 45, right? 00:04:58.540 --> 00:05:00.450 So x is equal to 4 degrees. 00:05:00.450 --> 00:05:01.690 And that's choice A. 00:05:01.690 --> 00:05:03.230 That also makes sense intuitively. 00:05:03.230 --> 00:05:06.970 4 degrees is a very small amount of degrees, so that 00:05:06.970 --> 00:05:09.190 makes sense if you were going from the center. 00:05:09.190 --> 00:05:12.270 I know what I just drew just probably confused you. 00:05:12.270 --> 00:05:14.290 Hopefully, you get the idea. 00:05:14.290 --> 00:05:16.620 Figure out how long the b's are and then what fraction 00:05:16.620 --> 00:05:19.430 that is of the entire circumference and then that's 00:05:19.430 --> 00:05:23.120 what fraction it is of 360 degrees. 00:05:23.120 --> 00:05:25.830 Problem 19. 00:05:25.830 --> 00:05:29.090 The cost of maintenance on an automobile increases 00:05:29.090 --> 00:05:31.190 each year by 10%. 00:05:31.190 --> 00:05:34.340 And Andrew paid $300 this year for maintenance on his 00:05:34.340 --> 00:05:35.080 automobile. 00:05:35.080 --> 00:05:36.110 Fine. 00:05:36.110 --> 00:05:40.620 If the cost c for maintenance on Andrew's automobile n years 00:05:40.620 --> 00:05:44.680 from now is given by the function, so c of n is equal 00:05:44.680 --> 00:05:52.230 to 300x to the n, what is the value of x? 00:05:52.230 --> 00:05:55.860 So they say every year the cost increases by 10%. 00:05:55.860 --> 00:05:59.130 So what does it mean to increase by 10%. 00:05:59.130 --> 00:06:03.150 Let me say the cost in year 1, let's say the cost in year 1 00:06:03.150 --> 00:06:06.350 is going to be-- what's the cost in year 2? 00:06:06.350 --> 00:06:11.040 It's going to be the cost of year 1 increased by 10%. 00:06:11.040 --> 00:06:17.710 So plus 0.1 times the cost in year 1, which is equal to 1.1 00:06:17.710 --> 00:06:20.410 times the cost in year 1. 00:06:20.410 --> 00:06:26.320 The cost in year 3 is equal to 1.1 times the cost in year 2, 00:06:26.320 --> 00:06:28.830 which is the same thing as what? 00:06:28.830 --> 00:06:35.630 1.1 times the cost of year 2, which is this. 00:06:35.630 --> 00:06:40.620 1.1 times 1.1, cost of year 1, which is the same 00:06:40.620 --> 00:06:45.210 thing as 1.1 squared. 00:06:45.210 --> 00:06:49.900 So every year that you go out, you're essentially just taking 00:06:49.900 --> 00:06:53.070 1.1 to that power of how many years you're going out. 00:06:53.070 --> 00:06:55.240 Actually, I should probably make this cost of year 0, make 00:06:55.240 --> 00:06:58.620 this cost of year 1, 0, 0, 0. 00:06:58.620 --> 00:07:00.956 This is probably confusing you, but cost of year 2, so 00:07:00.956 --> 00:07:05.020 then the exponents at least match up: 1 and then 0. 00:07:05.020 --> 00:07:06.500 Then the exponents match up. 00:07:06.500 --> 00:07:11.450 The cost of year 2 is 1.1 squared times cost of year 0. 00:07:11.450 --> 00:07:13.790 So as you can see, all we're doing when we're doing this, 00:07:13.790 --> 00:07:15.290 this is kind of an exponential growth problem. 00:07:15.290 --> 00:07:17.540 But they're just saying what is this base? 00:07:17.540 --> 00:07:19.440 But when you increase something by 10%, you're 00:07:19.440 --> 00:07:22.080 essentially multiplying it by 1., so 00:07:22.080 --> 00:07:23.510 that's all they're asking. 00:07:23.510 --> 00:07:26.330 So that's choice C. 00:07:26.330 --> 00:07:27.580 Next problem. 00:07:33.490 --> 00:07:40.341 They drew this parallelogram, which I have not drawn well. 00:07:40.341 --> 00:07:41.670 It gets the point. 00:07:49.910 --> 00:07:57.060 A, B, C, D. 00:07:57.060 --> 00:07:59.720 If the five line segments in the figure above are all 00:07:59.720 --> 00:08:02.930 congruent-- OK, so the five lines, these are all 00:08:02.930 --> 00:08:05.940 congruent, so they're all equal to each other. 00:08:05.940 --> 00:08:08.530 all of those sides, what is the ratio of the 00:08:08.530 --> 00:08:10.990 length of AC, not drawn? 00:08:10.990 --> 00:08:14.870 So they want to know AC, so that's this length. 00:08:14.870 --> 00:08:20.560 They want to know AC to length BD. 00:08:20.560 --> 00:08:21.160 All right. 00:08:21.160 --> 00:08:24.270 So all of those lines are congruent. 00:08:24.270 --> 00:08:24.480 You know what? 00:08:24.480 --> 00:08:25.590 Let's just pick a number. 00:08:25.590 --> 00:08:27.290 That's what I find to be easiest when 00:08:27.290 --> 00:08:28.260 they tell us it all. 00:08:28.260 --> 00:08:32.669 So let's just say that this is 1, this is 1, this 00:08:32.669 --> 00:08:34.990 is 1, this is 1. 00:08:34.990 --> 00:08:37.510 This whole thing is going to be 1. 00:08:37.510 --> 00:08:41.590 So how long is this and this? 00:08:41.590 --> 00:08:44.830 This will bisect this line, right? 00:08:44.830 --> 00:08:48.230 I think we know that about parallelograms that the two 00:08:48.230 --> 00:08:51.450 lines bisect each other, the two diagonals. 00:08:51.450 --> 00:08:56.466 So this is going to be 1/2, and this is going to be 1/2. 00:08:56.466 --> 00:08:58.730 And there's also something else we know. 00:08:58.730 --> 00:09:05.900 We know that this is 60 degrees, this is 60 degrees, 00:09:05.900 --> 00:09:11.710 that this whole angle is 60 degrees, this whole angle is 00:09:11.710 --> 00:09:12.280 60 degrees. 00:09:12.280 --> 00:09:13.010 And how do we know that? 00:09:13.010 --> 00:09:14.770 Before we drew the green line, we knew that both of those 00:09:14.770 --> 00:09:17.980 triangles are equilateral triangles. 00:09:17.980 --> 00:09:21.610 So if this whole angle was 60 degrees, then this angle right 00:09:21.610 --> 00:09:24.410 here is going to be 30 degrees. 00:09:24.410 --> 00:09:26.870 And I think you see where I'm going with this. 00:09:26.870 --> 00:09:30.980 This is a 30-60-90 triangle where this is 1-- that's the 00:09:30.980 --> 00:09:32.610 hypotenuse-- this is 1/2. 00:09:32.610 --> 00:09:35.960 And then what is the side opposite the 60-degree side, 00:09:35.960 --> 00:09:36.915 which is this? 00:09:36.915 --> 00:09:41.896 The square root of 3 times this side right here. 00:09:41.896 --> 00:09:45.520 The square root of 3/2. 00:09:45.520 --> 00:09:46.250 And so would this one. 00:09:46.250 --> 00:09:50.500 This one would also be square root of 3/2. 00:09:50.500 --> 00:09:52.350 I know maybe it's a little confusing. 00:09:52.350 --> 00:09:54.360 Actually, I'm about to run out of time, so I will continue 00:09:54.360 --> 00:09:56.110 this in the next video.

SAT Prep: Test 7 Section 5 Part 5

https://www.youtube.com/watch?v=A-qer1ChzXk

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WEBVTT Kind: captions Language: en 00:00:00.870 --> 00:00:01.510 Welcome back. 00:00:01.510 --> 00:00:03.130 I was running out of time in the last video and I was 00:00:03.130 --> 00:00:05.630 afraid that I was jumbling up the problem just to get in 00:00:05.630 --> 00:00:06.700 under the wire. 00:00:06.700 --> 00:00:08.780 So I just drew the triangle. 00:00:08.780 --> 00:00:10.950 Actually, let me do it over since I have all the time in 00:00:10.950 --> 00:00:13.380 the world now. 00:00:13.380 --> 00:00:16.660 Because I really want you to get this. 00:00:16.660 --> 00:00:19.720 So the original parallelogram looks like this. 00:00:28.560 --> 00:00:29.820 This is a diagonal. 00:00:29.820 --> 00:00:35.450 And they tell us that all of these sides are congruent, so 00:00:35.450 --> 00:00:38.070 that means that they're all the same length. 00:00:38.070 --> 00:00:40.500 So this side is equal to the side is equal to this side, 00:00:40.500 --> 00:00:42.050 this side is equal to this side. 00:00:42.050 --> 00:00:43.930 So that also tells you these are both equilateral 00:00:43.930 --> 00:00:46.730 triangles, all the sides are equals so all these angles, 00:00:46.730 --> 00:00:52.830 this must be 60, this must be 60, this must be 60, this must 00:00:52.830 --> 00:00:59.390 be 60, 60 and 60. 00:00:59.390 --> 00:01:01.770 Now what else do we know? 00:01:01.770 --> 00:01:12.180 They want us to figure out the ratio of this diagonal, the 00:01:12.180 --> 00:01:17.020 long diagonal to this shorter diagonal right here. 00:01:17.020 --> 00:01:19.300 We know the shorter diagonals of length 1. 00:01:19.300 --> 00:01:23.260 And with parallelograms, you might already kind of know 00:01:23.260 --> 00:01:25.830 this, and actually we could prove this to you. 00:01:25.830 --> 00:01:29.490 Since all of these lines are congruent, we know 00:01:29.490 --> 00:01:32.010 that-- we know what? 00:01:32.010 --> 00:01:34.130 What is this angle right here? 00:01:34.130 --> 00:01:38.130 What is half of-- this line is going to bisect 00:01:38.130 --> 00:01:39.990 both of these angles. 00:01:39.990 --> 00:01:40.470 How do we know? 00:01:40.470 --> 00:01:44.040 Well if it didn't bisect it, all the 00:01:44.040 --> 00:01:45.420 sides wouldn't be congruent. 00:01:45.420 --> 00:01:50.670 So we know that this side is 30, this side is 30, this is 00:01:50.670 --> 00:01:53.080 30, this is 30. 00:01:53.080 --> 00:01:58.500 And we also know that for any parallelogram-- actually for 00:01:58.500 --> 00:02:03.940 any rhombus-- the two soon. diagonals are going to be 00:02:03.940 --> 00:02:05.980 perpendicular bisectors of each other. 00:02:05.980 --> 00:02:08.940 It's not true for any parallelogram, it's only true 00:02:08.940 --> 00:02:11.940 when the sides are all congruent. 00:02:11.940 --> 00:02:14.750 In this situation, and we already know this because this 00:02:14.750 --> 00:02:16.830 is a 30 degree angle, this is a 60 degree angle, so what 00:02:16.830 --> 00:02:18.200 does this have to be? 00:02:18.200 --> 00:02:20.496 This has to be a 90 degree angle because they have 00:02:20.496 --> 00:02:22.470 to add up to 180. 00:02:22.470 --> 00:02:24.100 So let's just pick numbers. 00:02:24.100 --> 00:02:26.570 Let's say that this side is 1, this side is 1, this side is 00:02:26.570 --> 00:02:27.990 1, this side is 1. 00:02:27.990 --> 00:02:30.770 Then this entire diagonal here would be 1, half of it would 00:02:30.770 --> 00:02:33.540 be 1/2, this would be 1/2, this would be 1/2. 00:02:33.540 --> 00:02:35.490 If we didn't know it was 1/2, we could just say well this is 00:02:35.490 --> 00:02:36.770 a 30, 60, 90 triangle. 00:02:36.770 --> 00:02:40.230 So the 30 degree side is 1/2 of the hypotenuse. 00:02:40.230 --> 00:02:42.730 And then what else do we know about 30, 60, 90 triangles? 00:02:42.730 --> 00:02:48.030 The 60 degree side is squares of 3 times the shorter side. 00:02:48.030 --> 00:02:52.180 So this side is going to be square root of 3 over 2. 00:02:52.180 --> 00:02:54.990 And all we did is we realized that this is 00:02:54.990 --> 00:02:56.310 a 30, 60, 90 triangle. 00:02:56.310 --> 00:02:58.320 And then we were using the principle of 30, 60, 90 00:02:58.320 --> 00:03:01.240 triangle to figure out that the side opposite the 60 00:03:01.240 --> 00:03:04.880 degree side is square root of 3 over 2. 00:03:04.880 --> 00:03:07.770 If that side is square root of 3 over 2, then this is also 00:03:07.770 --> 00:03:09.810 square root of 3 over 2. 00:03:09.810 --> 00:03:11.650 So what's this entire length? 00:03:11.650 --> 00:03:16.310 It'd be square root of 3 over 2 plus square root of 3 over 00:03:16.310 --> 00:03:20.300 2, which is equal to square root of 3. 00:03:20.300 --> 00:03:22.260 You add two halves of anything you get the whole. 00:03:22.260 --> 00:03:23.760 So the square roots of 3. 00:03:23.760 --> 00:03:27.020 So the purple diagonal is square roots of 3, and what is 00:03:27.020 --> 00:03:30.640 this-- this was d and c. 00:03:30.640 --> 00:03:31.890 This is ac. 00:03:34.880 --> 00:03:38.540 So we know that ac is equal to the square root of 3. 00:03:38.540 --> 00:03:45.120 And we also know that d-- I'm sorry, this is a b-- db is 00:03:45.120 --> 00:03:45.700 equal to 1. 00:03:45.700 --> 00:03:46.920 We just defined it as 1. 00:03:46.920 --> 00:03:51.220 So the ratio of ac to bd or db is going to be square 00:03:51.220 --> 00:03:53.420 root of 3 to 1. 00:03:53.420 --> 00:03:55.580 And that is choice B. 00:03:55.580 --> 00:03:57.910 I hope I didn't confuse you. 00:03:57.910 --> 00:04:00.360 Next section-- actually, we just finished that section. 00:04:00.360 --> 00:04:01.820 I'll see you

SAT Prep: Test 7 Section 2 Part 1

https://www.youtube.com/watch?v=k3Wd-9bZCgw

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WEBVTT Kind: captions Language: en 00:00:01.000 --> 00:00:03.050 We're now starting test 7. 00:00:03.050 --> 00:00:04.280 And we're on section two. 00:00:04.280 --> 00:00:08.039 Let's see, problem 1 is on page 774. 00:00:08.039 --> 00:00:19.450 So they say set x-- x is equal to this-- 30, 31, 32, 33. 00:00:19.450 --> 00:00:21.320 I haven't read the problem yet, I'm just 00:00:21.320 --> 00:00:22.900 drawing what they drew. 00:00:22.900 --> 00:00:33.000 This is 32, 33, 34, 35 and 36. 00:00:33.000 --> 00:00:34.600 Sets x and y are shown above. 00:00:34.600 --> 00:00:37.850 How many numbers in set x are also in set y? 00:00:37.850 --> 00:00:41.140 Well hopefully it's fairly straightforward. 00:00:41.140 --> 00:00:42.650 Is 30 in set y? 00:00:42.650 --> 00:00:43.356 No. 00:00:43.356 --> 00:00:44.840 Then is 31 in set y? 00:00:44.840 --> 00:00:45.380 No. 00:00:45.380 --> 00:00:47.040 32 is, right? 00:00:47.040 --> 00:00:48.660 And so is 33. 00:00:48.660 --> 00:00:50.320 1, 2, the answer is 2. 00:00:50.320 --> 00:00:53.090 That's choice A. 00:00:53.090 --> 00:00:54.370 Not too tough, huh? 00:00:54.370 --> 00:00:56.550 All right, problem number 2. 00:00:56.550 --> 00:01:03.630 If Peg traveled 10 miles in 2 hours, and Linda traveled 00:01:03.630 --> 00:01:07.880 twice as far in 1/2 the time-- So Linda 00:01:07.880 --> 00:01:09.280 traveled twice as far. 00:01:09.280 --> 00:01:10.950 So twice as far is 10 miles. 00:01:10.950 --> 00:01:13.170 So she traveled 20 miles, right? 00:01:13.170 --> 00:01:15.760 Times 2, that's all I did. 00:01:15.760 --> 00:01:18.810 And they say, in 1/2 the time, so she traveled 20 miles in 1 00:01:18.810 --> 00:01:20.050 hour, right? 00:01:20.050 --> 00:01:21.700 I just do the 1/2 here. 00:01:21.700 --> 00:01:22.320 20 miles in 1 hour. 00:01:22.320 --> 00:01:24.220 What was Linda's average speed in miles per hour? 00:01:24.220 --> 00:01:26.270 She did 20 miles in 1 hour. 00:01:26.270 --> 00:01:31.170 Well that's 20 miles per hour. 00:01:31.170 --> 00:01:33.230 And that's choice C. 00:01:33.230 --> 00:01:36.320 Also not a difficult problem. 00:01:36.320 --> 00:01:38.880 Not that you should feel bad if you had it wrong, it's just 00:01:38.880 --> 00:01:41.860 all about practice. 00:01:41.860 --> 00:01:44.600 It's all about actually reading the problem. 00:01:44.600 --> 00:01:46.190 I know that might seem a bit obvious, 00:01:46.190 --> 00:01:47.860 but you'd be surprised. 00:01:47.860 --> 00:01:50.110 You've even seen me make a couple of mistakes by not 00:01:50.110 --> 00:01:51.910 reading the problem properly. 00:01:51.910 --> 00:01:53.530 OK, problem 3. 00:01:53.530 --> 00:01:56.670 My cousin marked this up with a black marker, so I only hope 00:01:56.670 --> 00:01:57.870 that what I see is what you see. 00:01:57.870 --> 00:02:01.610 So let's see, x is equal to-- she did this when she was in 00:02:01.610 --> 00:02:02.680 sixth grade. 00:02:02.680 --> 00:02:05.400 So I'm very proud of her that she was taking the SAT in 00:02:05.400 --> 00:02:05.985 sixth grade. 00:02:05.985 --> 00:02:10.684 But anyway, x is equal to k times k minus 2. 00:02:10.684 --> 00:02:13.550 Then x plus 1 is equal to what? 00:02:13.550 --> 00:02:14.400 [COUGH] 00:02:14.400 --> 00:02:18.300 Excuse me. x plus 1 is equal to what? 00:02:18.300 --> 00:02:21.200 So let's just add 1 to both sides of this equation. 00:02:21.200 --> 00:02:27.560 So x plus 1 is equal to k times k minus 2 plus 1, right? 00:02:27.560 --> 00:02:30.110 I just added 1 to both sides of that equation, and that 00:02:30.110 --> 00:02:36.330 equals k squared minus 2k plus 1. 00:02:36.330 --> 00:02:40.195 And that is choice-- if I can read this properly-- 00:02:40.195 --> 00:02:43.390 that is choice C. 00:02:43.390 --> 00:02:45.730 k squared minus 2k plus 1. 00:02:45.730 --> 00:02:47.580 Problem number 4. 00:02:47.580 --> 00:02:49.240 I should have drank water before starting this video, 00:02:49.240 --> 00:02:54.732 but I will move forward and complete what I have begun. 00:02:54.732 --> 00:02:59.160 So they draw a graph that looks something like that. 00:02:59.160 --> 00:03:05.370 And then there's a line that goes something like this. 00:03:05.370 --> 00:03:07.450 And then they tell us-- what do they tell us? 00:03:07.450 --> 00:03:11.070 They tell us, well it looks like, it intersects at this 00:03:11.070 --> 00:03:12.270 point, at y equals 1. 00:03:12.270 --> 00:03:14.520 So this is the point, 0,1, right? 00:03:14.520 --> 00:03:16.750 x is 0, so this is the x-axis. 00:03:16.750 --> 00:03:18.590 This is the y-axis. 00:03:18.590 --> 00:03:21.420 The figure above shows the graph of a line of y equals ax 00:03:21.420 --> 00:03:27.040 plus b where a and b are constants. 00:03:27.040 --> 00:03:29.400 Which of the following best represents the graph of the 00:03:29.400 --> 00:03:34.910 line 2a x plus b? 00:03:34.910 --> 00:03:36.500 So let's draw it. 00:03:36.500 --> 00:03:37.580 I'm not even going to look at all the choices. 00:03:37.580 --> 00:03:38.580 What would this look like? 00:03:38.580 --> 00:03:40.420 So it's going to have the same y intercept, right? 00:03:40.420 --> 00:03:42.740 It's going to intersect at the same point. 00:03:42.740 --> 00:03:43.630 That's the y intercept. 00:03:43.630 --> 00:03:45.960 Let me draw in this peach color. 00:03:45.960 --> 00:03:48.530 So it's going to intersect right here. 00:03:48.530 --> 00:03:51.650 And its slope is 2 times the slope of the previous one. 00:03:51.650 --> 00:03:53.860 So it's going to be even a steeper slope. 00:03:53.860 --> 00:03:55.750 So the line's going to look something like this. 00:03:55.750 --> 00:03:57.960 It's going to be twice as steep, so it's going 00:03:57.960 --> 00:03:59.210 to look like this. 00:04:02.090 --> 00:04:02.230 All right? 00:04:02.230 --> 00:04:05.060 And all I just have is that same y intercept, right? 00:04:05.060 --> 00:04:08.500 Because the b is still d, but they've doubled the slope, so 00:04:08.500 --> 00:04:10.070 it's going to go up even faster. 00:04:12.840 --> 00:04:16.130 And so, if you look at all the choices, we can immediately 00:04:16.130 --> 00:04:18.660 rule out all of the choices that don't 00:04:18.660 --> 00:04:19.720 have the same y intercept. 00:04:19.720 --> 00:04:22.110 And the only choices that have the same y intercept are 00:04:22.110 --> 00:04:24.140 choices B and C, right? 00:04:24.140 --> 00:04:26.060 Only those that intersect at 0, 1. 00:04:26.060 --> 00:04:32.100 And then between B and C, B is steeper so B is going to be 00:04:32.100 --> 00:04:33.080 our answer, right? 00:04:33.080 --> 00:04:35.230 C is actually less steep than this green line. 00:04:39.100 --> 00:04:42.040 Choice C looks something like this. 00:04:42.040 --> 00:04:45.080 So its slope is less than A. 00:04:45.080 --> 00:04:47.150 So it's not going to be choice C, so it has to be choice B. 00:04:47.150 --> 00:04:50.370 B goes through the same y intercept and it is steeper 00:04:50.370 --> 00:04:53.050 than our original green line. 00:04:53.050 --> 00:04:56.450 So that's the answer, choice B. 00:04:56.450 --> 00:05:01.350 Next problem, problem 5. 00:05:04.770 --> 00:05:07.460 OK, they drew us a triangle. 00:05:07.460 --> 00:05:09.302 That's one side. 00:05:09.302 --> 00:05:10.552 That's another side. 00:05:12.790 --> 00:05:13.990 They have drawn this-- whoops. 00:05:13.990 --> 00:05:15.930 I can't-- OK, there you go. 00:05:15.930 --> 00:05:18.516 That's not as nice looking as I would have liked, but you 00:05:18.516 --> 00:05:19.090 get the point. 00:05:19.090 --> 00:05:20.040 That's a right triangle. 00:05:20.040 --> 00:05:23.400 They say that this is x. 00:05:23.400 --> 00:05:26.940 And the figure above the perimeter of the triangle is 4 00:05:26.940 --> 00:05:32.400 plus 2 root 2. 00:05:32.400 --> 00:05:34.260 What is the value of x? 00:05:34.260 --> 00:05:36.250 So what is the value of this side? 00:05:36.250 --> 00:05:39.000 Well Pythagorean theorem, the value of that side is the 00:05:39.000 --> 00:05:44.010 square root of x squared plus x squared. 00:05:44.010 --> 00:05:45.600 So that equals, what? 00:05:45.600 --> 00:05:49.450 The square root of 2x squared. 00:05:49.450 --> 00:05:50.760 You can take the x out, right? 00:05:50.760 --> 00:05:52.065 Square root of x squared is just x. 00:05:52.065 --> 00:05:54.120 So that equals x square root of 2. 00:05:54.120 --> 00:05:56.060 And they actually even tell you that on the first page 00:05:56.060 --> 00:05:59.630 of-- if you go back to page 774, they tell you, you have a 00:05:59.630 --> 00:06:01.850 45, 45, 90 triangle, it's s square roots of 2. 00:06:01.850 --> 00:06:04.280 But I like to solve it every time. 00:06:04.280 --> 00:06:05.910 So this is x, this is x, this is x square root of 2. 00:06:05.910 --> 00:06:06.490 So what's the perimeter? 00:06:06.490 --> 00:06:12.330 It's x plus x plus x root 2. 00:06:12.330 --> 00:06:16.080 So it's 2x plus x root 2. 00:06:16.080 --> 00:06:17.650 That's the perimeter, right? 00:06:17.650 --> 00:06:20.910 And they also tell us that, that equals 4 plus 00:06:20.910 --> 00:06:22.830 2 root to 2, right? 00:06:22.830 --> 00:06:24.090 That's for the perimeter. 00:06:24.090 --> 00:06:25.355 So you could just do pattern matching. 00:06:25.355 --> 00:06:28.200 You'd say well, this looks a lot like x is equal to 2. 00:06:28.200 --> 00:06:31.450 And it is, because if x is equal to 2, this becomes a 2, 00:06:31.450 --> 00:06:34.070 and then this becomes a 4, right? 00:06:34.070 --> 00:06:36.560 So that's our answer, x is equal to 2. 00:06:36.560 --> 00:06:37.910 And that is choice A. 00:06:40.840 --> 00:06:42.090 Next problem. 00:06:50.480 --> 00:06:51.490 Let me switch colors. 00:06:51.490 --> 00:06:51.960 OK. 00:06:51.960 --> 00:06:53.640 There's always something for me to draw. 00:06:53.640 --> 00:06:54.760 Well before I draw it, let me read the question. 00:06:54.760 --> 00:06:57.040 The scores on Tuesday's history test for 16 students 00:06:57.040 --> 00:06:58.710 are shown on the table above. 00:06:58.710 --> 00:07:03.180 Sam was the only student absent on Tuesday. 00:07:03.180 --> 00:07:04.640 He'll take the test next week. 00:07:04.640 --> 00:07:08.070 If Sam receives a 95 on the test, what will be the median 00:07:08.070 --> 00:07:10.690 score for the test? 00:07:10.690 --> 00:07:11.570 OK. 00:07:11.570 --> 00:07:13.990 So they give us a score and number of students. 00:07:13.990 --> 00:07:14.850 So I'm actually going to write all this, 00:07:14.850 --> 00:07:19.800 score and then number. 00:07:19.800 --> 00:07:30.000 And so 1 kid got 100; two kids got a 95; 90, four kids got 00:07:30.000 --> 00:07:44.810 it; 85, 1 kid; 80, 3 kids; 75, 2 kids; 70, 2 kids; 65, no 00:07:44.810 --> 00:07:49.250 kids; and then 60, 1 kid. 00:07:49.250 --> 00:07:52.670 And then Sam comes along and gets a 95 on the exam, right? 00:07:52.670 --> 00:07:54.680 So before, there's only 2 kids who got a 95. 00:07:54.680 --> 00:07:56.950 Now Sam comes along and there's going to be 3 kids who 00:07:56.950 --> 00:07:58.640 got a 95 on the exam. 00:07:58.640 --> 00:08:01.250 So to figure out the median, all I do is I list out all the 00:08:01.250 --> 00:08:03.080 numbers, and then I pick the middle number. 00:08:03.080 --> 00:08:04.600 That's all the median is. 00:08:04.600 --> 00:08:08.480 So one kid got 100. 00:08:08.480 --> 00:08:13.380 3 kids got a 95: 95, 95, 95. 00:08:13.380 --> 00:08:19.160 Four kids got a 90: 1, 2, 3, 4. 00:08:19.160 --> 00:08:21.260 One kid got an 85. 00:08:21.260 --> 00:08:25.320 Three kids got an 80: 80, 80, 80. 00:08:25.320 --> 00:08:29.370 Two kids got a 75. 00:08:29.370 --> 00:08:31.860 Two kids got a 70. 00:08:31.860 --> 00:08:33.590 And one kid got a 60. 00:08:33.590 --> 00:08:34.350 So how many does that total? 00:08:34.350 --> 00:08:35.340 I could have just added this. 00:08:35.340 --> 00:08:36.750 That's actually what I should have done. 00:08:36.750 --> 00:08:42.010 But it's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 00:08:42.010 --> 00:08:44.760 13, 14, 15, 16, 17. 00:08:44.760 --> 00:08:45.990 So there's 17 kids. 00:08:45.990 --> 00:08:49.230 So the median is going to be the number that has eight 00:08:49.230 --> 00:08:50.370 above it and eight below it. 00:08:50.370 --> 00:08:56.220 So 1, 2, 3, 4, 5, 6, 7, 8, Bam! 00:08:56.220 --> 00:08:59.840 That's the median, because there's 1, 2, 3, 4, 5, 6-- oh, 00:08:59.840 --> 00:09:00.810 no, no, that's not the median. 00:09:00.810 --> 00:09:03.340 Sorry, sorry, sorry, this is the median: 85. 00:09:03.340 --> 00:09:07.130 Because there's 1, 2, 3, 4, 5, 6, 7, 8 above it. 00:09:07.130 --> 00:09:12.410 And there are 1, 2, 3, 4, 5, 6, 7, 8 below it. 00:09:12.410 --> 00:09:14.240 So the answer is 85. 00:09:14.240 --> 00:09:19.770 Before Sam came in there, then the median actually would be-- 00:09:19.770 --> 00:09:21.650 you'd have one number less here, so you'd 00:09:21.650 --> 00:09:22.290 have to move it up. 00:09:22.290 --> 00:09:24.230 So it'd probably be 90, or it'd be an average. 00:09:24.230 --> 00:09:26.410 But anyway, we know that the median is 85. 00:09:26.410 --> 00:09:28.950 And that is choice C. 00:09:28.950 --> 00:09:30.850 See you in the next video.

SAT Prep: Test 7 Section 2 Part 2

https://www.youtube.com/watch?v=FhUj7sF00VM

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https://www.youtube.com/api/timedtext?v=FhUj7sF00VM&ei=YmeUZbzaMeuip-oPzJ6NmAc&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=6E047533458BD99EE07D79CB61932145A19D592D.C82316D534F667E3CBBC8211AADD8FE1AE94CB7C&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.820 --> 00:00:02.360 We're on problem number 7. 00:00:02.360 --> 00:00:05.100 Ahmed has two containers of-- oh, look at that, they're 00:00:05.100 --> 00:00:07.070 doing ethnic names now. 00:00:07.070 --> 00:00:08.930 For the first time, someone on the SAT sounds like they might 00:00:08.930 --> 00:00:09.730 be related to me. 00:00:09.730 --> 00:00:13.860 Anyway, Ahmed has containers of two different sizes. 00:00:13.860 --> 00:00:17.210 The total capacity of 16 containers of 00:00:17.210 --> 00:00:20.760 one size is x gallons. 00:00:20.760 --> 00:00:24.250 So let's see, so it's container 1 size 00:00:24.250 --> 00:00:25.700 and container 2 size. 00:00:25.700 --> 00:00:31.990 So it's 16 containers of size 1, so 16 times size 1 is equal 00:00:31.990 --> 00:00:33.240 to x gallons. 00:00:37.680 --> 00:00:40.940 And the total capacity of 8 containers on the other side 00:00:40.940 --> 00:00:43.520 is also x gallons. 00:00:43.520 --> 00:00:47.170 So 8 times c2 do is also x gallons. 00:00:50.350 --> 00:00:51.770 And x is greater than 0, of course. 00:00:51.770 --> 00:00:55.480 You can't have negative gallons. 00:00:55.480 --> 00:01:00.030 In terms of x, what is the capacity, in gallons, of each 00:01:00.030 --> 00:01:03.939 of the larger containers? 00:01:03.939 --> 00:01:05.010 Let me see what it is. 00:01:05.010 --> 00:01:07.160 Ahmed has containers of two different sizes. 00:01:07.160 --> 00:01:10.700 The total capacity of 16 containers of 1 size is x 00:01:10.700 --> 00:01:12.240 gallons, right. 00:01:12.240 --> 00:01:16.060 And the total capacity of 8 containers of the other size 00:01:16.060 --> 00:01:17.740 is also x gallons. 00:01:17.740 --> 00:01:19.980 And x is greater than 0. 00:01:19.980 --> 00:01:25.070 In terms of x, what is the capacity, in gallons, of each 00:01:25.070 --> 00:01:26.470 of the larger containers? 00:01:26.470 --> 00:01:26.800 OK. 00:01:26.800 --> 00:01:29.200 So all we have to know is about the larger containers. 00:01:29.200 --> 00:01:32.450 Oh, the trick here is which one is the larger container? 00:01:32.450 --> 00:01:34.630 Right, that is the trick to this question. 00:01:34.630 --> 00:01:39.140 So you really need 8 of the second containers to get x. 00:01:39.140 --> 00:01:42.050 We need 16 of this container to get to x. 00:01:42.050 --> 00:01:45.010 So this is a larger container, right? 00:01:45.010 --> 00:01:47.310 And if we wanted to know the exact size, you divide both 00:01:47.310 --> 00:01:52.190 sides by 8, you get container 2 is equal to x/8 gallons. 00:01:52.190 --> 00:01:54.430 And you could have figured out container 1 is 00:01:54.430 --> 00:01:59.220 equal to x/16 gallons. 00:01:59.220 --> 00:02:01.090 And something divided by 8 is going to be bigger than 00:02:01.090 --> 00:02:01.935 something divided by 16. 00:02:01.935 --> 00:02:02.770 So this is our answer. 00:02:02.770 --> 00:02:05.450 This is the larger container, x divided by 8, 00:02:05.450 --> 00:02:07.180 that's choice D. 00:02:07.180 --> 00:02:08.850 Got to read these problems carefully. 00:02:08.850 --> 00:02:11.260 Next question. 00:02:11.260 --> 00:02:12.510 Whoops. 00:02:15.090 --> 00:02:18.170 Problem 8. 00:02:18.170 --> 00:02:22.300 Rectangle a, b, c, d. 00:02:22.300 --> 00:02:24.160 I'm still getting over the fact that they had the name 00:02:24.160 --> 00:02:25.740 Ahmed in the SAT. 00:02:25.740 --> 00:02:29.660 Anyway, rectangle A, B, C, D. 00:02:29.660 --> 00:02:32.330 One day, they'll have a Salman in there too and I will be 00:02:32.330 --> 00:02:34.910 very proud. 00:02:34.910 --> 00:02:35.830 That's my full name. 00:02:35.830 --> 00:02:37.680 I go by Sal. 00:02:37.680 --> 00:02:39.970 Some people think that I'm not proud of my ethnicity, but I 00:02:39.970 --> 00:02:41.470 explained that's what my mother called me. 00:02:41.470 --> 00:02:43.020 But my full name is Salman. 00:02:43.020 --> 00:02:46.300 It's actually Salman Khan, which is the name of an Indian 00:02:46.300 --> 00:02:49.360 actor that gets more web hits than I do. 00:02:49.360 --> 00:02:50.490 But that's life. 00:02:50.490 --> 00:02:52.380 OK, problem eight. 00:02:52.380 --> 00:02:55.490 Rectangle A, B, C, D, lies in the x, y coordinate plane, so 00:02:55.490 --> 00:02:57.360 the sides are not parallel to the axes. 00:02:57.360 --> 00:03:00.615 Well, let me clear this. 00:03:00.615 --> 00:03:02.450 I have to focus. 00:03:02.450 --> 00:03:03.040 OK. 00:03:03.040 --> 00:03:04.560 It's not parallel to the axes. 00:03:04.560 --> 00:03:07.130 What is the product of the slopes of all four 00:03:07.130 --> 00:03:09.410 sides of A, B, C, D? 00:03:09.410 --> 00:03:10.250 Oh, this is fun. 00:03:10.250 --> 00:03:10.922 OK. 00:03:10.922 --> 00:03:12.040 So let me draw it. 00:03:12.040 --> 00:03:17.300 So it looks like that, like that, like 00:03:17.300 --> 00:03:22.610 that, like that, roughly. 00:03:22.610 --> 00:03:23.970 I didn't draw it perfectly, but these are 00:03:23.970 --> 00:03:25.220 right angles, right? 00:03:27.730 --> 00:03:28.020 OK. 00:03:28.020 --> 00:03:33.860 So let's say the slope of this line, right here, is m, right? 00:03:33.860 --> 00:03:35.910 Well this line's going to be parallel to this line, so this 00:03:35.910 --> 00:03:38.140 is also going to have a slope of m. 00:03:38.140 --> 00:03:40.670 And this is something you should just memorize. 00:03:40.670 --> 00:03:43.980 The slope of a perpendicular line-- so the slope 00:03:43.980 --> 00:03:46.320 perpendicular to this-- is a negative inverse. 00:03:46.320 --> 00:03:49.090 So the inverse of m is 1/m, and it's going to be the 00:03:49.090 --> 00:03:49.630 negative inverse. 00:03:49.630 --> 00:03:53.060 So the slope of this line right here is negative 1/m. 00:03:53.060 --> 00:03:54.020 And that's something that you should just 00:03:54.020 --> 00:03:55.820 memorize, for life. 00:03:55.820 --> 00:03:57.450 And, of course, this one is parallel to this one. 00:03:57.450 --> 00:03:58.735 It's also perpendicular to these two, so 00:03:58.735 --> 00:04:01.220 it's also minus 1/m. 00:04:01.220 --> 00:04:03.130 So they want to know the product of all of these. 00:04:03.130 --> 00:04:12.330 So you get m times minus 1/m times m times minus 1/m. 00:04:12.330 --> 00:04:16.930 m times minus 1/m, this is minus 1 times minus 1, 00:04:16.930 --> 00:04:18.810 which equals 1. 00:04:18.810 --> 00:04:22.430 And that is choice D. 00:04:22.430 --> 00:04:23.780 Next problem. 00:04:23.780 --> 00:04:26.315 This problem is all based on, did you know that the slope of 00:04:26.315 --> 00:04:28.505 a perpendicular line is a negative inverse? 00:04:33.510 --> 00:04:34.250 Invert colors. 00:04:34.250 --> 00:04:35.600 All right. 00:04:35.600 --> 00:04:38.080 Problem number 9. 00:04:38.080 --> 00:04:41.210 An hour long television program includes 20 minutes of 00:04:41.210 --> 00:04:42.390 commercials. 00:04:42.390 --> 00:04:46.230 What fraction of the hour long program was not commercials? 00:04:46.230 --> 00:04:50.110 So if it had 20 minutes of commercials, it must have had 00:04:50.110 --> 00:04:55.700 40 minutes not commercials, right? 00:04:55.700 --> 00:05:01.140 So 40 minutes over an hour is the fraction that's not 00:05:01.140 --> 00:05:01.640 commercials. 00:05:01.640 --> 00:05:04.850 So that's 4/6, which equals 2/3. 00:05:04.850 --> 00:05:05.690 That's the answer. 00:05:05.690 --> 00:05:06.720 That's it. 00:05:06.720 --> 00:05:08.930 Problem 10. 00:05:08.930 --> 00:05:14.740 If the product of 0.3 and a number is equal to 1-- 0.3 and 00:05:14.740 --> 00:05:17.055 a number, the product of 0.3 and a number, I'll say x is 00:05:17.055 --> 00:05:17.555 equal to 1. 00:05:17.555 --> 00:05:19.170 So the product of 0.3 and x is equal to 00:05:19.170 --> 00:05:21.050 1-- what is the number? 00:05:21.050 --> 00:05:22.635 Divide both sides by 0.3. 00:05:22.635 --> 00:05:26.800 x is equal to 1 divided by 0.3. 00:05:26.800 --> 00:05:28.890 And what is that equal to? 00:05:28.890 --> 00:05:30.500 I don't know, can you use a calculator on this? 00:05:33.730 --> 00:05:36.770 You could multiply the top and the bottom by 10. 00:05:36.770 --> 00:05:42.510 That is equal to 10/3, which is equal to 3 and 1/3, which 00:05:42.510 --> 00:05:45.310 is equal to-- well, you could just write 10/3, actually. 00:05:45.310 --> 00:05:46.520 Multiply the top and the bottom of it. 00:05:46.520 --> 00:05:49.560 Well, that's also 3.333 repeating 00:05:49.560 --> 00:05:50.490 over, and over again. 00:05:50.490 --> 00:05:53.470 So you could do it that way too. 00:05:53.470 --> 00:05:56.050 Next problem. 00:05:56.050 --> 00:05:56.930 Did I do that right? 00:05:56.930 --> 00:05:57.670 Is equal to what? 00:05:57.670 --> 00:05:57.980 Right. 00:05:57.980 --> 00:06:00.630 That's right. 00:06:00.630 --> 00:06:06.570 Problem 11. 00:06:06.570 --> 00:06:08.565 Let-- oh, I like when they define these new operations. 00:06:11.120 --> 00:06:18.990 OK, this is a fun one, x y, z can be defined as-- so that 00:06:18.990 --> 00:06:26.150 means x to the y minus z to the y. 00:06:26.150 --> 00:06:29.600 For all positive integers x, y, and z, what is the value of 00:06:29.600 --> 00:06:35.200 triangle 10, 3, 5? 00:06:35.200 --> 00:06:35.540 OK. 00:06:35.540 --> 00:06:36.590 So what's x? 00:06:36.590 --> 00:06:38.400 x is this term. 00:06:38.400 --> 00:06:43.220 So it's 10 to the y-- y is this term, 10 to the third-- 00:06:43.220 --> 00:06:45.580 minus z, z is this term. 00:06:45.580 --> 00:06:47.110 I'm just pattern matching. 00:06:47.110 --> 00:06:48.920 So that z is this. 00:06:48.920 --> 00:06:52.270 And then my z to the y, y is 3. 00:06:52.270 --> 00:06:53.030 That's all it is. 00:06:53.030 --> 00:06:55.110 So it's 10 to the third minus 5 to the third. 00:06:55.110 --> 00:06:58.650 10 to the third is 10 times 10 times 10, which is 1,000, 00:06:58.650 --> 00:07:02.100 minus 5 times 5 times 5, which is 125. 00:07:02.100 --> 00:07:02.890 And so that's what? 00:07:02.890 --> 00:07:04.730 That's 875. 00:07:04.730 --> 00:07:06.680 And we are done. 00:07:06.680 --> 00:07:07.930 Next problem. 00:07:15.890 --> 00:07:17.400 OK, draw a rectangle. 00:07:17.400 --> 00:07:19.040 Let's see, they have a rectangle that looks 00:07:19.040 --> 00:07:19.740 something like that. 00:07:19.740 --> 00:07:27.485 And then there's a line there. 00:07:27.485 --> 00:07:40.940 And these are points P, Q, R, S, T, and U. 00:07:40.940 --> 00:07:46.430 And the figure above P, Q, S, T is a rectangle. 00:07:46.430 --> 00:07:49.200 And U, R, S, T is a square. 00:07:49.200 --> 00:07:50.100 So this is a square. 00:07:50.100 --> 00:07:50.650 Fair enough. 00:07:50.650 --> 00:07:52.150 So that means this side's equal. 00:07:52.150 --> 00:07:54.720 P, U is equal to 5. 00:07:54.720 --> 00:07:56.420 So this is equal to 5. 00:07:56.420 --> 00:07:59.110 And U, T is a positive integer. 00:07:59.110 --> 00:08:01.090 U, T is an integer, So that's interesting. 00:08:01.090 --> 00:08:02.790 We know it's an integer, and we know it can't be negative. 00:08:02.790 --> 00:08:07.800 You can't have a negative side of a square in our universe. 00:08:07.800 --> 00:08:14.420 If the area of P, Q, S, T must be more than 10-- so area is 00:08:14.420 --> 00:08:17.600 greater than 10 and less than 30. 00:08:17.600 --> 00:08:20.610 Area is less than 30, so we can write this way. 00:08:20.610 --> 00:08:30.710 Area less than 30-- what is one possible value of U, T? 00:08:30.710 --> 00:08:34.090 Well let's just say that U, T is x, right? 00:08:34.090 --> 00:08:36.309 And this is a square, so this is x, and this 00:08:36.309 --> 00:08:38.429 is x as well, right? 00:08:38.429 --> 00:08:42.490 And so what is the area of this big thing? 00:08:42.490 --> 00:08:45.130 It's going to be this side which is what? 00:08:45.130 --> 00:08:50.880 5 plus x times this side, which is x. 00:08:50.880 --> 00:08:52.430 So that's going to be the area. 00:08:52.430 --> 00:08:57.050 So it's going to be 5x plus x squared. 00:08:57.050 --> 00:08:58.500 And so that's the area. 00:08:58.500 --> 00:09:03.860 And it has to be between these two spaces, right? 00:09:03.860 --> 00:09:05.090 So we could try out some numbers. 00:09:05.090 --> 00:09:09.130 Let's see, if x is 1 what's the area? 00:09:09.130 --> 00:09:11.240 It's 5 plus 1. 00:09:11.240 --> 00:09:12.650 So then the area is 7. 00:09:12.650 --> 00:09:14.850 No, that's not greater than 10. 00:09:14.850 --> 00:09:17.130 What happens if x is equal to 2? 00:09:17.130 --> 00:09:20.150 Then we have 5 times of x is 2. 00:09:20.150 --> 00:09:23.760 You have 7 times 2, which is 14. 00:09:23.760 --> 00:09:25.690 And that satisfies these conditions. 00:09:25.690 --> 00:09:28.230 14 is greater than 10 and less than 30. 00:09:28.230 --> 00:09:29.190 So that's it. 00:09:29.190 --> 00:09:30.100 You didn't even have to do x. 00:09:30.100 --> 00:09:31.610 You could have just said well, what if x is, you could have 00:09:31.610 --> 00:09:32.950 tried 1, and just tried 2. 00:09:32.950 --> 00:09:36.020 And you could say 5 plus 2 is 7 times 2 is 14. 00:09:36.020 --> 00:09:37.900 That meets my conditions, and I am done. 00:09:37.900 --> 00:09:39.240 That's all you've got to do. 00:09:39.240 --> 00:09:41.150 I'll see you in the next video.

SAT Prep: Test 7 Section 2 Part 3

https://www.youtube.com/watch?v=lb3TmicdsCU

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https://www.youtube.com/api/timedtext?v=lb3TmicdsCU&ei=YmeUZdf6MtW2vdIPppGMwAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=95CA19C3E42310762589962685E7F55E9C537CFF.314D08588C1180AAE99831DBDCEDE0DBB901F8B3&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:01.080 --> 00:00:04.250 We're on problem 13. 00:00:04.250 --> 00:00:07.140 A company sells boxes of balloons in which the balloons 00:00:07.140 --> 00:00:09.260 are red, green, or blue. 00:00:09.260 --> 00:00:12.030 Luann purchased-- that doesn't sound like a 00:00:12.030 --> 00:00:13.020 name related to me. 00:00:13.020 --> 00:00:16.090 Luann purchased a box of balloons in which 1/3 00:00:16.090 --> 00:00:17.340 of them were red. 00:00:20.670 --> 00:00:24.250 If there were 1/2 as many green balloons in the box as 00:00:24.250 --> 00:00:29.190 red ones-- so green is equal to 1/2 red. 00:00:29.190 --> 00:00:30.250 Fair enough. 00:00:30.250 --> 00:00:32.332 Green is equal to 1/2 red. 00:00:32.332 --> 00:00:34.380 There's 1/2 as many green balloons in the box as red 00:00:34.380 --> 00:00:43.350 ones, and 18 balloons were blue, how many balloons were 00:00:43.350 --> 00:00:44.600 in the box? 00:00:47.140 --> 00:00:48.550 OK, so this is fascinating. 00:00:48.550 --> 00:00:52.120 So 1/3 are red. 00:00:52.120 --> 00:00:55.050 And then the green is 1/2 as many as the red, right? 00:00:55.050 --> 00:00:58.370 So if 1/3 of them are red, how may are going to be green? 00:00:58.370 --> 00:01:00.270 Well, 1/2 of the 1/3, right? 00:01:00.270 --> 00:01:07.080 So we know that 1/6 are going to be green. 00:01:07.080 --> 00:01:08.330 How did I get 1/6? 00:01:10.500 --> 00:01:12.860 Let's say there's 10 red, right? 00:01:12.860 --> 00:01:15.950 And 10 is 1/3, then there's going to be-- let's say 00:01:15.950 --> 00:01:17.280 there's 30 balloons. 00:01:17.280 --> 00:01:19.400 1/3 is red, so then it's 10. 00:01:19.400 --> 00:01:22.500 So green's going to be 1/2 of that, so it'd be 5/30 or 1/6. 00:01:22.500 --> 00:01:25.710 So whatever the number is, 1/3 are red, 1/2 of 00:01:25.710 --> 00:01:27.350 that, or 1/6, are green. 00:01:27.350 --> 00:01:29.180 And there are 18 blue. 00:01:29.180 --> 00:01:30.290 So what can we do now? 00:01:30.290 --> 00:01:31.480 Well, let's figure out what fraction 00:01:31.480 --> 00:01:33.020 would have to be blue. 00:01:33.020 --> 00:01:35.630 We know that 1/3 are red 1/6 are green. 00:01:35.630 --> 00:01:36.760 What's left over? 00:01:36.760 --> 00:01:41.110 So what's 1 minus 1/3 minus 1/6? 00:01:41.110 --> 00:01:44.480 This will tell us how many blue, what fraction of the 00:01:44.480 --> 00:01:45.870 balloons have to be blue. 00:01:45.870 --> 00:01:49.860 So that equals-- let's make 6 the common denominator, right? 00:01:49.860 --> 00:01:53.020 1 is equal to 6/6 minus 1/3. 00:01:53.020 --> 00:01:56.180 That's equal to 2/6 minus 1/6. 00:01:56.180 --> 00:02:00.170 So that's 6 minus 2 minus 1. 00:02:00.170 --> 00:02:03.130 So that's 3/6 are blue, right? 00:02:03.130 --> 00:02:05.010 And I just subtracted the fraction that are red and I 00:02:05.010 --> 00:02:08.240 subtracted the fraction that are green from the whole. 00:02:08.240 --> 00:02:13.020 And so I get 3/6, or 1/2, are blue. 00:02:13.020 --> 00:02:15.070 So 1/2 of the balloons in the container, or in 00:02:15.070 --> 00:02:16.430 the box, are blue. 00:02:16.430 --> 00:02:19.300 And they want to know how many balloons are in the box? 00:02:19.300 --> 00:02:21.950 Well, there are 18 blue, and that's 1/2 of all of the 00:02:21.950 --> 00:02:23.270 balloons, right? 00:02:23.270 --> 00:02:28.520 So 18 is equal to 1/2 of all of the balloons. 00:02:28.520 --> 00:02:31.380 Multiply both sides by 2, you get 36 is equal to x. 00:02:31.380 --> 00:02:32.410 And you know that. 00:02:32.410 --> 00:02:34.570 18 is 1/2 of something, then the total number 00:02:34.570 --> 00:02:36.000 of balloons is 36. 00:02:36.000 --> 00:02:36.810 That's our answer. 00:02:36.810 --> 00:02:45.710 Next problem, 14. 00:02:45.710 --> 00:02:51.460 The three distinct points P, Q, and R lie on line L, OK? 00:02:51.460 --> 00:02:54.080 The four distinct points S, T, U, V lie on a different line 00:02:54.080 --> 00:02:55.360 that is parallel to L. 00:02:55.360 --> 00:02:57.080 What is the total number of different lines that can be 00:02:57.080 --> 00:03:03.610 drawn so that each line contains exactly two of the 00:03:03.610 --> 00:03:04.860 seven points. 00:03:07.500 --> 00:03:09.060 OK, I see what they're saying. 00:03:09.060 --> 00:03:11.330 So let's draw the first line, P, Q, and R. 00:03:14.770 --> 00:03:21.560 So let me just draw the two lines first. So the first line 00:03:21.560 --> 00:03:23.150 is line P, Q, and R. 00:03:28.860 --> 00:03:31.735 And then the second line is S, T, U, and V. 00:03:31.735 --> 00:03:38.080 So then we have S, T, U, and V. 00:03:38.080 --> 00:03:40.640 And this line is parallel to this line, right? 00:03:40.640 --> 00:03:42.540 They're parallel, so they're never going to 00:03:42.540 --> 00:03:44.420 intersect each other. 00:03:44.420 --> 00:03:47.170 OK, what is the total number of different lines that can be 00:03:47.170 --> 00:03:50.040 drawn so that each line intersects exactly two of the 00:03:50.040 --> 00:03:50.710 seven points? 00:03:50.710 --> 00:03:54.300 So you can't even intersect three of this. 00:03:54.300 --> 00:03:57.950 Well, you can, only if you go through that line. 00:03:57.950 --> 00:03:59.860 So what is the total number of different lines that can be 00:03:59.860 --> 00:04:06.520 drawn so that each line contains exactly two of the 00:04:06.520 --> 00:04:06.800 seven points? 00:04:06.800 --> 00:04:07.390 Right. 00:04:07.390 --> 00:04:08.020 So that's interesting. 00:04:08.020 --> 00:04:09.670 So you can't count these lines, right? 00:04:09.670 --> 00:04:11.030 Because these lines have three. 00:04:11.030 --> 00:04:13.590 This line has three of the seven points, and this line 00:04:13.590 --> 00:04:15.040 has four of the seven points. 00:04:15.040 --> 00:04:17.160 So those can't be it, because it says exactly two. 00:04:17.160 --> 00:04:19.000 You can't have even three of the points. 00:04:19.000 --> 00:04:19.920 Sp what are they? 00:04:19.920 --> 00:04:26.260 Well, P can go to four points, right? 00:04:26.260 --> 00:04:28.890 I mean, I could just count them out. 00:04:28.890 --> 00:04:33.310 P can go to four points, Q can be connected with four points, 00:04:33.310 --> 00:04:35.360 and R can be connected with four points, so 00:04:35.360 --> 00:04:36.120 it should be 12. 00:04:36.120 --> 00:04:37.750 And if you don't know what I'm saying, let me 00:04:37.750 --> 00:04:39.230 just draw it out. 00:04:39.230 --> 00:04:43.380 P could be this line, one, two. 00:04:43.380 --> 00:04:46.160 Sorry, that one wasn't drawn well. 00:04:46.160 --> 00:04:53.400 One, two, three, four. 00:04:53.400 --> 00:05:03.300 One, two, three, four, and then one, two, three. 00:05:03.300 --> 00:05:05.240 Well, that's kind of a nice-looking shape there. 00:05:05.240 --> 00:05:09.540 So 4 plus 4 plus 4 is 12, so there's 12 possible lines that 00:05:09.540 --> 00:05:12.880 intersect exactly two of these seven points. 00:05:12.880 --> 00:05:19.585 Next problem, problem 15. 00:05:19.585 --> 00:05:21.430 I'm still using the line tool. 00:05:21.430 --> 00:05:23.260 OK, problem 15. 00:05:23.260 --> 00:05:31.150 If 2 to the x plus 2 to the x plus 2 to the x plus-- how 00:05:31.150 --> 00:05:31.990 many of these are there? 00:05:31.990 --> 00:05:35.090 There's four of them-- plus 2 to the x is equal to 2 the 00:05:35.090 --> 00:05:39.400 seventh, what is the value of x? 00:05:39.400 --> 00:05:40.790 So how many of these are there? 00:05:40.790 --> 00:05:43.210 There's one, two, three, four, which I had to figure out 00:05:43.210 --> 00:05:44.720 while I was drawing it. 00:05:44.720 --> 00:05:46.430 So there's four. 00:05:46.430 --> 00:05:48.990 We're essentially adding to the x four times, so this is 00:05:48.990 --> 00:05:53.520 the same thing as saying 4 times 2 to the x, right? 00:05:53.520 --> 00:05:55.540 Because we have 2 to the x four times: 00:05:55.540 --> 00:05:56.720 one, two, three, four. 00:05:56.720 --> 00:05:59.450 So this is the same thing as this: 4 times 2 to the x. 00:05:59.450 --> 00:06:03.490 And that equals 2 to the seventh. 00:06:03.490 --> 00:06:07.460 What's 4 written as base 2? 00:06:07.460 --> 00:06:09.740 That's the same thing as 2 squared, right? 00:06:09.740 --> 00:06:11.063 Whenever you see these problems and you have two 00:06:11.063 --> 00:06:13.310 different bases, try to see if you can convert them all to 00:06:13.310 --> 00:06:13.730 the same base. 00:06:13.730 --> 00:06:17.810 So that's 2 squared plus-- 2 squared times 2 to the x is 00:06:17.810 --> 00:06:19.930 equal to 2 to the seventh. 00:06:19.930 --> 00:06:22.380 2 squared times 2 to the x, that's the same thing as 2 to 00:06:22.380 --> 00:06:24.850 the x plus 2. 00:06:24.850 --> 00:06:27.230 That equals 2 to the seventh. 00:06:27.230 --> 00:06:29.660 So x plus 2 must equal 7. 00:06:29.660 --> 00:06:31.600 x plus 2 is equal to 7. 00:06:31.600 --> 00:06:33.190 x is equal to 5. 00:06:33.190 --> 00:06:34.440 And we are done. 00:06:34.440 --> 00:06:43.030 Next problem, problem 16. 00:06:43.030 --> 00:06:46.560 Each of five people had a blank card on which they wrote 00:06:46.560 --> 00:06:48.280 a positive integer. 00:06:48.280 --> 00:06:52.350 If the average of these integers is 15, what is the 00:06:52.350 --> 00:06:53.990 greatest possible integers that can be 00:06:53.990 --> 00:06:56.260 on one of the cards? 00:06:56.260 --> 00:06:57.720 This is fascinating. 00:06:57.720 --> 00:07:02.930 So essentially, they're saying you have five positive 00:07:02.930 --> 00:07:05.310 integers and their average is 15. 00:07:05.310 --> 00:07:07.450 What is the greatest possible integer that could be among 00:07:07.450 --> 00:07:09.780 these cards? 00:07:09.780 --> 00:07:13.500 So think of it this way: the sum of the five integers is 00:07:13.500 --> 00:07:14.760 going to be what? 00:07:14.760 --> 00:07:19.340 So let's say it's x1 plus x2 plus x3 00:07:19.340 --> 00:07:23.090 plus x4 plus x5, right? 00:07:23.090 --> 00:07:27.470 All of them over 5 is equal to 15, right? 00:07:27.470 --> 00:07:29.060 That's what they told us. 00:07:29.060 --> 00:07:33.940 The average of the t numbers is 15, so the sum x1 plus x2 00:07:33.940 --> 00:07:37.960 plus x3 plus x4 plus x5 is equal to what? 00:07:37.960 --> 00:07:40.940 5 times 15, that's 75. 00:07:40.940 --> 00:07:43.640 5 times 10 is 50 plus-- OK, that's 75. 00:07:43.640 --> 00:07:46.620 So the sum of the integers are going to be 75. 00:07:46.620 --> 00:07:51.510 And so we want to know what the largest one of these, the 00:07:51.510 --> 00:07:52.900 greatest possible integer here. 00:07:52.900 --> 00:07:54.540 So let's just say that this is what we're 00:07:54.540 --> 00:07:55.240 trying to figure out. 00:07:55.240 --> 00:07:58.530 Let's try to maximize this number here, x5. 00:07:58.530 --> 00:08:01.670 If we want this number to be as large as possible, these 00:08:01.670 --> 00:08:05.080 numbers have to be as small as possible, right? 00:08:05.080 --> 00:08:07.430 And you could subtract these numbers from the other side. 00:08:07.430 --> 00:08:14.770 You could say x5 is equal to 75 minus x1 minus x2 minus x3 00:08:14.770 --> 00:08:16.720 minus x4, right? 00:08:16.720 --> 00:08:18.970 And we're going to try to maximize this number. 00:08:18.970 --> 00:08:19.880 And then we're going to say that's going to be the 00:08:19.880 --> 00:08:21.160 largest. 00:08:21.160 --> 00:08:23.280 So if this is the largest, we want to subtract as small a 00:08:23.280 --> 00:08:24.950 number as possible here, here, here. 00:08:24.950 --> 00:08:26.420 And what are the constraints? 00:08:26.420 --> 00:08:29.860 They have to be positive integers. 00:08:29.860 --> 00:08:32.510 So each of these numbers have to be greater than zero and 00:08:32.510 --> 00:08:33.320 have to be integers. 00:08:33.320 --> 00:08:35.430 So we want them to be as small as possible, so 00:08:35.430 --> 00:08:36.710 let's make them 1. 00:08:36.710 --> 00:08:41.780 So let's say it's 75 minus 1 minus 1 minus 1 minus 1. 00:08:41.780 --> 00:08:45.350 That's 75 minus 4, which equals 71. 00:08:45.350 --> 00:08:47.730 So that's the greatest possible value 00:08:47.730 --> 00:08:50.240 of one of the integers. 00:08:50.240 --> 00:08:53.320 And I have a minute left and do problems. No, I'd better do 00:08:53.320 --> 00:08:54.350 it in another video. 00:08:54.350 --> 00:08:56.300 So I'll see you in the next video.

SAT Prep: Test 7 Section 2 Part 4

https://www.youtube.com/watch?v=UDyFevRLKb8

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https://www.youtube.com/api/timedtext?v=UDyFevRLKb8&ei=YmeUZcvwMtf4mLAP0qypsAU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=CE450B802B4F86268DA6BE46E3DF3069EC59B695.7DD2966337169CDB3C02E33865456BEEEE01F40F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.260 --> 00:00:05.820 We are on problem 17. 00:00:05.820 --> 00:00:08.290 Alice and Corinne stand back-to-back. 00:00:08.290 --> 00:00:11.620 They each take 10 steps in opposite directions away from 00:00:11.620 --> 00:00:13.060 each other and stop. 00:00:13.060 --> 00:00:15.740 Alison turns around walks towards Corinne and reaches 00:00:15.740 --> 00:00:17.090 her in 17 steps. 00:00:17.090 --> 00:00:19.720 OK, so this is interesting. 00:00:19.720 --> 00:00:23.300 So they're back-to-back. 00:00:23.300 --> 00:00:24.775 Let's say Alice goes this way. 00:00:24.775 --> 00:00:27.130 And let's say A is for Alice steps. 00:00:27.130 --> 00:00:33.740 So 10 Alice steps, we don't know how long Alice steps are. 00:00:33.740 --> 00:00:38.540 And then Corinne had gone this way and taken 10 Corinne 00:00:38.540 --> 00:00:39.520 steps, right? 00:00:39.520 --> 00:00:41.930 C is, I don't know how long she takes per step, two feet? 00:00:41.930 --> 00:00:42.970 Who knows? 00:00:42.970 --> 00:00:45.850 But C is the distance of Corinne steps in my little 00:00:45.850 --> 00:00:47.410 world right here. 00:00:47.410 --> 00:00:49.700 OK, so they took ten steps in opposite directions from each 00:00:49.700 --> 00:00:50.240 other and stopped. 00:00:50.240 --> 00:00:52.730 Alison turns around and walks towards Corinne and reaches 00:00:52.730 --> 00:00:53.990 her in 17 steps. 00:00:53.990 --> 00:00:56.030 So this is where Corinne is now, right? 00:00:56.030 --> 00:01:07.380 So this whole distance here is 17 Alice steps. 00:01:07.380 --> 00:01:11.010 The length of one of Alice steps is how many times the 00:01:11.010 --> 00:01:12.920 length of one of Corinne's steps? 00:01:12.920 --> 00:01:15.580 And they say, all of Alice's steps are the same length, and 00:01:15.580 --> 00:01:17.990 all of Corinne's steps are the same length. 00:01:17.990 --> 00:01:19.390 Well we know a couple of things. 00:01:19.390 --> 00:01:22.680 We know to get to here would have been 10 00:01:22.680 --> 00:01:23.920 Alice steps, right? 00:01:23.920 --> 00:01:26.790 This would have been 10 Alice steps. 00:01:26.790 --> 00:01:30.370 So essentially, in the same distance that Corinne took 10 00:01:30.370 --> 00:01:32.090 steps, Alice took what? 00:01:32.090 --> 00:01:34.330 She took 7 steps, right? 00:01:34.330 --> 00:01:36.500 She took 10 to get to the middle, and then 7 more to get 00:01:36.500 --> 00:01:37.476 to where Corinne is. 00:01:37.476 --> 00:01:39.870 So this is 7 Alice steps. 00:01:39.870 --> 00:01:44.470 So 7 times Alice step length is equal to 10 times Corinne 00:01:44.470 --> 00:01:45.830 step length, right? 00:01:45.830 --> 00:01:47.610 I'm just saying 7a is equal to 10c. 00:01:47.610 --> 00:01:48.740 And what do we want to know? 00:01:48.740 --> 00:01:52.280 We want to know the length of one of Alice's steps is how 00:01:52.280 --> 00:01:53.840 many times the length of one of Corinne's steps? 00:01:53.840 --> 00:01:54.770 So we want to solve for a. 00:01:54.770 --> 00:01:57.786 So Alice's step is equal to-- divide both sides by 7-- is 00:01:57.786 --> 00:02:01.520 equal to 10/7 times a Corinne step. 00:02:01.520 --> 00:02:04.460 And that's our answer, 10/7. 00:02:04.460 --> 00:02:13.230 Next problem, problem 18. 00:02:13.230 --> 00:02:20.390 Let the function f be defined by f of x is equal to x 00:02:20.390 --> 00:02:24.950 squared plus 18. 00:02:24.950 --> 00:02:33.710 If m is a positive number such that f of 2m is equal to 2 f 00:02:33.710 --> 00:02:37.960 of m, what is the value of m? 00:02:37.960 --> 00:02:39.360 A lot of people get intimidated by these function 00:02:39.360 --> 00:02:40.780 problems, but in some ways they're kind of the most 00:02:40.780 --> 00:02:42.490 straightforward problems. You just have to sit 00:02:42.490 --> 00:02:43.670 and evaluate them. 00:02:43.670 --> 00:02:45.250 So what's f of 2m? 00:02:45.250 --> 00:02:47.670 So everywhere we see an x, we put a 2m. 00:02:47.670 --> 00:02:53.730 So it's 2m squared plus 18, right? 00:02:53.730 --> 00:02:55.115 Wherever I saw an x, I put a 2m. 00:02:55.115 --> 00:02:58.475 And that equals 2 times f of m. 00:02:58.475 --> 00:03:00.200 So wherever I see an x, I put an m. 00:03:00.200 --> 00:03:05.360 So that's m squared plus 18. 00:03:05.360 --> 00:03:06.870 Now I just simplify. 00:03:06.870 --> 00:03:14.510 2m squared, that's equal to 4m squared plus 18 is equal to 2m 00:03:14.510 --> 00:03:16.680 squared plus 36. 00:03:16.680 --> 00:03:18.760 I just distributed the 2. 00:03:18.760 --> 00:03:22.100 So you subtract 2m squared from both sides, you get 2m 00:03:22.100 --> 00:03:24.870 squared plus 18 is equal to 36. 00:03:24.870 --> 00:03:27.060 I subtracted this from both sides. 00:03:27.060 --> 00:03:29.590 Subtract 18 from both sides, you get 2m 00:03:29.590 --> 00:03:32.090 squared is equal to 18. 00:03:32.090 --> 00:03:38.110 Divide both sides by 2, you get m squared is equal to 9. 00:03:38.110 --> 00:03:43.180 And then you get m is equal to-- if you just did this 00:03:43.180 --> 00:03:44.850 straight off, you'd get plus or minus 3. 00:03:44.850 --> 00:03:46.920 But they tell us that m is a positive number, so we know 00:03:46.920 --> 00:03:48.780 that m is equal to 3. 00:03:48.780 --> 00:03:50.162 And we are done. 00:03:50.162 --> 00:03:51.280 And that's it. 00:03:51.280 --> 00:03:54.440 I'll see you in the next section.

SAT Prep: Test 6 Section 9 Part 1

https://www.youtube.com/watch?v=pplnmRsNDuY

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https://www.youtube.com/api/timedtext?v=pplnmRsNDuY&ei=YmeUZab9Na25mLAPrYiLiAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5622FBE7E9B7007D0048D350FE1B8EF5C6354673.7AD26CDF6239BE0F3F9FCEA1617623130BF76C56&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.660 --> 00:00:03.490 We're in the last section of test 6. 00:00:03.490 --> 00:00:06.170 Section 9, page 743. 00:00:06.170 --> 00:00:09.660 Problem number 1. 00:00:09.660 --> 00:00:12.900 For which of the following values of m will the value of 00:00:12.900 --> 00:00:20.130 3m minus 1 be greater than 10? 00:00:20.130 --> 00:00:22.570 So we add 1 to both sides. 00:00:22.570 --> 00:00:26.960 3m is going to be greater than 11. 00:00:26.960 --> 00:00:29.050 Divide both sides by 3, and you don't have to switch the 00:00:29.050 --> 00:00:30.530 inequality because 3 is positive. 00:00:30.530 --> 00:00:34.650 So you get m is going to be greater than 11/3. 00:00:34.650 --> 00:00:36.580 And 11/3, this is what? 00:00:36.580 --> 00:00:39.480 This is equal to 3 and 2/3. 00:00:39.480 --> 00:00:41.880 So anything greater than 3, really, any integer 00:00:41.880 --> 00:00:42.790 greater than 3. 00:00:42.790 --> 00:00:44.310 If you look at all the choices, the only one that's 00:00:44.310 --> 00:00:50.040 greater than 3 and 2/3 is choice A, 4. 00:00:50.040 --> 00:00:52.620 Next problem. 00:00:52.620 --> 00:00:58.650 If a times k is equal to k-- sorry, if a times k is equal 00:00:58.650 --> 00:01:05.050 to a for all values of a, what is the value of k? 00:01:05.050 --> 00:01:06.590 And the reason why they have to say all values of a is 00:01:06.590 --> 00:01:09.890 because if a was 0, then that would work for any k. 00:01:09.890 --> 00:01:12.230 But if it's for all values of k, you divide both sides of 00:01:12.230 --> 00:01:17.110 this equation by a, you get k is equal to a/a 00:01:17.110 --> 00:01:18.840 which is equal to 1. 00:01:18.840 --> 00:01:21.790 So k would have be equal to 1, which is D. 00:01:21.790 --> 00:01:22.430 And you knew that. 00:01:22.430 --> 00:01:23.790 You didn't have to do any algebra. 00:01:23.790 --> 00:01:26.980 If I multiply something times some number and I get the 00:01:26.980 --> 00:01:30.470 original thing, that number's going to be 1. 00:01:30.470 --> 00:01:31.720 Problem 3. 00:01:34.814 --> 00:01:36.064 Image. 00:01:39.980 --> 00:01:42.370 So let me draw what they've drawn. 00:01:42.370 --> 00:01:45.400 So I have one parallel line like that. 00:01:45.400 --> 00:01:47.310 I'm assuming they're parallel, because they look parallel in 00:01:47.310 --> 00:01:50.060 the picture. 00:01:50.060 --> 00:01:54.510 And then there's a line that comes down like this. 00:01:54.510 --> 00:01:57.266 There's another line that goes up like that. 00:01:57.266 --> 00:01:59.580 And what do they tell us about it? 00:01:59.580 --> 00:02:03.280 They tell us that this is x degrees right here. 00:02:03.280 --> 00:02:06.870 This is z degrees. 00:02:06.870 --> 00:02:08.330 This is y degrees. 00:02:08.330 --> 00:02:09.389 This is line l. 00:02:09.389 --> 00:02:11.500 This is line m. 00:02:11.500 --> 00:02:13.795 In the figure above, l is parallel to m. 00:02:17.390 --> 00:02:22.655 If x is equal to 80 degrees-- so this is 80 degrees-- and y 00:02:22.655 --> 00:02:29.850 is equal to 70 degrees, what is the value of z? 00:02:29.850 --> 00:02:32.860 So the trick here is just to see the opposite angles. 00:02:32.860 --> 00:02:35.800 In fact, the information that they're parallel lines is 00:02:35.800 --> 00:02:37.970 actually just extra information to start making 00:02:37.970 --> 00:02:39.870 you do corresponding angles and all that 00:02:39.870 --> 00:02:40.850 kind of fancy stuff. 00:02:40.850 --> 00:02:42.380 Which you could do. 00:02:42.380 --> 00:02:44.950 But the really easy thing is to realize that x, this angle, 00:02:44.950 --> 00:02:47.020 is the same thing as this angle, which is 00:02:47.020 --> 00:02:49.130 going to be 80 degrees. 00:02:49.130 --> 00:02:49.400 Right? 00:02:49.400 --> 00:02:52.090 Because they're opposite angles of intersecting lines. 00:02:52.090 --> 00:02:54.170 And this-- if that's 70, then this angle is 00:02:54.170 --> 00:02:56.260 also going to be 70. 00:02:56.260 --> 00:02:59.840 And now 80 plus 70 plus z has to equal 180, because they're 00:02:59.840 --> 00:03:02.095 all in the same triangle together. 00:03:02.095 --> 00:03:07.920 So 80 plus 70 plus z is equal to 180. 00:03:07.920 --> 00:03:11.660 150 plus z is equal to 180. 00:03:11.660 --> 00:03:14.750 Subtract 150 from both sides, you get z 00:03:14.750 --> 00:03:17.260 is equal to 30 degrees. 00:03:17.260 --> 00:03:20.080 The other way you could have done it is you could have used 00:03:20.080 --> 00:03:21.990 the information about the parallel lines. 00:03:21.990 --> 00:03:24.350 And you could have said, well, the corresponding angle here 00:03:24.350 --> 00:03:26.360 is here, so that this is also 80 degrees. 00:03:26.360 --> 00:03:27.920 And you could have said the corresponding angle here is 00:03:27.920 --> 00:03:30.080 also here, so this is 70 degrees. 00:03:30.080 --> 00:03:31.760 And then you could have said all three of these are 00:03:31.760 --> 00:03:34.970 supplementary, and that they would have to add up to 180. 00:03:34.970 --> 00:03:37.290 And you'd get z equals 30 either way. 00:03:37.290 --> 00:03:38.540 And that is choice A. 00:03:41.820 --> 00:03:43.070 Next problem. 00:03:47.570 --> 00:03:50.690 Problem 4. 00:03:50.690 --> 00:03:53.890 The scenic route from Mia's home to her office is 5 00:03:53.890 --> 00:03:56.000 kilometers longer than the direct route. 00:03:56.000 --> 00:03:57.600 So I'll call S for scenic. 00:03:57.600 --> 00:04:02.650 S is equal to the direct route plus 4. 00:04:02.650 --> 00:04:05.375 The scenic route is 4 miles longer than the direct route, 00:04:05.375 --> 00:04:07.030 or 4 kilometers. 00:04:07.030 --> 00:04:09.460 When she goes by the scenic route and returns by the 00:04:09.460 --> 00:04:13.360 direct route, the round trip is 35 kilometers. 00:04:13.360 --> 00:04:14.150 So that means what? 00:04:14.150 --> 00:04:18.010 The distance of the scenic route plus the distance of the 00:04:18.010 --> 00:04:26.240 direct route is equal to 35 kilometers. 00:04:26.240 --> 00:04:28.550 How many kilometers is the direct route? 00:04:28.550 --> 00:04:31.010 So we want to solve for this. 00:04:31.010 --> 00:04:40.600 So this top equation, we can rewrite this as S minus D is 00:04:40.600 --> 00:04:43.420 equal to 4. 00:04:43.420 --> 00:04:45.580 And actually just to make things-- well, let me write 00:04:45.580 --> 00:04:46.270 that down right here. 00:04:46.270 --> 00:04:51.580 So you get S minus D is equal to 4. 00:04:51.580 --> 00:04:54.240 And we can multiply this equation by negative 1, so you 00:04:54.240 --> 00:04:56.840 cancel out the S's instead of the D's, but since it's so 00:04:56.840 --> 00:04:58.560 convenient already, let's just cancel out the D's and solve 00:04:58.560 --> 00:04:59.410 for the scenic route. 00:04:59.410 --> 00:05:00.610 The S's. 00:05:00.610 --> 00:05:03.080 So this is just a system with two 00:05:03.080 --> 00:05:04.110 equations and two unknowns. 00:05:04.110 --> 00:05:06.180 So let's add these two equations. 00:05:06.180 --> 00:05:11.740 So you get 2S-- D plus D is 0-- is equal to 00:05:11.740 --> 00:05:18.430 35 plus 4 is 39. 00:05:18.430 --> 00:05:19.450 The numbers look strange. 00:05:19.450 --> 00:05:25.770 The scenic distance is 39/2. 00:05:25.770 --> 00:05:26.360 Right? 00:05:26.360 --> 00:05:28.390 That's 39/2 kilometers. 00:05:28.390 --> 00:05:29.640 And what is that equal to? 00:05:29.640 --> 00:05:31.190 If I were to write that as a mixed number. 00:05:31.190 --> 00:05:31.840 Let's see. 00:05:31.840 --> 00:05:35.625 38 is-- that's 19 and 1/2 kilometers. 00:05:39.000 --> 00:05:40.620 A little less than 20 kilometers. 00:05:40.620 --> 00:05:42.060 That's the scenic route. 00:05:42.060 --> 00:05:44.950 The direct route is 4 less than that, right? 00:05:44.950 --> 00:05:47.380 We said the scenic route is 4 more than the direct route. 00:05:47.380 --> 00:05:49.890 So the direct route is going to be that minus 4. 00:05:49.890 --> 00:05:51.200 So what's 19 minus 4? 00:05:51.200 --> 00:05:53.760 It's 15, and then you still have that half there. 00:05:53.760 --> 00:05:55.990 15 and 1/2 kilometers. 00:05:55.990 --> 00:05:58.370 And I don't see that choice there, so I 00:05:58.370 --> 00:05:59.890 must have made a mistake. 00:05:59.890 --> 00:06:05.200 Let me do it here, just to see where-- I 00:06:05.200 --> 00:06:06.270 don't see that choice. 00:06:06.270 --> 00:06:09.240 The scenic route from Mia's home to her office is 5 00:06:09.240 --> 00:06:11.450 kilometers longer than the direct route. 00:06:11.450 --> 00:06:12.350 Right. 00:06:12.350 --> 00:06:16.170 The scenic route-- we could even say scenic route minus-- 00:06:16.170 --> 00:06:18.100 oh, sorry, it's 5 kilometers longer. 00:06:18.100 --> 00:06:19.020 Sorry. 00:06:19.020 --> 00:06:20.180 This is 5. 00:06:20.180 --> 00:06:21.730 That was my mistake. 00:06:21.730 --> 00:06:23.020 5 kilometers longer. 00:06:23.020 --> 00:06:25.470 Scenic minus direct is 5. 00:06:25.470 --> 00:06:27.120 They normally don't give weird numbers like that. 00:06:27.120 --> 00:06:29.350 So this is 5. 00:06:29.350 --> 00:06:31.150 This is 40. 00:06:31.150 --> 00:06:33.360 This is 40/2. 00:06:33.360 --> 00:06:34.540 This is 20. 00:06:34.540 --> 00:06:35.940 Hope I'm not confusing you. 00:06:35.940 --> 00:06:38.280 And then if you're going to be 5 less than 20, it's going to 00:06:38.280 --> 00:06:40.220 be 15 kilometers, right? 00:06:40.220 --> 00:06:43.030 I hope you see what I just did. 00:06:43.030 --> 00:06:45.140 I wrote 4 kilometers longer instead of 5. 00:06:45.140 --> 00:06:48.250 That was my mistake, which you should avoid. 00:06:48.250 --> 00:06:49.500 Next problem. 00:06:53.350 --> 00:06:54.120 Let me switch colors. 00:06:54.120 --> 00:06:58.880 I think that yellow induces careless mistakes. 00:06:58.880 --> 00:06:59.950 5. 00:06:59.950 --> 00:07:03.650 A complete cycle of a traffic light takes 80 seconds. 00:07:03.650 --> 00:07:10.030 During each cycle, the light is green for 40 seconds. 00:07:10.030 --> 00:07:17.260 Amber-- I guess that's red-- amber for 10 seconds. 00:07:17.260 --> 00:07:18.380 Oh, I guess amber is yellow. 00:07:18.380 --> 00:07:19.400 I don't know my colors. 00:07:19.400 --> 00:07:22.690 And red for 30 seconds. 00:07:22.690 --> 00:07:25.020 And the whole cycle takes 80 seconds. 00:07:25.020 --> 00:07:27.500 If you add those up. 00:07:27.500 --> 00:07:30.170 At a randomly chosen time, what is the probability that 00:07:30.170 --> 00:07:32.630 the light will not be red? 00:07:32.630 --> 00:07:34.470 So the probability that the light will not be red is the 00:07:34.470 --> 00:07:37.980 probability that it's going to be green or amber. 00:07:37.980 --> 00:07:43.730 And it's green or amber for 40 plus 10, for 50 seconds out of 00:07:43.730 --> 00:07:45.570 a total of 80 seconds, right? 00:07:45.570 --> 00:07:48.290 So the probability that when you randomly walk up to the 00:07:48.290 --> 00:07:51.260 light, that it's going to be green or amber-- or not red-- 00:07:51.260 --> 00:07:53.370 is going to be 50 seconds. 00:07:53.370 --> 00:07:57.180 50/80, which is 5/8. 00:07:57.180 --> 00:08:01.030 And that is choice B. 00:08:01.030 --> 00:08:02.280 Next problem. 00:08:04.520 --> 00:08:06.430 Problem 6. 00:08:06.430 --> 00:08:09.310 For a certain hot water heater, the increase in 00:08:09.310 --> 00:08:12.640 heating expenses is directly proportional to the increase 00:08:12.640 --> 00:08:14.680 in the water temperature setting. 00:08:14.680 --> 00:08:19.960 If heating expenses increase by $24 when the water 00:08:19.960 --> 00:08:23.460 temperature is increased by 20 degrees Fahrenheit-- so 00:08:23.460 --> 00:08:33.780 expense goes up by 24 when temperature is increased by 20 00:08:33.780 --> 00:08:37.630 degrees Fahrenheit. 00:08:37.630 --> 00:08:40.159 By how much will heating expenses increase when the 00:08:40.159 --> 00:08:43.030 water temperature setting is increased by 15 degrees 00:08:43.030 --> 00:08:43.900 Fahrenheit? 00:08:43.900 --> 00:08:46.210 So this is just a ratio problem. 00:08:46.210 --> 00:08:52.160 So you say, well, the expenses go up $24 when we have a 00:08:52.160 --> 00:08:54.990 20-degree increase in temperature. 00:08:54.990 --> 00:08:58.340 So then how many dollars will it increase when you have a 00:08:58.340 --> 00:09:00.780 15-degree increase? 00:09:00.780 --> 00:09:04.280 The denominator in both cases was the change in temperature. 00:09:04.280 --> 00:09:06.160 And then this is the change in expense. 00:09:06.160 --> 00:09:08.410 The triangle means change, right? 00:09:08.410 --> 00:09:10.920 So you just have to make sure that the ratios are consistent 00:09:10.920 --> 00:09:11.830 and let's just multiply. 00:09:11.830 --> 00:09:15.990 You get 20-- actually, and so we don't have to get big 00:09:15.990 --> 00:09:18.420 numbers, let me reduce this fraction. 00:09:18.420 --> 00:09:19.350 Divide the top and the bottom. 00:09:19.350 --> 00:09:24.500 This is the same thing as 6/5 is equal to x/15. 00:09:24.500 --> 00:09:25.920 This is easier to multiply now. 00:09:25.920 --> 00:09:30.820 You get 5x is equal to 15 times 6. 00:09:30.820 --> 00:09:32.400 Divide both sides by 5. 00:09:32.400 --> 00:09:37.070 You get x is equal to 15 times 6 divided by 5. 00:09:37.070 --> 00:09:40.460 That's 3, that's 1, that's 18. 00:09:40.460 --> 00:09:45.240 So the change in expense will be $18, and that's choice B. 00:09:45.240 --> 00:09:47.060 See you in the next video.

SAT Prep: Test 6 Section 9 Part 2

https://www.youtube.com/watch?v=ewzoWPBLG3g

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https://www.youtube.com/api/timedtext?v=ewzoWPBLG3g&ei=ZWeUZb2AJtOtp-oP2PeUoAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249813&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=AF2965274308BA07A0A403C015B630EDB5F8D6F2.AB82871A799DCAB06F73677A6ECB8593C7082D5F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.660 --> 00:00:04.440 We are on problem 7. 00:00:04.440 --> 00:00:05.705 Let me draw what they have drawn. 00:00:08.670 --> 00:00:11.810 They drew a right triangle. 00:00:11.810 --> 00:00:14.380 The right triangle looks something like this. 00:00:17.840 --> 00:00:18.990 And then they drew another triangle. 00:00:18.990 --> 00:00:21.150 And it looks like an isosceles triangle, although we can't 00:00:21.150 --> 00:00:25.600 assume anything in this life. 00:00:28.290 --> 00:00:31.795 Nothing comes for free. 00:00:31.795 --> 00:00:34.950 So this is x. 00:00:34.950 --> 00:00:37.410 This is y. 00:00:37.410 --> 00:00:40.560 u, v, w. 00:00:40.560 --> 00:00:41.590 Those are the angles. 00:00:41.590 --> 00:00:44.790 In the triangles above, what is the average-- arithmetic 00:00:44.790 --> 00:00:51.860 mean-- of u, v, w, x, and y? 00:00:51.860 --> 00:00:55.310 So you have to just add up the sum of the angles and then 00:00:55.310 --> 00:00:56.420 divide by the number of angles. 00:00:56.420 --> 00:01:03.020 So we want to take u plus v plus w plus x plus y, and then 00:01:03.020 --> 00:01:04.239 divide by the number of angles. 00:01:04.239 --> 00:01:08.030 So there's one, two, three, four, five angles. 00:01:08.030 --> 00:01:09.630 First question. 00:01:09.630 --> 00:01:12.250 What is u plus v plus w? 00:01:14.750 --> 00:01:17.065 Well, they form the angles in a triangle, so they're going 00:01:17.065 --> 00:01:19.740 to add up to 180 degrees. 00:01:19.740 --> 00:01:21.820 And what's x plus y? 00:01:21.820 --> 00:01:24.680 Well, it's a right triangle. 00:01:24.680 --> 00:01:27.875 And all three of these angles have to add up to 180. 00:01:27.875 --> 00:01:33.080 x plus y plus this 90 have to add up to 180. 00:01:33.080 --> 00:01:36.480 So x plus y is going to be equal to 90. 00:01:36.480 --> 00:01:38.580 Just subtract 90 from both sides. 00:01:38.580 --> 00:01:42.200 So x plus y is equal to 90. 00:01:42.200 --> 00:01:46.640 So if we take the sum of the top, 180 plus 90, that's 270. 00:01:46.640 --> 00:01:48.650 Divided by 5. 00:01:48.650 --> 00:01:56.920 5 goes into 270-- 5, 25, 7, 20, 54. 00:01:56.920 --> 00:01:59.036 So the average is 54 degrees. 00:01:59.036 --> 00:02:03.180 That's the arithmetic mean of all of those angles. 00:02:03.180 --> 00:02:04.430 And that is choice E. 00:02:06.980 --> 00:02:08.230 Next problem. 00:02:10.636 --> 00:02:14.050 I just inadvertently switched back to my bad-luck yellow. 00:02:14.050 --> 00:02:14.920 All right. 00:02:14.920 --> 00:02:16.870 Problem 8. 00:02:16.870 --> 00:02:25.510 Looks like they've drawn-- It looks like a ray. 00:02:25.510 --> 00:02:28.642 I'm going to put an arrow on one side. 00:02:28.642 --> 00:02:31.860 Let me switch colors. 00:02:31.860 --> 00:02:38.120 They say that this is x to the third. 00:02:38.120 --> 00:02:42.940 This is x squared. 00:02:42.940 --> 00:02:46.040 This is x. 00:02:46.040 --> 00:02:48.970 If x, x squared, and x to the third lie on a number line in 00:02:48.970 --> 00:02:51.090 the order shown above, which of the following could be the 00:02:51.090 --> 00:02:52.680 value of x? 00:02:52.680 --> 00:02:57.910 So the first thing we know, x has got to be-- well, the 00:02:57.910 --> 00:03:00.240 first thing we know, that x is a positive number. 00:03:00.240 --> 00:03:01.940 How do I know that? 00:03:01.940 --> 00:03:06.330 Because if x was not a positive number-- well, let me 00:03:06.330 --> 00:03:07.690 think about that. 00:03:07.690 --> 00:03:08.590 Right. 00:03:08.590 --> 00:03:10.810 x has to be a positive number. 00:03:10.810 --> 00:03:11.770 Because-- let me put it this way. 00:03:11.770 --> 00:03:18.270 If any negative number-- let's say it's negative 1/2. 00:03:18.270 --> 00:03:25.090 If you square negative 1/2, you get positive 1/4. 00:03:25.090 --> 00:03:29.080 So for any negative number, the square is going to be 00:03:29.080 --> 00:03:31.420 bigger than the number itself. 00:03:31.420 --> 00:03:34.550 So we know that x is greater than 0. 00:03:34.550 --> 00:03:36.210 That's one thing we know. 00:03:36.210 --> 00:03:37.670 What else do we know? 00:03:37.670 --> 00:03:40.410 We also know that x is going to be less than 1. 00:03:40.410 --> 00:03:42.000 How do I know that? 00:03:42.000 --> 00:03:44.640 Because x squared is less than x. 00:03:44.640 --> 00:03:47.680 And that only applies for numbers less than 1. 00:03:47.680 --> 00:03:50.870 Because 1/2 squared is equal to 1/4, and 1/4 00:03:50.870 --> 00:03:52.280 is less than 1/2. 00:03:52.280 --> 00:03:53.990 So we also know that x is less than 1. 00:03:53.990 --> 00:03:56.520 So we just have to find a choice where x is greater than 00:03:56.520 --> 00:03:59.240 0 and less than 1. 00:03:59.240 --> 00:04:03.390 And if we look at the choices, greater than 0 and less than 00:04:03.390 --> 00:04:06.510 1, there's only one choice that that applies to. 00:04:06.510 --> 00:04:08.800 And that's 3/4. 00:04:08.800 --> 00:04:10.730 Choice C. 00:04:10.730 --> 00:04:12.420 You might want to say 3/2, but that's 1 and 1/2. 00:04:12.420 --> 00:04:13.670 That's greater than 1. 00:04:13.670 --> 00:04:14.820 The first two choices are negative. 00:04:14.820 --> 00:04:17.339 The last two choices are greater than 1-- or greater 00:04:17.339 --> 00:04:18.640 than or equal to 1. 00:04:18.640 --> 00:04:20.660 So we know it's choice C. 00:04:20.660 --> 00:04:21.910 Next problem. 00:04:26.900 --> 00:04:29.080 And if you didn't want to do it in an abstract way, and if 00:04:29.080 --> 00:04:30.640 you have a lot of time, if you're really fast on the 00:04:30.640 --> 00:04:31.560 problems you do know how to do, you can 00:04:31.560 --> 00:04:32.460 always try out the numbers. 00:04:32.460 --> 00:04:33.710 But that takes longer. 00:04:36.590 --> 00:04:39.210 They drew a coordinate system right here. 00:04:42.950 --> 00:04:46.150 And this is my y-axis. 00:04:46.150 --> 00:04:48.910 This is the x-axis. 00:04:48.910 --> 00:04:52.100 And then they draw a line that goes through the origin. 00:05:00.530 --> 00:05:06.420 And then they tell us that this point right here is the 00:05:06.420 --> 00:05:08.215 point 1 comma 3. 00:05:08.215 --> 00:05:10.160 So this is 1. 00:05:10.160 --> 00:05:12.120 This is 3. 00:05:12.120 --> 00:05:13.560 And what do they want us to do with this? 00:05:13.560 --> 00:05:16.420 In the figure above, line l passes through the origin. 00:05:16.420 --> 00:05:18.600 This is line l. 00:05:18.600 --> 00:05:20.710 What is the value of k/h? 00:05:20.710 --> 00:05:23.820 Oh, and then they label another point here. 00:05:23.820 --> 00:05:29.200 They say this is h comma k. 00:05:29.200 --> 00:05:33.090 So what is the value of k divided by h? 00:05:33.090 --> 00:05:41.300 So the y value over the x value, right? 00:05:41.300 --> 00:05:43.890 And I'm going to tell you right now, the value of it is 00:05:43.890 --> 00:05:46.680 going to be the slope of this line. 00:05:46.680 --> 00:05:47.930 And how do we know that? 00:05:50.580 --> 00:05:55.980 If you wanted to figure out the slope from the origin to 00:05:55.980 --> 00:05:57.930 this point, what would it be? 00:05:57.930 --> 00:06:00.910 The origin of course is 0, 0. 00:06:00.910 --> 00:06:02.340 So it would be change in y. 00:06:02.340 --> 00:06:04.710 Change in y is 0 minus k. 00:06:04.710 --> 00:06:05.740 You wouldn't have to do this on the SAT. 00:06:05.740 --> 00:06:07.840 I just want to show you that it would be the slope. 00:06:07.840 --> 00:06:14.060 And then change in x would be 0 minus h. 00:06:14.060 --> 00:06:17.060 And that of course is equal to minus k over minus h. 00:06:17.060 --> 00:06:20.360 The negatives cancel out, and that equals k/h. 00:06:20.360 --> 00:06:22.520 So they're essentially just asking us, what is the slope 00:06:22.520 --> 00:06:23.630 of this line? 00:06:23.630 --> 00:06:24.770 Well, that's easy enough, because it 00:06:24.770 --> 00:06:26.120 gives us another point. 00:06:26.120 --> 00:06:29.530 And so we can say, well, we could do this rise over run. 00:06:29.530 --> 00:06:31.350 So here, what is the change in y? 00:06:31.350 --> 00:06:34.410 Change in y is equal to 3 minus 0. 00:06:34.410 --> 00:06:36.180 I'm taking this as kind of the initial point. 00:06:36.180 --> 00:06:37.660 3 minus 0. 00:06:37.660 --> 00:06:40.560 Change in x is 1 minus 0. 00:06:40.560 --> 00:06:43.900 So change in y over change in x is equal to 3. 00:06:43.900 --> 00:06:45.460 And that's our slope, and that's our answer. 00:06:45.460 --> 00:06:48.760 So the answer is A. 00:06:48.760 --> 00:06:52.290 Next problem. 00:06:52.290 --> 00:06:57.820 Problem 10. 00:06:57.820 --> 00:07:00.410 They're saying the absolute value of m minus 00:07:00.410 --> 00:07:03.130 3 is equal to 5. 00:07:03.130 --> 00:07:08.850 They're also telling us that the absolute value of k plus 7 00:07:08.850 --> 00:07:10.655 is equal to 15. 00:07:10.655 --> 00:07:14.640 In the equations above, m is less than 0 and k 00:07:14.640 --> 00:07:15.890 is less than 0. 00:07:20.190 --> 00:07:24.410 What is the value of m minus k? 00:07:24.410 --> 00:07:27.230 So m is less than 0, right? 00:07:27.230 --> 00:07:30.450 So there's two possibilities here. 00:07:30.450 --> 00:07:32.660 This first equation, if we didn't see that, it could mean 00:07:32.660 --> 00:07:33.390 two different things. 00:07:33.390 --> 00:07:38.790 You could say m minus 3 is equal to 5 or m minus 3 is 00:07:38.790 --> 00:07:41.050 equal to negative 5. 00:07:41.050 --> 00:07:44.050 This one would solve to m is equal to 8, and this one would 00:07:44.050 --> 00:07:47.840 solve to m is equal to minus 2. 00:07:47.840 --> 00:07:52.390 Well, we know m is not 8, because m is less than 0. 00:07:52.390 --> 00:07:54.510 And to solve this, I just added 3 to both sides. 00:07:54.510 --> 00:07:56.690 So we know that m is minus 2. 00:07:56.690 --> 00:07:59.610 Similarly, let's take that second equation. 00:07:59.610 --> 00:08:07.220 That tells us that k plus 7 is equal to 15, or k plus 7 is 00:08:07.220 --> 00:08:09.210 equal to minus 15. 00:08:09.210 --> 00:08:12.010 Subtract 7 from both sides of this, you get k equals 8. 00:08:12.010 --> 00:08:14.270 Subtract 7 from both sides of this, you get k is 00:08:14.270 --> 00:08:16.810 equal to minus 22. 00:08:16.810 --> 00:08:20.530 And they tell us that k is also less than 0. 00:08:20.530 --> 00:08:22.980 So we know that this is k. 00:08:22.980 --> 00:08:27.980 And they want us to know what m minus k is. 00:08:27.980 --> 00:08:32.390 Well, m is minus 2, and k is minus 22. 00:08:32.390 --> 00:08:36.422 So minus 2 minus minus 22. 00:08:36.422 --> 00:08:40.090 So that's the same thing as minus 2 plus 22. 00:08:40.090 --> 00:08:43.030 And that is of course 20. 00:08:43.030 --> 00:08:44.920 And that is choice E. 00:08:48.030 --> 00:08:50.620 I will see you in the next video.

SAT Prep: Test 6 Section 9 Part 3

https://www.youtube.com/watch?v=PMFJCEOWVmc

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en

WEBVTT Kind: captions Language: en 00:00:00.900 --> 00:00:02.240 We are on problem number 11. 00:00:02.240 --> 00:00:04.040 I wrote out that chart, although you 00:00:04.040 --> 00:00:05.145 probably can't read it. 00:00:05.145 --> 00:00:08.460 So they say, according to the table above, car engine oil 00:00:08.460 --> 00:00:14.800 with a rating of 5w flows how many times as fast as a car 00:00:14.800 --> 00:00:18.100 engine oil with a rating of 20w? 00:00:18.100 --> 00:00:21.970 So how much faster is 5w versus 20w? 00:00:21.970 --> 00:00:24.650 So let's read the chart. 00:00:24.650 --> 00:00:28.430 It says 10w is half as fast as 5w. 00:00:28.430 --> 00:00:31.860 So 5w is twice as fast as 10w. 00:00:31.860 --> 00:00:37.340 So 5w is 2 times 10w. 00:00:37.340 --> 00:00:42.740 Then they tell us that 15w is half as fast as 10w. 00:00:42.740 --> 00:00:46.190 So 10w is double the speed of 15w. 00:00:46.190 --> 00:00:47.040 Actually, let's do it that way. 00:00:47.040 --> 00:00:50.000 10w is double the speed of 15w, right? 00:00:50.000 --> 00:00:53.980 And then 15w is double the speed of 20w. 00:00:53.980 --> 00:00:55.250 So you have two doublings. 00:00:55.250 --> 00:00:58.710 So 10w is going to be 4 times as fast. It's 2 times as fast 00:00:58.710 --> 00:01:01.040 as this, which is 2 times as fast as this. 00:01:01.040 --> 00:01:05.230 So 10w is 4 times faster than 20w. 00:01:05.230 --> 00:01:13.540 So 10w is 4 times 20w. 00:01:13.540 --> 00:01:16.180 And 5w is twice 10w. 00:01:16.180 --> 00:01:20.820 So 5w is going to be 8 times 20w. 00:01:20.820 --> 00:01:23.460 And really, the most confusing thing about this is the fact 00:01:23.460 --> 00:01:27.180 that these labels for these ratings involve numbers that 00:01:27.180 --> 00:01:29.380 aren't proportional to actually how much faster they 00:01:29.380 --> 00:01:30.130 are than each other. 00:01:30.130 --> 00:01:36.230 You could actually change this to x, y, z, and then it might 00:01:36.230 --> 00:01:37.700 be a little bit easier. 00:01:37.700 --> 00:01:39.180 But that's all there is to it. 00:01:39.180 --> 00:01:40.830 This is twice as fast as this, which is 00:01:40.830 --> 00:01:42.340 twice as fast as this. 00:01:42.340 --> 00:01:45.980 And if you were at 5w up here, so 5w is twice as fast as 00:01:45.980 --> 00:01:47.940 this, which is twice as fast as this, which is twice as 00:01:47.940 --> 00:01:48.610 fast as that. 00:01:48.610 --> 00:01:51.000 So you're doubling three times, so it's 8 times as 00:01:51.000 --> 00:01:54.600 fast. And that's choice C. 00:01:54.600 --> 00:01:55.850 Next problem. 00:02:00.747 --> 00:02:02.050 Let me draw that. 00:02:05.487 --> 00:02:07.195 We have a line like that. 00:02:09.775 --> 00:02:11.025 Straight line at the bottom. 00:02:15.085 --> 00:02:17.420 Then we have a line. 00:02:17.420 --> 00:02:18.880 Looks something like that. 00:02:24.020 --> 00:02:34.350 And this is P, A, Q, B, R. 00:02:34.350 --> 00:02:35.860 This is line m. 00:02:35.860 --> 00:02:38.290 This is line l. 00:02:38.290 --> 00:02:39.610 Figure not drawn to the scale. 00:02:39.610 --> 00:02:40.590 Sure. 00:02:40.590 --> 00:02:44.790 In the figure above, points P, A, and B are equally 00:02:44.790 --> 00:02:46.590 spaced on line l. 00:02:46.590 --> 00:02:48.110 Equally spaced. 00:02:48.110 --> 00:02:50.470 So that means that the distance from P to A is equal 00:02:50.470 --> 00:02:51.660 to the distance from A to B. 00:02:51.660 --> 00:02:52.940 They're equally spaced. 00:02:52.940 --> 00:02:55.560 That's good to know. 00:02:55.560 --> 00:02:59.500 And points P, Q, and R are equally spaced on line m. 00:02:59.500 --> 00:03:00.450 OK. 00:03:00.450 --> 00:03:02.030 So P, Q, and R are equally spaced. 00:03:02.030 --> 00:03:03.580 It didn't look like it the way I drew it. 00:03:03.580 --> 00:03:06.310 But so this is equal to this. 00:03:09.430 --> 00:03:17.630 If PB is equal to 4-- so that entire thing is equal to 4. 00:03:17.630 --> 00:03:18.850 So we immediately know that this is 2 00:03:18.850 --> 00:03:20.110 and this is 2, right? 00:03:20.110 --> 00:03:23.110 Because they're equally spaced. 00:03:23.110 --> 00:03:24.940 PR is equal to 6. 00:03:24.940 --> 00:03:29.020 So this entire thing is equal to 6. 00:03:29.020 --> 00:03:33.410 So we know that this is 3 and this is 3. 00:03:33.410 --> 00:03:35.445 And AQ is 4. 00:03:41.760 --> 00:03:44.760 What is the perimeter of quadrilateral QABR? 00:03:47.510 --> 00:03:49.015 Let's see, where is Q? 00:03:49.015 --> 00:03:50.265 QABR. 00:03:52.110 --> 00:03:53.660 So we know this side. 00:03:53.660 --> 00:03:55.700 We know this side. 00:03:55.700 --> 00:03:56.880 We know this side. 00:03:56.880 --> 00:04:00.290 We just have to figure out this side. 00:04:03.615 --> 00:04:06.970 And I'll tell you right now that this is going to be a 00:04:06.970 --> 00:04:09.145 similar triangle problem, I believe. 00:04:11.940 --> 00:04:14.530 Because these triangles are going to be proportional to 00:04:14.530 --> 00:04:15.340 each other. 00:04:15.340 --> 00:04:16.590 How do I know that? 00:04:21.769 --> 00:04:23.260 I'm going to draw two triangles for you. 00:04:23.260 --> 00:04:31.120 You have this big triangle, and then you have 00:04:31.120 --> 00:04:32.370 this smaller triangle. 00:04:37.150 --> 00:04:38.790 And the big triangle is similar 00:04:38.790 --> 00:04:40.030 to the smaller triangle. 00:04:40.030 --> 00:04:40.980 And how do I know that? 00:04:40.980 --> 00:04:42.960 Well, they share an angle here. 00:04:42.960 --> 00:04:48.730 and we know that the bigger triangle, two of its sides are 00:04:48.730 --> 00:04:52.620 just double the sides of the other two angles. 00:04:52.620 --> 00:04:53.600 Or-- I'm sorry. 00:04:53.600 --> 00:04:58.080 We know that, for example, this side on the big triangle 00:04:58.080 --> 00:05:02.610 is double this side on the small triangle, right? 00:05:02.610 --> 00:05:02.950 Similar. 00:05:02.950 --> 00:05:06.725 We know this big side on the big triangle is double this 00:05:06.725 --> 00:05:08.980 side on the small triangle. 00:05:08.980 --> 00:05:11.760 So two sides are proportional to each other. 00:05:11.760 --> 00:05:12.620 We have one common angle. 00:05:12.620 --> 00:05:15.550 We know that this is a similar triangle. 00:05:15.550 --> 00:05:18.840 So we know that this side is going to be double this side. 00:05:18.840 --> 00:05:21.390 So this side is going to be 8. 00:05:21.390 --> 00:05:22.980 And you could have probably guessed that even if you 00:05:22.980 --> 00:05:24.530 didn't want to prove that they're similar triangles. 00:05:24.530 --> 00:05:25.640 So this is 8. 00:05:25.640 --> 00:05:27.840 The perimeter that we had to figure out to begin with-- 00:05:27.840 --> 00:05:31.440 this is the quadrilateral right here. 00:05:31.440 --> 00:05:38.230 It's going to be 8 plus 3 plus 4 plus 2. 00:05:38.230 --> 00:05:39.280 And what is that? 00:05:39.280 --> 00:05:41.140 That's 8 plus-- 17. 00:05:41.140 --> 00:05:41.460 Right? 00:05:41.460 --> 00:05:42.860 6, 9, right. 00:05:42.860 --> 00:05:43.580 17. 00:05:43.580 --> 00:05:45.770 And that is choice E. 00:05:45.770 --> 00:05:47.020 Next problem. 00:05:52.880 --> 00:05:55.330 Problem 13. 00:05:55.330 --> 00:05:55.580 OK. 00:05:55.580 --> 00:05:57.810 13 and 14 refer to these functions. 00:05:57.810 --> 00:06:04.260 So g of n is equal to n squared plus n. 00:06:04.260 --> 00:06:10.060 h of n is equal to n squared minus n. 00:06:10.060 --> 00:06:10.760 So the first problem. 00:06:10.760 --> 00:06:15.920 They want to know what g of 5 minus h of 4 is. 00:06:15.920 --> 00:06:16.910 So g of 5. 00:06:16.910 --> 00:06:19.700 Let's stick in 5 into g. 00:06:19.700 --> 00:06:23.280 So you get 5 squared plus 5. 00:06:23.280 --> 00:06:25.220 I just replace 5 where n is. 00:06:25.220 --> 00:06:27.960 And then we're going to subtract h of n. 00:06:27.960 --> 00:06:30.030 So what's h of n? 00:06:30.030 --> 00:06:31.175 Oh, sorry, h of 4. 00:06:31.175 --> 00:06:32.570 It looked like an n. 00:06:32.570 --> 00:06:34.160 And h of 4. 00:06:34.160 --> 00:06:38.675 So h of 4 is 4 squared minus 4. 00:06:38.675 --> 00:06:39.710 And so let's simplify this. 00:06:39.710 --> 00:06:46.750 That's 25 plus 5 minus 16 plus 4. 00:06:46.750 --> 00:06:47.370 So that's what? 00:06:47.370 --> 00:06:53.620 30 minus 16 plus 4. 00:06:53.620 --> 00:07:00.710 30 minus 16 is 14, plus 4, so it's 18. 00:07:00.710 --> 00:07:03.680 So that is choice D. 00:07:03.680 --> 00:07:04.000 All right. 00:07:04.000 --> 00:07:06.115 Problem 14 also applies to this. 00:07:06.115 --> 00:07:07.470 So I'll do it right here. 00:07:07.470 --> 00:07:11.855 Which of the following is equivalent to h of m plus 1? 00:07:16.210 --> 00:07:18.430 So we essentially just, everywhere we see an n here, 00:07:18.430 --> 00:07:20.170 we put an m plus 1. 00:07:20.170 --> 00:07:23.715 So h of m plus 1 is equal to-- we see an n, so we put an m 00:07:23.715 --> 00:07:24.890 plus 1 there. 00:07:24.890 --> 00:07:31.890 m plus 1 squared minus-- instead of an n-- m plus 1. 00:07:31.890 --> 00:07:32.780 Multiply that out. 00:07:32.780 --> 00:07:41.510 You get m squared plus 2m plus 1, and then minus m minus 1. 00:07:41.510 --> 00:07:42.190 Let's see. 00:07:42.190 --> 00:07:45.870 So this 1 and this 1 cancel out. 00:07:45.870 --> 00:07:47.680 And then 2m minus m. 00:07:47.680 --> 00:07:52.410 So you're left with m squared minus m. 00:07:54.920 --> 00:08:00.730 So h of m plus 1 is equal to m squared minus m. 00:08:00.730 --> 00:08:05.050 But isn't that the same thing as h of m? 00:08:05.050 --> 00:08:07.140 Reminds me of that store. 00:08:07.140 --> 00:08:09.970 That's H&M, not h of m, although that would be a fun 00:08:09.970 --> 00:08:11.530 name for a store as well. 00:08:11.530 --> 00:08:12.600 h of m. 00:08:12.600 --> 00:08:13.380 Well, what's h of m? 00:08:13.380 --> 00:08:14.590 You put an m where you see an n. 00:08:14.590 --> 00:08:17.300 So that also equals m squared minus m. 00:08:17.300 --> 00:08:18.720 So the answer is h of m. 00:08:18.720 --> 00:08:20.970 And let me now look at the choices. 00:08:20.970 --> 00:08:23.060 And that is not there. 00:08:23.060 --> 00:08:24.920 So I have made a mistake. 00:08:27.610 --> 00:08:31.790 m squared plus 1. 00:08:31.790 --> 00:08:34.710 They're saying, which of the following is equal to h of m-- 00:08:34.710 --> 00:08:38.700 let me clear things. 00:08:38.700 --> 00:08:39.760 OK. 00:08:39.760 --> 00:08:42.120 So they say, which of the following is equivalent to h 00:08:42.120 --> 00:08:43.799 of m plus 1? 00:08:47.475 --> 00:08:51.130 And that is equal to h of m, right? 00:08:51.130 --> 00:08:51.440 Right. 00:08:51.440 --> 00:09:02.440 That is equal to m plus 1 squared minus m plus 1. 00:09:02.440 --> 00:09:09.420 That's m squared plus 2m plus 1, minus m, minus 1. 00:09:09.420 --> 00:09:13.520 So that's m squared-- plus m. 00:09:13.520 --> 00:09:14.200 Oh, sorry. 00:09:14.200 --> 00:09:15.250 Plus m. 00:09:15.250 --> 00:09:16.200 m squared plus m. 00:09:16.200 --> 00:09:18.440 And then this cancels out. 00:09:18.440 --> 00:09:24.190 So h of m plus 1 is equal to m squared plus m. 00:09:24.190 --> 00:09:24.570 Right? 00:09:24.570 --> 00:09:26.220 And what's m squared plus m? 00:09:26.220 --> 00:09:29.910 Well, that's the same thing as g of m. 00:09:29.910 --> 00:09:33.880 Because take g of n, or take g of m and that's also m 00:09:33.880 --> 00:09:35.810 squared plus m. 00:09:35.810 --> 00:09:39.370 So that is choice A. 00:09:39.370 --> 00:09:41.940 Before I cleared it, my mistake was when I did 2m 00:09:41.940 --> 00:09:45.040 minus m, I got, for some silly reason, minus m instead of 00:09:45.040 --> 00:09:45.650 positive m. 00:09:45.650 --> 00:09:46.480 Should be a positive m. 00:09:46.480 --> 00:09:47.940 That was my mistake. 00:09:47.940 --> 00:09:49.190 I'll see you in the next video.

SAT Prep: Test 6 Section 9 Part 4

https://www.youtube.com/watch?v=LeNa9s0rdj8

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https://www.youtube.com/api/timedtext?v=LeNa9s0rdj8&ei=Z2eUZYrbHIG7vdIP6P-QsAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249815&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=158FE79E44EBA9869CE61F4920F5D0429FAFF7E0.C3D0EC4EDCF2A9209BED6E9CDBF22B6CC9318479&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.780 --> 00:00:04.890 We are on problem 15. 00:00:04.890 --> 00:00:07.795 A store charges $28 for a type of sweater. 00:00:10.880 --> 00:00:15.800 This price is 40% more than it costs the store to buy one of 00:00:15.800 --> 00:00:16.980 these sweaters. 00:00:16.980 --> 00:00:22.370 So $28 is 40% more than what it costs. 00:00:22.370 --> 00:00:29.750 So that equals the cost plus 40% of the cost. Right? 00:00:29.750 --> 00:00:32.090 0.4 is the same thing as 40%, right? 00:00:32.090 --> 00:00:34.390 It's the cost plus 40% of cost. That's 00:00:34.390 --> 00:00:35.890 what 40% more means. 00:00:35.890 --> 00:00:38.300 Another way, you could kind of immediately say, well, that's 00:00:38.300 --> 00:00:41.840 just the same thing as 1.4 times the cost. And that's 00:00:41.840 --> 00:00:43.560 kind of shorthand. 00:00:43.560 --> 00:00:46.410 You might immediately go to the step, 1.4 times the cost 00:00:46.410 --> 00:00:47.800 OK, let's see what it says. 00:00:47.800 --> 00:00:51.110 At the end-of-season sale, store employees can purchase 00:00:51.110 --> 00:00:56.900 any remaining sweaters at 30% off the store's cost. So the 00:00:56.900 --> 00:01:01.600 sale price-- let me call that s-- is equal to the cost minus 00:01:01.600 --> 00:01:03.910 0.3 times the cost. Right? 00:01:03.910 --> 00:01:07.820 30% off of the cost. This is 30% of the cost. You take that 00:01:07.820 --> 00:01:10.390 off of the original cost. And that is equal to, of course, 00:01:10.390 --> 00:01:13.885 0.7 times the cost. This is the sale price. 00:01:13.885 --> 00:01:18.210 It's 0.7 times the cost. How much would it cost an employee 00:01:18.210 --> 00:01:23.680 to purchase a sweater of this type at this sale? 00:01:23.680 --> 00:01:24.720 So let's see. 00:01:24.720 --> 00:01:26.020 So how much would it cost? 00:01:26.020 --> 00:01:27.490 So we could figure out what the cost 00:01:27.490 --> 00:01:29.070 is from this equation. 00:01:29.070 --> 00:01:35.900 We have 28 is equal to 1.4 times the cost. Cost is equal 00:01:35.900 --> 00:01:38.670 to 28 divided by 1.4. 00:01:38.670 --> 00:01:40.280 I just divide both sides by 1.4. 00:01:40.280 --> 00:01:42.240 I switched them around. 00:01:42.240 --> 00:01:46.190 Let's see, 28 divided by 14 would be 2, right? 00:01:46.190 --> 00:01:50.000 28 divided by 1.4 is 20. 00:01:50.000 --> 00:01:50.900 So that makes sense. 00:01:50.900 --> 00:01:52.170 And you could go the other way. 00:01:52.170 --> 00:01:53.520 40% more than 20. 00:01:53.520 --> 00:01:55.110 40% of 20 is 8. 00:01:55.110 --> 00:01:56.540 You add 8 to 20, you get 28. 00:01:56.540 --> 00:01:57.160 So that's right. 00:01:57.160 --> 00:01:58.930 The cost is $20. 00:01:58.930 --> 00:02:03.000 And now the sale is 0.7 times the cost. So sale is equal to 00:02:03.000 --> 00:02:06.450 0.7 times the cost. And that equals what? 00:02:06.450 --> 00:02:12.060 20 times 0.7 is $14. 00:02:12.060 --> 00:02:13.690 Because 20 times 7 is 140. 00:02:13.690 --> 00:02:15.710 You have one decimal place, $14. 00:02:15.710 --> 00:02:19.430 So that is choice B. 00:02:19.430 --> 00:02:21.220 Next problem. 00:02:21.220 --> 00:02:22.246 Problem 16. 00:02:22.246 --> 00:02:23.780 And then we're done with this section. 00:02:27.310 --> 00:02:30.810 In rectangle ABCD-- let me draw rectangle ABCD, even 00:02:30.810 --> 00:02:33.110 though they haven't. 00:02:33.110 --> 00:02:41.350 OK, rectangle-- whoops-- A, B, C, D. 00:02:41.350 --> 00:02:45.640 In rectangle ABCD, E is the midpoint of BC. 00:02:45.640 --> 00:02:46.960 So let me draw E. 00:02:51.230 --> 00:02:54.930 So this distance is equal to that distance. 00:02:54.930 --> 00:03:04.725 If the area of quadrilateral ABED is 2/3, what is the area 00:03:04.725 --> 00:03:06.060 of the entire rectangle? 00:03:06.060 --> 00:03:06.620 Fascinating. 00:03:06.620 --> 00:03:10.505 So they're making another quadrilateral within this. 00:03:10.505 --> 00:03:12.260 And I'll just draw it. 00:03:12.260 --> 00:03:13.880 Let me do it in another color. 00:03:13.880 --> 00:03:16.520 So they're saying the area of this thing, ABED. 00:03:27.670 --> 00:03:31.470 Area of that thing is 2/3. 00:03:31.470 --> 00:03:32.790 And so they want you to figure out the area 00:03:32.790 --> 00:03:33.540 of the whole thing. 00:03:33.540 --> 00:03:35.750 And so this is just a pure visualization problem. 00:03:35.750 --> 00:03:39.410 Because what I can do is I can break up this rectangle in a 00:03:39.410 --> 00:03:40.710 interesting way. 00:03:40.710 --> 00:03:43.480 Let me do it with this brown color. 00:03:43.480 --> 00:03:50.920 If I wanted to break up this rectangle, I could break it up 00:03:50.920 --> 00:03:52.780 into four equal triangles, right? 00:03:57.100 --> 00:03:58.810 You see here, 1, 2, 3, 4. 00:03:58.810 --> 00:04:00.020 And how do I know they're equal? 00:04:00.020 --> 00:04:01.570 Because E is the midpoint. 00:04:01.570 --> 00:04:05.730 We know that this side is equal to this side, because E 00:04:05.730 --> 00:04:07.210 is the midpoint. 00:04:07.210 --> 00:04:13.680 We know that this is equal to this, which is equal to this. 00:04:13.680 --> 00:04:16.339 So we know that all four of these rectangles are going to 00:04:16.339 --> 00:04:17.450 be the same. 00:04:17.450 --> 00:04:20.120 Now, they tell us that the area of three-- of this 00:04:20.120 --> 00:04:22.060 rectangle, this rectangle, and this rectangle-- right? 00:04:22.060 --> 00:04:23.680 That's ABED. 00:04:23.680 --> 00:04:26.920 They tell us three of the rectangles-- sorry, three of 00:04:26.920 --> 00:04:28.250 the triangles. 00:04:28.250 --> 00:04:29.550 Have I been saying rectangles the whole time? 00:04:29.550 --> 00:04:30.350 These are triangles. 00:04:30.350 --> 00:04:32.550 ABEA, these are triangles. 00:04:32.550 --> 00:04:38.180 So this rectangle is made up of four equal triangles. 00:04:38.180 --> 00:04:43.910 We know that three of the triangles-- this quadrilateral 00:04:43.910 --> 00:04:46.290 here-- three of the triangles, so I'll 00:04:46.290 --> 00:04:47.490 call them three triangles. 00:04:47.490 --> 00:04:51.590 So I'll call it 3 times the area of each 00:04:51.590 --> 00:04:53.402 triangle is equal to what? 00:04:53.402 --> 00:04:56.290 It is equal to 2/3. 00:04:56.290 --> 00:04:58.640 And we know that because they told us that the area of this 00:04:58.640 --> 00:05:00.040 plus this, plus this is 2/3. 00:05:00.040 --> 00:05:02.970 This quadrilateral is 2/3. 00:05:02.970 --> 00:05:04.810 So what's the area of each triangle? 00:05:04.810 --> 00:05:09.160 The area of each triangle is going to be 2/3 divided by 3, 00:05:09.160 --> 00:05:10.840 which is 2/9, right? 00:05:10.840 --> 00:05:12.900 2/3 times 1/3. 00:05:12.900 --> 00:05:15.430 Area of each triangle is 2/9. 00:05:15.430 --> 00:05:20.230 So what's going to be the area of this entire rectangle? 00:05:20.230 --> 00:05:23.080 Well, it's going to be this area-- that we just know is 00:05:23.080 --> 00:05:27.980 2/3-- plus the area of one more triangle right here. 00:05:27.980 --> 00:05:29.240 Plus the area of this triangle. 00:05:29.240 --> 00:05:31.170 And then we get the whole rectangle. 00:05:31.170 --> 00:05:33.230 So plus 2/9. 00:05:33.230 --> 00:05:37.280 And this equals-- let's see, common denominator is 9. 00:05:37.280 --> 00:05:39.290 2/3 becomes 6/9. 00:05:39.290 --> 00:05:44.720 And then you have that plus the 2/9, and this becomes 8/9. 00:05:44.720 --> 00:05:48.290 And that is choice C. 00:05:48.290 --> 00:05:49.100 And we are done. 00:05:49.100 --> 00:05:51.220 So this was a pure visualization problem. 00:05:51.220 --> 00:05:53.930 You just had to see that the area they gave you was three 00:05:53.930 --> 00:05:56.270 equal triangles, and they essentially just want you to 00:05:56.270 --> 00:05:57.750 add the fourth onto it and figure out what 00:05:57.750 --> 00:05:59.160 the total area is. 00:05:59.160 --> 00:06:01.110 I'll see you in the next section.

SAT Prep: Test 6 Section 7 Part 3

https://www.youtube.com/watch?v=wXQrhgpMI0o

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https://www.youtube.com/api/timedtext?v=wXQrhgpMI0o&ei=YmeUZdewNee7p-oPyciU0Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=84FAD2D4C493D0FBDC38F333B6D1973D18B9E2A9.3B6A8561C9096A4EFD28E85E8E835CE78619485F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.930 --> 00:00:01.520 Welcome back. 00:00:01.520 --> 00:00:03.610 I tried to start doing problem number 10 in the last video, 00:00:03.610 --> 00:00:04.800 but I realized I was running out of time, so 00:00:04.800 --> 00:00:06.240 let me start over. 00:00:06.240 --> 00:00:07.520 Problem number 10. 00:00:07.520 --> 00:00:10.180 The Smith Metals Company old machine makes 00:00:10.180 --> 00:00:15.191 300 bolts per hour. 00:00:15.191 --> 00:00:23.410 Its new machine makes 450 bolts per hour. 00:00:23.410 --> 00:00:28.080 If both machines begin running at the same time, how many 00:00:28.080 --> 00:00:31.420 minutes will it take the two machines to make a 00:00:31.420 --> 00:00:33.330 total of 900 bolts? 00:00:33.330 --> 00:00:34.560 So the important thing to realize is 00:00:34.560 --> 00:00:36.630 that they said minutes. 00:00:36.630 --> 00:00:39.580 So we could convert both of these rates to minutes now, or 00:00:39.580 --> 00:00:42.120 we could say how many hours is it going to take, and then 00:00:42.120 --> 00:00:47.180 convert that to minutes after we have our answer. 00:00:47.180 --> 00:00:49.220 Actually, let's do it the second way, let's say how many 00:00:49.220 --> 00:00:51.510 hours and then convert that to minutes. 00:00:51.510 --> 00:00:58.560 So let's say we want to produce 900 bolts. 00:00:58.560 --> 00:01:00.280 And how much are we going to produce in each hour? 00:01:00.280 --> 00:01:02.800 Well, they're both running at the same time, right? 00:01:02.800 --> 00:01:08.810 So in every hour, we're going to produce 300 plus 450 bolts. 00:01:08.810 --> 00:01:18.260 We're going to produce 750 bolts per hour. 00:01:18.260 --> 00:01:21.500 Times, let's say x hours. 00:01:21.500 --> 00:01:24.530 The units might confuse you, so just leave out the units. 00:01:24.530 --> 00:01:28.740 This is how many hours it takes to produce 900 bolts, so 00:01:28.740 --> 00:01:31.460 you divide both sides by 750. 00:01:31.460 --> 00:01:36.620 You get x is equal to 900/750. 00:01:36.620 --> 00:01:38.350 Let's see what I can do here. 00:01:38.350 --> 00:01:44.040 See, if I divide the top and the bottom by 30, the top will 00:01:44.040 --> 00:01:53.390 become 30 over-- and then the bottom, 75 divided by 3 is 00:01:53.390 --> 00:01:57.540 20-- 75 divided by 3 is 25. 00:01:57.540 --> 00:02:00.280 So 30/25. 00:02:00.280 --> 00:02:02.890 Then I could-- let's see, 5 is a common factor. 00:02:02.890 --> 00:02:04.550 I can do it all in one fell swoop. 00:02:04.550 --> 00:02:06.590 So that's 6/5. 00:02:06.590 --> 00:02:10.520 So it's going to take 6/5 hours. 00:02:10.520 --> 00:02:11.680 That's how long it's going to take us. 00:02:11.680 --> 00:02:13.470 How many minutes is that? 00:02:13.470 --> 00:02:15.910 Every hour is 1 minute-- I mean, sorry, 00:02:15.910 --> 00:02:17.420 every hour is 60 minutes. 00:02:17.420 --> 00:02:18.560 It's getting late. 00:02:18.560 --> 00:02:20.700 So 6/5 hours. 00:02:20.700 --> 00:02:23.660 You just have to multiply it by 60 to get how many minutes 00:02:23.660 --> 00:02:29.070 is equal to-- see, you can cancel this 5, make this a 12. 00:02:29.070 --> 00:02:34.670 You get 6 times 12 is 72 minutes. 00:02:34.670 --> 00:02:37.730 And that is choice B. 00:02:37.730 --> 00:02:38.980 Next problem. 00:02:41.920 --> 00:02:45.050 I've been using this yellow a while, let me switch. 00:02:45.050 --> 00:02:47.110 Problem 11. 00:02:47.110 --> 00:02:49.390 The table above gives the values of the linear function 00:02:49.390 --> 00:02:51.620 g for selected values of t. 00:02:51.620 --> 00:02:54.130 Which of the following defines g? 00:02:54.130 --> 00:03:01.570 OK, so they say t and they say g of t. 00:03:01.570 --> 00:03:08.280 They go from negative 1, 0, 1, 2, let's see, it's 00:03:08.280 --> 00:03:16.450 4, 2, 0, minus 2. 00:03:16.450 --> 00:03:19.090 So the one thing I always look at is what g of 0 is because 00:03:19.090 --> 00:03:20.080 that tends to be interesting. 00:03:20.080 --> 00:03:21.700 Especially when I look at all of the choices. 00:03:21.700 --> 00:03:24.270 All of the choices are of this form, they're all of the form 00:03:24.270 --> 00:03:26.220 m times t plus B. 00:03:26.220 --> 00:03:28.290 Where m is the slope-- if you're familiar with linear 00:03:28.290 --> 00:03:30.060 equations, you're familiar with this form. 00:03:30.060 --> 00:03:34.290 And so when t equals 0, g of t tells you what the y-intercept 00:03:34.290 --> 00:03:36.040 is going to be, right? 00:03:36.040 --> 00:03:43.130 So let's see, g of 0 is equal to 2. 00:03:43.130 --> 00:03:46.710 So that tells us that this equation g of t is going to be 00:03:46.710 --> 00:03:51.620 equal to the slope times t plus 2, right? 00:03:51.620 --> 00:03:56.170 Because when t was 0, all we had left with was 2. 00:03:56.170 --> 00:03:59.190 And so immediately, we can cancel out all but the last 00:03:59.190 --> 00:04:00.915 two choices. 00:04:00.915 --> 00:04:05.470 So the last two choices, choice D is g of t is equal to 00:04:05.470 --> 00:04:07.460 minus t plus 2. 00:04:07.460 --> 00:04:10.910 And then the last choice is g of t is equal to 00:04:10.910 --> 00:04:13.650 minus 2t plus 2. 00:04:13.650 --> 00:04:15.050 Let's see which one of these works, we can 00:04:15.050 --> 00:04:16.670 try out some numbers. 00:04:16.670 --> 00:04:19.450 So what happens when t is negative 1? 00:04:19.450 --> 00:04:24.760 When t is negative 1, this expression becomes negative 1 00:04:24.760 --> 00:04:25.950 times negative. 00:04:25.950 --> 00:04:29.550 Negative negative 1 is positive 1, so this becomes 3. 00:04:29.550 --> 00:04:31.080 That's not right. 00:04:31.080 --> 00:04:33.740 This one becomes negative 2 times negative 1 is 00:04:33.740 --> 00:04:36.630 positive 2, plus 2. 00:04:36.630 --> 00:04:39.600 So this becomes 4. 00:04:39.600 --> 00:04:41.830 So we can immediately cancel this one out because it 00:04:41.830 --> 00:04:45.930 didn't-- here, for this g of t, g of negative 1 equaled 3, 00:04:45.930 --> 00:04:47.980 and they tell us right here it's supposed to equal 4. 00:04:47.980 --> 00:04:48.760 This one worked. 00:04:48.760 --> 00:04:50.300 And this is kind of the only one that still works. 00:04:50.300 --> 00:04:52.750 It had a 2 for the y-intercept, and when you 00:04:52.750 --> 00:04:54.660 evaluate it for just even the first point, you 00:04:54.660 --> 00:04:55.650 got the right answer. 00:04:55.650 --> 00:04:57.230 So that's the answer, the answer is E. 00:04:59.760 --> 00:05:01.010 Next problem. 00:05:06.650 --> 00:05:08.540 OK, survey results. 00:05:08.540 --> 00:05:10.150 I guess I should draw this. 00:05:10.150 --> 00:05:14.220 I haven't read the question, but it's probably important. 00:05:14.220 --> 00:05:16.440 Let's see, there's about five squares that way. 00:05:19.630 --> 00:05:33.950 So that means I have to draw four lines, that's 1, 2, 3, 4. 00:05:33.950 --> 00:05:36.720 And then eight lines I have to draw. 00:05:36.720 --> 00:05:39.790 1-- that's always the hardest part, just drawing these 00:05:39.790 --> 00:05:49.320 diagrams-- 2, 3, 4-- and you're learning how to count-- 00:05:49.320 --> 00:05:59.230 5, 6, 7-- almost there-- and 8. 00:05:59.230 --> 00:06:00.500 All righty. 00:06:00.500 --> 00:06:02.560 And then they say, these are the grades-- 00:06:02.560 --> 00:06:04.440 the y-axis is grade. 00:06:04.440 --> 00:06:08.990 Grade 9, 10, 11, 12. 00:06:08.990 --> 00:06:12.060 The x-axis is distance to school in miles. 00:06:12.060 --> 00:06:16.450 1, 2, 3, 4, 5, 6, 7, 8. 00:06:16.450 --> 00:06:17.390 And these are the points. 00:06:17.390 --> 00:06:20.220 1 comma 10 is right here. 00:06:20.220 --> 00:06:23.330 2 comma 9. 00:06:23.330 --> 00:06:25.770 2 comma 11. 00:06:25.770 --> 00:06:29.260 3 comma 10. 00:06:29.260 --> 00:06:32.380 3 comma 12. 00:06:32.380 --> 00:06:36.440 4 comma-- let's see, 4 is at 10 and 11. 00:06:36.440 --> 00:06:40.290 5-- they have one point at 11. 00:06:40.290 --> 00:06:45.640 6 has three points right here, 10, 11, and 12. 00:06:45.640 --> 00:06:47.080 Let's see. 00:06:47.080 --> 00:06:50.770 There's a point here, here, here. 00:06:50.770 --> 00:06:52.470 And then a point here and here. 00:06:52.470 --> 00:06:54.710 Now we can start the problem. 00:06:54.710 --> 00:06:57.490 The results of a survey of 16 students at Thompson High 00:06:57.490 --> 00:06:59.680 School are given in the grid above. 00:06:59.680 --> 00:07:03.300 It shows the distance to the nearest mile that students at 00:07:03.300 --> 00:07:05.180 various grade levels travel to school. 00:07:05.180 --> 00:07:07.510 So this is miles. 00:07:07.510 --> 00:07:08.760 And this is grade. 00:07:10.740 --> 00:07:13.430 According to the grid, which of the following is true? 00:07:13.430 --> 00:07:15.010 So I'll just read them out. 00:07:15.010 --> 00:07:17.180 A, there's only one student who travels 00:07:17.180 --> 00:07:18.605 two miles to school. 00:07:18.605 --> 00:07:20.060 Let's see, two miles. 00:07:20.060 --> 00:07:22.050 False, there's two students. 00:07:22.050 --> 00:07:24.500 There is this guy and this guy. 00:07:24.500 --> 00:07:25.600 So it's not A. 00:07:25.600 --> 00:07:28.540 Choice B, half of the students travel less than 00:07:28.540 --> 00:07:31.500 four miles to school. 00:07:31.500 --> 00:07:34.250 So that's-- less than four miles is everyone to the left 00:07:34.250 --> 00:07:35.450 of this line, right? 00:07:35.450 --> 00:07:38.490 And this is actually 1, 2, 3, 4, 5. 00:07:38.490 --> 00:07:43.800 5 out of 16 is not half, so we know it's not choice B. 00:07:43.800 --> 00:07:47.640 C, more 12th graders than 11th graders travel six miles or 00:07:47.640 --> 00:07:49.890 more to school. 00:07:49.890 --> 00:07:54.560 So they're saying more 12th graders than 11th graders. 00:07:54.560 --> 00:07:57.110 So six miles or more. 00:07:57.110 --> 00:08:00.530 So let's see, six miles or more is anything to the right 00:08:00.530 --> 00:08:01.310 of this line, right? 00:08:01.310 --> 00:08:03.090 That's six miles or more. 00:08:03.090 --> 00:08:05.670 There are three 12th graders. 00:08:05.670 --> 00:08:07.220 And how many 11th graders are there? 00:08:09.850 --> 00:08:13.140 There are two 11th graders. 00:08:13.140 --> 00:08:14.770 I think that is correct. 00:08:14.770 --> 00:08:18.280 More 12th graders than 11th graders travel six or more 00:08:18.280 --> 00:08:19.540 miles to school. 00:08:19.540 --> 00:08:24.430 Six or more miles, three 12th graders, two 11th graders. 00:08:24.430 --> 00:08:28.420 That's our answer, our answer is C. 00:08:28.420 --> 00:08:31.000 Next problem. 00:08:31.000 --> 00:08:34.950 I don't know if I'll have time for this one, I'll try. 00:08:34.950 --> 00:08:37.520 Problem 13. 00:08:37.520 --> 00:08:42.429 How many positive three digit integers have the hundreds 00:08:42.429 --> 00:08:47.960 digit equal to 3 and the units digit is equal to 4. 00:08:47.960 --> 00:08:51.190 So it's going to be like 3 blank 4. 00:08:51.190 --> 00:08:53.410 So how many numbers are here? 00:08:53.410 --> 00:08:57.630 Well, how many digits can we stick in for that? 00:08:57.630 --> 00:09:01.240 Well, we could put a 0, a 1, 2, 3, 4, 5, 00:09:01.240 --> 00:09:03.940 6, 7, 8, or 9 there. 00:09:03.940 --> 00:09:06.240 We could put any of those in that middle spot. 00:09:06.240 --> 00:09:09.500 And there are 10 digits we can put there, so there are 10 00:09:09.500 --> 00:09:10.230 possibilities. 00:09:10.230 --> 00:09:12.640 There are 10 positive three digit integers that have the 00:09:12.640 --> 00:09:16.060 hundreds digit equal to 3 and the units digit equal to 4. 00:09:16.060 --> 00:09:17.900 That's choice A. 00:09:17.900 --> 00:09:20.740 That's one of those problems that you question yourself 00:09:20.740 --> 00:09:23.140 because it seems maybe even too easy. 00:09:23.140 --> 00:09:25.420 I'll see you in the next video.

SAT Prep: Test 6 Section 7 Part 2

https://www.youtube.com/watch?v=j9iKQnnAsgI

vtt

https://www.youtube.com/api/timedtext?v=j9iKQnnAsgI&ei=YmeUZaimNoiOp-oP-aOH0AI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=897700AE73F890956D18989C395826A3894BBAA8.2AAA66262C60C947C8D87D78A2FEAA56521C379F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:03.020 We are on problem number six. 00:00:06.020 --> 00:00:09.760 If there is no waste, how many square yards of carpeting is 00:00:09.760 --> 00:00:12.960 needed to cover a rectangular floor that is 00:00:12.960 --> 00:00:15.460 12 feet by 18 feet? 00:00:15.460 --> 00:00:18.700 So I can just draw it like that. 00:00:18.700 --> 00:00:22.170 And it's 12 feet by 18 feet. 00:00:22.170 --> 00:00:24.660 And the trick here is, they're not asking how many feet, 00:00:24.660 --> 00:00:28.140 they're asking how many square yards of carpeting. 00:00:28.140 --> 00:00:30.510 So the easiest thing to do really at this point, is to 00:00:30.510 --> 00:00:34.640 convert the dimensions of the room to yards. 00:00:34.640 --> 00:00:37.040 And they tell us 3 feet make a yard. 00:00:37.040 --> 00:00:41.800 So 18 feet, that's the same thing as what? 00:00:41.800 --> 00:00:44.720 Divide by 3, and you get, that's 6 yards. 00:00:44.720 --> 00:00:47.020 6 yards is 18 feet. 00:00:47.020 --> 00:00:50.500 And you divide by 3, you get 4 yards. 00:00:50.500 --> 00:00:52.860 I always find the arithmetic the hardest part. 00:00:52.860 --> 00:00:54.530 4 yards is equal to 12 feet, right? 00:00:54.530 --> 00:00:56.500 4 times 3 is equal to 12. 00:00:56.500 --> 00:01:02.090 So now this is a 6 by 4, so 6 yards times 4 yards is equal 00:01:02.090 --> 00:01:05.129 to 24 square yards. 00:01:05.129 --> 00:01:07.740 That's choice C. 00:01:07.740 --> 00:01:09.940 And the mistake, if you got this wrong, you might have 00:01:09.940 --> 00:01:12.610 just-- well one, you might have multiplied 12 times 18 00:01:12.610 --> 00:01:15.520 and gotten a very large number and said, 00:01:15.520 --> 00:01:16.180 oh, that's the answer. 00:01:16.180 --> 00:01:17.820 But then your answer would've been in feet. 00:01:17.820 --> 00:01:19.910 Then the other mistake you might have done is you might 00:01:19.910 --> 00:01:22.550 have multiplied 12 times 18-- you might done this, 12 times 00:01:22.550 --> 00:01:26.270 18 divided by 3. 00:01:26.270 --> 00:01:28.100 Because you said, oh well, 3 feet is equal to a yard. 00:01:28.100 --> 00:01:31.860 And that is a mistake because you're now converting square 00:01:31.860 --> 00:01:34.150 feet to square yards. 00:01:34.150 --> 00:01:38.790 And it's actually-- you should have divided by 9 because 00:01:38.790 --> 00:01:42.410 there's actually 9 square feet per square yard. 00:01:42.410 --> 00:01:43.090 Why is that? 00:01:43.090 --> 00:01:46.560 Because if I had-- let's say this is a 1 00:01:46.560 --> 00:01:48.060 square yard, right? 00:01:48.060 --> 00:01:49.700 That means it's going to be 3 feet on that side, 3 00:01:49.700 --> 00:01:50.490 feet on that side. 00:01:50.490 --> 00:01:53.640 So it's actually 9 square feet. 00:01:53.640 --> 00:01:56.770 So if you figured out the area in feet first, you had to 00:01:56.770 --> 00:01:59.450 divide by 9, not by 3. 00:01:59.450 --> 00:02:02.630 That probably was the most common mistake, assuming that 00:02:02.630 --> 00:02:04.640 you realized that you had to convert units. 00:02:04.640 --> 00:02:05.890 Next problem. 00:02:08.289 --> 00:02:10.990 Problem seven. 00:02:10.990 --> 00:02:13.550 A certain scale only registers weights that are 00:02:13.550 --> 00:02:15.290 greater than 6 pounds. 00:02:15.290 --> 00:02:16.540 OK, fair enough. 00:02:16.540 --> 00:02:18.680 A person who wanted to know the weights of a puppy, a 00:02:18.680 --> 00:02:22.210 kitten and a bunny-- very cute-- weighed them in pairs 00:02:22.210 --> 00:02:24.720 and got the following results. 00:02:24.720 --> 00:02:27.010 Kitten plus bunny weighed 7 pounds. 00:02:31.090 --> 00:02:33.376 Kitten plus puppy is 8. 00:02:37.290 --> 00:02:39.230 And bunny plus puppy is 9. 00:02:43.490 --> 00:02:46.700 This is by far the cutest problem I've ever done. 00:02:46.700 --> 00:02:48.640 What is the weight of the puppy? 00:02:48.640 --> 00:02:53.180 So we want to solve for P. 00:02:53.180 --> 00:02:55.460 Well, if we're going to solve for P, the best thing we could 00:02:55.460 --> 00:02:59.730 do is we want to use these two equations. 00:02:59.730 --> 00:03:02.350 But we have three-- if we just look at these two equations, 00:03:02.350 --> 00:03:04.780 we have three unknowns with two equations. 00:03:04.780 --> 00:03:10.460 So let's use this equation to substitute into this equation. 00:03:10.460 --> 00:03:12.370 So let's say we want to replace this K. 00:03:12.370 --> 00:03:15.970 So we can use this top equation to say that K is 00:03:15.970 --> 00:03:19.240 equal to 7 minus B. 00:03:19.240 --> 00:03:23.530 So if we substitute that into this equation, we'll get-- so 00:03:23.530 --> 00:03:28.720 let me just-- this is 7 minus B, so we'll get 7 minus B-- 00:03:28.720 --> 00:03:33.600 and I'm just rewriting this top equation, that's this-- 00:03:33.600 --> 00:03:36.850 plus P is equal to 8. 00:03:36.850 --> 00:03:42.480 And then this bottom equation still is B plus 00:03:42.480 --> 00:03:44.450 P is equal to 9. 00:03:44.450 --> 00:03:47.410 I could add the 7 or subtract 7 from both sides right now, 00:03:47.410 --> 00:03:49.900 but I'm just going to actually add the equations, because I 00:03:49.900 --> 00:03:51.090 like how these B's are going to cancel out. 00:03:51.090 --> 00:03:53.820 I get excited about canceling variables. 00:03:53.820 --> 00:03:57.000 So-- actually, no let me subtract 7 from both sides 00:03:57.000 --> 00:03:59.900 first. So if we take 7 from here, than this 8 00:03:59.900 --> 00:04:00.860 becomes a 1, right? 00:04:00.860 --> 00:04:02.760 I just subtracted 7 from both sides. 00:04:02.760 --> 00:04:05.740 So negative B plus P is equal to 1. 00:04:05.740 --> 00:04:07.420 Now I just add these two equations. 00:04:07.420 --> 00:04:08.550 And what's minus B plus B? 00:04:08.550 --> 00:04:14.680 It's 0 plus P plus P is 2P is equal to 1 plus 9 is 10. 00:04:14.680 --> 00:04:16.100 2P equals 10. 00:04:16.100 --> 00:04:17.950 P is equal to 5. 00:04:17.950 --> 00:04:18.730 Choice D. 00:04:18.730 --> 00:04:22.530 5 pounds, that's how much the puppy weighs. 00:04:22.530 --> 00:04:25.430 Which shows us the bunny weighs 2 pounds, and the 00:04:25.430 --> 00:04:26.450 kitten weighs 3 pounds. 00:04:26.450 --> 00:04:29.520 Which is-- it looks like they actually thought about how 00:04:29.520 --> 00:04:32.450 much a puppy, a bunny and a kitten would actually weigh, 00:04:32.450 --> 00:04:34.960 which I have to give them credit for. 00:04:34.960 --> 00:04:37.570 Next problem. 00:04:37.570 --> 00:04:40.520 Problem eight. 00:04:40.520 --> 00:04:47.490 On a blueprint, 1/4 inch represents 16 00:04:47.490 --> 00:04:49.730 feet in real life. 00:04:49.730 --> 00:04:53.510 If a driveway is 40 feet long, what is its length in inches 00:04:53.510 --> 00:04:54.890 on the map? 00:04:54.890 --> 00:04:59.850 So a driveway-- that's a driveway, I don't know, that's 00:04:59.850 --> 00:05:01.260 my driveway. 00:05:01.260 --> 00:05:07.890 And it's 40 feet long. 00:05:07.890 --> 00:05:10.460 So how many of these units is it going to be? 00:05:14.720 --> 00:05:18.880 Well, actually-- we could do it a bunch of different ways. 00:05:18.880 --> 00:05:22.530 We could say 1/4 of an inch is equal to 16 feet. 00:05:22.530 --> 00:05:25.500 If we multiply both sides of this relationship by 4, you 00:05:25.500 --> 00:05:31.910 get 1 inch is equal to 64 feet. 00:05:31.910 --> 00:05:34.226 1 inch-- I just multiplied both sides of this 00:05:34.226 --> 00:05:35.930 relationship by 4. 00:05:35.930 --> 00:05:38.930 So if you want to figure out how many inches this would 00:05:38.930 --> 00:05:42.640 represent, you would say, well, x is how many inches it 00:05:42.640 --> 00:05:48.730 would represent over 1 inch is equal to this length, 00:05:48.730 --> 00:05:52.050 40 feet, over 64. 00:05:52.050 --> 00:05:53.790 It's the same fraction. 00:05:53.790 --> 00:05:59.360 And of course, if you divide by 1, this cancels out. 00:05:59.360 --> 00:06:02.250 So you're left with x, which is its representation in 00:06:02.250 --> 00:06:05.950 inches, is equal to 40/64. 00:06:05.950 --> 00:06:10.020 Well you could divide the top and the bottom by 8, so that 00:06:10.020 --> 00:06:12.880 becomes 5/8. 00:06:12.880 --> 00:06:14.220 40 divided by 5. 00:06:14.220 --> 00:06:15.840 64 divided by 8. 00:06:15.840 --> 00:06:18.140 x is equal to 5/8. 00:06:18.140 --> 00:06:20.970 And that is choice B. 00:06:20.970 --> 00:06:21.820 Next problem. 00:06:21.820 --> 00:06:22.900 There's a bunch of ways you could have don e it. 00:06:22.900 --> 00:06:27.090 This is the way that occurred to me. 00:06:27.090 --> 00:06:31.590 Next problem, problem nine. 00:06:31.590 --> 00:06:36.140 In the xy-coordinate system, p comma 0 is one of the points 00:06:36.140 --> 00:06:38.840 of intersection of the graphs-- OK, so they're giving 00:06:38.840 --> 00:06:40.150 us two graphs. 00:06:40.150 --> 00:06:45.360 y is equal to minus x squared plus 9, and the other graph is 00:06:45.360 --> 00:06:51.120 y is equal to x squared minus 9. 00:06:51.120 --> 00:06:58.950 If p is positive, what is the value of p? 00:06:58.950 --> 00:07:00.360 So they're saying that the coordinate where they 00:07:00.360 --> 00:07:03.970 intersect is p comma 0. 00:07:03.970 --> 00:07:05.115 So we want to figure out the 00:07:05.115 --> 00:07:10.320 x-coordinate where they intersect. 00:07:10.320 --> 00:07:13.650 So the easy way-- you know, they're tempting you to add 00:07:13.650 --> 00:07:16.090 these equations and do all sorts of things, but we know 00:07:16.090 --> 00:07:19.260 that they intersect at the point y is equal to 0. 00:07:19.260 --> 00:07:21.050 They're telling us that, so we don't have to do all this 00:07:21.050 --> 00:07:21.590 fancy stuff. 00:07:21.590 --> 00:07:23.470 We don't have to figure out where they intersect. 00:07:23.470 --> 00:07:25.060 They tell us it intersects at y equals 0. 00:07:25.060 --> 00:07:29.190 So y equals 0, what's x in either of these situations? 00:07:29.190 --> 00:07:31.030 Well, let's take this second equation. 00:07:31.030 --> 00:07:34.020 0 is equal to x squared minus 9. 00:07:34.020 --> 00:07:38.280 Add 9 to both sides, you get x squared is equal to 9. 00:07:38.280 --> 00:07:41.400 I added 9 and switched the sides. 00:07:41.400 --> 00:07:44.680 You get x is equal to plus or minus 3. 00:07:44.680 --> 00:07:48.350 And they tell us, that if p is positive, what is 00:07:48.350 --> 00:07:49.530 the value of p? 00:07:49.530 --> 00:07:52.180 Well, it has to be plus 3 then. 00:07:52.180 --> 00:07:55.160 And that is choice A. 00:07:55.160 --> 00:07:56.390 And you could have used the top equation. 00:07:56.390 --> 00:08:00.390 You could have said 0 is equal to negative x squared plus 9. 00:08:00.390 --> 00:08:01.200 You would've gotten the same thing. 00:08:01.200 --> 00:08:03.600 You would've gotten x squared is equal to 9, x is equal to 00:08:03.600 --> 00:08:05.120 plus or minus 3. 00:08:05.120 --> 00:08:08.080 You would've gotten the same thing either way. 00:08:08.080 --> 00:08:09.150 Next problem. 00:08:09.150 --> 00:08:10.053 I don't know if I'm going to have time to 00:08:10.053 --> 00:08:10.840 do it in this video. 00:08:10.840 --> 00:08:13.930 I might have to do it in the next. 00:08:13.930 --> 00:08:16.370 Problem ten. 00:08:16.370 --> 00:08:19.540 The Smith Metal Company's old machine makes 00:08:19.540 --> 00:08:21.100 300 bolts per hour. 00:08:21.100 --> 00:08:24.610 300-- I'll call it b per h. 00:08:24.610 --> 00:08:30.220 Its new machine makes 450 bolts per hour. 00:08:30.220 --> 00:08:34.890 If both machines begin running at the same time, how many 00:08:34.890 --> 00:08:38.880 minutes will it take the two machines to make a 00:08:38.880 --> 00:08:41.539 total of 900 bolts? 00:08:41.539 --> 00:08:42.220 So how many minutes? 00:08:42.220 --> 00:08:44.120 So let's say m for minutes. 00:08:44.120 --> 00:08:46.480 So the total number of bolts they're going to produce-- so 00:08:46.480 --> 00:08:56.380 bolts is going to be equal-- so this is per hour, right? 00:08:56.380 --> 00:08:57.790 These are both per hour. 00:08:57.790 --> 00:09:01.400 So let's figure it out minutes, because they're 00:09:01.400 --> 00:09:03.420 asking us how many minutes will it take. 00:09:03.420 --> 00:09:06.860 So if you make 300 bolts in an hour, you're going to make 00:09:06.860 --> 00:09:08.740 1/60 that in a minute. 00:09:08.740 --> 00:09:11.920 So what's 1/60 of 300? 00:09:11.920 --> 00:09:16.910 Well, 300 divided by 60 is 5 bolts per minute. 00:09:16.910 --> 00:09:23.390 And this one-- what's 450 divided by 60? 00:09:23.390 --> 00:09:24.340 Actually, I'm going to run out of time. 00:09:24.340 --> 00:09:25.590 I'm going to do this problem in the next video. 00:09:25.590 --> 00:09:27.320 I'll see you soon.

SAT Prep: Test 6 Section 7 Part 1

https://www.youtube.com/watch?v=dkN6eSd2AIA

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https://www.youtube.com/api/timedtext?v=dkN6eSd2AIA&ei=YmeUZe6kM9C2vdIPlvWXwAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B128F2EF329F2BC8343D0669E5C20DE27C5D2CDD.9885A2C8AFDFEF3F9D71671FF2D53CF1654600C0&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.850 --> 00:00:04.480 I just had dinner so I am well fed, and I am ready to do more 00:00:04.480 --> 00:00:08.100 SAT problems. So we're on section seven, on page-- 00:00:08.100 --> 00:00:11.660 excuse me, maybe I should drink some more water-- 00:00:11.660 --> 00:00:13.940 section seven of the sixth test, on page 00:00:13.940 --> 00:00:16.890 733, problem one. 00:00:16.890 --> 00:00:21.110 In a certain game, points are assigned to every word. 00:00:21.110 --> 00:00:24.740 Each q, x and z in a word-- it sounds like Scrabble-- is 00:00:24.740 --> 00:00:26.420 worth 5 points. 00:00:26.420 --> 00:00:28.840 And all other letters are worth 1 point each. 00:00:28.840 --> 00:00:34.320 So q, x and z are 5 points each, and the other ones are 1 00:00:34.320 --> 00:00:34.840 point each. 00:00:34.840 --> 00:00:36.320 What is the sum of the points 00:00:36.320 --> 00:00:37.970 assigned to the word exquisite? 00:00:37.970 --> 00:00:43.070 Well this is just-- exquisite. 00:00:48.940 --> 00:00:52.190 So let's see, which of these does it have? 00:00:52.190 --> 00:00:53.420 It has an x, right? 00:00:53.420 --> 00:00:55.920 So that's worth 5 points. 00:00:55.920 --> 00:00:58.030 Does it have any other of these? 00:00:58.030 --> 00:01:01.200 No-- well, it has a q, right? 00:01:01.200 --> 00:01:03.190 That's worth 5 points. 00:01:03.190 --> 00:01:05.830 It doesn't have a z, right? 00:01:05.830 --> 00:01:07.440 And so how many one-pointers are there? 00:01:07.440 --> 00:01:14.390 There's 1, 2, 3, 4, 5, 6, 7. 00:01:14.390 --> 00:01:15.930 So 7 one-pointers. 00:01:15.930 --> 00:01:21.050 So that's 7 points, plus 5 plus 5 plus 10 is equal to 17. 00:01:21.050 --> 00:01:21.700 So that's B. 00:01:21.700 --> 00:01:23.680 So that's really just to make sure that you don't make 00:01:23.680 --> 00:01:26.030 careless mistakes, that problem. 00:01:26.030 --> 00:01:27.950 Next problem. 00:01:27.950 --> 00:01:31.230 If you've ever played Scrabble, that problem would 00:01:31.230 --> 00:01:33.010 be a joke for you. 00:01:33.010 --> 00:01:35.060 Problem two. 00:01:35.060 --> 00:01:42.965 If 2x minus 10 is equal to 20, then x minus 5 is what? 00:01:45.890 --> 00:01:49.510 So the big discovery here is-- you could solve for x, you 00:01:49.510 --> 00:01:53.210 could say 2x is equal to 30, x is equal to 15, and say 15 00:01:53.210 --> 00:01:54.660 minus 5 equals 10. 00:01:54.660 --> 00:01:56.920 Or you could say, well this is the same thing as 2 times x 00:01:56.920 --> 00:02:00.960 minus 10 is equal to 20, divide both sides by 2, x 00:02:00.960 --> 00:02:06.490 minus 10 is equal to 10. 00:02:06.490 --> 00:02:10.550 Oh, sorry, maybe that's why you shouldn't do it this way. 00:02:10.550 --> 00:02:14.040 2 times x minus 5, x minus 5 is equal to 10. 00:02:14.040 --> 00:02:15.700 So that's the other way you could have done it, you just 00:02:15.700 --> 00:02:18.690 factor out a 2, you get 2 times x minus 5 is 20, divide 00:02:18.690 --> 00:02:21.000 both sides by 2, you get x minus 5 is 10. 00:02:21.000 --> 00:02:22.910 The other way is obviously to say 2x minus 00:02:22.910 --> 00:02:24.680 10 is equal to 20. 00:02:24.680 --> 00:02:25.760 Add 10 to both sides. 00:02:25.760 --> 00:02:28.010 2x is equal to 30. 00:02:28.010 --> 00:02:29.570 x is equal to 15. 00:02:29.570 --> 00:02:33.930 So then x minus 5 is 15 minus 5, which is equal to 10. 00:02:33.930 --> 00:02:35.510 I don't know which way might be faster for you. 00:02:35.510 --> 00:02:37.730 This way might-- you have to do less thinking, and you just 00:02:37.730 --> 00:02:38.837 kind of chug through it, so you can just 00:02:38.837 --> 00:02:39.700 kind of speed along. 00:02:39.700 --> 00:02:42.510 But this way's a few less steps, but maybe you might 00:02:42.510 --> 00:02:44.130 make a careless mistake like what I just did. 00:02:44.130 --> 00:02:46.120 But anyway, let's move on. 00:02:46.120 --> 00:02:49.230 Hopefully, either way you know how to do it. 00:02:49.230 --> 00:02:51.520 Problem three. 00:02:51.520 --> 00:02:54.120 I'll do it right here. 00:02:54.120 --> 00:02:57.360 If t represents an odd integer, which of the 00:02:57.360 --> 00:03:00.270 following represents an even integer? 00:03:00.270 --> 00:03:01.960 So t is an odd integer. 00:03:01.960 --> 00:03:05.360 And we can do this kind of abstractly, but let's just 00:03:05.360 --> 00:03:06.600 pick a number. 00:03:06.600 --> 00:03:09.280 Let's just say that t is equal to-- what's an odd integer? 00:03:09.280 --> 00:03:12.120 Well, 3, 3 is an odd integer. 00:03:12.120 --> 00:03:14.050 So let's just say t equals 3-- they didn't say, we could have 00:03:14.050 --> 00:03:15.700 picked t equals 7, who knows. 00:03:15.700 --> 00:03:18.250 So then just go through the choices and say, well, which 00:03:18.250 --> 00:03:20.840 one of those is going to be an even integer? 00:03:20.840 --> 00:03:22.820 Choice A is t plus 2. 00:03:22.820 --> 00:03:26.800 Well 3 plus 2 is 5, that is still odd, so that's not the 00:03:26.800 --> 00:03:27.980 right answer. 00:03:27.980 --> 00:03:30.960 B, 2t minus 1. 00:03:30.960 --> 00:03:35.980 Well that's 6 minus 1, that's 5, that's odd. 00:03:35.980 --> 00:03:44.800 Choice C is 3t minus 2, that's 9 minus 2, which is still 7, 00:03:44.800 --> 00:03:47.380 so that's still odd. 00:03:47.380 --> 00:03:53.700 Choice D is 3t plus 2, that's 9 plus 2, 00:03:53.700 --> 00:03:55.090 which is equal to 11. 00:03:55.090 --> 00:03:56.710 And I'm just saying t is 3. 00:03:56.710 --> 00:03:57.690 That's still odd. 00:03:57.690 --> 00:03:59.700 So it's probably going to be E. 00:03:59.700 --> 00:04:06.690 5t plus 1, and 15 plus 1 is 16, and that's even. 00:04:06.690 --> 00:04:08.280 So that's our answer. 00:04:08.280 --> 00:04:12.160 So sometimes I find it easy if you just pick a number. 00:04:12.160 --> 00:04:14.110 The other way is if you are familiar with even and odd 00:04:14.110 --> 00:04:16.339 numbers, you can just say, well, the only way to go from 00:04:16.339 --> 00:04:27.050 an odd to an even number, is you either have to multiply by 00:04:27.050 --> 00:04:35.140 an even soon. number, or add an odd number. 00:04:35.140 --> 00:04:37.960 Those are the only ways you can go from an odd number to 00:04:37.960 --> 00:04:39.260 an even number. 00:04:39.260 --> 00:04:42.570 So looking at 5t plus 1, you could say, well when I 00:04:42.570 --> 00:04:45.990 multiply an odd number by an odd number, I'm going to get 00:04:45.990 --> 00:04:48.190 another odd number, right? 00:04:48.190 --> 00:04:50.470 So this number is odd. 00:04:50.470 --> 00:04:52.850 When I take an odd number and I add 1 to it, 00:04:52.850 --> 00:04:54.560 I get an even number. 00:04:54.560 --> 00:04:58.830 So you could say 5t plus 1 is definitely going 00:04:58.830 --> 00:04:59.800 to be an even number. 00:04:59.800 --> 00:05:04.300 And it worked with the odd number that we picked for t. 00:05:04.300 --> 00:05:05.550 Next problem. 00:05:10.530 --> 00:05:13.466 OK, so they drew a picture here. 00:05:13.466 --> 00:05:20.270 So I will draw a picture here as best as I can. 00:05:20.270 --> 00:05:22.670 That's one triangle they drew, this is the other one. 00:05:22.670 --> 00:05:24.110 Looks something like this. 00:05:28.480 --> 00:05:30.500 Like that. 00:05:30.500 --> 00:05:36.160 And then this-- they say this is A, B, C. 00:05:36.160 --> 00:05:42.200 This is D, E, F. 00:05:42.200 --> 00:05:47.100 These sides are 4, 8, and 9. 00:05:47.100 --> 00:05:47.870 Now what are they saying? 00:05:47.870 --> 00:05:52.850 For the triangles above, the perimeter of ABC equals the 00:05:52.850 --> 00:05:53.880 perimeter of DEF. 00:05:53.880 --> 00:05:55.340 OK, it equals that. 00:05:55.340 --> 00:05:58.810 If ABC is equilateral, what is the length of AB? 00:05:58.810 --> 00:06:01.040 OK, so it's equilateral, right? 00:06:01.040 --> 00:06:03.390 So let's just say that AB is equal to x. 00:06:03.390 --> 00:06:06.180 Well then, so is BC and AC, because they told us, this is 00:06:06.180 --> 00:06:07.350 an equilateral triangle. 00:06:07.350 --> 00:06:09.330 All the sides are equal. 00:06:09.330 --> 00:06:11.430 And what is the perimeter of ABC then? 00:06:11.430 --> 00:06:15.350 It's going to be x plus x plus x, and that's just 3x. 00:06:15.350 --> 00:06:17.270 And they say that that's the same thing as the perimeter of 00:06:17.270 --> 00:06:18.620 this triangle. 00:06:18.620 --> 00:06:19.520 What's the perimeter of this triangle? 00:06:19.520 --> 00:06:23.110 It's 4 plus 8 plus 9. 00:06:23.110 --> 00:06:28.750 So 3x is equal to--what's 12 plus 9-- is 21. 00:06:28.750 --> 00:06:33.150 x is equal to 7, and that is choice C, and we are done. 00:06:33.150 --> 00:06:34.400 Next problem. 00:06:39.430 --> 00:06:42.100 They have drawn a diagram here that I will 00:06:42.100 --> 00:06:44.790 now attempt to draw. 00:06:44.790 --> 00:06:45.550 OK. 00:06:45.550 --> 00:06:50.316 And then, let's see they draw a dividing line, looks like 00:06:50.316 --> 00:06:52.690 it's about 1/4 of the circle. 00:06:52.690 --> 00:06:54.880 And then they have another one that's like a little less than 00:06:54.880 --> 00:06:56.660 1/4, like that. 00:06:56.660 --> 00:07:00.820 Then they have one that's like that, roughly. 00:07:00.820 --> 00:07:05.565 Then it goes like that. 00:07:05.565 --> 00:07:07.640 Then it goes like that. 00:07:07.640 --> 00:07:10.240 And then they fill in the values too. 00:07:10.240 --> 00:07:14.480 So this is K, it's 15%. 00:07:14.480 --> 00:07:18.420 J is 25%. 00:07:18.420 --> 00:07:21.630 O-- let's just call that other, is O. 00:07:21.630 --> 00:07:25.570 Other is 20%. 00:07:25.570 --> 00:07:28.410 N is 10%. 00:07:28.410 --> 00:07:35.090 M is 15% and L is 15%. 00:07:35.090 --> 00:07:38.380 And these are sales of jeans in 2001, and the letters are 00:07:38.380 --> 00:07:39.370 the brands. 00:07:39.370 --> 00:07:41.740 The circle graph above represents all the jeans that 00:07:41.740 --> 00:07:43.950 were sold by a retail store in 2001 00:07:43.950 --> 00:07:46.070 according to their brands. 00:07:46.070 --> 00:07:49.480 If the store sold 900 pairs of jeans-- so this is the total 00:07:49.480 --> 00:07:53.050 of all of these, is 900 pairs of jeans. 00:07:53.050 --> 00:07:58.620 If the store sold 900 pairs of jeans other than brands J, K, 00:07:58.620 --> 00:08:01.930 L, M and N, how may did it sell together? 00:08:01.930 --> 00:08:04.600 OK, so this isn't the total, this is the other. 00:08:04.600 --> 00:08:09.320 This is essentially this category, right? 00:08:09.320 --> 00:08:12.220 And other is equal to 20% of the total. 00:08:12.220 --> 00:08:13.370 And that's what they want to ask us, how 00:08:13.370 --> 00:08:15.030 many did we sell together? 00:08:15.030 --> 00:08:22.160 So 20% of the total times t for total-- let me just say t 00:08:22.160 --> 00:08:25.200 for total-- is equal to 900. 00:08:25.200 --> 00:08:28.840 And you could also write this as 1/5. 00:08:28.840 --> 00:08:29.790 Or 0.2. 00:08:29.790 --> 00:08:33.320 1/5 of the total is equal to 900. 00:08:33.320 --> 00:08:36.490 Multiply both sides by 5. 00:08:36.490 --> 00:08:37.330 That cancels out. 00:08:37.330 --> 00:08:41.839 So you get t is equal to 4,500 pairs of jeans. 00:08:41.839 --> 00:08:45.830 And that is choice E. 00:08:45.830 --> 00:08:48.310 Next problem. 00:08:48.310 --> 00:08:50.220 Oh, I thought this problem might apply to this pie graph, 00:08:50.220 --> 00:08:51.985 but no, they're talking about carpets, not jeans. 00:08:51.985 --> 00:08:53.950 So I'll do the next problem in the next video. 00:08:53.950 --> 00:08:55.730 I'll see you

SAT Prep: Test 6 Section 7 Part 4

https://www.youtube.com/watch?v=kmP97_diMVM

vtt

https://www.youtube.com/api/timedtext?v=kmP97_diMVM&ei=YmeUZaO-OOOjvdIPmI-IoAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249811&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=864AD836A14F565A19C09BC74274D165F71F4F4D.B36FF4902E0292E82EC59C0173F4198CB35A9460&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.080 --> 00:00:04.360 We are on problem number 14. 00:00:04.360 --> 00:00:06.095 See, what did they draw here? 00:00:06.095 --> 00:00:10.850 That's the y-axis, that's the x-axis. 00:00:10.850 --> 00:00:13.255 They drew a line that looks something like this. 00:00:13.255 --> 00:00:18.100 Let's see, negative 3, so it goes something like this. 00:00:22.580 --> 00:00:23.940 That's close enough. 00:00:23.940 --> 00:00:28.090 And then they're saying that this is the point negative 3. 00:00:28.090 --> 00:00:29.820 This is the point y equals negative 1. 00:00:29.820 --> 00:00:31.760 This is of course the y-axis. 00:00:31.760 --> 00:00:33.610 This is the x-axis. 00:00:33.610 --> 00:00:36.790 The figure above shows a graph of the line of y equals mx 00:00:36.790 --> 00:00:39.000 plus b, where m and b are constants. 00:00:39.000 --> 00:00:43.160 Which of the following best represents the graph of y is 00:00:43.160 --> 00:00:48.050 equal to minus 3mx plus b? 00:00:48.050 --> 00:00:53.120 Well, if this yellow line-- this is y equals mx plus b. 00:00:57.380 --> 00:01:00.230 Well, what is b, first of all? b is just the y-intercept. 00:01:00.230 --> 00:01:02.770 So the y-intercept is right here, y equals negative 1. 00:01:02.770 --> 00:01:05.160 So b is negative 1. 00:01:05.160 --> 00:01:06.240 And what's the slope? 00:01:06.240 --> 00:01:12.660 Well, when we run, that's what-- 1, 2, 3 units. 00:01:12.660 --> 00:01:16.000 Change in x is 3. 00:01:16.000 --> 00:01:17.390 What is the rise? 00:01:17.390 --> 00:01:20.000 The rise, we go down, so it's negative 1. 00:01:20.000 --> 00:01:21.430 We go down 1. 00:01:21.430 --> 00:01:25.150 So the change in y is equal to negative 1. 00:01:25.150 --> 00:01:27.500 So the slope is negative 1/3. 00:01:27.500 --> 00:01:29.650 So the equation of this line is y is equal to 00:01:29.650 --> 00:01:34.740 minus 1/3 x minus 1. 00:01:34.740 --> 00:01:37.560 So if you want to know what minus 3mx plus b is, what's 00:01:37.560 --> 00:01:41.560 minus 3 times this m? 00:01:41.560 --> 00:01:45.800 What's minus 3 times minus 1/3? 00:01:45.800 --> 00:01:47.970 Well, the negatives cancel out, so you just get-- let me 00:01:47.970 --> 00:01:52.000 switch to purple, because that's that problem-- minus 3 00:01:52.000 --> 00:01:54.070 times-- this is m, right? 00:01:54.070 --> 00:01:55.540 This is m. 00:01:55.540 --> 00:01:59.820 So minus 3 times minus 1/3, that just equals 1. 00:01:59.820 --> 00:02:04.070 So it equals 1x plus b. 00:02:04.070 --> 00:02:04.570 What's b? 00:02:04.570 --> 00:02:06.760 It's minus 1. 00:02:06.760 --> 00:02:09.729 So the graph we're looking for is x minus 1. 00:02:09.729 --> 00:02:12.190 And so if we wanted to draw that, the y-intercept would be 00:02:12.190 --> 00:02:13.700 the same, it would be right there. 00:02:13.700 --> 00:02:15.330 And then you just have a slope of 1. 00:02:15.330 --> 00:02:19.830 Slope of 1 looks-- well, that's as close as I could do 00:02:19.830 --> 00:02:21.670 to drawing it, right-- that's a slope of 1. 00:02:21.670 --> 00:02:27.140 Your change in x should be the same as your change in y. 00:02:27.140 --> 00:02:30.630 So if you look at all the choices, first of all, it's 00:02:30.630 --> 00:02:32.410 pretty obvious it's D, I think. 00:02:32.410 --> 00:02:34.360 And they make it very explicit that the slope is 1 and the 00:02:34.360 --> 00:02:36.460 y-intercept is negative 1. 00:02:36.460 --> 00:02:38.850 And that's choice D. 00:02:38.850 --> 00:02:40.100 Next problem. 00:02:43.051 --> 00:02:45.750 I get excited when there's something to draw, because we 00:02:45.750 --> 00:02:47.790 can get through them faster. 00:02:47.790 --> 00:02:50.120 15. 00:02:50.120 --> 00:02:56.120 If the volume of a cube is 8, what is the shortest distance 00:02:56.120 --> 00:02:59.610 from the center of the cube to the base of the cube? 00:02:59.610 --> 00:03:01.220 Fascinating. 00:03:01.220 --> 00:03:03.880 So the volume is 8, so what are the 00:03:03.880 --> 00:03:05.130 dimensions of this cube? 00:03:08.270 --> 00:03:11.890 It's a cube, so all the sides are the same, right? 00:03:11.890 --> 00:03:15.420 So x times x times x is x to the third, and that equals 8. 00:03:15.420 --> 00:03:17.810 So the dimensions-- what to the third power is 8? 00:03:17.810 --> 00:03:19.000 You should know that. 00:03:19.000 --> 00:03:19.930 2, right? 00:03:19.930 --> 00:03:21.720 2 times 2 times 2 is 8. 00:03:21.720 --> 00:03:25.360 So the dimensions of the cube are 2 by 2 by 2. 00:03:25.360 --> 00:03:28.450 So let's take that cross section where the center is. 00:03:36.610 --> 00:03:41.490 And if I were to draw it, so that's just-- so the center of 00:03:41.490 --> 00:03:43.410 the cube is right here. 00:03:43.410 --> 00:03:45.400 And we know the dimensions of this cross section. 00:03:45.400 --> 00:03:48.020 It's 2, 2, 2, 2. 00:03:48.020 --> 00:03:48.280 Right? 00:03:48.280 --> 00:03:51.290 I just sliced the cube right there. 00:03:51.290 --> 00:03:52.830 And I just want to make sure I'm doing this problem. 00:03:52.830 --> 00:03:54.940 From the center of the cube to the base of the cube-- so the 00:03:54.940 --> 00:04:00.145 base of the cube is going to be-- if I were to draw the 00:04:00.145 --> 00:04:02.740 cube, this would have been the base of the cube down here. 00:04:02.740 --> 00:04:06.490 So this is the base of the cube. 00:04:06.490 --> 00:04:08.820 So they're essentially just asking us, what's this 00:04:08.820 --> 00:04:11.250 distance right here? 00:04:11.250 --> 00:04:11.870 Well that's easy. 00:04:11.870 --> 00:04:13.300 This is the center, so it's right in the 00:04:13.300 --> 00:04:14.940 middle, so it's 1. soon. 00:04:14.940 --> 00:04:18.230 It's 1 to the bottom, it's 1 to the roof of the cube, it's 00:04:18.230 --> 00:04:19.709 1 to each of the sides. 00:04:19.709 --> 00:04:20.910 So it's 1 away. 00:04:20.910 --> 00:04:23.590 So that is answer A. 00:04:27.150 --> 00:04:28.400 Next problem. 00:04:34.060 --> 00:04:37.390 Problem 16. 00:04:37.390 --> 00:04:46.440 If y is equal to 5x cubed over z, what happens to the value 00:04:46.440 --> 00:04:50.952 of y when both x and z are doubled? 00:04:53.800 --> 00:04:58.330 So the way I think about it is, y is equal to 5x cubed 00:04:58.330 --> 00:04:59.670 divided by z. 00:04:59.670 --> 00:05:03.030 Actually do you know what the very easiest way to do this 00:05:03.030 --> 00:05:04.490 problem is? 00:05:04.490 --> 00:05:06.720 Substitute numbers for x and z. 00:05:06.720 --> 00:05:10.140 So let's say that x is equal to 1 and z equals to 1. 00:05:10.140 --> 00:05:12.190 So in this situation, y is equal to what? 00:05:15.220 --> 00:05:21.790 5 times 1 cubed divided by 1, well that's just equal to 5. 00:05:21.790 --> 00:05:22.880 Now they're saying, what happens 00:05:22.880 --> 00:05:24.030 if x and z are doubled? 00:05:24.030 --> 00:05:25.490 So let's take the second situation. 00:05:25.490 --> 00:05:28.450 x is equal to 2, z is equal to 2. 00:05:28.450 --> 00:05:35.910 Now, y is equal to 5 times 2 cubed over 2. 00:05:35.910 --> 00:05:38.630 And that is equal to what? 00:05:38.630 --> 00:05:42.920 Well, 2 cubed divided by 2, this 2 cancels with the cubed. 00:05:42.920 --> 00:05:43.760 You get 2 squared. 00:05:43.760 --> 00:05:48.620 So it's just 5 times 2 squared, which is 5 times 4, 00:05:48.620 --> 00:05:50.100 which is 20. 00:05:50.100 --> 00:05:57.310 So y went from 5 to 20, so y increased by a factor of 4. 00:05:57.310 --> 00:05:58.810 And that is choice E. 00:05:58.810 --> 00:06:02.280 y is multiplied by 4. 00:06:02.280 --> 00:06:04.860 These questions can be surprisingly confusing because 00:06:04.860 --> 00:06:06.710 you're like, oh, you have these variables, do you make x 00:06:06.710 --> 00:06:09.530 into 2x and z into 2z, and then solve for-- 00:06:09.530 --> 00:06:10.430 just try out numbers. 00:06:10.430 --> 00:06:12.700 They're not putting any restrictions on you, so just 00:06:12.700 --> 00:06:17.260 substitute x and z, 1 and 1, and then double them. 00:06:17.260 --> 00:06:20.490 Next problem. 00:06:20.490 --> 00:06:27.690 The SAT, speed matters more than maybe rigor. 00:06:27.690 --> 00:06:28.880 All right. 00:06:28.880 --> 00:06:31.840 Problem 17. 00:06:31.840 --> 00:06:38.340 Luke purchased an automobile for $5,000. 00:06:38.340 --> 00:06:39.680 And the value of the automobile 00:06:39.680 --> 00:06:43.420 decreased by 20% per year. 00:06:43.420 --> 00:06:46.470 The value in dollars of the automobile n years from the 00:06:46.470 --> 00:06:50.100 date of the purchase is given by the function v, where v of 00:06:50.100 --> 00:06:59.660 n is equal to 5,000 times 4/5, which is the same thing as 00:06:59.660 --> 00:07:01.480 0.8, to the nth power. 00:07:01.480 --> 00:07:02.460 Fair enough. 00:07:02.460 --> 00:07:06.350 How many years from the date of the purchase will the value 00:07:06.350 --> 00:07:09.150 of the automobile be $3,200? 00:07:09.150 --> 00:07:11.400 Well, they're saying the value is $3,200, so we 00:07:11.400 --> 00:07:12.730 just solve for n. 00:07:12.730 --> 00:07:15.060 v of how many years is equal to $3,200? 00:07:15.060 --> 00:07:23.200 3,200 is equal to 5,000 times 4/5 to the n. 00:07:23.200 --> 00:07:30.380 Divide both sides by 5,000, you get 3,200/5,000 is equal 00:07:30.380 --> 00:07:35.860 to 4/5 to the n. 00:07:35.860 --> 00:07:38.690 And then, let's see, what's 32/5,000? 00:07:38.690 --> 00:07:41.710 You can delete these 0's. 00:07:41.710 --> 00:07:44.760 Let's see, 16/25. 00:07:44.760 --> 00:07:54.540 That equals-- so 16/25 is equal to 4/5 to the n. 00:07:54.540 --> 00:07:57.120 And we should immediately-- well, hopefully you'd 00:07:57.120 --> 00:08:01.810 recognize is that 16 is 4 squared, and 25 is 5 squared, 00:08:01.810 --> 00:08:09.970 so 16/25 is equal to 4/5 squared. 00:08:09.970 --> 00:08:10.970 4 squared is 16. 00:08:10.970 --> 00:08:13.310 5 squared is 25. 00:08:13.310 --> 00:08:15.780 So n is equal to 2. 00:08:15.780 --> 00:08:16.930 2 years. 00:08:16.930 --> 00:08:18.210 And that is choice B. 00:08:22.030 --> 00:08:24.260 Let's see if I have time for-- no, we have a couple more 00:08:24.260 --> 00:08:25.300 problems. I'll do the last three 00:08:25.300 --> 00:08:26.910 problems in the next video. 00:08:26.910 --> 00:08:29.240 I will see you

SAT Prep: Test 6 Section 7 Part 5

https://www.youtube.com/watch?v=0CIn1_M-BoY

vtt

https://www.youtube.com/api/timedtext?v=0CIn1_M-BoY&ei=YmeUZYPAM_W4vdIPu8KLkA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2034764BAF6CF23E6116ECFE974401765D34255F.2997B0E6BE556B1E941EF1BE8A838C54CB81B4D6&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:02.660 The home stretch, we're on problem 18. 00:00:02.660 --> 00:00:06.570 And this diagram actually looks like something that this 00:00:06.570 --> 00:00:09.410 little tool I'm using is well-suited to draw. 00:00:09.410 --> 00:00:13.410 Let me try-- I'll draw the one in the back first, so I can do 00:00:13.410 --> 00:00:16.470 different colors and everything. 00:00:16.470 --> 00:00:21.370 So I'll do the one in the back in yellow, and it looks 00:00:21.370 --> 00:00:23.010 something like this. 00:00:23.010 --> 00:00:28.100 Goes like this, then it switches over like that. 00:00:28.100 --> 00:00:31.790 And then I'll draw the next one, this orange color, and it 00:00:31.790 --> 00:00:36.200 looks something-- no, I'll do it in a more drastic color. 00:00:36.200 --> 00:00:37.600 So it does something like this. 00:00:37.600 --> 00:00:42.330 It goes like this, crosses over, dips down, then it goes 00:00:42.330 --> 00:00:43.760 over again. 00:00:43.760 --> 00:00:48.660 And then finally, this last one, and it looks 00:00:48.660 --> 00:00:51.550 something like this. 00:00:51.550 --> 00:00:55.530 Crosses over and it goes like that, very pretty. 00:00:55.530 --> 00:01:00.990 Now let me switch to a smaller-- OK. 00:01:00.990 --> 00:01:03.030 Why don't we do white, I never write in white. 00:01:03.030 --> 00:01:06.910 In the figure above--so they say this is the start. 00:01:06.910 --> 00:01:08.320 And this is step 1. 00:01:11.645 --> 00:01:14.770 This is step 2. 00:01:14.770 --> 00:01:15.670 OK, what is this? 00:01:15.670 --> 00:01:18.610 In the figure above, three wires are braided. 00:01:18.610 --> 00:01:22.580 That is, by starting in the order A, B, C-- OK, so this is 00:01:22.580 --> 00:01:30.890 A, B, C-- and then the order changes to B-- right, because 00:01:30.890 --> 00:01:33.730 the yellow line is B-- A, C. 00:01:33.730 --> 00:01:37.350 And then we get B, C, A. 00:01:37.350 --> 00:01:37.770 Right? 00:01:37.770 --> 00:01:40.580 There's just the order of the strings, left to right. 00:01:40.580 --> 00:01:43.380 Or the braids, or the wires. 00:01:43.380 --> 00:01:44.580 OK. 00:01:44.580 --> 00:01:47.040 That is start with the order A, B, C, the outer left wire 00:01:47.040 --> 00:01:48.670 is brought over wire B, right? 00:01:48.670 --> 00:01:52.380 This is brought over wire B, to the middle position forming 00:01:52.380 --> 00:01:53.610 the order shown in step 1. 00:01:53.610 --> 00:01:58.350 Than the outer right wire C, this one, is brought to the 00:01:58.350 --> 00:02:00.750 new middle position shown in step 2, and so on. 00:02:00.750 --> 00:02:03.850 Alternately bringing each new left and each new right wire 00:02:03.850 --> 00:02:04.770 to the middle. 00:02:04.770 --> 00:02:09.449 At what numbered step does the braid first repeat the 00:02:09.449 --> 00:02:13.070 original order A, B, C? 00:02:13.070 --> 00:02:16.910 So what you do is, you first bring the left over the 00:02:16.910 --> 00:02:18.865 middle, then you bring the right over the middle, then 00:02:18.865 --> 00:02:20.730 you bring the left over the middle, then you bring the 00:02:20.730 --> 00:02:22.310 right over the middle. 00:02:22.310 --> 00:02:24.470 So at step 3, what do we have to do? 00:02:24.470 --> 00:02:27.750 So step 1 we brought the left over the middle. 00:02:27.750 --> 00:02:29.840 So we went from A, B, C. 00:02:29.840 --> 00:02:32.720 You essentially switch the left and the middle. 00:02:32.720 --> 00:02:35.330 And then step 2, you switch the right and the middle. 00:02:35.330 --> 00:02:39.420 So step 3, we switch B and C again. 00:02:39.420 --> 00:02:41.790 Because we're back on the left-hand side. 00:02:41.790 --> 00:02:44.930 So you get C, B, A. 00:02:44.930 --> 00:02:46.765 And then step 4, you're going to switch to 00:02:46.765 --> 00:02:48.690 the right-hand side. 00:02:48.690 --> 00:02:52.170 Step 3 we switched these two. 00:02:52.170 --> 00:02:54.470 Now step 4, we're going to switch these two. 00:02:54.470 --> 00:02:57.930 So you get C, A, B. 00:02:57.930 --> 00:03:02.020 Then in step 5, we're back switched on this side. 00:03:02.020 --> 00:03:05.180 So then you get A, C, B. 00:03:05.180 --> 00:03:07.380 And then step 6, you're going to switch to the right-hand 00:03:07.380 --> 00:03:07.895 side again. 00:03:07.895 --> 00:03:10.610 So you get A, B, C. 00:03:10.610 --> 00:03:12.045 So by step 6, we repeat. 00:03:12.045 --> 00:03:13.890 So that's choice D. 00:03:13.890 --> 00:03:16.030 The hard part here, as far as I'm concerned, is just 00:03:16.030 --> 00:03:18.070 understanding the problem and their diagram. 00:03:18.070 --> 00:03:19.460 And then just seeing the pattern that you're just 00:03:19.460 --> 00:03:20.910 switching the letters. 00:03:20.910 --> 00:03:23.490 First you switch the left and the middle letter. 00:03:23.490 --> 00:03:24.740 Then you switch the right and the middle letter. 00:03:24.740 --> 00:03:26.050 Then you switch the left and the middle letter. 00:03:26.050 --> 00:03:27.610 Then you switch the right and the middle letter. 00:03:27.610 --> 00:03:29.810 Until you get back to A, B, C. 00:03:29.810 --> 00:03:32.900 Next problem. 00:03:32.900 --> 00:03:34.860 White I don't think is colorful enough. 00:03:34.860 --> 00:03:36.690 Let me do magenta. 00:03:36.690 --> 00:03:39.450 19. 00:03:39.450 --> 00:03:42.360 In a set of 11 different numbers-- they're different-- 00:03:42.360 --> 00:03:44.920 which of the following cannot affect the 00:03:44.920 --> 00:03:46.170 value of the median? 00:03:49.370 --> 00:03:54.670 So it's a set of 11 different numbers, they're different, so 00:03:54.670 --> 00:03:56.210 let's say it's 1 through 11. 00:03:56.210 --> 00:04:02.450 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. 00:04:02.450 --> 00:04:05.300 That's 11 different numbers. 00:04:05.300 --> 00:04:06.660 And in this case, what's the median? 00:04:06.660 --> 00:04:09.010 Well we have 5 on this side, 5-- the median is this, this 00:04:09.010 --> 00:04:10.610 is the middle number, right? 00:04:10.610 --> 00:04:12.000 6 is the middle number. 00:04:12.000 --> 00:04:14.350 Doubling each number-- well, if you double every number, 00:04:14.350 --> 00:04:17.700 the median's going to become 12, so that's not right. 00:04:17.700 --> 00:04:20.060 Increasing each number by 10-- once again, if you increase 00:04:20.060 --> 00:04:23.280 all of these numbers by 10, this number's going to be 16, 00:04:23.280 --> 00:04:25.380 so you would've changed the median. 00:04:25.380 --> 00:04:30.060 Increasing the smallest number only. 00:04:30.060 --> 00:04:33.690 Well, what if you increase the smallest number to, 00:04:33.690 --> 00:04:35.430 I don't know, 12? 00:04:35.430 --> 00:04:37.540 So then this 12 would go all the way on 00:04:37.540 --> 00:04:39.610 the other side, right? 00:04:39.610 --> 00:04:42.840 And then the median would move to 7 because you would have to 00:04:42.840 --> 00:04:45.480 have 5 below the median and 5 above the median. 00:04:45.480 --> 00:04:46.980 So you can increase that smallest number. 00:04:46.980 --> 00:04:49.830 But if you increase it by a large enough amount, it kind 00:04:49.830 --> 00:04:51.930 of goes into the greater than the median column, and then 00:04:51.930 --> 00:04:54.780 the median would have to shift to the right. 00:04:54.780 --> 00:04:59.060 Try it out, write out the list from 2 to 12, and then the 00:04:59.060 --> 00:05:00.480 median becomes 7. 00:05:00.480 --> 00:05:03.080 OK, so we know that choice B is not right. 00:05:03.080 --> 00:05:06.777 Choice C, increasing-- oh no, that was choice C-- choice D, 00:05:06.777 --> 00:05:10.640 decreasing the largest number only-- well, the same 00:05:10.640 --> 00:05:11.390 argument can apply. 00:05:11.390 --> 00:05:15.330 We could take the 11 and decrease it down to, I don't 00:05:15.330 --> 00:05:17.690 know, decrease it to 0. 00:05:17.690 --> 00:05:20.050 You decrease it to 0, then that'll-- and you could list 00:05:20.050 --> 00:05:22.450 them all out, 0 through 10-- and then the median number 00:05:22.450 --> 00:05:25.700 will shift to 5, so that will change it. 00:05:25.700 --> 00:05:29.340 E, increasing the largest number only. 00:05:29.340 --> 00:05:31.360 I can tell you that's the answer from deductive 00:05:31.360 --> 00:05:35.210 reasoning, but if we just take this 11 and we make it one 00:05:35.210 --> 00:05:39.740 billion, does that change the fact that they're five numbers 00:05:39.740 --> 00:05:41.880 larger than 6 and five numbers less than 6? 00:05:41.880 --> 00:05:42.990 Ignore the 0. 00:05:42.990 --> 00:05:44.590 Five numbers less than 6? 00:05:44.590 --> 00:05:45.370 No. 00:05:45.370 --> 00:05:47.140 I can make it into a trillion. 00:05:47.140 --> 00:05:50.350 I can change it to any number, but the fact remains that I 00:05:50.350 --> 00:05:52.475 have the same amount of numbers larger than 6 00:05:52.475 --> 00:05:54.340 as I do below 6. 00:05:54.340 --> 00:05:59.410 So choice E cannot affect the value of the median. 00:05:59.410 --> 00:06:01.110 E. 00:06:01.110 --> 00:06:02.360 Next problem. 00:06:06.000 --> 00:06:09.340 OK, something for me to draw. 00:06:09.340 --> 00:06:11.710 Let's see, they have a 1/4 circle, so I'll just draw a 00:06:11.710 --> 00:06:15.500 big circle, and I'll just focus on a 1/4 of it. 00:06:22.390 --> 00:06:24.810 Close enough. 00:06:24.810 --> 00:06:27.760 That looks like a 1/4 of a circle. 00:06:27.760 --> 00:06:30.690 They say that-- oh, and there's a rectangle there too, 00:06:30.690 --> 00:06:32.520 fascinating. 00:06:32.520 --> 00:06:35.770 So this rectangle in this 1/4 circle, so this rectangle 00:06:35.770 --> 00:06:37.940 could look like this. 00:06:37.940 --> 00:06:41.640 I know theirs looks taller, but I think this will do. 00:06:41.640 --> 00:06:43.605 And then I think I'm almost done. 00:06:43.605 --> 00:06:47.510 I think I'm there, I think I can. 00:06:47.510 --> 00:06:49.930 And then they shade in some stuff, so if they shaded it, I 00:06:49.930 --> 00:06:51.920 will shade it. 00:06:51.920 --> 00:06:54.270 A suitably tasteful color. 00:06:57.300 --> 00:07:00.500 OK, so I have shaded what they have shaded. 00:07:00.500 --> 00:07:02.220 And now, what are they saying? 00:07:02.220 --> 00:07:15.710 They're saying that this is R, C, T, B, S, A. 00:07:15.710 --> 00:07:20.060 And then they tell us that is a 90 degree angle. 00:07:20.060 --> 00:07:22.980 And then they tell us that this distance, from here to 00:07:22.980 --> 00:07:26.390 here, from all the way to the top-- so it's essentially the 00:07:26.390 --> 00:07:29.700 radius of the circle-- is 6. 00:07:29.700 --> 00:07:31.000 The radius of the circle is 6. 00:07:31.000 --> 00:07:37.410 In the figure above, arc SBT is 1/4 of a circle with center 00:07:37.410 --> 00:07:38.390 R and radius 6. 00:07:38.390 --> 00:07:39.350 Fair enough. 00:07:39.350 --> 00:07:46.130 If the length plus the width of rectangle ABCR is 8-- so 00:07:46.130 --> 00:07:49.740 the length plus the width, so ABCR. 00:07:49.740 --> 00:07:52.100 So this is the width, this is the length. 00:07:52.100 --> 00:07:57.390 So L plus W is equal to 8. 00:07:57.390 --> 00:08:01.220 What is the perimeter of the shaded region? 00:08:01.220 --> 00:08:02.920 Fascinating. 00:08:02.920 --> 00:08:04.420 Let's do it step by step. 00:08:04.420 --> 00:08:07.140 First we can just figure out what this part of the 00:08:07.140 --> 00:08:08.310 perimeter is. 00:08:08.310 --> 00:08:11.440 And that's the easiest part, right? 00:08:11.440 --> 00:08:13.380 Because what's the perimeter of the whole circle? 00:08:13.380 --> 00:08:15.450 Well, it's the circumference of the whole circle. 00:08:15.450 --> 00:08:18.170 Circumference is equal to 2 pi r. 00:08:18.170 --> 00:08:20.210 This is the whole circle we're talking about. 00:08:20.210 --> 00:08:25.000 So that's equal to 2 pi times 6, so that's equal to 12 pi. 00:08:25.000 --> 00:08:26.950 This is the circumference of the whole circle, so the 00:08:26.950 --> 00:08:30.880 circumference of this arc, of this piece, is going to be 1/4 00:08:30.880 --> 00:08:33.610 of that, because it's 1/4 of the whole circle. 00:08:33.610 --> 00:08:37.860 So it's 1/4 of 12 pi, so that's 3 pi. 00:08:37.860 --> 00:08:40.690 Now, what we need to be able to figure out is L and W, 00:08:40.690 --> 00:08:43.559 because if we can figure out L and W, we can figure out 00:08:43.559 --> 00:08:47.600 everything else about this circle. 00:08:47.600 --> 00:08:49.720 And actually, I'm going to continue this problem in the 00:08:49.720 --> 00:08:52.430 next video, because I think it might get a little involved, 00:08:52.430 --> 00:08:53.890 and I don't want you to get too confused. 00:08:53.890 --> 00:08:55.140 I'll see you in the next video.

SAT Prep: Test 6 Section 7 Part 6

https://www.youtube.com/watch?v=aEvm-V7A5s8

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https://www.youtube.com/api/timedtext?v=aEvm-V7A5s8&ei=YmeUZaG7NafAmLAPlP2bgAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=87FAE34610646E17680F345722F3F4E13EC88D5B.2E8DB5156434C6D2B69B1917FFBC8C5419678204&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.790 --> 00:00:02.850 All right, this is a fun problem, so it'll get a video 00:00:02.850 --> 00:00:03.850 on its own. 00:00:03.850 --> 00:00:05.730 I started in the last video, but I'm going to start over 00:00:05.730 --> 00:00:07.970 just so we get it all in one take. 00:00:07.970 --> 00:00:13.480 So they say, in the figure above, arc SBT is 1/4 of a 00:00:13.480 --> 00:00:17.620 circle with center R and radius 6. 00:00:17.620 --> 00:00:20.590 And they actually tell-- they draw this distance, which is 00:00:20.590 --> 00:00:22.160 the radius. 00:00:22.160 --> 00:00:24.490 This distance is 6, right? 00:00:24.490 --> 00:00:26.725 So this distance would also be 6, right? 00:00:32.130 --> 00:00:35.580 If the length plus the width of rectangle ABCR is 8-- so 00:00:35.580 --> 00:00:37.300 this is ABCR. 00:00:37.300 --> 00:00:39.440 So the length plus the width is 8. 00:00:39.440 --> 00:00:44.640 So let's call this L and this is W. 00:00:44.640 --> 00:00:47.660 Actually, this should be-- well, it doesn't matter, this 00:00:47.660 --> 00:00:49.150 should be height, but whatever. 00:00:49.150 --> 00:00:51.160 We know that these two sides are equal to 8. 00:00:51.160 --> 00:00:56.040 So we know that L plus W is equal to 8. 00:00:56.040 --> 00:00:56.940 Fair enough. 00:00:56.940 --> 00:01:01.750 Then the perimeter of the shaded region is-- that's this 00:01:01.750 --> 00:01:04.590 perimeter right here, so there's a lot of pieces to it. 00:01:04.590 --> 00:01:06.490 So let's start with what I would call 00:01:06.490 --> 00:01:11.080 the low hanging fruit. 00:01:11.080 --> 00:01:14.320 So the low hanging fruit, in my opinion, is the 00:01:14.320 --> 00:01:16.780 length of this arc. 00:01:16.780 --> 00:01:18.030 Arc SBT. 00:01:20.440 --> 00:01:22.420 What's the length of that? 00:01:22.420 --> 00:01:25.140 Well, it's going to be 1/4 the circumference of the entire 00:01:25.140 --> 00:01:25.600 circle, right? 00:01:25.600 --> 00:01:28.840 Because this arc represents-- what we drew is 1/4 of the 00:01:28.840 --> 00:01:31.450 circle, so this length is going to be 1/4 of the 00:01:31.450 --> 00:01:32.600 circumference. 00:01:32.600 --> 00:01:34.270 So the length of the circumference-- so the 00:01:34.270 --> 00:01:36.690 circumference of, let me see, the circumference of the 00:01:36.690 --> 00:01:39.510 circle, is what? 00:01:39.510 --> 00:01:42.710 It's 2 pi r. 00:01:42.710 --> 00:01:43.760 r is 6, right? 00:01:43.760 --> 00:01:49.760 So 2 pi times 6, is equal to 12 pi. 00:01:49.760 --> 00:01:53.920 If the circumference of the circle is 12 pi, then we could 00:01:53.920 --> 00:02:00.070 call the length of the arc, right-- let's call this LA-- 00:02:00.070 --> 00:02:03.270 the length of the arc is going to be what? 00:02:03.270 --> 00:02:08.370 It's going to be 1/4 the circumference 00:02:08.370 --> 00:02:10.350 of the whole circle. 00:02:10.350 --> 00:02:14.040 1/4 times 12 pi is equal to 3 pi. 00:02:17.520 --> 00:02:18.680 Good enough. 00:02:18.680 --> 00:02:21.040 Now what else can we try to figure out? 00:02:21.040 --> 00:02:25.390 Well, the second lowest hanging fruit is actually the 00:02:25.390 --> 00:02:26.640 length of this line. 00:02:26.640 --> 00:02:27.820 Which you're probably saying, well, they give me no 00:02:27.820 --> 00:02:29.920 information, maybe I have to use the Pythagorean theorem 00:02:29.920 --> 00:02:32.810 with W and L, but it's really just a visual trick, because 00:02:32.810 --> 00:02:34.770 I'll tell you that you could actually just look at this 00:02:34.770 --> 00:02:38.000 graph and figure out what the length from A to C is. 00:02:38.000 --> 00:02:39.510 And I'll give you a hint. 00:02:39.510 --> 00:02:42.860 What is the length from R to B? 00:02:42.860 --> 00:02:45.780 What is that length? 00:02:45.780 --> 00:02:48.480 Well isn't R to B a radius of the circle? 00:02:48.480 --> 00:02:48.710 Right? 00:02:48.710 --> 00:02:49.780 It's going from the center to the edge of 00:02:49.780 --> 00:02:51.070 circle, so it's a radius. 00:02:51.070 --> 00:02:53.560 So RB has length 6. 00:02:53.560 --> 00:02:58.590 And RB is a diagonal of this rectangle just like AC is, so 00:02:58.590 --> 00:03:01.490 they're symmetric, so this is also going to be equal to 6. 00:03:01.490 --> 00:03:03.530 And you could look at it a bunch of different ways, you 00:03:03.530 --> 00:03:06.030 could say that W squared plus L squared is 00:03:06.030 --> 00:03:07.290 equal to this, AC squared. 00:03:07.290 --> 00:03:10.660 Well, W squared, this is W as well, right, you could also 00:03:10.660 --> 00:03:12.570 say that W squared plus L squared is equal 00:03:12.570 --> 00:03:13.790 to this side squared. 00:03:13.790 --> 00:03:15.290 So that's why they're equal to each other. 00:03:15.290 --> 00:03:21.660 So we know length of AC is equal to 6, as well. 00:03:21.660 --> 00:03:23.250 Right? 00:03:23.250 --> 00:03:24.120 We figured out that. 00:03:24.120 --> 00:03:26.160 So we have two pieces left. 00:03:26.160 --> 00:03:31.150 We just have that piece, and that piece. 00:03:31.150 --> 00:03:33.750 What's the length of this piece? 00:03:33.750 --> 00:03:35.760 Well, this whole distance is 6. 00:03:35.760 --> 00:03:38.110 We figured that out because that's a radius. 00:03:38.110 --> 00:03:40.260 And this piece is L, right? 00:03:40.260 --> 00:03:40.830 That's L. 00:03:40.830 --> 00:03:44.150 So that piece is 6 minus L. 00:03:44.150 --> 00:03:45.810 Similarly, what's this piece? 00:03:45.810 --> 00:03:48.440 By the same logic, that's 6 minus W, right? 00:03:48.440 --> 00:03:50.740 This is W, the whole thing is 6, so this 00:03:50.740 --> 00:03:53.040 leftover is 6 minus W. 00:03:53.040 --> 00:03:59.680 So if we want the perimeter of this entire shaded region, 00:03:59.680 --> 00:04:05.150 it's this perimeter of this brown arc, which is 3 pi. 00:04:05.150 --> 00:04:09.010 So let me say, perimeter is equal to 3 pi. 00:04:09.010 --> 00:04:13.040 Plus the length of AC, plus 6. 00:04:13.040 --> 00:04:17.240 Plus this piece, plus 6 minus L. 00:04:17.240 --> 00:04:22.200 Plus this piece, plus 6 minus W. 00:04:22.200 --> 00:04:24.030 And let's simplify that a bit. 00:04:24.030 --> 00:04:28.020 So let's see, that equals 3 pi, and then we have three 00:04:28.020 --> 00:04:32.690 6's, so plus 18, minus L, minus W. 00:04:32.690 --> 00:04:41.190 And this is the same thing as 3 pi plus 18 minus L plus W. 00:04:41.190 --> 00:04:42.440 Fascinating. 00:04:42.440 --> 00:04:45.470 And really, I kind of just bumbled my way here. 00:04:45.470 --> 00:04:47.620 You know the problem is solvable, and that's better 00:04:47.620 --> 00:04:50.430 than most mathematicians have going for them, and so you 00:04:50.430 --> 00:04:52.660 just kind of have to bumble away with whatever information 00:04:52.660 --> 00:04:52.940 they give you. 00:04:52.940 --> 00:04:55.820 The real trick here, I think, is recognizing that AC is the 00:04:55.820 --> 00:04:57.490 same length as RB. 00:04:57.490 --> 00:04:59.380 But once you get to this point, you have to now employ 00:04:59.380 --> 00:04:59.970 the second trick. 00:04:59.970 --> 00:05:02.270 You have to realize that minus L minus W, well that looks a 00:05:02.270 --> 00:05:04.850 lot like this L plus W up here. 00:05:04.850 --> 00:05:08.000 And this is really kind of what separates the adults from 00:05:08.000 --> 00:05:09.910 the children, I guess. 00:05:09.910 --> 00:05:13.830 So we know that L plus W is 8, so just substitute here, 8. 00:05:13.830 --> 00:05:19.980 So the perimeter is 3 pi, plus 18, minus 8. 00:05:19.980 --> 00:05:27.090 That equals 3 pi plus 10, and that is choice B. 00:05:27.090 --> 00:05:28.960 That was exciting. 00:05:28.960 --> 00:05:30.940 I'll see you in the next section.

SAT Prep: Test 6 Section 3 Part 5

https://www.youtube.com/watch?v=PH92iJRiUgg

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https://www.youtube.com/api/timedtext?v=PH92iJRiUgg&ei=YmeUZbDSNoG5mLAPhMSbuAU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8A41D8E3EF1A7000D969D10DA17A074E4D98DBF2.92A1FA374B528E8B4BB8854C0CB9E22191A929B5&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.210 --> 00:00:04.610 We're on problem 16. 00:00:04.610 --> 00:00:06.960 And they want to know what is the area of the shaded square? 00:00:06.960 --> 00:00:10.590 Let me shade it because they call it shaded square. 00:00:10.590 --> 00:00:13.100 So what's that area? 00:00:13.100 --> 00:00:15.030 So how can we do it? 00:00:15.030 --> 00:00:19.180 Well, it's a bit of a trick, but what you need to realize 00:00:19.180 --> 00:00:26.880 is, you could draw-- Let me draw some things that might-- 00:00:26.880 --> 00:00:28.610 This is actually a really fun problem. 00:00:28.610 --> 00:00:30.900 Let me draw some interesting rectangles here. 00:00:34.162 --> 00:00:38.020 Let me draw a rectangle there. 00:00:38.020 --> 00:00:44.640 Let me draw another rectangle there to there. 00:00:44.640 --> 00:00:48.226 This really is just kind of a brain teaser. 00:00:48.226 --> 00:00:50.510 Draw another rectangle from there to there. 00:00:56.680 --> 00:01:06.930 And then let me draw another rectangle from there to there. 00:01:10.410 --> 00:01:14.720 OK, so let me ask you a couple of things. 00:01:14.720 --> 00:01:19.980 One: what is the distance of the base of this rectangle 00:01:19.980 --> 00:01:20.820 right here? 00:01:20.820 --> 00:01:22.090 This distance. 00:01:22.090 --> 00:01:25.540 Well, we know this entire-- whoops, wrong tool-- we know 00:01:25.540 --> 00:01:28.490 this entire distance is 3, right? 00:01:28.490 --> 00:01:32.310 Because they tell us-- oh, they don't even tell us. 00:01:32.310 --> 00:01:33.910 They tell us that this distance is this 1. 00:01:33.910 --> 00:01:36.300 So we know that this is a cube, right? 00:01:36.300 --> 00:01:38.110 That this distance is 3. 00:01:38.110 --> 00:01:41.180 So this distance is 1. 00:01:41.180 --> 00:01:45.580 And then this distance down here is 3. 00:01:45.580 --> 00:01:48.445 And they tell us that this is a shaded square, right? 00:01:48.445 --> 00:01:51.770 So all the sides of this inner shape are also 00:01:51.770 --> 00:01:53.510 going to be the same. 00:01:53.510 --> 00:01:56.280 So if this is 1, this is 2, this is also 00:01:56.280 --> 00:01:59.220 going to be 2, right? 00:01:59.220 --> 00:02:02.970 That's going to be 2, that's going to be 1, that's going to 00:02:02.970 --> 00:02:05.780 be 1, this is going to be 2, 1. 00:02:05.780 --> 00:02:08.580 I think you get when I'm saying, right? 00:02:08.580 --> 00:02:11.830 And what is the distances of the smaller 00:02:11.830 --> 00:02:14.520 square inside of here? 00:02:14.520 --> 00:02:18.940 Well, if this distance is 1, this distance is 1 from here 00:02:18.940 --> 00:02:22.870 to here, this distance is also going to be 1 00:02:22.870 --> 00:02:24.640 because 1 plus 1 is 2. 00:02:24.640 --> 00:02:25.660 That's 1. 00:02:25.660 --> 00:02:27.200 Same reason, you can make the same argument. 00:02:27.200 --> 00:02:30.410 This is 1, this is 1, this is 1. 00:02:30.410 --> 00:02:34.280 So the area of this small square inside, what's the area 00:02:34.280 --> 00:02:35.740 of that small square? 00:02:35.740 --> 00:02:39.370 What's the area of the magenta square that I just filled in? 00:02:39.370 --> 00:02:41.270 Well that area is going to be 1. 00:02:41.270 --> 00:02:41.610 right? 00:02:41.610 --> 00:02:43.240 The magenta square is 1. 00:02:43.240 --> 00:02:49.420 So now all we have to figure out is the area of these 4 00:02:49.420 --> 00:02:50.630 yellow triangles. 00:02:50.630 --> 00:02:54.360 And what are the areas of each of those yellow triangles? 00:02:54.360 --> 00:02:58.300 Well, each of those yellow triangles are-- they have a 00:02:58.300 --> 00:03:02.140 dimension, on the long side, the dimension is 2. 00:03:02.140 --> 00:03:03.570 Let me find a good color. 00:03:03.570 --> 00:03:06.450 On the long side, the dimension is 2, right? 00:03:06.450 --> 00:03:10.220 This distance right here is 2. 00:03:10.220 --> 00:03:13.610 And on the short side, the distance is 1. 00:03:13.610 --> 00:03:16.950 So its area, each yellow triangle is 2 times 1 times 00:03:16.950 --> 00:03:17.850 1/2, right? 00:03:17.850 --> 00:03:19.890 Because area of a triangle is area equals 00:03:19.890 --> 00:03:22.160 1/2 base times height. 00:03:22.160 --> 00:03:28.880 So area of each of those triangles is going to be 1/2 00:03:28.880 --> 00:03:34.950 times 2 times 1, so that equals 1, right? 00:03:34.950 --> 00:03:37.390 So the area of this triangle is 1, the area of this 00:03:37.390 --> 00:03:39.006 triangle is 1, the area of this triangle is 1, the area 00:03:39.006 --> 00:03:40.020 of this triangle is 1. 00:03:40.020 --> 00:03:42.800 And then the area of the square inside is also 1. 00:03:42.800 --> 00:03:45.540 So it's 1 plus 1 plus 1 plus 1 plus 1. 00:03:45.540 --> 00:03:47.360 The total area is 5. 00:03:47.360 --> 00:03:50.920 Area equals 5, that was pretty neat. 00:03:50.920 --> 00:03:54.990 And the big thing you need to realize is that you can split 00:03:54.990 --> 00:03:59.205 up the square on the inside like this. 00:03:59.205 --> 00:04:03.200 Let me think if there was another way. 00:04:03.200 --> 00:04:04.440 That's the way that it occurred to me. 00:04:04.440 --> 00:04:07.130 Maybe there's another way, maybe I'm over-complicating 00:04:07.130 --> 00:04:07.900 it, I don't know. 00:04:07.900 --> 00:04:10.100 But it looks nice. 00:04:10.100 --> 00:04:12.850 Next problem. 00:04:12.850 --> 00:04:16.899 For all positive integers, j and k, let j-- I like these 00:04:16.899 --> 00:04:20.680 where they define new math operations-- j square with an 00:04:20.680 --> 00:04:26.780 r inside of it, k, be defined as the whole number remainder 00:04:26.780 --> 00:04:29.070 when j is divided by k. 00:04:29.070 --> 00:04:34.320 So this equals remainder when j is divided by k. 00:04:34.320 --> 00:04:36.680 If you're familiar with the concept of a modulus, that's 00:04:36.680 --> 00:04:39.890 essentially what this operation is. 00:04:39.890 --> 00:04:47.760 If 13-- I guess we'll call this remainder operation-- k 00:04:47.760 --> 00:04:53.010 is equal to 2, what is the value of k? 00:04:53.010 --> 00:04:55.780 And they tell us in the beginning that j and k, all of 00:04:55.780 --> 00:04:57.870 these things have to be positive numbers, right? 00:04:57.870 --> 00:05:01.790 So this is essentially saying that when I divide 13 by some 00:05:01.790 --> 00:05:12.510 positive number, I get a remainder of 2. 00:05:12.510 --> 00:05:16.160 Well, what number when I divide it into 13 is 2? 00:05:16.160 --> 00:05:19.620 I guess another way to think about it is 13 minus 2 is a 00:05:19.620 --> 00:05:22.040 multiple of this number, right? 00:05:22.040 --> 00:05:24.050 So you could say, what's 13 minus 2? 00:05:24.050 --> 00:05:25.270 Well it's 11. 00:05:25.270 --> 00:05:30.170 So 11 is a multiple of k, and actually 11 should be k 00:05:30.170 --> 00:05:36.310 because if 11's a multiple of k, and 11's not a multiple of 00:05:36.310 --> 00:05:38.090 much, right? 00:05:38.090 --> 00:05:42.320 Eleven's only a multiple of 1 and 11, so k has to be 11. 00:05:42.320 --> 00:05:43.510 And you can test it out. 00:05:43.510 --> 00:05:45.050 I mean, if you get confused, just try 00:05:45.050 --> 00:05:46.300 out different numbers. 00:05:48.500 --> 00:05:51.460 13 divided by 11 is equal to one, right? 00:05:51.460 --> 00:05:53.450 Because 11 goes in 13 one time. 00:05:53.450 --> 00:05:57.020 Remainder 2. 00:05:57.020 --> 00:05:57.920 Which satisfies this. 00:05:57.920 --> 00:06:02.830 13 remainder 11 is equal to 2. 00:06:02.830 --> 00:06:04.080 Next problem. 00:06:06.810 --> 00:06:09.270 Problem 18. 00:06:09.270 --> 00:06:11.720 The average of the test scores of a class of 00:06:11.720 --> 00:06:14.060 p students is 70. 00:06:14.060 --> 00:06:17.190 The average is 70, so what do we know about that? 00:06:17.190 --> 00:06:21.130 We know that if we were to add up all of the p students-- 00:06:21.130 --> 00:06:23.170 well, actually, let me read the rest of the problem before 00:06:23.170 --> 00:06:24.670 I do anything. 00:06:24.670 --> 00:06:27.150 And the average of the test scores of a class of n 00:06:27.150 --> 00:06:29.620 students is 92. 00:06:29.620 --> 00:06:32.150 When the scores of both classes are combined, the 00:06:32.150 --> 00:06:34.450 average score is 86. 00:06:34.450 --> 00:06:36.750 What is the value of p/n? 00:06:36.750 --> 00:06:43.880 OK, so what is the sum of the p student scores? 00:06:43.880 --> 00:06:47.720 So their average is 70, so the sum of all of their scores is 00:06:47.720 --> 00:06:51.810 going to be 70p, right? 00:06:51.810 --> 00:06:53.320 You can take the average and multiply it by 00:06:53.320 --> 00:06:54.440 the number of students. 00:06:54.440 --> 00:06:58.260 And then you get the sum of all of their scores, right? 00:06:58.260 --> 00:07:00.300 And if that doesn't make a lot of sense, think about it. 00:07:00.300 --> 00:07:01.375 What's the definition of average? 00:07:01.375 --> 00:07:10.920 It's the sum divided by p is equal to the average, right? 00:07:10.920 --> 00:07:15.520 So if you multiply both sides by p, you get the sum is equal 00:07:15.520 --> 00:07:18.940 to p times the average. 00:07:18.940 --> 00:07:21.690 In this case, the average is 70. 00:07:21.690 --> 00:07:24.590 So 70 times p is the sum of the first class, and then 00:07:24.590 --> 00:07:26.560 what's the sum of the second class? 00:07:26.560 --> 00:07:30.810 It's going to be 92n for the same exact reason. 00:07:30.810 --> 00:07:34.430 And now, how many total students are we averaging? 00:07:34.430 --> 00:07:37.380 Well there are p in this sum and there are n in this sum, 00:07:37.380 --> 00:07:41.640 so it's p plus n, and they tell us that this average 00:07:41.640 --> 00:07:43.370 score is 86. 00:07:43.370 --> 00:07:46.410 So we set up our equation and now let's solve. 00:07:46.410 --> 00:07:53.040 Let's multiply both sides by p plus n, so you get 70p plus 00:07:53.040 --> 00:08:03.410 92n is equal to 86p plus 86n, right? p plus n times 86, 00:08:03.410 --> 00:08:05.360 distribute the 86. 00:08:05.360 --> 00:08:06.180 Now what do we want to do? 00:08:06.180 --> 00:08:09.650 We eventually want to figure out p/n, so let's put the n's 00:08:09.650 --> 00:08:11.020 on the left-hand side. 00:08:11.020 --> 00:08:15.220 So I'm going to subtract 86n from both sides, so you get 00:08:15.220 --> 00:08:22.710 70p plus-- 92 minus 86 is 6n is equal to 86p, right? 00:08:22.710 --> 00:08:25.590 I subtracted this from the other side. 00:08:25.590 --> 00:08:29.690 And then let's subtract 70p from both sides, so you get 6n 00:08:29.690 --> 00:08:32.360 is equal to 16p, right? 00:08:32.360 --> 00:08:34.140 86 minus 70. 00:08:34.140 --> 00:08:35.965 And then we want to know p/n. 00:08:40.309 --> 00:08:46.620 So we divide both sides by n, you get 6 is equal to 16p/n 00:08:46.620 --> 00:08:48.920 and now divide both sides by 16, and you get 00:08:48.920 --> 00:08:53.090 6/16 is equal to p/n. 00:08:53.090 --> 00:08:56.300 And, of course, we can reduced 6/16. 00:08:56.300 --> 00:09:00.030 6/16 is the same thing as 3/8. 00:09:00.030 --> 00:09:03.040 That is our answer for p/n. 00:09:03.040 --> 00:09:05.170 I'll see you in the next section.

SAT Prep: Test 6 Section 3 Part 1

https://www.youtube.com/watch?v=Zee_TGgEBsg

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https://www.youtube.com/api/timedtext?v=Zee_TGgEBsg&ei=YmeUZaezNqqep-oPnOWPMA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=1B49D5BCEEEF523BF9C8B28378E1CE960EFF606D.D15F4E51F0F11101BA2A8BC9AB3F6115CAF34704&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.760 --> 00:00:04.200 We're now on test number 6 in section 3. 00:00:04.200 --> 00:00:06.140 Let's just get started. 00:00:06.140 --> 00:00:08.250 Problem number 1. 00:00:08.250 --> 00:00:13.560 Which of the following numbers is between 1/5 and 1/4? 00:00:13.560 --> 00:00:15.170 So when you look at the choices, you immediately see 00:00:15.170 --> 00:00:17.340 that all of the numbers are decimals. 00:00:17.340 --> 00:00:19.010 So let's just convert these to decimals. 00:00:19.010 --> 00:00:20.290 1/5 is what? 00:00:20.290 --> 00:00:22.320 It's 0.2. 00:00:22.320 --> 00:00:25.270 And 1/4 is 0.25. 00:00:25.270 --> 00:00:29.390 So we need a number that's larger-- so let me call it x-- 00:00:29.390 --> 00:00:32.170 than 0.2 and less than 0.25. 00:00:32.170 --> 00:00:35.980 And if you look at the choices, d is that answer. 00:00:35.980 --> 00:00:38.840 And if you didn't know off hand that 1/5 is 0.2, you 00:00:38.840 --> 00:00:42.890 could divide 5 into 1.0. 00:00:42.890 --> 00:00:46.850 5 goes into 10 two times. 00:00:46.850 --> 00:00:50.450 2 times 10 is 10, 0, and then, of course, the decimal you 00:00:50.450 --> 00:00:52.590 bring up, so it's 0.2 times. 00:00:52.590 --> 00:00:54.540 And just do the same thing with 1/4. 00:00:54.540 --> 00:00:55.970 Next problem. 00:00:55.970 --> 00:00:58.630 Problem 2. 00:00:58.630 --> 00:01:02.150 The following are coordinates of points in the xy plane. 00:01:02.150 --> 00:01:05.800 Which of these points is nearest the origin? 00:01:05.800 --> 00:01:08.690 So if we look at all of the choices-- well, let me 00:01:08.690 --> 00:01:10.110 actually write all the choices down. 00:01:10.110 --> 00:01:15.020 We have a, a says 0, minus 1/2. 00:01:17.680 --> 00:01:20.160 b is-- I was thinking whether I should draw these for you, 00:01:20.160 --> 00:01:21.540 but it's good to learn how to do this without 00:01:21.540 --> 00:01:25.540 drawing-- 0, 1/2. 00:01:25.540 --> 00:01:32.400 c is 1/2, minus 1/2. 00:01:32.400 --> 00:01:38.550 d is 1/2, 1/2. 00:01:38.550 --> 00:01:45.530 And e is minus 1, minus 1. 00:01:45.530 --> 00:01:50.110 So if we look at all the choices, we didn't move at all 00:01:50.110 --> 00:01:51.520 along the x-axis here, right? 00:01:51.520 --> 00:01:52.560 Because we're still at x equals 0. 00:01:52.560 --> 00:01:54.740 We want to figure out how far it is away from 0. 00:01:54.740 --> 00:01:57.130 So we didn't move at all from the x-axis and we went one 00:01:57.130 --> 00:02:00.510 below the y-axis here, to minus one. 00:02:00.510 --> 00:02:03.070 Clearly b is closer than a because b is 00:02:03.070 --> 00:02:05.220 only 1/2 away, right? 00:02:05.220 --> 00:02:08.320 Once again, we didn't move on the x-axis, and we only moved 00:02:08.320 --> 00:02:10.470 1/2 away on the y-axis. 00:02:10.470 --> 00:02:11.680 So a is not our choice. 00:02:11.680 --> 00:02:13.350 So far, b is our best contender. 00:02:13.350 --> 00:02:17.790 It's only 1/2 away from the origin. 00:02:17.790 --> 00:02:22.720 c is 1/2 away in the x direction and 1/2 away in the 00:02:22.720 --> 00:02:25.720 y direction, and if you do the math, if you use the 00:02:25.720 --> 00:02:27.365 Pythagorean theorem, you'll see that, of course, is going 00:02:27.365 --> 00:02:29.080 to be further than 1/2. 00:02:29.080 --> 00:02:31.270 And let me just draw a little bit of a graphic for you just 00:02:31.270 --> 00:02:34.240 so you get the intuition. 00:02:34.240 --> 00:02:35.550 How I can do that without drawing it. 00:02:35.550 --> 00:02:36.800 So this point is 0, 1/2. 00:02:42.630 --> 00:02:44.200 So that point is here. 00:02:44.200 --> 00:02:48.450 That's our best contender right now, 0, 1/2. 00:02:48.450 --> 00:02:52.480 The point 1/2, negative 1/2 is here. 00:02:52.480 --> 00:02:56.570 See, this is 1/2 minus 1/2 here. 00:03:01.160 --> 00:03:03.450 So its distance from the origin is 00:03:03.450 --> 00:03:06.450 this line right here. 00:03:06.450 --> 00:03:08.980 And that difference is definitely going to be longer 00:03:08.980 --> 00:03:09.730 than this distance. 00:03:09.730 --> 00:03:11.170 How do I know? 00:03:11.170 --> 00:03:13.890 Because this distance is the same thing as this distance, 00:03:13.890 --> 00:03:17.800 it's the same thing as 1/2 to the negative side, right? 00:03:17.800 --> 00:03:19.570 And if you do the Pythagorean theorem, we 00:03:19.570 --> 00:03:20.340 could figure it out. 00:03:20.340 --> 00:03:24.420 This distance is 1/2 half as well. 00:03:24.420 --> 00:03:28.700 This is going to be the square root of 1/2 squared plus 1/2 00:03:28.700 --> 00:03:31.950 squared, so 1/4 plus 1/4, which is 1/2. 00:03:31.950 --> 00:03:35.020 So it's the square root of 1/2, which is 1 over the 00:03:35.020 --> 00:03:37.190 square root of 2. 00:03:37.190 --> 00:03:39.550 Which is the same thing as two square roots of 2 over 2. 00:03:39.550 --> 00:03:42.750 Any way you look at it, it is a bigger number than 1/2. 00:03:42.750 --> 00:03:46.120 When you take a square root of a fraction, you're going to 00:03:46.120 --> 00:03:47.680 get something bigger than the fraction, right? 00:03:47.680 --> 00:03:49.250 So this is essentially the square root of 1/2. 00:03:49.250 --> 00:03:51.810 So it's going to be bigger than 1/2. 00:03:51.810 --> 00:03:53.930 So we know that this isn't the choice. 00:03:53.930 --> 00:03:57.005 And d is the same distance as c, d is up here, it's just in 00:03:57.005 --> 00:03:59.740 the positive quadrant, right? 00:03:59.740 --> 00:04:02.190 This distance is 1/2 and then this distance is 1/2. 00:04:02.190 --> 00:04:07.320 So d, for the same reasons, is further then b. 00:04:07.320 --> 00:04:09.910 And e is the furthest of them all, right? 00:04:09.910 --> 00:04:12.890 It's one away in two directions, so it's out here 00:04:12.890 --> 00:04:14.520 someplace relative to this origin, so 00:04:14.520 --> 00:04:15.330 that's not the choice. 00:04:15.330 --> 00:04:18.190 So b is definitely the closest, and it's only 1/2 00:04:18.190 --> 00:04:20.610 units away from the origin. 00:04:20.610 --> 00:04:21.860 Next problem. 00:04:25.040 --> 00:04:28.120 I will change colors for variety. 00:04:28.120 --> 00:04:29.370 Problem 3. 00:04:31.730 --> 00:04:33.330 OK, they drew a lot here. 00:04:33.330 --> 00:04:36.090 Let me see if I can draw the same thing they did. 00:04:36.090 --> 00:04:37.746 So they have a horizontal line. 00:04:40.920 --> 00:04:43.380 I want to draw just like the way they did it. 00:04:43.380 --> 00:04:51.730 And so then there's 1, 2, 3, 4, 5. 00:04:51.730 --> 00:04:53.660 I'm telling you, drawing it is the hardest part. 00:04:56.180 --> 00:05:04.110 And they have something like this, like that, 00:05:04.110 --> 00:05:09.310 and then like that. 00:05:09.310 --> 00:05:12.410 And then the next two don't go all the way through, they go 00:05:12.410 --> 00:05:14.090 kind of halfway. 00:05:14.090 --> 00:05:15.050 That one goes like that. 00:05:15.050 --> 00:05:17.100 And that one. 00:05:17.100 --> 00:05:19.090 That's my best shot. 00:05:19.090 --> 00:05:25.670 And they say that this is x degrees, x degrees, x degrees, 00:05:25.670 --> 00:05:28.980 x degrees, x degrees, x degrees, x degrees. 00:05:28.980 --> 00:05:32.940 And they say that this big one right here is y degrees. 00:05:32.940 --> 00:05:37.180 In the figure above, what is the value of y? 00:05:37.180 --> 00:05:40.880 Well, the first thing I would do is figure out what the 00:05:40.880 --> 00:05:42.080 value of x is. 00:05:42.080 --> 00:05:43.840 And how do I do that? 00:05:43.840 --> 00:05:46.470 Well, you could make a whole circle or whatever, but you 00:05:46.470 --> 00:05:52.010 could say look, all of these x's here, how many is that? 00:05:52.010 --> 00:05:55.470 1, 2, 3, 4, 5, x's right? 00:05:55.470 --> 00:05:58.660 These five x's are collectively supplementary to 00:05:58.660 --> 00:05:59.350 each other, right? 00:05:59.350 --> 00:06:01.590 They're going to add up to 180 degrees. 00:06:01.590 --> 00:06:07.240 So we know that 5x is equal to 180 degrees. 00:06:07.240 --> 00:06:12.230 And that x is equal to-- how many times does 5 go into 180? 00:06:12.230 --> 00:06:17.740 5 goes into 150 30 times, and so there's another 30, so 00:06:17.740 --> 00:06:20.210 it'll go into it 36 times. 00:06:20.210 --> 00:06:22.680 So x equal to 36 degrees, you just divide 00:06:22.680 --> 00:06:24.470 180 by 5 to get that. 00:06:24.470 --> 00:06:27.780 So how do we figure out y? 00:06:27.780 --> 00:06:30.820 Well, the easy way to do this is say well, if you go into 00:06:30.820 --> 00:06:38.180 the other side of the circle, that's also going to be equal 00:06:38.180 --> 00:06:40.470 to 180 degrees, right? 00:06:40.470 --> 00:06:41.910 So we have two x's here. 00:06:45.510 --> 00:06:48.160 And then we have y, right? 00:06:48.160 --> 00:06:54.090 So you could say y plus 2x is equal to 180. 00:06:54.090 --> 00:06:55.290 I know what x is. 00:06:55.290 --> 00:06:59.990 x is 36 degrees, so y plus 2 times 36 is 00:06:59.990 --> 00:07:03.170 72 is equal to 180. 00:07:03.170 --> 00:07:12.340 y is equal to 180 minus 72, which is 108 degrees. 00:07:12.340 --> 00:07:15.360 Now, if you wanted to do this really, really, really fast, 00:07:15.360 --> 00:07:17.080 what you could have said is well, you know what? 00:07:17.080 --> 00:07:19.440 If I kept drawing these x's all the way 00:07:19.440 --> 00:07:21.730 around the circle, right? 00:07:21.730 --> 00:07:27.420 So let's say that I had lines like this. 00:07:27.420 --> 00:07:30.820 Say this line actually just continues on like this. 00:07:30.820 --> 00:07:35.310 And let's say that this line continued on like this. 00:07:35.310 --> 00:07:38.790 And we know that this is kind of a visualization exercise, 00:07:38.790 --> 00:07:42.290 but you'd have known that this is x, this is x, this is x. 00:07:42.290 --> 00:07:43.620 And then how many x's total are there? 00:07:43.620 --> 00:07:46.790 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 00:07:46.790 --> 00:07:49.020 There would have been 10 x's, right? 00:07:49.020 --> 00:07:51.630 And we're going all the way around the circle, so 10x 00:07:51.630 --> 00:07:54.090 would equal 360. 00:07:54.090 --> 00:07:55.760 x is equal to 36. 00:07:55.760 --> 00:07:58.380 And then a really easy thing, you could've said well y is 3 00:07:58.380 --> 00:08:00.150 of these x's, right? 00:08:00.150 --> 00:08:01.870 1, 2, 3. 00:08:01.870 --> 00:08:04.190 y is three of those x's, so three times 00:08:04.190 --> 00:08:06.650 36 is equal to 108. 00:08:06.650 --> 00:08:07.740 Either way would've been fine. 00:08:07.740 --> 00:08:09.970 I think the way we did it the first time, well, it didn't 00:08:09.970 --> 00:08:11.880 take too long, but if you can do it faster, 00:08:11.880 --> 00:08:13.832 that all the better. 00:08:13.832 --> 00:08:16.490 Because at some point, the SAT really just 00:08:16.490 --> 00:08:18.770 becomes a speed exam. 00:08:18.770 --> 00:08:20.020 Problem number 4. 00:08:22.800 --> 00:08:34.130 If 6,565 is equal to 65 times x plus 1, then x equals? 00:08:34.130 --> 00:08:36.280 So the real trick here is being able to divide fast. 00:08:36.280 --> 00:08:42.710 Divide both sides by 65, you get 65 goes into 6,565 goes 00:08:42.710 --> 00:08:48.200 into 65 one time, 1 time 65, 0, 0, 6, goes into it 0 times. 00:08:48.200 --> 00:08:51.410 Get a 0, 65, goes into 65 one time. 00:08:51.410 --> 00:08:54.560 So you divide both sides by 65, you get 101 is 00:08:54.560 --> 00:08:56.920 equal to x plus 1. 00:08:56.920 --> 00:09:00.180 Subtract 1 from both sides, you get 100 is equal to x. 00:09:00.180 --> 00:09:01.960 This is just kind of a speed problem. 00:09:01.960 --> 00:09:04.500 Maybe they want to make you have a careless mistake. 00:09:04.500 --> 00:09:05.920 The answer is c. 00:09:05.920 --> 00:09:07.880 I'll see you in the next video.

SAT Prep: Test 6 Section 3 Part 3

https://www.youtube.com/watch?v=QPKMZHYBcTc

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https://www.youtube.com/api/timedtext?v=QPKMZHYBcTc&ei=YmeUZbfsNe68mLAP05mAoAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249811&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=7704C193A617F9B162586344E0417FE2F2B721B0.2D87C7B879F5889D8D36A2822EE859146E9BCB0E&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.470 --> 00:00:02.930 Welcome back. 00:00:02.930 --> 00:00:05.020 I don't like to rush too much on these videos. 00:00:05.020 --> 00:00:07.060 I just wanted to continue the next one. 00:00:07.060 --> 00:00:09.170 So what we did, I just wrote out everything they tell us in 00:00:09.170 --> 00:00:10.040 the problem. 00:00:10.040 --> 00:00:12.780 We figured out x is equal to y, so they must both be 00:00:12.780 --> 00:00:14.520 45-degree angles. 00:00:14.520 --> 00:00:17.910 And since y is opposite of this angle, it must be 45. 00:00:17.910 --> 00:00:20.690 And since this angle, this angle, and this angle are the 00:00:20.690 --> 00:00:23.450 same triangle and this one's 45 and this one's 90, this 00:00:23.450 --> 00:00:25.410 must also equal 45. 00:00:25.410 --> 00:00:28.070 And what I said is we need to figure out this green line. 00:00:28.070 --> 00:00:30.280 And I made the claim, and I think you'll believe me, that 00:00:30.280 --> 00:00:31.930 this green line is the same length of the 00:00:31.930 --> 00:00:33.410 line that I just drew. 00:00:33.410 --> 00:00:36.400 This peach line, right? 00:00:36.400 --> 00:00:39.610 And also, what else do we know about this larger triangle? 00:00:39.610 --> 00:00:41.505 Well, its hypotenuse is here. 00:00:41.505 --> 00:00:45.480 Its hypotenuse is this magenta line right there. 00:00:45.480 --> 00:00:47.920 And let me redraw it here. 00:00:47.920 --> 00:00:49.390 So its hypotenuse is this magenta line. 00:00:49.390 --> 00:00:51.120 What's the length of the magenta line? 00:00:51.120 --> 00:00:52.550 It's 4 plus 8. 00:00:52.550 --> 00:00:54.480 They gave us that information. 00:00:54.480 --> 00:00:56.510 So it's 12. 00:00:56.510 --> 00:01:01.360 And we want to figure out this peach line. 00:01:01.360 --> 00:01:04.720 I'm just re-drawing it so that you know that we're clear of 00:01:04.720 --> 00:01:08.360 all of the mess, right? 00:01:08.360 --> 00:01:10.400 I'm saying this green line is the same as this peach line, 00:01:10.400 --> 00:01:12.620 and I just re-drew this peach line here. 00:01:12.620 --> 00:01:15.010 What do we also know about this triangle? 00:01:15.010 --> 00:01:18.870 Well, if this angle up here is 45, what's this angle here? 00:01:18.870 --> 00:01:20.410 That's also going to be 45, right? 00:01:20.410 --> 00:01:24.590 Because these two lines are perpendicular, so this is 45. 00:01:24.590 --> 00:01:26.870 And we also know that this angle x is this angle here, so 00:01:26.870 --> 00:01:27.850 that's also 45. 00:01:27.850 --> 00:01:30.020 Or you can figure it out any other way. 00:01:30.020 --> 00:01:31.440 So there we go. 00:01:31.440 --> 00:01:33.770 We have a 45-45-90 triangle. 00:01:33.770 --> 00:01:34.710 We know its hypotenuse. 00:01:34.710 --> 00:01:37.650 Can we figure out the sides? 00:01:37.650 --> 00:01:39.500 Well we know the sides are going to be equal, right? 00:01:39.500 --> 00:01:40.860 Because the base angles are equal. 00:01:40.860 --> 00:01:43.175 We also know that in a 45-45-90 triangle 00:01:43.175 --> 00:01:44.610 the sides are equal. 00:01:44.610 --> 00:01:47.270 In fact, you can even look at the beginning of the book, 00:01:47.270 --> 00:01:50.130 they even define a 45-45-90 triangle. 00:01:50.130 --> 00:01:53.540 But we'll work it out just so you see it worked out without 00:01:53.540 --> 00:01:55.690 having to memorize. 00:01:55.690 --> 00:01:59.310 So x squared-- this is just Pythagorean theorem-- plus x 00:01:59.310 --> 00:02:00.380 squared is equal to the hypotenuse 00:02:00.380 --> 00:02:03.500 squared is equal to 144. 00:02:03.500 --> 00:02:07.290 2x squared is equal to 144. 00:02:07.290 --> 00:02:11.410 x squared is equal to 72. 00:02:11.410 --> 00:02:14.140 x is equal to the square root of 72. 00:02:14.140 --> 00:02:16.530 So what's the square root of 72? 00:02:16.530 --> 00:02:19.780 The square root of 72 is the square root of 36 times 2, 00:02:19.780 --> 00:02:23.440 which is the same thing as the square root of 36 times the 00:02:23.440 --> 00:02:25.320 square root of 2. 00:02:25.320 --> 00:02:27.090 And what does that turn out to be? 00:02:27.090 --> 00:02:30.050 Well, the square root of 36 is 6, right? 00:02:30.050 --> 00:02:36.290 So our answer is 6 square roots of 2. 00:02:36.290 --> 00:02:40.900 So that is choice b. 00:02:40.900 --> 00:02:42.150 Next problem. 00:02:49.410 --> 00:02:50.660 Problem 8. 00:02:53.100 --> 00:02:55.410 The price of ground coffee bean is d 00:02:55.410 --> 00:02:57.980 dollars for 8 ounces. 00:02:57.980 --> 00:03:04.710 So d dollars for 8 ounces. 00:03:04.710 --> 00:03:10.810 And each ounce makes c cups of brewed coffee. 00:03:10.810 --> 00:03:18.130 So one ounce makes c cups of brewed coffee. 00:03:18.130 --> 00:03:30.920 In terms of c and d, what is the dollar cost of the ground 00:03:30.920 --> 00:03:34.190 coffee beans required to make one cup? 00:03:34.190 --> 00:03:36.320 We just want to make one cup and we say how much 00:03:36.320 --> 00:03:38.950 does that cost us? 00:03:38.950 --> 00:03:40.110 OK. 00:03:40.110 --> 00:03:45.470 So let's see how many cups we can make with d dollars, OK? 00:03:45.470 --> 00:03:47.790 So d dollars, we would get 8 ounces. 00:03:47.790 --> 00:03:53.190 So d dollars, we get 8 ounces. 00:03:53.190 --> 00:03:57.320 And then each ounce, you get c cups, right? 00:03:57.320 --> 00:04:00.540 One ounce turns into c cups, so 8 ounces will 00:04:00.540 --> 00:04:02.020 turn into 8c cups. 00:04:06.220 --> 00:04:08.550 But we only want one cup, right? 00:04:08.550 --> 00:04:16.279 So we're saying d dollars is equal to 8c cups. 00:04:16.279 --> 00:04:19.060 But we only want one cup, so let's divide both sides of 00:04:19.060 --> 00:04:20.310 this by 8c. 00:04:24.070 --> 00:04:25.870 So what are we left with? 00:04:25.870 --> 00:04:32.890 We're left with d over 8c dollars is equal to one cup. 00:04:32.890 --> 00:04:37.010 And that is choice a. 00:04:37.010 --> 00:04:39.270 Not too bad, huh? 00:04:39.270 --> 00:04:40.520 Next problem. 00:04:44.350 --> 00:04:45.655 Problem number 9. 00:04:48.930 --> 00:04:57.250 If 10/a is equal to b/12, what is the value of ab? 00:04:57.250 --> 00:05:00.220 So this is just straight cross-multiplication. 00:05:00.220 --> 00:05:05.160 a times b is equal to 10 times 12, so ab is equal to 10 times 00:05:05.160 --> 00:05:07.290 12, which equals 120. 00:05:07.290 --> 00:05:08.120 We are done. 00:05:08.120 --> 00:05:09.670 120 is our answer. 00:05:09.670 --> 00:05:11.440 If you haven't learned to cross-multiply, it's just a 00:05:11.440 --> 00:05:14.880 quick way-- when you see two fractions equal each other, 00:05:14.880 --> 00:05:17.520 you can take the numerator of the left times the denominator 00:05:17.520 --> 00:05:18.570 of the right. 00:05:18.570 --> 00:05:20.710 And that equals the denominator of the left times 00:05:20.710 --> 00:05:21.880 the numerator of the right. 00:05:21.880 --> 00:05:22.820 And how do we know that? 00:05:22.820 --> 00:05:27.110 Well, we could do it step by step. 00:05:27.110 --> 00:05:30.680 You can multiply both sides times a. 00:05:30.680 --> 00:05:32.350 That cancels out and you're left with 10 is 00:05:32.350 --> 00:05:35.060 equal to ab over 12. 00:05:35.060 --> 00:05:37.950 And then you multiply both sides times 12, you get 120 is 00:05:37.950 --> 00:05:39.630 equal to ab. 00:05:39.630 --> 00:05:42.350 That just skips one step right there. 00:05:42.350 --> 00:05:43.600 Next problem. 00:05:47.660 --> 00:05:51.180 Problem 10. 00:05:51.180 --> 00:05:52.360 They give us a little sequence here. 00:05:52.360 --> 00:05:58.990 150, 30, 6, and then it keeps going. 00:05:58.990 --> 00:06:01.730 In the sequence above, each term after the first term is 00:06:01.730 --> 00:06:03.780 1/5 of the preceding term. 00:06:03.780 --> 00:06:06.310 What is the fifth term of the sequence? 00:06:06.310 --> 00:06:08.550 So 1/5 of the preceding term. 00:06:08.550 --> 00:06:11.260 So the next term is going to be 1/5 of this. 00:06:11.260 --> 00:06:15.260 So that's what, that's 6/5, right? 00:06:15.260 --> 00:06:17.040 I just multiplied 1/5 times that. 00:06:17.040 --> 00:06:20.100 So the next term is going to be 1/5 times this, so that's 00:06:20.100 --> 00:06:21.900 6/25, right? 00:06:21.900 --> 00:06:28.400 1/5 times 6/5 is equal to 6/25. 00:06:28.400 --> 00:06:33.080 So that is the fifth term. 00:06:33.080 --> 00:06:35.220 And you could write it as a fraction or you could write it 00:06:35.220 --> 00:06:37.020 as a decimal, I think you'd get 0.24 if 00:06:37.020 --> 00:06:39.370 you do it as a decimal. 00:06:39.370 --> 00:06:41.660 Next problem. 00:06:41.660 --> 00:06:43.600 I'm suspicious, that seemed too easy. 00:06:43.600 --> 00:06:51.930 Problem 11. 00:06:51.930 --> 00:06:55.520 Five points-- a, b, c, d, and e-- lie on a line, not 00:06:55.520 --> 00:06:57.540 necessarily in that order. 00:06:57.540 --> 00:07:03.170 ab has length 24. 00:07:03.170 --> 00:07:05.120 Point c is the midpoint of ab. 00:07:05.120 --> 00:07:07.590 So let's draw this and let's see if we can 00:07:07.590 --> 00:07:09.590 avoid messing up. 00:07:09.590 --> 00:07:11.450 OK. 00:07:11.450 --> 00:07:14.810 So let me draw a and b. 00:07:14.810 --> 00:07:22.770 So a, b. 00:07:22.770 --> 00:07:24.876 This is how I'd do it if I was taking it, taking 00:07:24.876 --> 00:07:27.230 the test. ab is 24. 00:07:27.230 --> 00:07:29.380 Point c is the midpoint of ab. 00:07:29.380 --> 00:07:31.620 So point c is right here. 00:07:31.620 --> 00:07:32.810 And it's the midpoint. 00:07:32.810 --> 00:07:37.140 So this is 12, and this is 12. 00:07:37.140 --> 00:07:39.040 So far so good? 00:07:39.040 --> 00:07:41.650 And point d is the midpoint of ac. 00:07:44.820 --> 00:07:48.540 So this distance is 6, and this distance is 6, right? 00:07:48.540 --> 00:07:49.330 It's getting smaller. 00:07:49.330 --> 00:07:50.910 I should have drawn this bigger, but I think you get 00:07:50.910 --> 00:07:51.710 what I'm saying. 00:07:51.710 --> 00:07:55.380 This is point d here in yellow. 00:07:55.380 --> 00:08:02.530 If the distance between d and e-- so this is d-- and the 00:08:02.530 --> 00:08:07.950 distance between d and e is 5, what is one possible distance 00:08:07.950 --> 00:08:09.000 between a and e? 00:08:09.000 --> 00:08:11.895 So they're essentially saying where could e be? 00:08:14.400 --> 00:08:17.520 So let me blow things up a little bit. 00:08:17.520 --> 00:08:19.160 Let me redraw this. 00:08:21.890 --> 00:08:25.180 And let me focus on a and c because that's the part that 00:08:25.180 --> 00:08:27.980 got all scrunched up. 00:08:27.980 --> 00:08:38.840 So if this is a, this is c, and they tell us that d is the 00:08:38.840 --> 00:08:40.090 midpoint of these two. 00:08:42.919 --> 00:08:47.970 And we figured out that a to c was 12 and a to d is 6, right? 00:08:47.970 --> 00:08:51.040 This distance is 6, which is the same as this distance, 00:08:51.040 --> 00:08:52.870 that's also 6. 00:08:52.870 --> 00:08:55.270 And then they tell us at the end, if the distance between d 00:08:55.270 --> 00:08:57.930 and e is 5, what is one possible distance 00:08:57.930 --> 00:08:59.240 between a and e? 00:08:59.240 --> 00:09:01.920 So we know that e is on this line as well. 00:09:01.920 --> 00:09:04.880 So there's two places where e could be. 00:09:04.880 --> 00:09:08.950 e could be 5 in this direction, it could be here. 00:09:08.950 --> 00:09:15.650 e could be there or e could be 5 in this direction where this 00:09:15.650 --> 00:09:19.540 is 5 and this is 5. 00:09:19.540 --> 00:09:24.390 So if e is here, what is the distance from a to e? 00:09:24.390 --> 00:09:25.810 Well, this distance from a to e is just 00:09:25.810 --> 00:09:27.240 going to be 1, right? 00:09:27.240 --> 00:09:31.160 You go one to e and then 5 to d, and that adds up to 6. 00:09:31.160 --> 00:09:37.260 So one possible distance for ae is 1. 00:09:37.260 --> 00:09:41.880 The other one is you go 6 to d and then you go 5 to e. 00:09:41.880 --> 00:09:44.510 So the other possibility is 11. 00:09:44.510 --> 00:09:46.680 And so, oh, this isn't multiple choice. 00:09:46.680 --> 00:09:49.070 You could fill out either one of those choices. 00:09:49.070 --> 00:09:50.830 See you in the next video.

SAT Prep: Test 6 Section 3 Part 2

https://www.youtube.com/watch?v=gWq0WpgBthI

vtt

https://www.youtube.com/api/timedtext?v=gWq0WpgBthI&ei=YmeUZeXWNfvMhcIPzN6coA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=09BFFA0900085D3D340282FD404C0B21B8129201.3DC562AA91C960D3F4A252E05F139B8796A840B6&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.530 --> 00:00:01.920 All right. 00:00:01.920 --> 00:00:04.930 We're on problem number five. 00:00:04.930 --> 00:00:08.052 OK, problem number five. 00:00:08.052 --> 00:00:17.170 If m to the x times m to the 7th is equal to 28. 00:00:17.170 --> 00:00:20.800 They also tell us that m to the 5th, and then that whole 00:00:20.800 --> 00:00:26.700 thing to the y power, is equal to m to the 15th. 00:00:26.700 --> 00:00:30.236 What is the value of x plus y? 00:00:30.236 --> 00:00:31.872 x plus y equals what? 00:00:31.872 --> 00:00:34.710 So this is really just a test of whether you know your 00:00:34.710 --> 00:00:36.100 exponent rules. 00:00:36.100 --> 00:00:39.680 So what is m to the x times m to the 7th? 00:00:39.680 --> 00:00:42.670 So your temptation might have been to say 7x, but no. 00:00:42.670 --> 00:00:45.970 Remember, when you multiply two numbers of the same base, 00:00:45.970 --> 00:00:48.620 and the base is m, you add the exponents. 00:00:48.620 --> 00:00:51.020 So it's m to the x plus 7. 00:00:51.020 --> 00:00:53.280 And they actually wanted to tempt you to do 7x because 00:00:53.280 --> 00:00:55.940 they actually use the number 28 here, which is a multiple 00:00:55.940 --> 00:00:58.540 of 7, so they were tempting you to do 7x. 00:00:58.540 --> 00:01:01.820 But we know that you add the exponents when you multiply m 00:01:01.820 --> 00:01:03.860 to the x times m to the 7th. 00:01:03.860 --> 00:01:09.060 And so we have m to the x times m to the 7th-- oh sorry, 00:01:09.060 --> 00:01:10.105 I wrote it down wrong. 00:01:10.105 --> 00:01:12.080 It equals m to the 28. 00:01:12.080 --> 00:01:13.990 Sorry, well you knew that. 00:01:13.990 --> 00:01:16.935 Anyway, m to the x times m to the 7th is equal to m to the x 00:01:16.935 --> 00:01:21.280 plus 7, and that equals m to the 28. 00:01:21.280 --> 00:01:25.660 So the exponents have to equal each other, so x plus 7 is 00:01:25.660 --> 00:01:27.860 equal to 28. 00:01:27.860 --> 00:01:31.060 x is equal to 21. 00:01:31.060 --> 00:01:32.340 Now let's go here. 00:01:32.340 --> 00:01:34.950 Here we have m to the 5th, and then we're raising 00:01:34.950 --> 00:01:37.080 that to the yth power. 00:01:37.080 --> 00:01:40.240 So here we actually would multiply the exponents, if you 00:01:40.240 --> 00:01:41.800 remember your exponent rules. 00:01:41.800 --> 00:01:48.160 So you have m to the 5y is equal to m to the 15th. 00:01:48.160 --> 00:01:50.610 So 5y must equal 15. 00:01:50.610 --> 00:01:56.710 5y is equal to 15, y is equal to 3. 00:01:56.710 --> 00:01:57.820 So what's x plus y? 00:01:57.820 --> 00:02:02.280 It's 21 plus 3, which equals 24. 00:02:02.280 --> 00:02:05.670 And that is choice d. 00:02:05.670 --> 00:02:06.920 Next problem. 00:02:10.430 --> 00:02:13.070 Looks like there's going to be some drawing 00:02:13.070 --> 00:02:14.676 for Sal in this problem. 00:02:14.676 --> 00:02:20.160 I'll try my best. Oh, I thought I was using-- undo. 00:02:20.160 --> 00:02:21.250 Edit undo. 00:02:21.250 --> 00:02:22.770 I thought I was using the line tool. 00:02:22.770 --> 00:02:24.510 OK. 00:02:24.510 --> 00:02:26.880 That's my y-axis, draw a straight line. 00:02:26.880 --> 00:02:28.570 I drew a straight line. 00:02:28.570 --> 00:02:32.270 That's my x-axis. 00:02:32.270 --> 00:02:33.540 And then what do they have here? 00:02:33.540 --> 00:02:44.730 They have 19-- let's see, 1985, '86, '87, 00:02:44.730 --> 00:02:51.300 '88, '89, then '90. 00:02:51.300 --> 00:02:57.240 Let's see, 1985 is, maybe, looks about 148,000. 00:02:57.240 --> 00:03:03.450 So if this is 150,000, 1985 is here. 00:03:03.450 --> 00:03:05.710 Let me just draw it all out. 00:03:05.710 --> 00:03:11.540 1985, then 1986 is up here at 160,000. 00:03:11.540 --> 00:03:13.096 This is 140,000. 00:03:13.096 --> 00:03:13.436 soon. 00:03:13.436 --> 00:03:17.370 So 1986 is up here at 160,000. 00:03:17.370 --> 00:03:24.280 1987 is a little bit less, 1987 looks like about 150,000, 00:03:24.280 --> 00:03:27.340 I don't know, eyeballing it at 156,000. 00:03:27.340 --> 00:03:33.870 1987 is here. 00:03:33.870 --> 00:03:44.270 1988 looks like a little bit above 140,000. 00:03:44.270 --> 00:03:48.410 1989 is between 130,000 and 120,000. 00:03:48.410 --> 00:03:50.950 All the work here is drawing what you hopefully already 00:03:50.950 --> 00:03:52.430 have in your book. 00:03:52.430 --> 00:03:54.080 So where was I? 00:03:54.080 --> 00:03:56.390 Oh sorry, this is 1988. 00:03:56.390 --> 00:04:00.120 1988 was a little bit better than 140,000. 00:04:00.120 --> 00:04:00.710 Ignore that one. 00:04:00.710 --> 00:04:09.620 1989 looks like about 126,000. 00:04:09.620 --> 00:04:13.310 Then 1990 is, I don't know, 112,000 or something. 00:04:13.310 --> 00:04:18.184 1990 is out here. 00:04:18.184 --> 00:04:23.350 And then if I were to connect the dots, look 00:04:23.350 --> 00:04:24.600 something like that. 00:04:30.230 --> 00:04:31.540 And then let's read the question. 00:04:31.540 --> 00:04:33.890 According to the graph above, which of the following is 00:04:33.890 --> 00:04:37.280 closest to the decrease per year in the number of homes 00:04:37.280 --> 00:04:41.610 sold between 1987 and 1990. 00:04:41.610 --> 00:04:44.890 So we're talking about from 1987, so it's from 00:04:44.890 --> 00:04:47.410 this year, to 1990. 00:04:47.410 --> 00:04:50.220 So how much did we decrease in the first year, right? 00:04:50.220 --> 00:04:52.070 The first year we decreased by this much. 00:04:52.070 --> 00:04:54.600 And I'm going to look in the book because I don't know how 00:04:54.600 --> 00:04:56.700 well I drew this graph. 00:04:56.700 --> 00:05:04.910 But in 1987, it looks like we're about 156,000. 00:05:04.910 --> 00:05:11.950 And then as we go to 1988, it looks like about 142,000 just 00:05:11.950 --> 00:05:13.620 eyeballing it, right? 00:05:18.130 --> 00:05:20.180 So this distance right here is what? 00:05:20.180 --> 00:05:23.710 It'd be 156,000 minus 142,000. 00:05:23.710 --> 00:05:27.010 About minus 14,000, right? 00:05:27.010 --> 00:05:29.966 That's how much we changed over this one year. 00:05:29.966 --> 00:05:32.860 And if I look at the choices, there already is choice c, 00:05:32.860 --> 00:05:34.920 which is 14,000, but let me just do one more 00:05:34.920 --> 00:05:36.460 just to make sure. 00:05:36.460 --> 00:05:39.180 So if we want to figure out from 1988 to 1989, so we said 00:05:39.180 --> 00:05:44.080 1988 is at 142,000, where is 1989? 00:05:44.080 --> 00:05:59.310 Looks like 128,000, I would say 128,000. 00:05:59.310 --> 00:06:04.270 And so 142,000 minus 128,000, that's also 14,000. 00:06:04.270 --> 00:06:08.060 And if we do the last one, that's 128,000, and then this 00:06:08.060 --> 00:06:13.110 looks something about 114,000 right here. 00:06:13.110 --> 00:06:15.660 So this is also 14,000. 00:06:15.660 --> 00:06:19.450 So the answer is c, 14,000. 00:06:19.450 --> 00:06:21.520 Next problem. 00:06:21.520 --> 00:06:22.770 Problem 7. 00:06:25.293 --> 00:06:30.450 you Here we have this diagram that shows up actually a lot 00:06:30.450 --> 00:06:31.675 when you're taking the SAT. 00:06:31.675 --> 00:06:34.510 I'll show you why. 00:06:34.510 --> 00:06:37.110 This is a very common thing you'll see. 00:06:39.950 --> 00:06:43.335 A line like that. 00:06:43.335 --> 00:06:46.790 A line like that. 00:06:46.790 --> 00:06:48.986 And then they connect it here. 00:06:48.986 --> 00:06:50.465 Now let's read the problem. 00:06:53.140 --> 00:06:55.560 I already have suspicion what they're going to ask. 00:06:55.560 --> 00:07:04.080 So this is a, b, c, d, and e. 00:07:04.080 --> 00:07:06.390 They tell us this is perpendicular. 00:07:06.390 --> 00:07:07.910 This is perpendicular. 00:07:07.910 --> 00:07:09.090 This is x. 00:07:09.090 --> 00:07:11.020 This is y. 00:07:11.020 --> 00:07:18.190 In the figure above, ae and cd are each particular to ce. 00:07:18.190 --> 00:07:19.450 Fair enough. 00:07:19.450 --> 00:07:26.920 If x is equal to y, the length of ab is 4, and the length of 00:07:26.920 --> 00:07:31.460 bd is 8, what is the length of ce? 00:07:31.460 --> 00:07:34.250 So we're trying to figure out-- let me draw ce, let's 00:07:34.250 --> 00:07:36.860 have our gold in a different color-- we're trying to figure 00:07:36.860 --> 00:07:39.950 out this line right there. 00:07:39.950 --> 00:07:42.700 And one thing I didn't write, they told us x is equal to y, 00:07:42.700 --> 00:07:44.700 so immediately that should hit a trigger. 00:07:44.700 --> 00:07:48.510 x is equal to y and this is a 90-degree angle, so we know 00:07:48.510 --> 00:07:56.490 that x plus y plus 90 is equal to 180, or that x plus y is 00:07:56.490 --> 00:07:57.300 equal to 90. 00:07:57.300 --> 00:08:00.960 And if they're equal to each other, x must equal y, which 00:08:00.960 --> 00:08:03.310 equals 45 degrees, right? 00:08:03.310 --> 00:08:08.510 So this is 45, this is 45, and if y is 45, what is this angle 00:08:08.510 --> 00:08:09.240 going to be? 00:08:09.240 --> 00:08:14.280 Well it's opposite to y, so this will also be 45 degrees. 00:08:14.280 --> 00:08:18.760 And this is a right angle too, so 45 plus 90 plus this angle 00:08:18.760 --> 00:08:21.800 has to equal 180, so what's this angle going to be? 00:08:21.800 --> 00:08:24.270 Well, that's also going to be 45 degrees. 00:08:24.270 --> 00:08:26.420 This also has to be a 45-45-90 triangle. 00:08:26.420 --> 00:08:29.010 If you ever see a triangle where one of the angles is 90 00:08:29.010 --> 00:08:31.650 and the other is 45, the other one has to be 45 because they 00:08:31.650 --> 00:08:33.530 add up to 180. 00:08:33.530 --> 00:08:37.199 And what we want to figure out is this length here. 00:08:37.199 --> 00:08:39.330 So I'm going to show you a little trick, and this trick-- 00:08:39.330 --> 00:08:41.190 well, it doesn't always work, it's not really a trick, it's 00:08:41.190 --> 00:08:44.800 just a visualization exercise, really-- we say, we want to 00:08:44.800 --> 00:08:46.750 figure out this length. 00:08:46.750 --> 00:08:48.690 Well that length, I'm going to say-- I'm going to make the 00:08:48.690 --> 00:08:51.250 claim, and I think you'll believe this claim is the same 00:08:51.250 --> 00:08:53.130 is this length, I'm essentially creating a 00:08:53.130 --> 00:08:57.920 rectangle-- is the same as that length, right? 00:08:57.920 --> 00:08:59.750 So if we figure out this length, we solve the problem, 00:08:59.750 --> 00:09:02.760 this light brown length. 00:09:02.760 --> 00:09:06.170 And we also know-- let me draw the bottom-- 00:09:06.170 --> 00:09:07.610 do we know the length? 00:09:07.610 --> 00:09:10.460 Actually, we don't even have to know this bottom line. 00:09:10.460 --> 00:09:12.180 We don't know the length of that bottom line. 00:09:12.180 --> 00:09:14.410 We need to figure out this length. 00:09:14.410 --> 00:09:17.690 We know since this line is parallel to this line. 00:09:17.690 --> 00:09:20.820 We know this is a 90-degree angle. 00:09:20.820 --> 00:09:23.740 Well, do we know the hypotenuse? 00:09:23.740 --> 00:09:24.280 Sure. 00:09:24.280 --> 00:09:27.120 The hypotenuse is 4 plus 8, it's this big red line. 00:09:27.120 --> 00:09:29.880 And let me draw the hypotenuse in a different color. 00:09:29.880 --> 00:09:34.090 The hypotenuse, right here, that's the hypotenuse. 00:09:34.090 --> 00:09:35.790 Actually, I'm running out of time, so I'm going to continue 00:09:35.790 --> 00:09:37.490 this in the next video. 00:09:37.490 --> 00:09:38.740 See

SAT Prep: Test 6 Section 3 Part 4

https://www.youtube.com/watch?v=i1aNc26PsOI

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https://www.youtube.com/api/timedtext?v=i1aNc26PsOI&ei=YmeUZfGONInKhcIPicG4gA8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=C8FAB9C70B0AF610F7B89A600DF34D77E67E6903.420AAA0EE1CAC570EAF7B135C18D054EC2C4CAFB&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.660 --> 00:00:03.040 We are on problem number 12. 00:00:03.040 --> 00:00:05.890 Problem 12. 00:00:05.890 --> 00:00:08.615 OK, what is the greatest of five consecutive integers if 00:00:08.615 --> 00:00:10.750 the sum of these integers is 185? 00:00:10.750 --> 00:00:11.630 So what is the greatest? 00:00:11.630 --> 00:00:15.160 Let's assume x is the greatest. And they're 00:00:15.160 --> 00:00:17.220 consecutive integers, so what's the second greatest? 00:00:17.220 --> 00:00:19.580 Well, that would be x minus 1. 00:00:19.580 --> 00:00:23.462 The third greatest would be x minus 2. 00:00:23.462 --> 00:00:24.960 x minus 3. 00:00:24.960 --> 00:00:26.050 x minus 4, right? 00:00:26.050 --> 00:00:28.590 These are five consecutive integers where x is the 00:00:28.590 --> 00:00:30.350 largest, right? 00:00:30.350 --> 00:00:35.460 If x was 150, x minus 4 would be-- these numbers would be 00:00:35.460 --> 00:00:39.820 150, 149, 148, 147, 146 like that, right? 00:00:39.820 --> 00:00:42.140 So these are consecutive integers, x is the largest. 00:00:42.140 --> 00:00:47.100 And they're telling us that the sum is 185. 00:00:47.100 --> 00:00:49.720 So the sum is equal to 185. 00:00:49.720 --> 00:00:50.850 So what is this sum? 00:00:50.850 --> 00:00:52.220 How do sum these up? 00:00:52.220 --> 00:00:54.100 So we have five x's, right? 00:00:54.100 --> 00:00:56.210 So get 5x. 00:00:56.210 --> 00:00:59.180 And then what's minus 1, minus 2, minus 3, minus 4? 00:00:59.180 --> 00:01:02.700 So 1 plus 2 is 3, 3 plus 3 is-- this is where I always 00:01:02.700 --> 00:01:07.730 mess up-- 3 plus 3 6, 6 plus 4 is 10, right? 00:01:07.730 --> 00:01:11.060 So you get 5x minus 10 is equal to 185. 00:01:11.060 --> 00:01:12.770 Add 10 to both sides. 00:01:12.770 --> 00:01:17.160 5x is equal to 195. 00:01:17.160 --> 00:01:19.390 And you could eyeball this, you could say well, how many 00:01:19.390 --> 00:01:26.230 times does 5 go into 200, and it's going to be 1 less than 00:01:26.230 --> 00:01:27.480 that, right? 00:01:27.480 --> 00:01:30.740 Five goes into 200, what, if it goes into it 40 times, 00:01:30.740 --> 00:01:34.470 it'll be 39 or you can divide. 00:01:34.470 --> 00:01:37.090 If you didn't want to do it that way, you could say 5 goes 00:01:37.090 --> 00:01:41.680 into 195, 3 times 5 is 15. 00:01:41.680 --> 00:01:43.600 45, 39. 00:01:43.600 --> 00:01:45.140 So that's our answer, x is 39. 00:01:45.140 --> 00:01:46.690 It's the largest of the consecutive numbers. 00:01:46.690 --> 00:01:53.510 So those numbers are 39, 38, 37, 36, 35. 00:01:53.510 --> 00:01:56.370 Those are our five consecutive numbers. 00:01:56.370 --> 00:01:57.620 Next problem. 00:01:59.990 --> 00:02:03.590 Problem 13. 00:02:03.590 --> 00:02:12.770 A salesman's monthly gross pay consists of $1,200 plus 20 00:02:12.770 --> 00:02:15.230 percent of the dollar amount of the sales. 00:02:15.230 --> 00:02:19.070 So plus 20 percent of the sales, s 00:02:19.070 --> 00:02:20.460 I'll say is for sales. 00:02:20.460 --> 00:02:29.430 If his gross pay for one month was $2,500, what was the 00:02:29.430 --> 00:02:32.230 dollar amount of the sales for that month? 00:02:32.230 --> 00:02:34.670 Well when I read it, I essentially set up this 00:02:34.670 --> 00:02:35.420 equation, right? 00:02:35.420 --> 00:02:38.570 His compensation is $1,200 plus 20 percent of the sales. 00:02:38.570 --> 00:02:40.765 And we know that that equals $2,500 in this month. 00:02:40.765 --> 00:02:43.250 So let's subtract $1,200 from both sides. 00:02:43.250 --> 00:02:51.890 So you get 0.2s is equal to $1,300, right? 00:02:51.890 --> 00:02:57.410 And so s is equal to $1,300 divided by 0.2. 00:02:57.410 --> 00:03:00.620 And the way I like to think about this, 0.2, instead of 00:03:00.620 --> 00:03:04.110 having to do this decimal, which you could do, is I could 00:03:04.110 --> 00:03:05.310 have rewritten this. 00:03:05.310 --> 00:03:07.740 20 percent is the same thing as 1/5, right? 00:03:07.740 --> 00:03:10.800 So it's also 1/5 s is equal to $1,300. 00:03:10.800 --> 00:03:11.790 And now this is easy. 00:03:11.790 --> 00:03:17.450 Multiply both sides by 5, you get s is equal to $1,300 times 00:03:17.450 --> 00:03:19.630 5, which equals what? 00:03:19.630 --> 00:03:24.660 Thirteen times 5 is 65, and add the two 0's. 00:03:24.660 --> 00:03:26.430 So s is equal to $6,500. 00:03:26.430 --> 00:03:29.730 So that's the monthly sales that month. 00:03:29.730 --> 00:03:32.260 I was a little redundant with the word "monthly" just now. 00:03:32.260 --> 00:03:33.510 Next problem. 00:03:37.930 --> 00:03:40.790 Problem 14. 00:03:40.790 --> 00:03:42.520 OK, so they've drawn a circle. 00:03:46.570 --> 00:03:56.050 And then, let's see, they've told us that 00:03:56.050 --> 00:04:00.850 this is, let's see. 00:04:00.850 --> 00:04:11.800 They tell us that this is 40 degrees. 00:04:11.800 --> 00:04:14.880 Let's see, Naomi makes silver jewelry. 00:04:14.880 --> 00:04:17.690 For one style of earring, she cuts wedges from a silver 00:04:17.690 --> 00:04:20.860 disk, which I guess is depicted here, as shown in the 00:04:20.860 --> 00:04:21.410 figure above. 00:04:21.410 --> 00:04:23.990 Each wedge makes a 40-degree angle at the 00:04:23.990 --> 00:04:25.380 center of the disk. 00:04:25.380 --> 00:04:28.910 If the weight of each uncut disk is a uniformly 00:04:28.910 --> 00:04:33.070 distributed 2.5 grams-- that's how much this whole silver 00:04:33.070 --> 00:04:35.950 disk-- I should've done it in grey instead of magenta 00:04:35.950 --> 00:04:40.940 because it's silver, but anyway-- how many grams does 00:04:40.940 --> 00:04:43.220 each wedge weigh? 00:04:43.220 --> 00:04:46.220 So we essentially just have to figure out what fraction of 00:04:46.220 --> 00:04:49.510 the entire circle is each wedge? 00:04:49.510 --> 00:04:52.860 Well, each wedge is 40 degrees. 00:04:52.860 --> 00:04:55.500 How many degrees are in the entire circle? 00:04:55.500 --> 00:04:57.880 Hopefully it's second nature to you right now, but there's 00:04:57.880 --> 00:05:01.160 360 degrees in an entire circle, right? 00:05:01.160 --> 00:05:05.070 So the wedge is what fraction of the entire circle? 00:05:05.070 --> 00:05:13.960 Each wedge is 40 over 360 of the entire circle. 00:05:13.960 --> 00:05:15.120 And that's what? 00:05:15.120 --> 00:05:17.460 That's equal to 1/9. 00:05:17.460 --> 00:05:19.600 Divide the top and the bottom by 40, right? 00:05:19.600 --> 00:05:22.150 Four goes into 36, right, 9 times. 00:05:22.150 --> 00:05:24.640 So each wedge is 1/9 of the entire circle, and I just did 00:05:24.640 --> 00:05:25.780 that based on the fact that it's 40 00:05:25.780 --> 00:05:28.890 degrees over 360 degrees. 00:05:28.890 --> 00:05:33.170 So if each wedge is 1/9 of the entire silver piece and the 00:05:33.170 --> 00:05:36.536 silver piece weights 2.5 grams, then each wedge will 00:05:36.536 --> 00:05:43.470 weigh 1/9 times 2.5 grams, right? 00:05:43.470 --> 00:05:51.120 And if we want to write 2.5 as a fraction, how do we do that? 00:05:51.120 --> 00:05:53.740 That might be convenient right now. 00:05:53.740 --> 00:05:55.190 Well, I'll tell you, that's the same thing. 00:05:55.190 --> 00:05:57.580 And you could multiply it out, and that's all fine, but you 00:05:57.580 --> 00:06:03.410 could also recognize that 2.5 can also be written as 9/4. 00:06:03.410 --> 00:06:05.860 Oh no, no it can't, sorry. 00:06:05.860 --> 00:06:09.410 2.5 can be written as 5/4. 00:06:09.410 --> 00:06:10.660 No. 00:06:12.350 --> 00:06:14.220 I'm pathetic. 00:06:14.220 --> 00:06:18.940 2.5 can be written as 10/4, right? 00:06:18.940 --> 00:06:21.990 Because four goes into 10 two times. 00:06:21.990 --> 00:06:23.140 Why am I even writing 10/4? 00:06:23.140 --> 00:06:25.260 See, my brain is malfunctioning. 00:06:25.260 --> 00:06:29.950 2.5 can be written as 5/2, which is 10/4. 00:06:29.950 --> 00:06:32.560 I was somehow thinking of the number 9, and I'm not going to 00:06:32.560 --> 00:06:35.530 make excuses for my deficits. 00:06:35.530 --> 00:06:36.950 Anyway. 00:06:36.950 --> 00:06:38.990 So what's 1/9 times 5/2? 00:06:38.990 --> 00:06:41.440 It's 5/18. 00:06:41.440 --> 00:06:44.500 So each wedge is going to weight 5/18 of a gram. 00:06:44.500 --> 00:06:48.930 The other option is you could have divided 2.5 by 9 and you 00:06:48.930 --> 00:06:51.940 would have gotten another answer with decimals. 00:06:51.940 --> 00:06:53.260 Either one would have worked because these 00:06:53.260 --> 00:06:55.770 are free answer questions. 00:06:55.770 --> 00:06:57.870 Next problem. 00:06:57.870 --> 00:06:59.075 Can you actually use a calculator on 00:06:59.075 --> 00:07:00.240 the SAT these days? 00:07:00.240 --> 00:07:04.250 If you can, that would've been an option as well. 00:07:04.250 --> 00:07:07.940 Problem 15. 00:07:07.940 --> 00:07:14.280 If x squared minus y squared is equal to 10 and x plus y is 00:07:14.280 --> 00:07:18.610 equal to 5, what is x minus y? 00:07:18.610 --> 00:07:21.700 Before I even read the rest of the question, when I saw this, 00:07:21.700 --> 00:07:23.540 it should be like your knee-jerk reaction when you 00:07:23.540 --> 00:07:24.080 take the SAT. 00:07:24.080 --> 00:07:26.500 When you see x squared minus y squared equals 10, well that's 00:07:26.500 --> 00:07:29.590 the same thing as x plus y times x minus 00:07:29.590 --> 00:07:31.100 y is equal to 10. 00:07:31.100 --> 00:07:32.850 You just factor this difference of squares 00:07:32.850 --> 00:07:33.700 immediately. 00:07:33.700 --> 00:07:34.860 You almost can do that before you read 00:07:34.860 --> 00:07:36.450 the rest of the question. 00:07:36.450 --> 00:07:38.430 And they tell us what's x plus y? 00:07:38.430 --> 00:07:40.210 x plus y is 5. 00:07:40.210 --> 00:07:43.950 So 5 times x minus y is equal to 10. 00:07:43.950 --> 00:07:49.320 Divide both sides by 5, you get x minus y is equal to 2. 00:07:49.320 --> 00:07:51.860 That's it. 00:07:51.860 --> 00:07:53.110 Next problem. 00:07:56.800 --> 00:08:01.400 So they draw us-- let me color. 00:08:01.400 --> 00:08:06.150 So they draw us a rectangle. 00:08:06.150 --> 00:08:08.840 And then within that rectangle there is another rectangle. 00:08:08.840 --> 00:08:10.030 It looks like this. 00:08:10.030 --> 00:08:12.180 Another square, really. 00:08:12.180 --> 00:08:15.340 And what I've drawn doesn't look that much like a square, 00:08:15.340 --> 00:08:16.770 but it's close enough, I think. 00:08:19.270 --> 00:08:21.000 You get the idea. 00:08:21.000 --> 00:08:24.050 And this is, I see, a right angle, right angle, right 00:08:24.050 --> 00:08:26.380 angle, right angle. 00:08:26.380 --> 00:08:29.790 They tell us that this distance is 2, 00:08:29.790 --> 00:08:32.110 this distance is 3. 00:08:32.110 --> 00:08:38.090 In the figure above, what is the area of the shaded square? 00:08:38.090 --> 00:08:40.159 So this is a bit of a trick. 00:08:40.159 --> 00:08:42.350 And actually, let me do it in the next video so I can kind 00:08:42.350 --> 00:08:44.610 of delve deeper into the trick. 00:08:44.610 --> 00:08:46.490 I'll see you in the next video.

SAT Prep: Test 5 Section 9 Part 3

https://www.youtube.com/watch?v=7D3ErvtvOaU

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https://www.youtube.com/api/timedtext?v=7D3ErvtvOaU&ei=YmeUZe7JM_Wdp-oPr6iteA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249810&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=3CEF08E8118861BED8EF40FEA998B79FD88348A8.79947B36B16118D2EF6A4E175116F7A5A9C0B321&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.940 --> 00:00:01.570 Welcome back. 00:00:01.570 --> 00:00:06.680 I'm on problem number 10. 00:00:06.680 --> 00:00:10.980 Phillip used 4 pieces of masking tape, each 6 inches 00:00:10.980 --> 00:00:25.020 long, to put up each of his posters. 00:00:27.650 --> 00:00:30.150 So this is per poster. 00:00:33.080 --> 00:00:34.060 Good enough. 00:00:34.060 --> 00:00:37.610 Phillip had a 300-foot roll of masking tape when he started. 00:00:37.610 --> 00:00:40.130 So he starts at 300 feet. 00:00:40.130 --> 00:00:42.760 I can already tell you that some unit conversion will 00:00:42.760 --> 00:00:45.180 happen, because they're talking about 6 inches here 00:00:45.180 --> 00:00:47.940 and they're talking about 300 feet here. 00:00:47.940 --> 00:00:51.850 If no tape was wasted, which of the following represents 00:00:51.850 --> 00:00:55.330 the number of feet--and they underline it-- of masking tape 00:00:55.330 --> 00:00:59.080 that was left on the roll after he put up the n posters? 00:00:59.080 --> 00:01:02.330 And they actually tell us that 12 inches are equal to a foot 00:01:02.330 --> 00:01:06.140 in case you aren't from this planet. 00:01:06.140 --> 00:01:07.715 So how do we do this? 00:01:12.520 --> 00:01:15.460 So he's going to put up n posters. 00:01:15.460 --> 00:01:17.630 So he's going to start off-- well, how much 00:01:17.630 --> 00:01:18.830 tape will he use? 00:01:18.830 --> 00:01:21.390 So for each of those n posters, how much will he use? 00:01:21.390 --> 00:01:26.980 He uses 4 pieces times 6 inches. 00:01:26.980 --> 00:01:28.300 But we want to go into feet. 00:01:28.300 --> 00:01:29.230 We want to know how many feet are left. 00:01:29.230 --> 00:01:31.170 So let's just convert immediately to feet. 00:01:31.170 --> 00:01:36.070 6 inches is equal to how many feet? 00:01:36.070 --> 00:01:37.980 Well, it's half a foot. 00:01:37.980 --> 00:01:39.160 6/12 inches. 00:01:39.160 --> 00:01:42.000 So it equals 1/2 foot. 00:01:42.000 --> 00:01:46.790 So he does 4 pieces for each poster, and each of those 00:01:46.790 --> 00:01:48.640 pieces is 1/2 foot. 00:01:48.640 --> 00:01:50.800 And now we're immediately in feet length. 00:01:50.800 --> 00:01:52.290 So this is how much he uses. 00:01:52.290 --> 00:01:55.790 So he will use-- so 4 times 1/2 is just 2. 00:01:55.790 --> 00:01:57.640 So he uses 2n feet. 00:02:00.400 --> 00:02:06.210 So if he starts with 300, the amount that he has left is 00:02:06.210 --> 00:02:08.485 what he started with minus what he used. 00:02:08.485 --> 00:02:10.900 He used 2n. 00:02:10.900 --> 00:02:14.360 So he starts with 300 feet minus 2n feet, so that's the 00:02:14.360 --> 00:02:15.350 expression. 00:02:15.350 --> 00:02:18.600 That's choice B. 00:02:18.600 --> 00:02:20.680 Next problem. 00:02:20.680 --> 00:02:24.990 Problem 11. 00:02:24.990 --> 00:02:28.870 I'll switch to magenta. 00:02:28.870 --> 00:02:32.450 In the x, y coordinate plane, line m is the reflection of 00:02:32.450 --> 00:02:35.320 line l about the x-axis. 00:02:35.320 --> 00:02:42.650 If the slope of m-- so m slope-- is equal to minus 4/5, 00:02:42.650 --> 00:02:45.300 what is the slope of l? 00:02:45.300 --> 00:02:48.000 So you should hopefully be able to do this on the real 00:02:48.000 --> 00:02:50.980 exam without having to draw it. 00:02:50.980 --> 00:02:53.330 Or you could actually just draw a really quick and dirty 00:02:53.330 --> 00:02:55.740 one, and that actually probably would 00:02:55.740 --> 00:02:56.990 do the job for you. 00:03:00.020 --> 00:03:01.450 So minus 4/5. 00:03:01.450 --> 00:03:05.460 That means for every 5-- and it's a 00:03:05.460 --> 00:03:06.510 reflection about the x-axis. 00:03:06.510 --> 00:03:08.805 So let's draw a line m. 00:03:12.870 --> 00:03:14.510 Now let's just assume that the origin's here. 00:03:14.510 --> 00:03:16.350 They don't tell us that, but they don't 00:03:16.350 --> 00:03:17.150 tell us it's not that. 00:03:17.150 --> 00:03:21.860 So that's zero, one, two, three, four, five. 00:03:21.860 --> 00:03:24.900 And this is one, two, three, four. 00:03:24.900 --> 00:03:25.850 Let me do it here. 00:03:25.850 --> 00:03:28.330 One, two, three, four. 00:03:28.330 --> 00:03:30.490 I just want to draw it so you understand. 00:03:30.490 --> 00:03:34.410 So 4, minus 4, this is 5. 00:03:34.410 --> 00:03:36.120 So we know line m. 00:03:36.120 --> 00:03:42.530 For every line m, for every 5 it goes to the right, 00:03:42.530 --> 00:03:43.450 it goes down 4. 00:03:43.450 --> 00:03:46.590 So this could be a legitimate line m right here. 00:03:46.590 --> 00:03:48.670 It could be like this. 00:03:48.670 --> 00:03:50.840 Line m could look like that. 00:03:54.170 --> 00:03:57.367 So a reflection about the x-axis, if I were to reflect 00:03:57.367 --> 00:04:00.520 it about the x-axis. 00:04:00.520 --> 00:04:04.160 This is the x-axis right here, so I just want to take its 00:04:04.160 --> 00:04:06.190 mirror image, or flip it over the x-axis. 00:04:06.190 --> 00:04:07.440 It would look like this. 00:04:11.900 --> 00:04:14.960 Oh, I thought I was using the line tool. 00:04:14.960 --> 00:04:16.210 It would look like this. 00:04:18.829 --> 00:04:20.950 I'm still not using the line tool. 00:04:20.950 --> 00:04:23.430 Now, I'm using the line tool. 00:04:23.430 --> 00:04:25.440 It would look like that, right? 00:04:25.440 --> 00:04:27.270 So what's the slope here? 00:04:27.270 --> 00:04:35.230 Well, for every 5 I go to the right, I move up 4. 00:04:35.230 --> 00:04:37.940 So change in y. 00:04:37.940 --> 00:04:40.790 Let me make sure this is line m, this is line l. 00:04:40.790 --> 00:04:47.350 Change in y over change in x for line l is equal to 00:04:47.350 --> 00:04:48.600 positive 4/5. 00:04:52.260 --> 00:04:55.000 It shouldn't take you that long to do it. 00:04:55.000 --> 00:04:57.720 One thing that you could just do as well, you could just 00:04:57.720 --> 00:04:58.530 draw a quick and dirty one. 00:04:58.530 --> 00:05:00.160 It's like, well, if I have something with a negative 00:05:00.160 --> 00:05:02.640 slope-- let's say I have a negative, really 00:05:02.640 --> 00:05:04.590 shallow slope like that. 00:05:04.590 --> 00:05:07.040 If I flip it, it's going to have the same slope, but it's 00:05:07.040 --> 00:05:08.620 going to be a positive slope, but it's still going to be 00:05:08.620 --> 00:05:12.260 shallow, so it's going to be the same magnitude. 00:05:12.260 --> 00:05:14.290 You'll flip the sign. 00:05:14.290 --> 00:05:16.100 So the answer is B, 4/5. 00:05:16.100 --> 00:05:18.920 But I did this just to give you the intuition. 00:05:18.920 --> 00:05:21.940 The next problem. 00:05:21.940 --> 00:05:24.550 Or to let you know why you got it wrong, if you got it wrong. 00:05:24.550 --> 00:05:24.960 Anyway. 00:05:24.960 --> 00:05:28.090 Problem number 12. 00:05:28.090 --> 00:05:37.900 If n is equal to 3p, for what value of p is n equal to p? 00:05:51.630 --> 00:05:53.380 This is kind of crazy. 00:05:53.380 --> 00:05:56.240 And at first, I was like, what are they saying? 00:05:56.240 --> 00:05:57.890 And then I read one of the choices. 00:05:57.890 --> 00:06:02.810 Because no matter what, n is equal to 3p. 00:06:02.810 --> 00:06:08.100 There's no circumstance-- well, oh, sorry. 00:06:08.100 --> 00:06:09.820 I was incorrect. 00:06:09.820 --> 00:06:12.350 There is a circumstance in which n is equal to 3p. 00:06:15.850 --> 00:06:19.070 Well, what's the circumstance? 00:06:19.070 --> 00:06:25.290 Well, you might initially say, well, as long as p is not 0, n 00:06:25.290 --> 00:06:28.580 is going to be exactly 3 times p. 00:06:28.580 --> 00:06:31.890 But then in our statement, I just told you the answer. 00:06:31.890 --> 00:06:33.750 They both can be 0. 00:06:33.750 --> 00:06:37.680 If p is 0, then 3 times 0 is 0. 00:06:37.680 --> 00:06:39.540 So there's no real algebra there. 00:06:39.540 --> 00:06:42.310 It's just kind of to realize that 0 is a choice. 00:06:42.310 --> 00:06:43.970 And if you looked at the choices, you'd immediately see 00:06:43.970 --> 00:06:45.160 choice A is 0. 00:06:45.160 --> 00:06:45.870 Try it out. 00:06:45.870 --> 00:06:49.180 You say, oh, well, if p is 0, then n is also 0. 00:06:49.180 --> 00:06:51.930 So then n would equal p. 00:06:51.930 --> 00:06:53.140 So next problem. 00:06:53.140 --> 00:06:56.310 Problem 13. 00:06:56.310 --> 00:06:58.920 That was one of those problems that in some ways are so easy 00:06:58.920 --> 00:07:00.700 that you waste time on it, making sure 00:07:00.700 --> 00:07:03.300 you didn't miss something. 00:07:03.300 --> 00:07:07.265 Let's draw what they drew. 00:07:07.265 --> 00:07:09.350 So we have a line like that. 00:07:09.350 --> 00:07:11.385 I have a line like that. 00:07:11.385 --> 00:07:15.030 Then I have a line like that. 00:07:15.030 --> 00:07:18.250 And this is line l. 00:07:18.250 --> 00:07:20.580 This is y degrees. 00:07:20.580 --> 00:07:23.430 This is line m. 00:07:23.430 --> 00:07:29.520 This is x degrees, and this is line n right here. 00:07:29.520 --> 00:07:32.440 In the figure above, if z-- oh, they tell us this is z 00:07:32.440 --> 00:07:34.670 right here. 00:07:34.670 --> 00:07:42.155 In the figure above, if z is equal to 30, what is the value 00:07:42.155 --> 00:07:44.670 of x plus y? 00:07:44.670 --> 00:07:46.720 x plus y is what? 00:07:46.720 --> 00:07:48.800 Well, what can we figure out? 00:07:48.800 --> 00:07:53.050 What do we know about this angle right here? 00:07:53.050 --> 00:07:55.130 It's supplementary to y, right? 00:07:55.130 --> 00:07:56.760 So this is kind of the angle game, but we're going to have 00:07:56.760 --> 00:07:59.260 a little bit more variables than normal. 00:07:59.260 --> 00:08:01.810 It's supplementary to y, so this is going to be 180 minus 00:08:01.810 --> 00:08:03.880 y because y plus this angle are going to 00:08:03.880 --> 00:08:05.610 have to equal 180. 00:08:05.610 --> 00:08:07.980 And for the exact same reason, this angle right here 00:08:07.980 --> 00:08:12.870 is 180 minus x. 00:08:12.870 --> 00:08:13.530 And what do we know? 00:08:13.530 --> 00:08:17.115 We know this angle plus this angle plus z is equal to 180. 00:08:17.115 --> 00:08:18.730 So let's write that down. 00:08:18.730 --> 00:08:29.230 This angle, 180 minus y, plus this angle, plus 180 minus x, 00:08:29.230 --> 00:08:33.419 plus z is equal to 180 because they're 00:08:33.419 --> 00:08:35.390 all in the same triangle. 00:08:35.390 --> 00:08:37.380 So let's try our best to simplify this. 00:08:37.380 --> 00:08:44.540 Well, we could immediately get rid of one of the 180's on 00:08:44.540 --> 00:08:47.610 that side, and that becomes 0. 00:08:47.610 --> 00:08:50.100 z is 30, right? 00:08:50.100 --> 00:08:51.490 So let's simplify it. 00:08:51.490 --> 00:08:58.990 We get minus y minus x, and then you have 180 plus 30, 00:08:58.990 --> 00:09:02.660 plus 210, is equal to 0. 00:09:02.660 --> 00:09:05.160 Now let's add x and y to both sides. 00:09:05.160 --> 00:09:06.100 I'm kind of skipping a step. 00:09:06.100 --> 00:09:08.440 You could add x to both sides and then add y to both sides. 00:09:08.440 --> 00:09:12.410 But if you add x and y to both sides, you get 210 is 00:09:12.410 --> 00:09:14.190 equal to x plus y. 00:09:14.190 --> 00:09:15.680 And that's the answer. 00:09:15.680 --> 00:09:17.710 They want to know what x plus y is. 00:09:17.710 --> 00:09:20.440 And that is choice D. 00:09:20.440 --> 00:09:22.060 And so the trick here is really saying, well, they 00:09:22.060 --> 00:09:23.370 only give us z. 00:09:23.370 --> 00:09:26.440 The only thing I know is that z is in a triangle with this 00:09:26.440 --> 00:09:28.220 angle and this angle. 00:09:28.220 --> 00:09:30.800 And let me express those two angles in terms of x and y, 00:09:30.800 --> 00:09:34.800 because they are supplementary to x and y. 00:09:34.800 --> 00:09:36.610 See you in the next video.

SAT Prep: Test 5 Section 9 Part 1

https://www.youtube.com/watch?v=NLBp-Tq3TS4

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en

WEBVTT Kind: captions Language: en 00:00:00.590 --> 00:00:04.840 We're on Test 5, Section 9, on page 679. 00:00:04.840 --> 00:00:09.530 Let's do some problems. Problem number 1. 00:00:09.530 --> 00:00:13.560 If 6 cars out of 10 on an assembly line are red, what is 00:00:13.560 --> 00:00:16.320 the probability that a car selected at random from the 00:00:16.320 --> 00:00:18.700 assembly line will be red? 00:00:18.700 --> 00:00:20.390 Well, 6 out of 10 are red. 00:00:20.390 --> 00:00:24.640 So if I take a random one, I have a 6 in 10 chance of 00:00:24.640 --> 00:00:26.640 getting a red car. 00:00:26.640 --> 00:00:30.310 If you look at the choices, there is no 6/10, but you can 00:00:30.310 --> 00:00:32.080 see that this can be reduced. 00:00:32.080 --> 00:00:37.010 If you divide the top and the bottom by 2, you get 3/5. 00:00:37.010 --> 00:00:40.330 So this is really just an exercise in reducing a 00:00:40.330 --> 00:00:44.370 fraction to its lowest common or simplified form, so the 00:00:44.370 --> 00:00:46.930 answer is B. 00:00:46.930 --> 00:00:48.965 Problem 2. 00:00:48.965 --> 00:00:51.240 Let me draw that diagram. 00:00:51.240 --> 00:00:52.520 So I have a triangle. 00:00:55.310 --> 00:00:57.460 Let me try my best to draw this. 00:01:01.500 --> 00:01:03.830 There's an altitude that comes down the center. 00:01:03.830 --> 00:01:06.380 Let's put a 2. 00:01:06.380 --> 00:01:08.000 And then what do they write? 00:01:08.000 --> 00:01:13.090 They write this angle right here is w. 00:01:13.090 --> 00:01:20.320 This is x, y, z over here. 00:01:20.320 --> 00:01:23.170 And they say, note: figure not drawn to scale. 00:01:23.170 --> 00:01:24.760 AB is equal to BC. 00:01:24.760 --> 00:01:26.395 Oh, I didn't draw the letters in. 00:01:26.395 --> 00:01:32.740 So A, B, C, and then D. 00:01:32.740 --> 00:01:35.330 So they say that AB is equal to BC. 00:01:40.050 --> 00:01:42.010 So immediately, a trigger should go off in your head. 00:01:42.010 --> 00:01:45.490 If two sides are equal, what does it say about the base 00:01:45.490 --> 00:01:47.180 angles or their corresponding angles? 00:01:47.180 --> 00:01:49.800 That means that those are also going to be equal. 00:01:49.800 --> 00:01:53.700 So if those two sides are equal, then we also know that 00:01:53.700 --> 00:01:55.410 these two angles are equal. 00:01:55.410 --> 00:01:57.480 That should immediately be a trigger in your head, that we 00:01:57.480 --> 00:01:59.330 know that w is equal to z. 00:01:59.330 --> 00:02:01.890 What's the next part of the information? 00:02:01.890 --> 00:02:06.900 And BD, this is BD, bisects AC. 00:02:06.900 --> 00:02:10.220 so bisects means it essentially intersects it at 00:02:10.220 --> 00:02:11.039 its midpoint. 00:02:11.039 --> 00:02:16.380 So that also tells us that this distance here from A to D 00:02:16.380 --> 00:02:20.080 is the same as the distance from D to C. 00:02:20.080 --> 00:02:24.320 Which of the following cannot be concluded? 00:02:24.320 --> 00:02:26.050 That w is equal to x. 00:02:31.110 --> 00:02:32.850 Actually, immediately, I don't even know how I 00:02:32.850 --> 00:02:33.470 can conclude that. 00:02:33.470 --> 00:02:34.920 Let's look at the other choices. 00:02:34.920 --> 00:02:36.190 w is equal to z. 00:02:36.190 --> 00:02:38.760 Well, w is equal to z can definitely be concluded. 00:02:38.760 --> 00:02:39.990 That's choice B. 00:02:39.990 --> 00:02:42.410 So we know that choice B is w is equal to z. 00:02:42.410 --> 00:02:44.050 We know that this isn't it because we concluded that 00:02:44.050 --> 00:02:45.770 immediately. 00:02:45.770 --> 00:02:49.020 Choice C is that x is equal to y. 00:02:51.640 --> 00:02:53.370 That actually can be concluded. 00:02:53.370 --> 00:02:54.090 How do we know that? 00:02:54.090 --> 00:03:00.730 Well, since this bisects this line and this is an isosceles 00:03:00.730 --> 00:03:03.560 triangle, even though that's not how I drew it. 00:03:03.560 --> 00:03:06.390 This triangle is completely symmetric. 00:03:06.390 --> 00:03:07.930 Maybe I should draw a little bit better. 00:03:07.930 --> 00:03:11.240 How do I know it's completely symmetric? 00:03:11.240 --> 00:03:14.860 Because this side is equal to this side, that is equal to 00:03:14.860 --> 00:03:17.980 that, and this side is equal to this side. 00:03:17.980 --> 00:03:20.290 And of course, this side is equal to that side. 00:03:20.290 --> 00:03:21.930 And actually, we can also conclude that this is a 00:03:21.930 --> 00:03:24.970 90-degree angle, because if this was anything other than a 00:03:24.970 --> 00:03:28.870 90-degree angle, this wouldn't be able to bisect at the base. 00:03:28.870 --> 00:03:34.340 So choice C, we actually can deduce that x is equal to y. 00:03:34.340 --> 00:03:36.060 So that's not the answer. 00:03:36.060 --> 00:03:39.880 Choice D is AD is equal to DC. 00:03:39.880 --> 00:03:41.120 Well, they essentially told us that one. 00:03:41.120 --> 00:03:45.460 They said BD bisects AC, so that's not our answer. 00:03:45.460 --> 00:03:47.100 Choice E. 00:03:47.100 --> 00:03:49.100 DB is perpendicular to AC. 00:03:49.100 --> 00:03:52.130 And I just said the triangle is completely symmetric. 00:03:52.130 --> 00:03:54.620 And given that it's an isosceles triangle, the only 00:03:54.620 --> 00:03:57.410 way that BD can bisect this bottom line is if it's 00:03:57.410 --> 00:03:59.220 perpendicular. 00:03:59.220 --> 00:04:01.570 So we can also rule out E. 00:04:01.570 --> 00:04:03.380 So our answer really is A. 00:04:03.380 --> 00:04:06.550 And really, there was no way that we could figure out. 00:04:06.550 --> 00:04:09.780 And a good way of thinking about that is actually we 00:04:09.780 --> 00:04:12.560 could make this point-- we could make the length of this 00:04:12.560 --> 00:04:16.339 line, this length BD, arbitrarily high because they 00:04:16.339 --> 00:04:17.670 gave us no information. 00:04:17.670 --> 00:04:20.600 The triangle could look like this. 00:04:20.600 --> 00:04:25.792 It could look like that, or it could look like this based on 00:04:25.792 --> 00:04:27.270 the information they gave us. 00:04:27.270 --> 00:04:32.610 So these angles could change arbitrarily. 00:04:32.610 --> 00:04:35.770 So we really don't know what those angles are. 00:04:35.770 --> 00:04:38.280 Next problem. 00:04:38.280 --> 00:04:39.530 Problem 3. 00:04:42.844 --> 00:04:44.420 Switch colors. 00:04:44.420 --> 00:04:46.214 About to sneeze. 00:04:46.214 --> 00:04:47.060 [SNEEZES] 00:04:47.060 --> 00:04:48.540 Excuse me. 00:04:48.540 --> 00:04:59.550 If 30% of m is 40, so 0.3 m, what is 15% of m? 00:04:59.550 --> 00:05:02.100 What is 0.15 of m? 00:05:02.100 --> 00:05:03.070 You could solve this. 00:05:03.070 --> 00:05:05.540 Divide both sides by 0.3, et cetera, et cetera. 00:05:05.540 --> 00:05:09.010 But the important thing to recognize is that 15%, or 00:05:09.010 --> 00:05:13.130 0.15, is half of 0.13. 00:05:13.130 --> 00:05:20.160 So if 15% is half of 0.3, or 30%, then 15% m is going to be 00:05:20.160 --> 00:05:24.460 half of whatever 30% m is, so it's going to be equal to 20. 00:05:24.460 --> 00:05:27.060 Another thing you could have done, if that's not completely 00:05:27.060 --> 00:05:28.770 obvious, take this top equation and 00:05:28.770 --> 00:05:30.760 multiply it by 1/2. 00:05:30.760 --> 00:05:35.415 Then you'll get 0.15m is equal to 20. 00:05:35.415 --> 00:05:37.560 And that's choice B. 00:05:37.560 --> 00:05:39.690 Problem 4. 00:05:39.690 --> 00:05:44.940 If n is any negative number, so n is less than 0, which of 00:05:44.940 --> 00:05:48.390 the following must be positive? 00:05:48.390 --> 00:05:51.110 So I can already tell you that the only way you can take a 00:05:51.110 --> 00:05:54.120 negative number and make it positive is either if you 00:05:54.120 --> 00:05:58.160 square it or if you multiply it by another negative number. 00:05:58.160 --> 00:05:59.150 So let's see. 00:05:59.150 --> 00:06:01.715 Or if you subtract it, actually, now that I look at 00:06:01.715 --> 00:06:02.670 the choices. 00:06:02.670 --> 00:06:05.335 So I could tell you an/2, that's definitely going to be 00:06:05.335 --> 00:06:06.710 a negative number. 00:06:06.710 --> 00:06:08.800 B is definitely going to be a negative number. 00:06:08.800 --> 00:06:10.890 You're multiplying by a positive. 00:06:10.890 --> 00:06:13.130 C, you're just adding something to it, so it still 00:06:13.130 --> 00:06:15.500 could be negative, if that's like negative 10 00:06:15.500 --> 00:06:16.820 or something, right? 00:06:16.820 --> 00:06:18.350 D, you're just subtracting from it, so you're just going 00:06:18.350 --> 00:06:19.780 to make it more negative. 00:06:19.780 --> 00:06:20.050 E. 00:06:20.050 --> 00:06:23.540 2 minus n. 00:06:23.540 --> 00:06:26.190 This will definitely result in a positive number. 00:06:26.190 --> 00:06:27.360 Why? 00:06:27.360 --> 00:06:31.190 Because let's say that you could express n as the product 00:06:31.190 --> 00:06:32.960 of a negative number. 00:06:32.960 --> 00:06:36.330 Let's say n is equal to minus p, where p is 00:06:36.330 --> 00:06:37.520 some positive number. 00:06:37.520 --> 00:06:40.050 You can represent any negative number like that. 00:06:40.050 --> 00:06:44.430 Then 2 minus n is the same thing as 2 minus minus p, 00:06:44.430 --> 00:06:47.730 which is the same thing as 2 plus p. 00:06:47.730 --> 00:06:50.730 And we said p is a positive number. 00:06:50.730 --> 00:06:51.510 So there we go. 00:06:51.510 --> 00:06:55.670 So choice number 4 is definitely E. 00:06:55.670 --> 00:06:57.410 And you could try it out with a number. 00:06:57.410 --> 00:07:00.400 Try it out with the number n equals negative 10, and I 00:07:00.400 --> 00:07:02.560 think it should work out. 00:07:02.560 --> 00:07:03.520 Next problem. 00:07:03.520 --> 00:07:06.530 Problem number 5. 00:07:06.530 --> 00:07:15.640 The ratio 1.2:1 is equal to which of the following ratios? 00:07:15.640 --> 00:07:17.990 So they give us a bunch of choices. 00:07:17.990 --> 00:07:21.390 1.2:1, so it's a little bit more than 1:1. 00:07:21.390 --> 00:07:23.610 So we could look at 1:2. 00:07:23.610 --> 00:07:27.750 No, that makes no sense, because 1 is smaller than 2. 00:07:27.750 --> 00:07:29.250 12:1. 00:07:29.250 --> 00:07:33.910 12 is 12 times 1, not 1.2 times 1. 00:07:33.910 --> 00:07:36.585 So you're just looking for a ratio where the first number 00:07:36.585 --> 00:07:38.660 is 1.2 times the second number. 00:07:38.660 --> 00:07:40.450 5:6, that's choice C. 00:07:40.450 --> 00:07:43.080 That doesn't work, because 5 is less than 6. 00:07:43.080 --> 00:07:43.680 Choice D. 00:07:43.680 --> 00:07:44.800 6:5. 00:07:44.800 --> 00:07:48.170 I think we hit it because 6 is just a little bit more than 5. 00:07:48.170 --> 00:07:51.840 And actually, if you wanted to confirm that, why don't you 00:07:51.840 --> 00:07:53.630 divide the top and the bottom by 5? 00:07:53.630 --> 00:07:56.900 So 5 goes into 6 how many times? 00:07:56.900 --> 00:08:00.490 5 goes into 6 one time. 00:08:00.490 --> 00:08:03.530 How many times does 5 go into 10? 00:08:03.530 --> 00:08:04.390 It's 2. 00:08:04.390 --> 00:08:06.200 5 goes into 6 1.2 times. 00:08:06.200 --> 00:08:09.280 So the answer is 6:5. 00:08:09.280 --> 00:08:11.080 Next problem. 00:08:11.080 --> 00:08:14.140 Let's see if I have time to do this. 00:08:14.140 --> 00:08:15.180 Clear image. 00:08:15.180 --> 00:08:16.680 Invert colors. 00:08:16.680 --> 00:08:20.750 The legend of a certain pictograph shows that this 00:08:20.750 --> 00:08:25.260 thing is equal to 5 million new homes. 00:08:25.260 --> 00:08:26.750 Approximately how many new homes are 00:08:26.750 --> 00:08:28.360 represented by the symbols? 00:08:28.360 --> 00:08:31.180 So they drew that 3 and 1/2 times. 00:08:31.180 --> 00:08:40.110 One, two, three, and then they draw half a house. 00:08:40.110 --> 00:08:41.900 That's half a house. 00:08:41.900 --> 00:08:43.270 So these three are going to be what? 00:08:43.270 --> 00:08:50.280 This is 3 times 5, which is equal to 15 million homes. 00:08:50.280 --> 00:08:51.560 And then this is half a house. 00:08:51.560 --> 00:08:53.430 That's what I tried to draw what they drew in the diagram. 00:08:53.430 --> 00:08:54.290 So half a house is what? 00:08:54.290 --> 00:08:55.620 2.5 million houses. 00:08:55.620 --> 00:08:58.880 It's just half of the 5 million. 00:08:58.880 --> 00:09:00.850 So it's 15 million plus 2.5. 00:09:00.850 --> 00:09:04.470 It equals 17.5 million homes. 00:09:04.470 --> 00:09:06.330 And that is choice D. 00:09:08.950 --> 00:09:11.380 You might be tempted to say that this represents half a 00:09:11.380 --> 00:09:13.840 million homes, but remember, this represents half of one of 00:09:13.840 --> 00:09:14.445 these pictures. 00:09:14.445 --> 00:09:17.600 And one of these pictures represents 5 million homes. 00:09:17.600 --> 00:09:19.140 So this is 2 and 1/2 million. 00:09:19.140 --> 00:09:23.040 This is 2 and 1/2, not 0.5. 00:09:23.040 --> 00:09:24.810 See you in the next video.