WEBVTT 00:00:00.080 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.050 Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:10.140 continue to offer high quality educational resources for free. 00:00:10.140 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.297 at ocw.mit.edu. 00:00:21.610 --> 00:00:24.360 PROFESSOR: I'm very interested in teaching in ways 00:00:24.360 --> 00:00:26.420 that really help you learn. 00:00:26.420 --> 00:00:33.640 And one of the most difficult things to learn 00:00:33.640 --> 00:00:37.960 is when you have to unlearn something you believe. 00:00:37.960 --> 00:00:40.250 So if you have learned a concept wrong 00:00:40.250 --> 00:00:43.630 in your previous education and learnings, 00:00:43.630 --> 00:00:46.973 your mind is really reluctant to let go of it. 00:00:46.973 --> 00:00:47.960 OK? 00:00:47.960 --> 00:00:52.140 And I have a wife who is a middle school science teacher. 00:00:52.140 --> 00:00:53.860 She also has a PhD in biology. 00:00:53.860 --> 00:00:56.750 But she really studies to understand kids 00:00:56.750 --> 00:00:58.050 and how they learn. 00:00:58.050 --> 00:01:00.500 And MIT students aren't much different from 6th graders, 00:01:00.500 --> 00:01:02.541 to tell you the truth, when it comes to learning. 00:01:02.541 --> 00:01:04.144 None of us are. 00:01:04.144 --> 00:01:05.935 But they're just a little more transparent. 00:01:09.558 --> 00:01:13.430 What you can do to help a person unlearn something they've 00:01:13.430 --> 00:01:15.960 learned wrong, the best time is when 00:01:15.960 --> 00:01:18.980 you run into something called a "discrepant event." 00:01:18.980 --> 00:01:20.685 So when you think you know the answer, 00:01:20.685 --> 00:01:23.850 and I'm asking you these questions, and you're wrong, 00:01:23.850 --> 00:01:25.730 really try to figure out what it is 00:01:25.730 --> 00:01:28.250 you believe that led you to that, 00:01:28.250 --> 00:01:31.372 and see if you can undo that in getting to the answer. 00:01:31.372 --> 00:01:32.830 So some of these concept questions, 00:01:32.830 --> 00:01:34.440 if you'd like to talk about them further, 00:01:34.440 --> 00:01:36.800 I think they're great for discussions in your recitation 00:01:36.800 --> 00:01:37.300 section. 00:01:37.300 --> 00:01:38.940 Look at Professor Gossard here. 00:01:38.940 --> 00:01:41.620 So talk about these things, and really try 00:01:41.620 --> 00:01:44.690 to get to the bottom of why you perhaps 00:01:44.690 --> 00:01:49.040 had a concept that wasn't quite up to what you needed. 00:01:49.040 --> 00:01:50.080 OK. 00:01:50.080 --> 00:01:53.670 We're going to move on. 00:01:53.670 --> 00:01:54.910 OK. 00:01:54.910 --> 00:01:57.850 Now we're going to get on to talking about velocities 00:01:57.850 --> 00:01:59.290 in rotating frames. 00:01:59.290 --> 00:02:02.690 And by velocities, I also mean derivatives 00:02:02.690 --> 00:02:04.320 of vectors in rotating frames. 00:02:22.230 --> 00:02:24.800 Just to give you a quick reminder of where we started 00:02:24.800 --> 00:02:27.900 this conversation, this is concept 00:02:27.900 --> 00:02:36.320 if you have three points-- A, B, and O in this fixed frame. 00:02:36.320 --> 00:02:44.240 This is r A with respect to O, B with respect to A, 00:02:44.240 --> 00:02:47.550 and B with respect to O, all vectors. 00:02:47.550 --> 00:02:49.900 And we're interested in computing 00:02:49.900 --> 00:02:54.150 the velocities of, say, point B. So we 00:02:54.150 --> 00:02:57.880 were talking about the derivative with respect 00:02:57.880 --> 00:03:01.650 to time of this vector. 00:03:01.650 --> 00:03:06.360 And we can make it up as a sum of the derivatives with respect 00:03:06.360 --> 00:03:07.930 to time of these other two vectors. 00:03:19.420 --> 00:03:21.370 But now there's a really important point. 00:03:21.370 --> 00:03:22.870 When you're taking time derivatives, 00:03:22.870 --> 00:03:26.450 you have to be explicit about what 00:03:26.450 --> 00:03:28.765 it's with respect to-- what frame of reference 00:03:28.765 --> 00:03:29.640 you're talking about. 00:03:29.640 --> 00:03:33.580 In this case we want to know the velocity with respect 00:03:33.580 --> 00:03:35.110 to this reference frame. 00:03:35.110 --> 00:03:37.520 So both of these derivatives have 00:03:37.520 --> 00:03:40.640 to be taken with respect to that frame. 00:03:40.640 --> 00:03:42.780 Well this one's easy. 00:03:42.780 --> 00:03:46.000 This is purely translational. 00:03:46.000 --> 00:03:48.980 It has nothing to do with rotations, or won't. 00:03:48.980 --> 00:03:53.910 We're going to think in terms of having an object out here. 00:03:53.910 --> 00:03:58.320 So maybe a rigid body to which we have attached a rotating 00:03:58.320 --> 00:04:02.660 frame at A. And I'll call this the rotating reference 00:04:02.660 --> 00:04:05.200 frame A x prime y prime. 00:04:05.200 --> 00:04:06.260 And it might be rotating. 00:04:06.260 --> 00:04:10.000 But this point is just the point with which we describe 00:04:10.000 --> 00:04:11.520 the translation of the body. 00:04:11.520 --> 00:04:13.150 So this one's pretty easy. 00:04:13.150 --> 00:04:16.029 This one's just the velocity of A with respect to O. 00:04:16.029 --> 00:04:21.800 But this one has complications. 00:04:21.800 --> 00:04:24.210 And I'm going to do it first, the answer to this, just 00:04:24.210 --> 00:04:27.110 intuitively-- appeal to your intuition 00:04:27.110 --> 00:04:29.710 about why this should be. 00:04:29.710 --> 00:04:33.910 So this is my merry-go-round. 00:04:33.910 --> 00:04:36.180 And I've set up my little coordinate system. 00:04:36.180 --> 00:04:39.200 This is my A, x prime y prime coordinate system. 00:04:39.200 --> 00:04:43.000 I've stuck my dog here on the x-axis. 00:04:43.000 --> 00:04:49.280 And if he's running some direction, 00:04:49.280 --> 00:04:53.550 he has a velocity with respect to this reference frame. 00:04:53.550 --> 00:04:56.210 If you're here at A, watch it, and you'll see him moving. 00:04:56.210 --> 00:04:59.400 And it will have nothing to do with the rotation. 00:04:59.400 --> 00:05:01.410 You will just see the dog moving. 00:05:01.410 --> 00:05:05.260 And if this is rotating, and you're sitting there rotating 00:05:05.260 --> 00:05:09.650 with it watching the dog, does the motion of the dog change? 00:05:09.650 --> 00:05:11.670 You just see it just the same way, right? 00:05:11.670 --> 00:05:12.260 OK. 00:05:12.260 --> 00:05:16.340 So there's one part of the motion of the dog that's 00:05:16.340 --> 00:05:18.340 with respect to this frame that has 00:05:18.340 --> 00:05:20.540 nothing to do with rotation. 00:05:20.540 --> 00:05:21.250 OK. 00:05:21.250 --> 00:05:27.050 So now let's say in an outside, and from an outside point 00:05:27.050 --> 00:05:31.800 of view, if we measure the velocity 00:05:31.800 --> 00:05:34.620 of the dog with respect to point A-- that's 00:05:34.620 --> 00:05:39.250 the one you can see-- and to observer outside says 00:05:39.250 --> 00:05:42.580 what's the velocity of dog with respect to A, 00:05:42.580 --> 00:05:48.275 will the answer differ if there's rotation? 00:05:52.190 --> 00:05:53.360 OK. 00:05:53.360 --> 00:05:55.590 Now, what we're going to find out 00:05:55.590 --> 00:05:58.580 is that there's actually two contributions. 00:05:58.580 --> 00:06:01.840 So the dog running with respect to this merry-go-round 00:06:01.840 --> 00:06:04.930 is one contribution to the velocity. 00:06:04.930 --> 00:06:07.410 There's another contribution that comes from the fact 00:06:07.410 --> 00:06:09.700 that you're sitting out there watching, 00:06:09.700 --> 00:06:12.180 and you see definitely the dog move when the thing rotates, 00:06:12.180 --> 00:06:12.880 right? 00:06:12.880 --> 00:06:13.380 OK. 00:06:13.380 --> 00:06:16.880 And if we only do that, if the only rotation-- 00:06:16.880 --> 00:06:19.430 and we'll do a case where the dog's not running, just 00:06:19.430 --> 00:06:21.280 sitting still-- what's the velocity 00:06:21.280 --> 00:06:24.165 of the dog with respect to point A due to the rotation? 00:06:27.990 --> 00:06:30.820 I mean you've done lots of problems like this. 00:06:30.820 --> 00:06:32.470 This is just pure rotation problem. 00:06:32.470 --> 00:06:35.980 You know the angular rate-- theta dot, or omega. 00:06:35.980 --> 00:06:39.300 What's the velocity of point B if point A is fixed 00:06:39.300 --> 00:06:41.730 and this thing rotates? 00:06:41.730 --> 00:06:42.780 r omega. 00:06:42.780 --> 00:06:43.900 You all know that. 00:06:43.900 --> 00:06:44.840 OK. 00:06:44.840 --> 00:06:50.540 And then we'll call the velocity of the dog relative to point 00:06:50.540 --> 00:06:54.280 A due to the fact that he is moving in this frame, 00:06:54.280 --> 00:06:58.040 we'll call that velocity to the dog. 00:06:58.040 --> 00:07:01.520 From the point of view of in the frame is a different quantity. 00:07:01.520 --> 00:07:04.622 They're both vectors, and you can add them together. 00:07:04.622 --> 00:07:06.205 So in fact the answer-- I'm just going 00:07:06.205 --> 00:07:09.720 to give you the answer to this problem-- the answer to this 00:07:09.720 --> 00:07:14.860 is the velocity of A with respect to O 00:07:14.860 --> 00:07:24.010 plus the derivative of r B/A with respect to time as seen 00:07:24.010 --> 00:07:29.660 from within the A xyz prime frame, 00:07:29.660 --> 00:07:32.710 plus a term here due to rotation. 00:07:35.540 --> 00:07:41.640 And we know that the magnitude of this answer is some r omega. 00:07:41.640 --> 00:07:43.045 We know it looks like r omega. 00:07:45.890 --> 00:07:48.760 So to interpret this as meaning the derivative 00:07:48.760 --> 00:07:55.500 of this vector-- this thing-- as if you were in the frame. 00:07:55.500 --> 00:07:57.450 And that accounts for the dog running. 00:07:57.450 --> 00:07:58.850 This is just the dog running. 00:08:05.350 --> 00:08:09.360 And mathematically, I actually find it easier 00:08:09.360 --> 00:08:12.310 to say what this piece means is this 00:08:12.310 --> 00:08:15.180 is the velocity of the dog with respect 00:08:15.180 --> 00:08:20.960 to A with omega equal to 0. 00:08:20.960 --> 00:08:21.870 It's the same thing. 00:08:21.870 --> 00:08:25.490 This is what you see if there's no rotation. 00:08:25.490 --> 00:08:27.100 That piece. 00:08:27.100 --> 00:08:28.980 And this is the piece with rotation. 00:08:28.980 --> 00:08:33.350 But I want you to think something through here. 00:08:38.500 --> 00:08:39.760 Come on. 00:08:39.760 --> 00:08:42.419 I'm still learning how to do this. 00:08:42.419 --> 00:08:42.919 OK. 00:08:46.830 --> 00:08:49.590 So I have my little reference frames like this. 00:08:49.590 --> 00:08:53.620 I'm going to rotate it in this direction, which given xyz 00:08:53.620 --> 00:08:55.570 ought to be in that direction. 00:08:55.570 --> 00:08:57.680 So the rotation rate, if it's going like this, 00:08:57.680 --> 00:09:00.100 I'm going to give it a constant value, 00:09:00.100 --> 00:09:05.620 is that the vector omega is some omega in the k hat direction, 00:09:05.620 --> 00:09:07.270 right? 00:09:07.270 --> 00:09:16.540 And I have on this body-- here's my wheel, 00:09:16.540 --> 00:09:23.770 and here's B, and here's A. And this is my little x-axis, 00:09:23.770 --> 00:09:25.230 my x-prime axis. 00:09:25.230 --> 00:09:31.240 And it has associated with it a lowercase i hat unit vector. 00:09:31.240 --> 00:09:32.910 And off in this direction then would 00:09:32.910 --> 00:09:37.860 be the y prime with a j hat unit vector. 00:09:37.860 --> 00:09:38.700 OK. 00:09:38.700 --> 00:09:42.340 So what's the vector r B/A? 00:09:42.340 --> 00:09:46.290 How would you write it in vector notation in components here? 00:09:49.750 --> 00:09:54.820 So this is r B/A from here to here. 00:09:54.820 --> 00:09:57.313 And what's its direction? 00:09:57.313 --> 00:09:59.240 AUDIENCE: i hat? 00:09:59.240 --> 00:10:00.249 PROFESSOR: i hat. 00:10:00.249 --> 00:10:01.540 And is it positive or negative? 00:10:06.180 --> 00:10:09.110 So this is the vector. 00:10:09.110 --> 00:10:15.210 At some magnitude r B/A in the positive i hat direction, 00:10:15.210 --> 00:10:17.860 the arrow goes, when you say r B/A, 00:10:17.860 --> 00:10:22.280 this is B with respect to A, the arrow goes with B. 00:10:22.280 --> 00:10:25.485 So this, it's positive r B/A i hat. 00:10:25.485 --> 00:10:29.690 And what is the velocity that we have discovered up here, 00:10:29.690 --> 00:10:32.350 the velocity of B with respect to A as 00:10:32.350 --> 00:10:35.980 measured inside of A xyz? 00:10:35.980 --> 00:10:39.180 That's this piece here. 00:10:39.180 --> 00:10:41.777 In vector terms? 00:10:41.777 --> 00:10:43.610 And we've figured out what its velocity was. 00:10:43.610 --> 00:10:45.320 It's some r omega. 00:10:45.320 --> 00:10:50.320 The r's r B/A-- the magnitude, the length of that r. 00:10:50.320 --> 00:10:54.040 Clearly got to be this length right here. 00:10:54.040 --> 00:10:57.800 And the omega is given. 00:10:57.800 --> 00:10:59.180 So what direction? 00:10:59.180 --> 00:11:00.896 You know that the magnitudes are omega. 00:11:00.896 --> 00:11:01.770 What's the direction? 00:11:07.764 --> 00:11:08.730 AUDIENCE: j hat? 00:11:08.730 --> 00:11:09.438 PROFESSOR: j hat. 00:11:09.438 --> 00:11:10.470 OK. 00:11:10.470 --> 00:11:13.360 So what we're saying is the velocity 00:11:13.360 --> 00:11:21.160 of B with respect to A as seen from inside this A frame is 00:11:21.160 --> 00:11:31.780 r B/A omega with respect to O in the j hat direction. 00:11:31.780 --> 00:11:35.996 Now this has a k associated with it. 00:11:35.996 --> 00:11:37.370 That has an i associated with it. 00:11:37.370 --> 00:11:39.550 And this has a j associated with it. 00:11:39.550 --> 00:11:43.164 So as a product r omega, what kind of product is it? 00:11:43.164 --> 00:11:44.330 What you know about vectors? 00:11:44.330 --> 00:11:45.630 What does it have to be? 00:11:45.630 --> 00:11:49.160 Is it a dot product between omega and r? 00:11:49.160 --> 00:11:51.242 Is it a cross product? 00:11:51.242 --> 00:11:52.960 I hear somebody saying cross product. 00:11:52.960 --> 00:11:53.880 But what's the order? 00:11:53.880 --> 00:11:56.380 So I want you to figure out-- just deduce 00:11:56.380 --> 00:11:59.230 what the vector notation is that gets 00:11:59.230 --> 00:12:01.749 you the right velocity-- the correct velocity-- 00:12:01.749 --> 00:12:03.415 correct magnitude and correct direction. 00:12:06.760 --> 00:12:11.150 So the answer is positive r omega j. 00:12:16.070 --> 00:12:17.550 AUDIENCE: Omega cross r? 00:12:17.550 --> 00:12:18.510 PROFESSOR: Omega. 00:12:18.510 --> 00:12:19.850 AUDIENCE: Cross to the r? 00:12:19.850 --> 00:12:20.891 PROFESSOR: Omega cross r. 00:12:20.891 --> 00:12:29.660 So I hear one person, omega with respect to O cross r B/A. 00:12:29.660 --> 00:12:32.220 And these are now both vectors. 00:12:32.220 --> 00:12:33.390 Anybody else? 00:12:33.390 --> 00:12:34.989 So if we just think about unit vector, 00:12:34.989 --> 00:12:36.030 actually, let's check it. 00:12:36.030 --> 00:12:37.880 This is k hat direction. 00:12:37.880 --> 00:12:40.170 This was i hat direction. 00:12:40.170 --> 00:12:44.330 And k cross i is positive j. 00:12:44.330 --> 00:12:45.015 So that works. 00:12:47.800 --> 00:12:51.980 So in general-- this is, in fact, the correct answer 00:12:51.980 --> 00:13:01.370 in general-- is that this term here is omega with respect 00:13:01.370 --> 00:13:09.260 to O cross r B/A. You're going to use that a lot. 00:13:09.260 --> 00:13:20.500 So to summarize, then-- while that's coming down, 00:13:20.500 --> 00:13:30.620 the velocity of this dog in a stationary frame where 00:13:30.620 --> 00:13:35.150 A is stationary for a moment is entirely due to the rotation, 00:13:35.150 --> 00:13:39.320 and it's omega cross r B/A. And now if the dog starts 00:13:39.320 --> 00:13:43.470 to run with respect to this reference frame, 00:13:43.470 --> 00:13:46.520 he has some velocity, which is the other piece 00:13:46.520 --> 00:13:50.400 is velocity of B with respect to A as seen from A-- 00:13:50.400 --> 00:13:52.610 or as you would compute, you'd just set momentarily, 00:13:52.610 --> 00:13:54.380 omega equal to zero. 00:13:54.380 --> 00:13:58.840 And finally then, the total formula for velocity. 00:13:58.840 --> 00:14:02.720 So the velocity of B with respect 00:14:02.720 --> 00:14:07.280 to O-- some point on a rotating reference frame-- 00:14:07.280 --> 00:14:15.696 is the velocity of the frame plus-- 00:14:15.696 --> 00:14:18.330 I won't write it as a derivative now. 00:14:18.330 --> 00:14:23.610 Plus the velocity of the point, but as 00:14:23.610 --> 00:14:29.180 seen from the point of view of somebody in the frame, 00:14:29.180 --> 00:14:37.780 plus omega with respect to O cross r B/A. 00:14:37.780 --> 00:14:40.000 So this is an important formula. 00:14:40.000 --> 00:14:43.370 You're going to use it a lot. 00:14:43.370 --> 00:14:45.640 Couple of subtle points in here. 00:14:45.640 --> 00:14:49.490 This one-- this thing from as seen from inside the frame. 00:14:49.490 --> 00:14:55.870 It's also the partial derivative of r B/A with respect 00:14:55.870 --> 00:15:00.570 to t when omega with respect to O equals zero. 00:15:00.570 --> 00:15:04.460 That's another way to say this. 00:15:04.460 --> 00:15:07.980 And this term is another subtlety. 00:15:07.980 --> 00:15:12.907 I was careful to write with respect to O. Yes? 00:15:12.907 --> 00:15:17.786 AUDIENCE: To make sure I'm not confused here on this vector 00:15:17.786 --> 00:15:18.286 right there. 00:15:21.709 --> 00:15:23.665 PROFESSOR: Yes. 00:15:23.665 --> 00:15:27.088 AUDIENCE: That leftmost term is the middle one there? 00:15:27.088 --> 00:15:28.570 Or the right one? 00:15:28.570 --> 00:15:30.810 PROFESSOR: So this was when we were working 00:15:30.810 --> 00:15:37.710 with just the velocity due to the motion of the dog 00:15:37.710 --> 00:15:40.490 due to rotation plus-- actually this 00:15:40.490 --> 00:15:44.066 was rotation-- what did I say? 00:15:44.066 --> 00:15:47.482 AUDIENCE: [INAUDIBLE] experiments. 00:15:47.482 --> 00:15:50.166 That the term on the left is the motion 00:15:50.166 --> 00:15:52.298 that you can see from the frame. 00:15:52.298 --> 00:15:53.923 From the right is the rotation that you 00:15:53.923 --> 00:15:56.270 don't see in the [INAUDIBLE], but needs to be [INAUDIBLE]. 00:15:56.270 --> 00:15:58.080 PROFESSOR: And I can't remember how I got to this point. 00:15:58.080 --> 00:15:59.710 But I think I was making a couple of points, 00:15:59.710 --> 00:16:00.860 and rammed them together. 00:16:00.860 --> 00:16:02.480 So this is the correct. 00:16:02.480 --> 00:16:04.690 Don't focus on this. 00:16:04.690 --> 00:16:07.410 This is the summary. 00:16:07.410 --> 00:16:12.380 This thing is now rotating, and translating, 00:16:12.380 --> 00:16:15.220 and the dog's running. 00:16:15.220 --> 00:16:17.400 And I want you to be able to write down 00:16:17.400 --> 00:16:20.590 the total velocity of the point B 00:16:20.590 --> 00:16:27.400 as seen from a fixed reference frame fixed on the ground. 00:16:27.400 --> 00:16:29.600 This is with respect to O. 00:16:29.600 --> 00:16:32.400 It has three terms-- the velocity 00:16:32.400 --> 00:16:40.280 of A, the velocity of the dog relative to the body 00:16:40.280 --> 00:16:43.180 it's on here, and finally, the component 00:16:43.180 --> 00:16:45.180 of the velocity-- contribution of the velocity 00:16:45.180 --> 00:16:49.630 that you see that's due to the rotation. 00:16:49.630 --> 00:16:54.360 Now this turns out to be a really powerful formula. 00:16:54.360 --> 00:16:55.560 It's a generalization. 00:16:55.560 --> 00:17:03.880 It's a special case of a very general formula. 00:17:03.880 --> 00:17:07.740 Let me catch up in my notes here. 00:17:07.740 --> 00:17:11.099 Because the following statement is true. 00:17:11.099 --> 00:17:35.950 So the time derivative of any vector 00:17:35.950 --> 00:17:44.080 that's defined in a rotating frame 00:17:44.080 --> 00:17:51.100 is given by-- and we'll call it just some vector 00:17:51.100 --> 00:17:54.095 A. It might be, for example, angular momentum. 00:17:54.095 --> 00:17:56.590 In fact we'll use this formula a lot 00:17:56.590 --> 00:17:58.940 when talking about angular momentum. 00:17:58.940 --> 00:18:04.825 The time derivative of this vector A-- 00:18:04.825 --> 00:18:09.260 whatever it represents-- position, velocity, 00:18:09.260 --> 00:18:17.130 angular momentum-- as seen from a reference frame xy-- O xyz, 00:18:17.130 --> 00:18:27.850 reference frame O-- is made up of the derivative of A from 00:18:27.850 --> 00:18:31.316 inside the frame plus-- 00:18:42.460 --> 00:18:45.150 So it's the same thing. 00:18:45.150 --> 00:18:49.720 This was the derivative of r B/A. Has two pieces-- the piece 00:18:49.720 --> 00:18:51.980 that came from the dog running and the piece that 00:18:51.980 --> 00:18:53.690 comes from rotation. 00:18:53.690 --> 00:18:56.680 Any vector, this piece is true. 00:18:56.680 --> 00:18:58.160 It's the derivative of the vector 00:18:58.160 --> 00:19:03.590 inside the frame in which it's defined plus omega with respect 00:19:03.590 --> 00:19:10.510 to the reference frame you want the derivative in cross A. Very 00:19:10.510 --> 00:19:11.390 powerful formula. 00:19:21.310 --> 00:19:25.090 This now is covered in a couple of different places 00:19:25.090 --> 00:19:28.140 in the readings, and in the handout that's 00:19:28.140 --> 00:19:30.560 posted called "Kinematics." 00:19:30.560 --> 00:19:32.570 So you should read by now the portion 00:19:32.570 --> 00:19:34.832 of the kinematics thing, at least on velocities. 00:19:34.832 --> 00:19:37.060 The second half of that kinematics handout 00:19:37.060 --> 00:19:38.160 is about acceleration. 00:19:38.160 --> 00:19:39.740 So we're going to use this formula 00:19:39.740 --> 00:19:41.560 to go from velocity to acceleration. 00:19:41.560 --> 00:19:43.420 I saw a couple hands up. 00:19:43.420 --> 00:19:44.383 Yes? 00:19:44.383 --> 00:19:47.281 AUDIENCE: So what is the difference between O sub 00:19:47.281 --> 00:19:49.213 xyz and A sub xyz? 00:19:49.213 --> 00:19:51.050 Where are they [INAUDIBLE]? 00:19:51.050 --> 00:19:51.780 PROFESSOR: OK. 00:19:51.780 --> 00:19:58.410 So this is a merry-go-round on a train, 00:19:58.410 --> 00:19:59.780 and you're looking down on it. 00:19:59.780 --> 00:20:02.950 And the train can move. 00:20:02.950 --> 00:20:05.550 And we fixed to the ground, not moving, 00:20:05.550 --> 00:20:12.030 a reference frame we call O. And I want to know in this case 00:20:12.030 --> 00:20:15.280 the velocity of the dog running around 00:20:15.280 --> 00:20:20.650 on this merry-go-round with respect to O. 00:20:20.650 --> 00:20:24.290 It's the sum of three vector contributions-- 00:20:24.290 --> 00:20:28.710 the velocity of the train, the velocity 00:20:28.710 --> 00:20:35.990 of the dog with respect to the merry-go-round-- this 00:20:35.990 --> 00:20:37.115 is velocity of the train. 00:20:40.110 --> 00:20:43.150 This is with respect to the merry-go-round. 00:20:43.150 --> 00:20:46.990 And this is the velocity that you see out there. 00:20:46.990 --> 00:20:52.620 So you're sitting in O. You are in this fixed frame. 00:20:52.620 --> 00:20:56.170 Which brings up another subtle point about velocities 00:20:56.170 --> 00:20:59.890 and inertial frames. 00:20:59.890 --> 00:21:04.540 If you are at any fixed point in an inertial frame-- 00:21:04.540 --> 00:21:06.710 you don't have to be at O; you can be right 00:21:06.710 --> 00:21:10.980 where you are-- this term is always the same. 00:21:10.980 --> 00:21:14.900 The velocity of this moving object relative to the frame 00:21:14.900 --> 00:21:17.540 is the same to any observer in the frame-- 00:21:17.540 --> 00:21:20.281 any fixed observer in the frame. 00:21:20.281 --> 00:21:20.780 OK? 00:21:23.290 --> 00:21:24.395 And finally, the u. 00:21:24.395 --> 00:21:27.200 But the u, the part of the contribution 00:21:27.200 --> 00:21:29.290 to the velocity due to the rotation of this thing, 00:21:29.290 --> 00:21:30.850 you can see it from out there, right? 00:21:30.850 --> 00:21:31.810 It moves. 00:21:31.810 --> 00:21:35.830 Well, the rotation contribution is this. 00:21:35.830 --> 00:21:38.370 The dog running around is this. 00:21:38.370 --> 00:21:42.300 And the movement of the train is the first term. 00:21:42.300 --> 00:21:44.240 OK? 00:21:44.240 --> 00:21:44.780 All right. 00:21:44.780 --> 00:21:47.620 And this is the general formula for the derivative 00:21:47.620 --> 00:21:54.280 of a vector in a translating, rotating frame. 00:21:54.280 --> 00:21:54.810 Yeah? 00:21:54.810 --> 00:22:00.605 AUDIENCE: [INAUDIBLE] vectors beyond the [INAUDIBLE] v 00:22:00.605 --> 00:22:02.858 of p with respect to the [INAUDIBLE]. 00:22:02.858 --> 00:22:04.316 PROFESSOR: I'm not following, here. 00:22:04.316 --> 00:22:07.232 AUDIENCE: Are any of the v, v of A, 00:22:07.232 --> 00:22:10.148 for example, or v of B with respect to A, are they 00:22:10.148 --> 00:22:11.245 all vectors? 00:22:11.245 --> 00:22:12.620 PROFESSOR: These are all vectors. 00:22:12.620 --> 00:22:15.190 And I've gotten a little careless 00:22:15.190 --> 00:22:18.340 about drawing my underlines when I haven't broken them down 00:22:18.340 --> 00:22:19.160 into components. 00:22:19.160 --> 00:22:20.590 This is a vector. 00:22:20.590 --> 00:22:22.400 And we can do this, because we're 00:22:22.400 --> 00:22:26.290 relying on formulas for the sums of vectors. 00:22:26.290 --> 00:22:28.260 So it's the vector. 00:22:28.260 --> 00:22:31.120 There's a vector describing the velocity of the train. 00:22:31.120 --> 00:22:34.590 For example, if it's moving in the capital I hat direction, 00:22:34.590 --> 00:22:36.350 you have to say that. 00:22:36.350 --> 00:22:41.240 And this may actually be the dog. 00:22:41.240 --> 00:22:45.180 At the moment you catch him, the frame's like this. 00:22:45.180 --> 00:22:51.070 And let's say he's running in the y direction. 00:22:51.070 --> 00:22:52.840 His direction at that instant in time 00:22:52.840 --> 00:22:56.730 is in the lowercase j hat direction 00:22:56.730 --> 00:22:58.420 relative to this frame, right? 00:22:58.420 --> 00:23:01.610 And that j hat direction has some angle 00:23:01.610 --> 00:23:04.040 with respect to this frame, which 00:23:04.040 --> 00:23:09.320 if you want to reduce these velocities down 00:23:09.320 --> 00:23:13.130 to the unit vectors in the inertial frame, 00:23:13.130 --> 00:23:13.930 you can do that. 00:23:13.930 --> 00:23:15.388 But you're going to have to account 00:23:15.388 --> 00:23:19.930 for the angles of this reference frame compared to that one. 00:23:19.930 --> 00:23:22.900 So cosine thetas and sine thetas and that kind of thing. 00:23:25.740 --> 00:23:34.440 One important point about this general expression 00:23:34.440 --> 00:23:38.290 for the derivative of a vector in a rotating frame, which 00:23:38.290 --> 00:23:41.250 therefore applies to this case, because this is just 00:23:41.250 --> 00:23:44.970 the specific example of derivative of a position 00:23:44.970 --> 00:23:47.420 vector giving you velocity. 00:23:47.420 --> 00:23:54.130 This term and this term-- these account 00:23:54.130 --> 00:23:56.190 for the change in length of the vector. 00:23:59.910 --> 00:24:03.100 So just to keep it real, in terms of velocities, 00:24:03.100 --> 00:24:05.530 you see a velocity that's due to the fact 00:24:05.530 --> 00:24:09.245 that the velocity vector itself is changing in length. 00:24:12.660 --> 00:24:15.540 Or in this case, the r vector is changing in length, 00:24:15.540 --> 00:24:18.670 so the velocity has something to do with the change in length. 00:24:18.670 --> 00:24:23.190 This term is due only to the fact that it is rotating. 00:24:23.190 --> 00:24:26.280 If the dog's not running, is his position vector 00:24:26.280 --> 00:24:28.790 changing in length? 00:24:28.790 --> 00:24:30.220 No. 00:24:30.220 --> 00:24:32.370 So this term here comes from taking 00:24:32.370 --> 00:24:35.140 this derivative of the position vector 00:24:35.140 --> 00:24:37.330 within the rotating frame. 00:24:37.330 --> 00:24:40.670 So this accounts for the change in length of that position 00:24:40.670 --> 00:24:42.570 vector. 00:24:42.570 --> 00:24:44.700 This accounts for the rotation. 00:24:44.700 --> 00:24:50.915 This term is the same as seen from in any frame. 00:24:50.915 --> 00:24:51.790 This is the subtlety. 00:24:54.900 --> 00:24:59.030 This term, you get the same answer if you see it from here. 00:24:59.030 --> 00:25:00.870 This velocity, this contribution. 00:25:00.870 --> 00:25:07.040 The dog's speed with respect to this frame it's in 00:25:07.040 --> 00:25:09.370 is the same to you out there in the fixed frame 00:25:09.370 --> 00:25:11.240 as it is to somebody sitting here 00:25:11.240 --> 00:25:14.460 in the frame moving with him. 00:25:14.460 --> 00:25:15.305 Important point. 00:25:19.110 --> 00:25:21.640 It'll become more important as you do problems. 00:25:21.640 --> 00:25:23.999 You need to remember that doesn't matter 00:25:23.999 --> 00:25:25.040 where you're calculating. 00:25:25.040 --> 00:25:27.440 You're given the velocity of the dog 00:25:27.440 --> 00:25:30.960 in this frame, and little i j k components, 00:25:30.960 --> 00:25:34.810 it is in magnitude exactly the same velocity as you would 00:25:34.810 --> 00:25:37.784 see it in the fixed frame. 00:25:37.784 --> 00:25:39.950 But then you just have to account for the right unit 00:25:39.950 --> 00:25:41.190 vectors and so forth. 00:25:41.190 --> 00:25:42.062 Yes. 00:25:42.062 --> 00:25:44.894 AUDIENCE: [INAUDIBLE] is the same as the frame, 00:25:44.894 --> 00:25:47.249 but the direction is changing? 00:25:47.249 --> 00:25:49.540 PROFESSOR: The direction is the same in any frame also. 00:25:49.540 --> 00:25:52.410 It's just that you have to decide which unit 00:25:52.410 --> 00:25:54.772 vectors to represent it in. 00:25:54.772 --> 00:25:56.230 And when you're in this frame, it's 00:25:56.230 --> 00:26:00.495 easiest to represent it in little i j k, right? 00:26:00.495 --> 00:26:01.870 But if you're in the fixed frame, 00:26:01.870 --> 00:26:05.450 you may want to actually know the velocity in the fixed 00:26:05.450 --> 00:26:08.584 frames unit vectors, and there's just geometric conversions 00:26:08.584 --> 00:26:09.500 that'll give you that. 00:26:09.500 --> 00:26:10.460 Yes. 00:26:10.460 --> 00:26:13.350 AUDIENCE: But if theta is like a function of time [INAUDIBLE] 00:26:13.350 --> 00:26:15.270 how would you use those? 00:26:15.270 --> 00:26:17.190 Does it just work the same way? 00:26:17.190 --> 00:26:20.320 PROFESSOR: So she says if theta is a function of time, 00:26:20.320 --> 00:26:24.220 if you were trying to convert-- this dog's running in the y 00:26:24.220 --> 00:26:28.430 direction, which is going in the master frame over there, 00:26:28.430 --> 00:26:31.410 is going to have a component in the negative capital I hat 00:26:31.410 --> 00:26:34.480 and positive capital J hat direction, right? 00:26:34.480 --> 00:26:38.020 And you could break down-- let's say his velocity vector is 00:26:38.020 --> 00:26:42.140 like this-- you could break it down into a component that's 00:26:42.140 --> 00:26:43.840 parallel to the ground, and a component 00:26:43.840 --> 00:26:45.470 perpendicular to the ground. 00:26:45.470 --> 00:26:47.630 And those then would be in that reference frame. 00:26:51.347 --> 00:26:53.430 You've all, I would think, have done some problems 00:26:53.430 --> 00:26:55.690 using polar coordinates. 00:26:55.690 --> 00:26:59.390 And the conversion between x and y and r and theta 00:26:59.390 --> 00:27:01.820 is you used cosine thetas and sine 00:27:01.820 --> 00:27:03.400 thetas to compute it, right? 00:27:03.400 --> 00:27:08.700 So read that handout on kinematics, 00:27:08.700 --> 00:27:16.940 because it proves this formula by doing just that. 00:27:16.940 --> 00:27:18.890 It puts everything in terms of thetas 00:27:18.890 --> 00:27:21.370 that are functions of time. 00:27:21.370 --> 00:27:23.580 And it's kind of a long, painful process, 00:27:23.580 --> 00:27:26.100 but you can do that, grind it out, 00:27:26.100 --> 00:27:28.520 and you will then, when you assemble the terms, 00:27:28.520 --> 00:27:29.980 you end up with this. 00:27:29.980 --> 00:27:31.760 And once you know this formula, it's 00:27:31.760 --> 00:27:36.720 far, far easier than to calculate 00:27:36.720 --> 00:27:38.889 all this in terms of thetas and theta dots, 00:27:38.889 --> 00:27:39.805 and sines and cosines. 00:27:44.730 --> 00:27:45.230 OK? 00:27:47.961 --> 00:27:48.460 All right. 00:27:48.460 --> 00:27:49.000 That's good. 00:27:49.000 --> 00:27:50.050 That's important stuff. 00:27:50.050 --> 00:27:52.997 We're going to make great use of these formulas 00:27:52.997 --> 00:27:54.080 over the rest of the term. 00:28:02.980 --> 00:28:06.660 So I'm going to just take a break from kinematics. 00:28:06.660 --> 00:28:09.480 It's time to talk a little bit about Newton's laws, 00:28:09.480 --> 00:28:12.010 because we really want to be able to do dynamics problems. 00:28:12.010 --> 00:28:14.780 And so this has been about describing the motion. 00:28:14.780 --> 00:28:19.124 And now we want to talk about the laws on which we depend 00:28:19.124 --> 00:28:20.790 in order to be able to calculate things, 00:28:20.790 --> 00:28:24.600 and to draw, write equations of motion and so forth. 00:28:24.600 --> 00:28:26.780 So any final questions about kinematics? 00:28:26.780 --> 00:28:29.520 Because I'm going to change topics. 00:28:29.520 --> 00:28:30.454 Yes. 00:28:30.454 --> 00:28:34.406 AUDIENCE: So for that equation to the [INAUDIBLE], then 00:28:34.406 --> 00:28:37.534 the acceleration of [INAUDIBLE] with respect 00:28:37.534 --> 00:28:39.840 to the reference frame, that's [INAUDIBLE] 00:28:39.840 --> 00:28:41.779 is just zero, right? 00:28:41.779 --> 00:28:43.820 PROFESSOR: You're asking about acceleration, now? 00:28:43.820 --> 00:28:44.710 Do you really mean? 00:28:44.710 --> 00:28:46.130 Is that what you mean? 00:28:46.130 --> 00:28:48.515 Because I haven't talked much about acceleration. 00:28:48.515 --> 00:28:51.490 AUDIENCE: I was just wondering, because also [INAUDIBLE] x 00:28:51.490 --> 00:28:53.367 to the velocity change. 00:28:53.367 --> 00:28:54.200 PROFESSOR: This one? 00:28:54.200 --> 00:28:54.825 AUDIENCE: Yeah. 00:28:54.825 --> 00:28:59.960 PROFESSOR: So this equation is true of any vector. 00:28:59.960 --> 00:29:02.240 This is basically vector calculus. 00:29:02.240 --> 00:29:04.910 It's true of any vector in a rotating frame. 00:29:04.910 --> 00:29:08.130 It's time derivative with respect to a fixed frame. 00:29:08.130 --> 00:29:10.880 It is the derivative you would see in the rotating 00:29:10.880 --> 00:29:14.760 frame, which is just the change of length of the vector, 00:29:14.760 --> 00:29:17.640 and the part due to the fact that you are making it rotate. 00:29:17.640 --> 00:29:21.490 And we'll do examples of this, I promise you, very soon. 00:29:21.490 --> 00:29:22.840 So you'll see how it's applied. 00:29:22.840 --> 00:29:23.340 OK? 00:29:26.540 --> 00:29:27.660 OK. 00:29:27.660 --> 00:29:30.700 So let's do a quick review of Newton. 00:29:50.940 --> 00:29:51.770 So Newton's Laws. 00:29:51.770 --> 00:29:54.270 He had three of them. 00:29:54.270 --> 00:29:56.900 And we're all pretty familiar with the second. 00:29:56.900 --> 00:29:59.515 The first, though, is called the law of inertia. 00:30:05.760 --> 00:30:07.780 And basically, the first law says, 00:30:07.780 --> 00:30:11.360 if an object is motionless in the absence of forces, 00:30:11.360 --> 00:30:12.030 what happens? 00:30:16.480 --> 00:30:17.410 Stays motionless. 00:30:17.410 --> 00:30:19.200 Or, if it's at some constant velocity 00:30:19.200 --> 00:30:24.340 in the absence of forces it stays at constant velocity, 00:30:24.340 --> 00:30:25.370 right? 00:30:25.370 --> 00:30:30.500 So these have been stated many ways over the years. 00:30:30.500 --> 00:30:33.450 I'm just going to try to come up with short ones for the board. 00:30:33.450 --> 00:31:05.000 So in the absence of forces an object-- 00:31:05.000 --> 00:31:07.580 a particle-- moves with constant velocity. 00:31:07.580 --> 00:31:10.740 In fact, Newton only talked about motions of particles, 00:31:10.740 --> 00:31:13.290 not as in little tiny things. 00:31:13.290 --> 00:31:17.160 Not about rigid bodies that have finite dimensions. 00:31:17.160 --> 00:31:21.810 He thought of the planets, appropriately, as particles. 00:31:21.810 --> 00:31:23.030 Second law. 00:31:23.030 --> 00:31:24.710 Second law, I won't write it all out. 00:31:24.710 --> 00:31:26.890 But basically F equals ma. 00:31:26.890 --> 00:31:30.350 Their vectors equals the time derivative 00:31:30.350 --> 00:31:31.547 of the linear momentum. 00:31:31.547 --> 00:31:33.130 That's what we know as the second law. 00:31:33.130 --> 00:31:38.030 The sum of all the external forces 00:31:38.030 --> 00:31:40.610 equals the mass times the acceleration. 00:31:40.610 --> 00:31:43.460 Another statement of the second law. 00:31:43.460 --> 00:31:44.646 OK? 00:31:44.646 --> 00:31:45.520 So you know that one. 00:31:45.520 --> 00:31:47.410 That's the one you're most familiar with. 00:31:47.410 --> 00:31:48.200 Third one. 00:31:48.200 --> 00:31:50.006 What's the third one? 00:31:50.006 --> 00:31:52.910 AUDIENCE: Every action has an equal and opposite reaction. 00:31:52.910 --> 00:31:56.660 PROFESSOR: So every action has an equal and opposite reaction. 00:31:56.660 --> 00:32:00.510 And I'm going to draw a picture for this one. 00:32:00.510 --> 00:32:02.800 So here's a particle. 00:32:02.800 --> 00:32:05.020 Here's another particle. 00:32:05.020 --> 00:32:07.070 This is particle 2. 00:32:07.070 --> 00:32:09.120 Particle 1. 00:32:09.120 --> 00:32:12.070 There's a force on this particle that's 00:32:12.070 --> 00:32:16.920 the force on 2 due to the presence of 1. 00:32:16.920 --> 00:32:20.000 They've each got a little gravity, little attraction 00:32:20.000 --> 00:32:20.740 from one another. 00:32:20.740 --> 00:32:27.540 And this one has a force on particle 1 due to particle 2. 00:32:27.540 --> 00:32:28.230 OK? 00:32:28.230 --> 00:32:29.313 They could be two planets. 00:32:32.590 --> 00:32:38.570 And basically what he said is that for every action, 00:32:38.570 --> 00:32:41.010 there's an equal and opposite reaction. 00:32:41.010 --> 00:32:47.210 That means that f 21 equals minus f 12. 00:32:50.120 --> 00:32:53.035 And this is called the strong form 00:32:53.035 --> 00:33:01.450 of Newton's third law, which the forces are equal, 00:33:01.450 --> 00:33:04.870 opposite, and collinear. 00:33:08.060 --> 00:33:12.160 They actually point exactly opposite one another. 00:33:12.160 --> 00:33:14.470 And this is true of mechanical systems. 00:33:14.470 --> 00:33:19.850 You get some rather interesting subtleties 00:33:19.850 --> 00:33:21.970 when you get into electromagnetic fields, 00:33:21.970 --> 00:33:23.897 and charged particles, and things like that. 00:33:23.897 --> 00:33:25.480 So you have to think really carefully. 00:33:25.480 --> 00:33:29.250 But for the mechanical systems, the strong form 00:33:29.250 --> 00:33:32.110 will suit us just fine. 00:33:32.110 --> 00:33:33.339 OK. 00:33:33.339 --> 00:33:34.380 Those are the three laws. 00:33:34.380 --> 00:33:41.540 Newton made one major condition for those to be true. 00:33:41.540 --> 00:33:45.690 What is the assumption that must be satisfied for these laws 00:33:45.690 --> 00:33:48.094 to be true? 00:33:48.094 --> 00:33:50.830 AUDIENCE: [INAUDIBLE] an inertial reference frame? 00:33:50.830 --> 00:33:55.190 PROFESSOR: I hear "an inertial reference frame," right? 00:33:55.190 --> 00:33:57.770 And that's what he assumed. 00:33:57.770 --> 00:34:00.990 You have to be in an inertial frame for these statements 00:34:00.990 --> 00:34:02.110 to be true. 00:34:02.110 --> 00:34:05.510 So then what I want to spend a few minutes talking about 00:34:05.510 --> 00:34:09.845 is basically what's an inertial frame. 00:34:09.845 --> 00:34:11.219 Because that's going to be really 00:34:11.219 --> 00:34:12.480 important in this subject. 00:34:12.480 --> 00:34:15.639 When you start getting on things that move and rotate, 00:34:15.639 --> 00:34:18.980 things sometimes are not inertial. 00:34:18.980 --> 00:34:21.639 So I'm going to ask you a quick question. 00:34:21.639 --> 00:34:24.090 So a reference frame. 00:34:24.090 --> 00:34:27.771 So we had our fixed frame up there, sitting on the ground, 00:34:27.771 --> 00:34:28.520 here on the Earth. 00:34:28.520 --> 00:34:32.139 Is the Earth an inertial reference frame, standing 00:34:32.139 --> 00:34:33.360 here observing things? 00:34:37.520 --> 00:34:38.540 I want a show of hands. 00:34:38.540 --> 00:34:39.820 I really want some participation here. 00:34:39.820 --> 00:34:41.940 How many of you think Earth's an inertial frame? 00:34:41.940 --> 00:34:42.845 Raise your hand. 00:34:42.845 --> 00:34:44.550 It's true. 00:34:44.550 --> 00:34:47.580 How many think it's not true? 00:34:47.580 --> 00:34:50.000 How many think it depends? 00:34:50.000 --> 00:34:52.250 On what? 00:34:52.250 --> 00:34:55.250 AUDIENCE: What you are focusing on. 00:34:55.250 --> 00:34:57.080 PROFESSOR: What the problem is. 00:34:57.080 --> 00:34:59.740 He says it depends on what you're focusing on. 00:34:59.740 --> 00:35:04.310 So it really depends on the sizes of the forces 00:35:04.310 --> 00:35:07.430 and the motion you're interested in. 00:35:07.430 --> 00:35:10.890 Can you give me an example in which the Earth cannot be 00:35:10.890 --> 00:35:14.730 assumed to be an inertial frame? 00:35:14.730 --> 00:35:16.240 Practical example? 00:35:16.240 --> 00:35:17.740 AUDIENCE: Oh. 00:35:17.740 --> 00:35:18.890 Not practical. 00:35:18.890 --> 00:35:20.874 PROFESSOR: Impractical. 00:35:20.874 --> 00:35:23.850 AUDIENCE: The rotation of Earth is slowing? 00:35:23.850 --> 00:35:26.167 The rotation of the Earth is slowing? 00:35:26.167 --> 00:35:28.000 PROFESSOR: Rotation of the earth is slowing. 00:35:28.000 --> 00:35:29.850 That's interesting. 00:35:29.850 --> 00:35:32.900 To account for that, you would definitely not be able to just 00:35:32.900 --> 00:35:34.370 to assume we're inertial. 00:35:34.370 --> 00:35:37.710 But what's an everyday example of where, 00:35:37.710 --> 00:35:40.954 if you're trying to solve a problem in this field, 00:35:40.954 --> 00:35:42.745 you couldn't make this inertial assumption? 00:35:45.896 --> 00:35:47.820 AUDIENCE: Rotation of the planets? 00:35:47.820 --> 00:35:49.700 PROFESSOR: Well, no, for the most part, 00:35:49.700 --> 00:35:50.870 you could get most of it. 00:35:50.870 --> 00:35:52.267 But, yeah. 00:35:52.267 --> 00:35:53.850 AUDIENCE: I think the weather, really, 00:35:53.850 --> 00:35:56.175 because the wind currents, or like the ocean 00:35:56.175 --> 00:35:58.970 currents are affected by the fact that the Earth is round. 00:35:58.970 --> 00:36:00.350 PROFESSOR: OK. 00:36:00.350 --> 00:36:01.800 That's a good one. 00:36:01.800 --> 00:36:05.480 In order to account for the circulation 00:36:05.480 --> 00:36:07.920 you have to take into account the Earth's motion. 00:36:07.920 --> 00:36:10.125 Let me give you an example. 00:36:10.125 --> 00:36:10.625 Clocks. 00:36:14.720 --> 00:36:18.200 Pendulum clocks. 00:36:18.200 --> 00:36:19.850 Does the speed of a clock change? 00:36:19.850 --> 00:36:24.550 Is it different at high noon from midnight? 00:36:29.691 --> 00:36:30.440 What do you think? 00:36:30.440 --> 00:36:31.930 Yes or no? 00:36:31.930 --> 00:36:34.080 How many think that the actual speed 00:36:34.080 --> 00:36:35.950 of a clock-- the length of a second-- 00:36:35.950 --> 00:36:38.220 would be different at noon on the Earth-- 00:36:38.220 --> 00:36:40.274 a pendulum clock-- from midnight? 00:36:40.274 --> 00:36:41.690 How many think that might be true? 00:36:41.690 --> 00:36:42.880 It's different? 00:36:42.880 --> 00:36:45.270 How many don't believe that it would be true? 00:36:45.270 --> 00:36:47.140 How many just not raising their hands? 00:36:47.140 --> 00:36:48.190 Come on you guys. 00:36:48.190 --> 00:36:49.130 Let's get with it. 00:36:49.130 --> 00:36:49.629 OK. 00:36:52.600 --> 00:36:55.590 So the effective acceleration of gravity 00:36:55.590 --> 00:36:58.800 that that pendulum feels is different at noon 00:36:58.800 --> 00:37:01.420 from midnight. 00:37:01.420 --> 00:37:07.590 And one reason is because at noon the sun's pulling away 00:37:07.590 --> 00:37:09.890 from the surface of the Earth, and at midnight 00:37:09.890 --> 00:37:12.664 the sun's gravity is pulling in the same direction 00:37:12.664 --> 00:37:14.080 as toward the center of the Earth. 00:37:14.080 --> 00:37:15.990 The total effective gravitational 00:37:15.990 --> 00:37:18.410 pull that the pendulum field changes 00:37:18.410 --> 00:37:21.489 due to the rotation of the Earth, what it feels. 00:37:21.489 --> 00:37:23.030 The sum of the forces on the pendulum 00:37:23.030 --> 00:37:25.200 includes the Earth's gravity and the sun's gravity. 00:37:25.200 --> 00:37:26.790 And the moon also does this. 00:37:26.790 --> 00:37:29.750 So there's daily variations in the speed 00:37:29.750 --> 00:37:33.160 of pendulum clocks just because of the Earth's 00:37:33.160 --> 00:37:35.640 rotation with respect to the sun and the moon. 00:37:35.640 --> 00:37:36.471 Yeah. 00:37:36.471 --> 00:37:39.560 AUDIENCE: Is that even great enough to be measurable? 00:37:39.560 --> 00:37:41.890 PROFESSOR: So she says, "Is that even great enough 00:37:41.890 --> 00:37:42.710 to be measurable?" 00:37:42.710 --> 00:37:44.160 So I have a friend, a guy named Hugh Hunt. 00:37:44.160 --> 00:37:45.950 He's a professor that teaches dynamics 00:37:45.950 --> 00:37:47.310 at Cambridge University. 00:37:47.310 --> 00:37:52.010 And he is the keeper of the Trinity College clock. 00:37:52.010 --> 00:37:53.990 Trinity College is where Newton was. 00:37:53.990 --> 00:37:56.620 So he took me to his clock one day up in this tower. 00:37:56.620 --> 00:37:58.580 And it's got about a two-meter pendulum on it. 00:37:58.580 --> 00:38:00.720 And he has got that clock to run so 00:38:00.720 --> 00:38:06.260 that it gains no more than one second per month. 00:38:06.260 --> 00:38:07.910 He's really tuned it up carefully. 00:38:07.910 --> 00:38:10.630 And the key to being able to do that 00:38:10.630 --> 00:38:15.672 is to do things on monthly averages. 00:38:15.672 --> 00:38:18.110 So he's got it tuned so that over a month 00:38:18.110 --> 00:38:20.540 it just barely gains a tiny bit. 00:38:20.540 --> 00:38:22.370 But if you measure it very carefully 00:38:22.370 --> 00:38:26.320 over the course of the day, it has amazingly large 00:38:26.320 --> 00:38:28.320 fluctuations. 00:38:28.320 --> 00:38:30.830 One of them is due to the thing I just described. 00:38:30.830 --> 00:38:34.460 So if you're really trying to keep close time, it matters. 00:38:34.460 --> 00:38:34.960 OK. 00:38:34.960 --> 00:38:37.176 Another one is gunnery. 00:38:37.176 --> 00:38:38.670 You're shooting long range. 00:38:38.670 --> 00:38:39.920 You're trying to hit a target. 00:38:39.920 --> 00:38:41.750 The fact that the earth rotates, you 00:38:41.750 --> 00:38:44.640 will not hit the target if you don't 00:38:44.640 --> 00:38:48.545 account for the effects that are caused by the Earth's rotation. 00:38:48.545 --> 00:38:49.920 And it's one of the first reasons 00:38:49.920 --> 00:38:52.997 that people got into understanding 00:38:52.997 --> 00:38:55.330 the importance of whether or not it's an inertial frame, 00:38:55.330 --> 00:38:59.430 was gunnery in the old days, from naval ships and so forth. 00:38:59.430 --> 00:39:00.100 OK. 00:39:00.100 --> 00:39:01.724 So we have three laws. 00:39:01.724 --> 00:39:04.140 I want to talk a little bit about the first and the third. 00:39:04.140 --> 00:39:05.940 We're going to use the second a lot. 00:39:05.940 --> 00:39:09.015 So the first law, most people think of the first law 00:39:09.015 --> 00:39:12.770 as being a special case of the second, right? 00:39:12.770 --> 00:39:15.950 It's just when there's no forces, nothing changes. 00:39:15.950 --> 00:39:19.780 But I think the first law is useful in its own right. 00:39:19.780 --> 00:39:22.720 And one of the reasons why it's called the law of inertia, 00:39:22.720 --> 00:39:26.180 it's the law that allows you to do a test 00:39:26.180 --> 00:39:27.930 to discover whether or not you're actually 00:39:27.930 --> 00:39:30.220 in an inertial frame. 00:39:30.220 --> 00:39:31.770 Useful to be able to do that. 00:39:31.770 --> 00:39:36.372 So I'm going to give you an example. 00:39:36.372 --> 00:39:38.330 And actually, I did want to ask you a question. 00:39:38.330 --> 00:39:43.310 So three possible answers to this question. 00:39:43.310 --> 00:39:49.730 An inertial frame can-- how to pose this? 00:39:49.730 --> 00:39:54.760 If you're in an inertial frame, can it be accelerating? 00:39:54.760 --> 00:39:59.480 If you're in an inertial frame, can it be rotating? 00:39:59.480 --> 00:40:01.460 Or if you're in an inertial frame, 00:40:01.460 --> 00:40:05.760 it can neither accelerate or rotate. 00:40:05.760 --> 00:40:08.290 So which of those three answers is the best answer 00:40:08.290 --> 00:40:11.940 for conditions to be in an inertial frame? 00:40:11.940 --> 00:40:14.650 Non-accelerating, non-rotating, or both? 00:40:14.650 --> 00:40:17.240 How many believe non-accelerating? 00:40:20.270 --> 00:40:21.230 Just non-accelerating? 00:40:21.230 --> 00:40:23.080 How many believe in just not rotating? 00:40:23.080 --> 00:40:25.130 How many believe both? 00:40:25.130 --> 00:40:26.310 Both is right. 00:40:26.310 --> 00:40:29.260 If you are in a frame which is rotating or accelerating 00:40:29.260 --> 00:40:31.270 it's not inertial. 00:40:31.270 --> 00:40:31.770 OK. 00:40:31.770 --> 00:40:34.420 So just rotation causes it not to be [INAUDIBLE]. 00:40:34.420 --> 00:40:35.940 So let's test that. 00:40:35.940 --> 00:40:40.070 I'm going to pick two cases quickly. 00:40:40.070 --> 00:40:44.820 Let's use this law of inertia to set up 00:40:44.820 --> 00:40:48.350 a test to see if a couple different frames are 00:40:48.350 --> 00:40:49.320 in fact inertial. 00:40:49.320 --> 00:40:51.750 So I've got a cart here. 00:40:58.580 --> 00:41:01.430 And I'm sitting here-- or you are. 00:41:01.430 --> 00:41:04.220 We're sitting on the cart. 00:41:04.220 --> 00:41:13.150 And this cart is accelerating. 00:41:17.210 --> 00:41:20.970 Acceleration of A with respect to O 00:41:20.970 --> 00:41:26.470 is, I'll call it a naught I hat. 00:41:26.470 --> 00:41:29.770 This cart's accelerating in that way, that direction, 00:41:29.770 --> 00:41:31.540 the positive I hat direction. 00:41:31.540 --> 00:41:32.410 OK. 00:41:32.410 --> 00:41:37.070 Now I'm sitting here on the cart-- not very sensitive. 00:41:37.070 --> 00:41:41.490 And I want to test whether or not I'm in an inertial frame. 00:41:41.490 --> 00:41:44.380 So let's pretend this cart, I've got an air table there, 00:41:44.380 --> 00:41:45.325 frictionless table. 00:41:45.325 --> 00:41:47.330 And I've got a hockey puck. 00:41:47.330 --> 00:41:50.730 And I set it down on the frictionless table 00:41:50.730 --> 00:41:52.960 and let it go. 00:41:52.960 --> 00:41:53.920 What happens? 00:41:53.920 --> 00:41:54.880 What do I observe? 00:41:57.327 --> 00:41:58.202 AUDIENCE: [INAUDIBLE] 00:42:01.650 --> 00:42:05.660 PROFESSOR: So the puck will accelerate, you're 00:42:05.660 --> 00:42:08.640 saying, towards me, right? 00:42:08.640 --> 00:42:10.835 AUDIENCE: [INAUDIBLE] 00:42:10.835 --> 00:42:11.460 PROFESSOR: Hmm. 00:42:11.460 --> 00:42:12.600 It'll move. 00:42:12.600 --> 00:42:15.490 So she says the puck will move towards you if you let it go. 00:42:15.490 --> 00:42:17.750 So is that an indication of whether or not 00:42:17.750 --> 00:42:18.950 you're in an inertial frame? 00:42:21.550 --> 00:42:23.360 So the test-- this is first law, now. 00:42:23.360 --> 00:42:25.690 So the test-- if you're in this frame, 00:42:25.690 --> 00:42:28.940 and you want to know whether or not this is an inertial frame, 00:42:28.940 --> 00:42:31.730 and I claim you can set this puck out there, 00:42:31.730 --> 00:42:36.024 if you're in an inertial frame, what should it do? 00:42:36.024 --> 00:42:36.910 Not move. 00:42:36.910 --> 00:42:39.830 If it moves, there's something going on. 00:42:39.830 --> 00:42:41.370 Something fishy, right? 00:42:41.370 --> 00:42:42.210 OK. 00:42:42.210 --> 00:42:46.530 So the acceleration-- and we'll call 00:42:46.530 --> 00:42:51.250 the puck at B. So the acceleration of B 00:42:51.250 --> 00:42:53.320 with respect to O-- and there's no rotation here. 00:42:53.320 --> 00:42:56.320 So the acceleration of B with respect to O, 00:42:56.320 --> 00:42:58.710 it can be written as the acceleration of A 00:42:58.710 --> 00:43:02.250 with respect to O, plus the acceleration of B 00:43:02.250 --> 00:43:11.280 with respect to A. 00:43:11.280 --> 00:43:14.990 Now this is O here. 00:43:14.990 --> 00:43:16.547 This is an inertial frame. 00:43:19.530 --> 00:43:22.390 And if it's an inertial frame, and there 00:43:22.390 --> 00:43:26.670 are no forces in the x direction acting on this puck, 00:43:26.670 --> 00:43:29.260 then I'm going to say that the sum of the forces on that puck 00:43:29.260 --> 00:43:32.100 equal zero. 00:43:32.100 --> 00:43:35.000 And therefore, what can I say about the acceleration 00:43:35.000 --> 00:43:37.440 of the puck? 00:43:37.440 --> 00:43:41.589 What's the acceleration of the puck as seen from O? 00:43:41.589 --> 00:43:42.380 It's gotta be zero. 00:43:42.380 --> 00:43:47.390 If no forces-- so the summation of the forces with respect 00:43:47.390 --> 00:43:51.190 to this O frame in the x direction, 00:43:51.190 --> 00:43:54.830 if there's zero, that implies the acceleration of B 00:43:54.830 --> 00:43:56.770 with respect to O is got to be zero. 00:44:01.730 --> 00:44:06.730 Knowing that, I can now solve for the acceleration of B 00:44:06.730 --> 00:44:09.610 with respect to A. And that's going 00:44:09.610 --> 00:44:13.315 to be minus the acceleration of A with respect to O. 00:44:13.315 --> 00:44:18.425 And that's minus a naught I hat. 00:44:23.070 --> 00:44:27.420 So you are correct in saying that it moves. 00:44:27.420 --> 00:44:31.390 But it actually accelerates. 00:44:31.390 --> 00:44:35.354 From your point of view, what you see sitting there in A, 00:44:35.354 --> 00:44:36.895 you were going to see this accelerate 00:44:36.895 --> 00:44:38.310 in the opposite direction. 00:44:38.310 --> 00:44:40.080 And that's a dead giveaway that you're not 00:44:40.080 --> 00:44:41.890 in an inertial frame. 00:44:41.890 --> 00:44:42.390 OK. 00:44:50.380 --> 00:44:53.910 Come on. 00:44:53.910 --> 00:44:55.270 OK. 00:44:55.270 --> 00:44:56.460 So a little harder problem. 00:45:05.550 --> 00:45:08.863 Now we've got our merry-go-round. 00:45:08.863 --> 00:45:10.220 OK. 00:45:10.220 --> 00:45:11.220 And it's fixed. 00:45:11.220 --> 00:45:12.395 Not on the train. 00:45:12.395 --> 00:45:14.220 It's just sitting here. 00:45:14.220 --> 00:45:17.160 But it can spin. 00:45:17.160 --> 00:45:20.340 And you're sitting at A. So you're 00:45:20.340 --> 00:45:24.040 up above this merry-go-round looking down on it. 00:45:24.040 --> 00:45:25.937 And it's rotating. 00:45:25.937 --> 00:45:27.520 Out there you're in an inertial frame. 00:45:27.520 --> 00:45:29.728 But now you come over to here, and you sit right here 00:45:29.728 --> 00:45:31.840 at the center. 00:45:31.840 --> 00:45:33.390 And if you're right at the center, 00:45:33.390 --> 00:45:36.200 you might not actually feel a thing, right? 00:45:36.200 --> 00:45:37.820 And there's no windows. 00:45:37.820 --> 00:45:43.330 So this could be a spaceship out there, slowly rolling over. 00:45:43.330 --> 00:45:46.450 So you're sitting inside of this system, 00:45:46.450 --> 00:45:49.200 no windows, right at the center, can't feel a thing. 00:45:49.200 --> 00:45:52.350 And I want you to construct a test-- be 00:45:52.350 --> 00:45:54.810 using the first law-- that'll tell you whether or not 00:45:54.810 --> 00:45:56.350 you're in an inertial frame. 00:45:56.350 --> 00:45:57.110 What might you do? 00:46:02.185 --> 00:46:04.310 AUDIENCE: Set a ball down on the ground [INAUDIBLE] 00:46:04.310 --> 00:46:05.556 roll off to the edge. 00:46:05.556 --> 00:46:06.430 PROFESSOR: All right. 00:46:06.430 --> 00:46:07.804 He says set a ball on the ground, 00:46:07.804 --> 00:46:09.330 and see if it rolls out to the edge. 00:46:09.330 --> 00:46:09.830 Right? 00:46:09.830 --> 00:46:12.010 See if it moves. 00:46:12.010 --> 00:46:14.302 AUDIENCE: Just [INAUDIBLE] is this [INAUDIBLE] 00:46:14.302 --> 00:46:16.510 under the effect of gravity, or is there [INAUDIBLE]? 00:46:19.350 --> 00:46:24.140 PROFESSOR: Let's really make it a merry-go-round. 00:46:24.140 --> 00:46:25.880 So there's gravity. 00:46:25.880 --> 00:46:27.770 So I want gravity to be useful here. 00:46:27.770 --> 00:46:29.630 It keeps the thing on the surface. 00:46:29.630 --> 00:46:31.400 Doesn't just go drifting off. 00:46:31.400 --> 00:46:33.760 So yeah, let's say we have gravity. 00:46:33.760 --> 00:46:35.260 The axis is vertical. 00:46:35.260 --> 00:46:38.430 You're sitting here, but you can't see outside, 00:46:38.430 --> 00:46:40.280 and you want to do this test. 00:46:40.280 --> 00:46:42.240 So do you agree if you set the ball down, 00:46:42.240 --> 00:46:44.030 you might learn something? 00:46:44.030 --> 00:46:44.830 OK. 00:46:44.830 --> 00:46:48.190 So you set the ball down. 00:46:53.990 --> 00:46:55.310 And you're here watching. 00:46:55.310 --> 00:46:57.050 And here's the ball. 00:46:57.050 --> 00:47:00.030 And let's say you've got a ball, and you actually 00:47:00.030 --> 00:47:02.380 have a string on the ball. 00:47:02.380 --> 00:47:03.490 Set it out there. 00:47:07.480 --> 00:47:09.800 So initially you've got this ball out there. 00:47:13.180 --> 00:47:15.710 And you're sitting here at the center. 00:47:15.710 --> 00:47:19.610 And this merry-go-round's going round and around. 00:47:19.610 --> 00:47:23.200 What can you sense that tells you 00:47:23.200 --> 00:47:25.570 that you're not in an inertial frame, 00:47:25.570 --> 00:47:27.050 if you're holding onto this string? 00:47:27.050 --> 00:47:30.130 Does the ball move, first of all? 00:47:32.376 --> 00:47:34.000 Would the ball move if I'm sitting here 00:47:34.000 --> 00:47:35.958 hanging onto the string, and I set it out there 00:47:35.958 --> 00:47:37.560 and set it down? 00:47:37.560 --> 00:47:38.480 No, it won't move. 00:47:38.480 --> 00:47:40.920 But what do you feel in the string? 00:47:40.920 --> 00:47:42.020 Tension in the string. 00:47:42.020 --> 00:47:42.580 OK. 00:47:42.580 --> 00:47:45.520 So now you've got indication number one 00:47:45.520 --> 00:47:47.811 that there's something fishy. 00:47:47.811 --> 00:47:48.310 OK. 00:47:48.310 --> 00:47:49.518 Now you let go of the string. 00:47:52.460 --> 00:47:55.270 What should happen? 00:47:55.270 --> 00:47:56.590 So you [INAUDIBLE] go out. 00:47:56.590 --> 00:47:57.420 All right. 00:47:57.420 --> 00:48:00.340 But now I'm going to ask you a little harder question. 00:48:00.340 --> 00:48:05.104 What direction should the ball travel in once you release it? 00:48:08.008 --> 00:48:09.460 AUDIENCE: Radially outwards. 00:48:09.460 --> 00:48:10.668 PROFESSOR: Radially outwards. 00:48:10.668 --> 00:48:13.790 I have one shot at radially outwards. 00:48:13.790 --> 00:48:14.720 Any other thoughts? 00:48:14.720 --> 00:48:16.136 AUDIENCE: From your point of view, 00:48:16.136 --> 00:48:18.136 it wouldn't seem to go in a straight line. 00:48:18.136 --> 00:48:21.270 It would seem to curve off to one side. 00:48:21.270 --> 00:48:25.110 PROFESSOR: So I have one postulate 00:48:25.110 --> 00:48:28.790 that it will curve-- it will go away and curve off, right? 00:48:28.790 --> 00:48:31.160 Which way would it curve? 00:48:31.160 --> 00:48:32.790 So you're saying not radially. 00:48:32.790 --> 00:48:34.052 You say it's going to curve. 00:48:34.052 --> 00:48:35.350 AUDIENCE: Opposite direction that you're spinning. 00:48:35.350 --> 00:48:36.683 PROFESSOR: Opposite to the spin. 00:48:36.683 --> 00:48:40.239 He says it will curve opposite to the spin. 00:48:40.239 --> 00:48:42.030 AUDIENCE: But only from your point of view. 00:48:42.030 --> 00:48:43.405 PROFESSOR: This is from the point 00:48:43.405 --> 00:48:47.619 of view on the merry-go-round. 00:48:47.619 --> 00:48:49.160 Point of view off the merry-go-round, 00:48:49.160 --> 00:48:51.790 might be easier to reason this. 00:48:51.790 --> 00:48:55.070 So now you're up above the merry-go-round in an inertial 00:48:55.070 --> 00:48:57.590 frame, up above this merry-go-round, just 00:48:57.590 --> 00:49:01.670 looking down, like sitting up in a tree and looking down on it. 00:49:01.670 --> 00:49:02.540 What do you see? 00:49:07.930 --> 00:49:11.360 AUDIENCE: [INAUDIBLE], but if you're on the merry-go-round 00:49:11.360 --> 00:49:12.405 then you [INAUDIBLE] 00:49:12.405 --> 00:49:13.280 PROFESSOR: All right. 00:49:13.280 --> 00:49:16.030 So this is an argument for if you're on the merry-go-round, 00:49:16.030 --> 00:49:18.070 you'll see radial motion. 00:49:18.070 --> 00:49:20.490 If you're in the tree, you'll see the curve. 00:49:20.490 --> 00:49:21.550 So we're going to do-- 00:49:21.550 --> 00:49:23.508 AUDIENCE: I would like to retract my statement. 00:49:26.100 --> 00:49:27.240 PROFESSOR: OK. 00:49:27.240 --> 00:49:27.740 All right. 00:49:27.740 --> 00:49:29.790 I think we'd better take a vote here. 00:49:29.790 --> 00:49:38.860 So the possible answers are from-- how do we frame this? 00:49:38.860 --> 00:49:43.840 From an inertial frame, looking down on it, 00:49:43.840 --> 00:49:47.660 answer A is it will go in a curved path. 00:49:47.660 --> 00:49:52.310 Answer B is it will go in a straight, radial line. 00:49:52.310 --> 00:49:57.900 Answer C is it does-- any other guesses? 00:49:57.900 --> 00:49:58.950 No other guesses. 00:49:58.950 --> 00:50:00.880 So it's the only two choices. 00:50:00.880 --> 00:50:03.670 It curves, or it goes in a straight, radial line. 00:50:03.670 --> 00:50:06.210 So how many vote for-- and everybody has to participate. 00:50:06.210 --> 00:50:09.060 How many vote for it goes a straight radial line? 00:50:09.060 --> 00:50:09.940 Let's have it. 00:50:09.940 --> 00:50:12.772 Straight out radial line from the point of view 00:50:12.772 --> 00:50:13.980 of the fixed reference frame. 00:50:13.980 --> 00:50:15.450 OK. 00:50:15.450 --> 00:50:16.220 A goodly number. 00:50:16.220 --> 00:50:16.780 OK. 00:50:16.780 --> 00:50:19.400 How many from the point of view of that fixed frame, looking 00:50:19.400 --> 00:50:22.050 down on it, will see it curve? 00:50:22.050 --> 00:50:23.900 How many vote for that? 00:50:23.900 --> 00:50:26.800 And how many didn't vote? 00:50:26.800 --> 00:50:28.460 Those are don't knows, huh? 00:50:28.460 --> 00:50:29.670 All right. 00:50:29.670 --> 00:50:33.130 So A and B are wrong. 00:50:35.640 --> 00:50:37.880 A and B are dead wrong. 00:50:37.880 --> 00:50:42.720 And you could have figured out what 00:50:42.720 --> 00:50:46.540 the answer is if you went right back to basics, back 00:50:46.540 --> 00:50:48.180 to Newton's Laws. 00:50:48.180 --> 00:50:56.170 When the string is released, the sum of the forces on the object 00:50:56.170 --> 00:50:57.760 are what? 00:50:57.760 --> 00:50:59.970 In the direction that it can travel. 00:50:59.970 --> 00:51:02.350 Still got gravity pushing on it, but it 00:51:02.350 --> 00:51:04.200 can't go in that direction. 00:51:04.200 --> 00:51:07.010 So if you did a free body diagram, 00:51:07.010 --> 00:51:12.145 what are the forces in that horizontal [? direct ?] plane 00:51:12.145 --> 00:51:13.020 that it's sitting on? 00:51:16.401 --> 00:51:17.642 I hear centrifugal force. 00:51:17.642 --> 00:51:18.475 I hear frictionless. 00:51:18.475 --> 00:51:20.260 Let's make the table frictionless 00:51:20.260 --> 00:51:22.250 so it can easy to move. 00:51:22.250 --> 00:51:23.110 No friction. 00:51:23.110 --> 00:51:25.186 I hear centrifugal force. 00:51:25.186 --> 00:51:25.940 AUDIENCE: Zero. 00:51:25.940 --> 00:51:29.368 PROFESSOR: He says zero. 00:51:29.368 --> 00:51:31.604 AUDIENCE: There's the force that you 00:51:31.604 --> 00:51:34.835 get from tangent of [? to ?] less perpendicular 00:51:34.835 --> 00:51:36.823 to the radial direction. 00:51:36.823 --> 00:51:39.308 PROFESSOR: What's perpendicular to the radial direction? 00:51:39.308 --> 00:51:42.290 AUDIENCE: [INAUDIBLE] it had to [INAUDIBLE] B traveling 00:51:42.290 --> 00:51:44.775 [INAUDIBLE] at one point had to have [INAUDIBLE] force 00:51:44.775 --> 00:51:49.506 with tangential to the circle. 00:51:49.506 --> 00:51:51.880 PROFESSOR: So you were the one that provided that for us. 00:51:51.880 --> 00:51:53.340 You're on the merry-go-round. 00:51:53.340 --> 00:51:55.680 You set it out there, and held the string. 00:51:55.680 --> 00:51:58.140 You set out there in some radius r. 00:51:58.140 --> 00:52:01.870 And so because when you set it down, you're already turning. 00:52:01.870 --> 00:52:04.790 So you just set it down with respect to the merry-go-round. 00:52:04.790 --> 00:52:05.556 It's not moving. 00:52:05.556 --> 00:52:06.430 You just set it down. 00:52:06.430 --> 00:52:08.020 OK? 00:52:08.020 --> 00:52:12.800 What are the forces-- so just draw the free-body diagram. 00:52:12.800 --> 00:52:14.940 Here's this puck. 00:52:14.940 --> 00:52:16.760 And you've released the string. 00:52:16.760 --> 00:52:21.400 And you got mg downwards, and you have a normal force 00:52:21.400 --> 00:52:22.440 upwards. 00:52:22.440 --> 00:52:26.270 And you have no friction. 00:52:26.270 --> 00:52:28.920 And I hear centrifugal force, but-- 00:52:28.920 --> 00:52:30.880 AUDIENCE: [INAUDIBLE] 00:52:30.880 --> 00:52:32.460 PROFESSOR: Well, that was a guess. 00:52:32.460 --> 00:52:36.430 Somebody said straight out, and the answer is no. 00:52:36.430 --> 00:52:38.632 And we're going to figure it out. 00:52:38.632 --> 00:52:40.590 First of all, we've got to sort out the forces. 00:52:40.590 --> 00:52:43.560 Well, what are the forces on this thing? 00:52:43.560 --> 00:52:45.640 So I heard centrifugal force. 00:52:45.640 --> 00:52:48.654 AUDIENCE: Yeah, I think the only force is the centrifugal force. 00:52:48.654 --> 00:52:52.450 But there was a velocity that's not caused by the force. 00:52:52.450 --> 00:52:55.943 So the velocity that you got was velocity that you told us about 00:52:55.943 --> 00:52:59.295 that [INAUDIBLE] that starts out-- 00:52:59.295 --> 00:53:00.420 PROFESSOR: You set it down. 00:53:00.420 --> 00:53:02.000 And once you set it down, it's there. 00:53:02.000 --> 00:53:02.996 And then? 00:53:02.996 --> 00:53:04.900 AUDIENCE: Yeah, your initial velocity 00:53:04.900 --> 00:53:06.804 from setting it out there. 00:53:06.804 --> 00:53:08.232 PROFESSOR: Yes. 00:53:08.232 --> 00:53:10.945 AUDIENCE: The change in direction-- the force only 00:53:10.945 --> 00:53:13.370 change direction of the ball. 00:53:13.370 --> 00:53:16.037 But once the string cuts, and the direction no longer 00:53:16.037 --> 00:53:18.705 changes, and the ball only goes the direction that 00:53:18.705 --> 00:53:21.620 is tangential to the circle. 00:53:21.620 --> 00:53:23.585 PROFESSOR: All right. 00:53:23.585 --> 00:53:25.040 The man has it right. 00:53:25.040 --> 00:53:26.490 You probably couldn't hear him. 00:53:26.490 --> 00:53:28.250 He says that when you let it go, it 00:53:28.250 --> 00:53:31.430 goes tangential to the circle. 00:53:31.430 --> 00:53:33.210 And that's true. 00:53:33.210 --> 00:53:35.710 The forces on in this direction-- 00:53:35.710 --> 00:53:39.290 in the horizontal direction-- once you release the object, 00:53:39.290 --> 00:53:41.940 there are no forces. 00:53:41.940 --> 00:53:46.530 Centrifugal force is a construction of convenience 00:53:46.530 --> 00:53:48.340 called a fictitious force. 00:53:48.340 --> 00:53:49.750 And we'll talk about that later. 00:53:49.750 --> 00:53:52.270 It is not a real force. 00:53:52.270 --> 00:53:55.380 It as a result of an acceleration. 00:53:55.380 --> 00:53:57.410 And it's the result of the acceleration when you 00:53:57.410 --> 00:53:59.010 are making it go in a circle. 00:53:59.010 --> 00:54:01.250 There is indeed-- that tension is 00:54:01.250 --> 00:54:03.760 what some people call the centrifugal force holding it 00:54:03.760 --> 00:54:04.260 there. 00:54:04.260 --> 00:54:07.980 But once you release it, that's no longer there. 00:54:07.980 --> 00:54:09.900 And if there are no external forces 00:54:09.900 --> 00:54:13.640 acting on the object in the horizontal direction, what's 00:54:13.640 --> 00:54:14.990 the mass times the acceleration? 00:54:14.990 --> 00:54:17.270 AUDIENCE: [INAUDIBLE] 00:54:17.270 --> 00:54:19.580 PROFESSOR: Therefore what's the acceleration? 00:54:19.580 --> 00:54:20.840 Zero. 00:54:20.840 --> 00:54:21.660 And its velocity. 00:54:21.660 --> 00:54:24.508 What's its velocity at that moment in time? 00:54:24.508 --> 00:54:25.436 AUDIENCE: [INAUDIBLE] 00:54:25.436 --> 00:54:28.164 PROFESSOR: Omega. 00:54:28.164 --> 00:54:29.580 R omega. 00:54:29.580 --> 00:54:31.700 So you have an R omega velocity. 00:54:31.700 --> 00:54:34.010 It's tangential to the motion. 00:54:34.010 --> 00:54:42.100 So let's make this my little x, and this my axes 00:54:42.100 --> 00:54:43.980 attached to the merry-go-round. 00:54:43.980 --> 00:54:48.270 This thing is moving in the j-hat direction 00:54:48.270 --> 00:54:54.430 at R B with respect to A in the j-hat direction. 00:54:54.430 --> 00:54:59.750 That is the velocity of B with respect to A 00:54:59.750 --> 00:55:02.778 at the moment you release it. 00:55:02.778 --> 00:55:03.670 OK? 00:55:03.670 --> 00:55:07.800 So it had better run off tangential to the circle 00:55:07.800 --> 00:55:09.110 at the moment of release. 00:55:09.110 --> 00:55:10.670 So let's test it. 00:55:10.670 --> 00:55:14.200 Now I want you to be my quality control person. 00:55:14.200 --> 00:55:16.519 If I smack somebody with this, I'm going to hurt them. 00:55:16.519 --> 00:55:17.060 AUDIENCE: No. 00:55:17.060 --> 00:55:18.780 PROFESSOR: Squeeze it. 00:55:18.780 --> 00:55:20.880 Soft and harmless, right? 00:55:20.880 --> 00:55:22.790 OK. 00:55:22.790 --> 00:55:24.960 So when should I release it if I want 00:55:24.960 --> 00:55:28.320 to hit the MIT sweatshirt sitting up there? 00:55:32.264 --> 00:55:34.110 When it's out here, right? 00:55:34.110 --> 00:55:35.375 I'll see if I can do it. 00:55:43.150 --> 00:55:46.044 All right. 00:55:46.044 --> 00:55:47.210 I'd better not try it again. 00:55:47.210 --> 00:55:50.166 I probably can't do it twice in a row. 00:55:50.166 --> 00:55:51.790 AUDIENCE: But that was a straight line. 00:55:51.790 --> 00:55:53.430 PROFESSOR: It was in a straight line. 00:55:53.430 --> 00:55:54.055 AUDIENCE: Yeah. 00:55:54.055 --> 00:55:55.490 So one of the guesses was-- 00:55:55.490 --> 00:55:59.210 PROFESSOR: Radial straight line was one guess. 00:55:59.210 --> 00:56:00.070 This is? 00:56:00.070 --> 00:56:01.558 AUDIENCE: Tangential straight line. 00:56:01.558 --> 00:56:02.550 PROFESSOR: Got it. 00:56:02.550 --> 00:56:06.703 So this tangential straight line is the right answer. 00:56:06.703 --> 00:56:08.152 AUDIENCE: [INAUDIBLE] the velocity 00:56:08.152 --> 00:56:11.540 is equal to the radius [INAUDIBLE]. 00:56:11.540 --> 00:56:13.375 PROFESSOR: I can't quite hear you. 00:56:13.375 --> 00:56:16.285 AUDIENCE: Here it says that the velocity of [INAUDIBLE] 00:56:16.285 --> 00:56:18.710 PROFESSOR: Ah, I left out the omega, didn't I? 00:56:18.710 --> 00:56:21.040 Sorry about that. 00:56:21.040 --> 00:56:26.820 So it is omega is the result of omega cross r B/A. 00:56:26.820 --> 00:56:31.446 So it's omega with respect to O, r B/A in the j direction. 00:56:31.446 --> 00:56:32.322 Yeah. 00:56:32.322 --> 00:56:35.214 AUDIENCE: [INAUDIBLE] the person on the merry-go-round 00:56:35.214 --> 00:56:36.657 does not see that. 00:56:36.657 --> 00:56:37.240 PROFESSOR: Ah. 00:56:37.240 --> 00:56:39.570 So actually I almost forgot this. 00:56:39.570 --> 00:56:41.290 She is talking about what does the person 00:56:41.290 --> 00:56:42.560 see on the merry-go-round? 00:56:45.442 --> 00:56:47.400 What do you actually see on the merry-go-round? 00:56:50.050 --> 00:56:51.330 Certainly doesn't go radial. 00:56:51.330 --> 00:56:53.800 We've proven that, right? 00:56:53.800 --> 00:56:57.670 But the person on the merry-go-round, 00:56:57.670 --> 00:56:59.890 is you're turning, and this thing is going off 00:56:59.890 --> 00:57:04.510 in a straight line with respect to the fixed frame, 00:57:04.510 --> 00:57:06.480 and you're turning away from it. 00:57:06.480 --> 00:57:10.900 So looking down on it, I release it here. 00:57:10.900 --> 00:57:13.490 It goes off in that direction. 00:57:13.490 --> 00:57:16.520 Your point of observation, you see it here. 00:57:16.520 --> 00:57:22.460 But now a short time later, when you've rotated to this point, 00:57:22.460 --> 00:57:25.200 and you're just keeping your eye on it, 00:57:25.200 --> 00:57:27.640 you see this thing start to move. 00:57:27.640 --> 00:57:30.850 But this is the spot that it was sitting on, 00:57:30.850 --> 00:57:32.930 which is now moved to here. 00:57:32.930 --> 00:57:35.530 But its position is now there. 00:57:35.530 --> 00:57:37.300 You see it moving away from you. 00:57:37.300 --> 00:57:42.580 And as you get up to, say, here, then it 00:57:42.580 --> 00:57:48.069 will have moved out a radius this far. 00:57:48.069 --> 00:57:50.110 Well, actually it's not-- you don't know how fast 00:57:50.110 --> 00:57:52.330 you're-- well, you're actually going exactly the same speed it 00:57:52.330 --> 00:57:52.830 is. 00:57:52.830 --> 00:57:57.610 So you've gone the arc length here. 00:57:57.610 --> 00:58:02.280 It's gone a quarter of a circle out to about there. 00:58:02.280 --> 00:58:03.360 Same point right here. 00:58:03.360 --> 00:58:07.090 But now you see it as being going off like that. 00:58:07.090 --> 00:58:10.040 To you, it's hooking off in the direction opposite 00:58:10.040 --> 00:58:11.800 to the direction of rotation. 00:58:11.800 --> 00:58:14.630 So the young man up there who described that early on 00:58:14.630 --> 00:58:16.910 was exactly right. 00:58:16.910 --> 00:58:21.177 So from your point of view, it goes mrrmm, like that. 00:58:21.177 --> 00:58:22.760 And then if you go all the way around, 00:58:22.760 --> 00:58:23.880 it'll appear to come back. 00:58:23.880 --> 00:58:27.370 It'll be further away, but you'll 00:58:27.370 --> 00:58:29.600 be back down to the point where you see it released. 00:58:29.600 --> 00:58:34.440 So it'll look like it goes from your point of view. 00:58:34.440 --> 00:58:35.970 OK. 00:58:35.970 --> 00:58:36.530 Good one. 00:58:44.410 --> 00:58:47.220 OK, we've got a few minutes left. 00:58:47.220 --> 00:58:50.507 I want to do something really important with the third law. 00:58:50.507 --> 00:58:52.340 So the first law's actually pretty important 00:58:52.340 --> 00:58:54.545 and very handy. 00:58:54.545 --> 00:58:56.670 The second law, we're going to make lots of use of. 00:58:56.670 --> 00:58:59.010 So I want to talk a minute about the third law. 00:58:59.010 --> 00:59:08.090 The third law is responsible for a law that we use, 00:59:08.090 --> 00:59:10.650 or an application that we use all the time. 00:59:10.650 --> 00:59:13.500 And let's see how this works out. 00:59:13.500 --> 00:59:15.070 So Newton's third law is the one that 00:59:15.070 --> 00:59:17.530 says F 21 is equal to F 12. 00:59:21.250 --> 00:59:24.170 So we have these two particles. 00:59:24.170 --> 00:59:31.040 And I called them 2 and 1. 00:59:31.040 --> 00:59:36.780 And I'll give this one mass 1, mass 2. 00:59:36.780 --> 00:59:39.650 And I'm going to say second law tells me 00:59:39.650 --> 00:59:50.250 that the sum of the forces, vectors, external forces, on 2 00:59:50.250 --> 00:59:54.130 is-- and this one, let's say it has things, 00:59:54.130 --> 00:59:55.830 active forces acting on it. 00:59:55.830 --> 00:59:57.940 F i's. 00:59:57.940 --> 01:00:02.120 And it has this little f 21. 01:00:02.120 --> 01:00:09.730 So the forces here are the external forces 01:00:09.730 --> 01:00:21.220 that are vectors, plus f, [? some ?] of the forces on 2. 01:00:21.220 --> 01:00:23.290 So forces on 2 due to 1. 01:00:23.290 --> 01:00:26.146 That's external influence of the other particle. 01:00:26.146 --> 01:00:27.930 OK? 01:00:27.930 --> 01:00:40.520 And this had better be equal to m 2 a 2, I'll call it. 01:00:40.520 --> 01:00:44.100 And with respect to some fixed, some inertial frame. 01:00:44.100 --> 01:00:49.210 So the sum of all these external forces on that particle 2 01:00:49.210 --> 01:00:51.460 had better be equal to the mass times the acceleration 01:00:51.460 --> 01:00:53.120 of particle 2. 01:00:53.120 --> 01:00:56.370 And the same thing can be said about particle 1. 01:00:56.370 --> 01:00:58.260 So I've really confused things here. 01:00:58.260 --> 01:01:00.910 There's 2, and there's 1. 01:01:00.910 --> 01:01:01.610 Boy. 01:01:01.610 --> 01:01:02.720 OK. 01:01:02.720 --> 01:01:05.210 Guys, help keep me honest here. 01:01:05.210 --> 01:01:08.300 So now that we can say the sum of the forces on 1 01:01:08.300 --> 01:01:16.560 is equal to these external applied forces, plus f 12-- 01:01:16.560 --> 01:01:19.610 the force caused by the other body. 01:01:19.610 --> 01:01:23.387 And those had better equal to m 1 a 1/0. 01:01:23.387 --> 01:01:23.886 OK. 01:01:27.540 --> 01:01:37.970 And these are also equal to the time derivative of m 2 v 2/0. 01:01:37.970 --> 01:01:40.990 That's the momentum, the time derivative of the momentum. 01:01:40.990 --> 01:01:43.930 It's fixed mass, so it's just the derivative velocity 01:01:43.930 --> 01:01:44.900 gives the acceleration. 01:01:44.900 --> 01:01:47.320 So these are clearly the same formula. 01:01:47.320 --> 01:01:53.880 And this is a time derivative of m 1 v 1 with respect to O dt. 01:01:53.880 --> 01:02:00.370 And we call that P 2 dot, P 1 dot. 01:02:00.370 --> 01:02:02.681 So these are statements that are applying second law 01:02:02.681 --> 01:02:03.805 to each of these particles. 01:02:08.280 --> 01:02:19.310 So if I want to compute the total momentum of the system. 01:02:19.310 --> 01:02:23.300 So the linear momentum, the total momentum of the system 01:02:23.300 --> 01:02:27.070 is going to be P 1 plus P 2. 01:02:27.070 --> 01:02:28.990 They're vectors. 01:02:28.990 --> 01:02:32.640 And I want to take the time derivative 01:02:32.640 --> 01:02:34.510 of that total momentum. 01:02:38.810 --> 01:02:40.950 Just the time derivative of a sum 01:02:40.950 --> 01:02:42.460 is the sum of the time derivatives. 01:02:42.460 --> 01:02:46.225 And so I get a P 1 dot plus a P 2 dot. 01:02:49.230 --> 01:02:50.600 But I know what those are. 01:02:50.600 --> 01:02:52.175 I have expressions for them. 01:02:55.920 --> 01:03:04.750 So this is F 1 external forces plus f 12 01:03:04.750 --> 01:03:14.790 plus the F 2 external forces plus f 21. 01:03:14.790 --> 01:03:15.700 These two things. 01:03:15.700 --> 01:03:18.570 This one plus this one, basically, 01:03:18.570 --> 01:03:23.410 is the sum of the two time derivatives of the momentum. 01:03:23.410 --> 01:03:26.380 But what's the sum of this term and that term? 01:03:26.380 --> 01:03:27.054 AUDIENCE: Zero. 01:03:27.054 --> 01:03:27.720 PROFESSOR: Zero. 01:03:31.560 --> 01:03:35.260 And that then allows you to say that this 01:03:35.260 --> 01:03:45.890 is the sum of the external forces on a system is 01:03:45.890 --> 01:03:49.272 equal to the time rate of change of the linear momentum 01:03:49.272 --> 01:03:49.855 of the system. 01:03:55.670 --> 01:03:56.670 OK. 01:03:56.670 --> 01:04:00.180 And that's basically this statement. 01:04:00.180 --> 01:04:03.160 And it has nothing to do with internal forces. 01:04:08.860 --> 01:04:13.050 And this allows you to say that the time 01:04:13.050 --> 01:04:16.930 rate of change of the linear momentum of a rigid body, 01:04:16.930 --> 01:04:19.720 for example. 01:04:19.720 --> 01:04:22.885 So a rigid body is made up of a whole mess 01:04:22.885 --> 01:04:25.430 of different particles. 01:04:25.430 --> 01:04:28.100 You could separate this into a whole bunch of little chunks, 01:04:28.100 --> 01:04:33.560 and treat each one of them as a particle which has connection, 01:04:33.560 --> 01:04:36.210 has forces with particles next to it. 01:04:36.210 --> 01:04:39.140 And all those internal forces-- so these 01:04:39.140 --> 01:04:43.390 are the internal forces-- all of those internal forces 01:04:43.390 --> 01:04:45.830 are equal and opposite and cancel. 01:04:45.830 --> 01:04:47.780 So it's really the third law that 01:04:47.780 --> 01:04:51.300 allows you to say if you have a system of particles 01:04:51.300 --> 01:04:54.690 that the time rate of change of the total momentum 01:04:54.690 --> 01:04:58.050 of the system is zero if there's no external forces acting 01:04:58.050 --> 01:04:58.550 on it. 01:04:58.550 --> 01:05:01.510 And that's where you get the conservation of momentum-- 01:05:01.510 --> 01:05:04.692 the law of conservation of momentum. 01:05:04.692 --> 01:05:05.600 OK. 01:05:05.600 --> 01:05:07.580 That's a consequence of the third law. 01:05:15.460 --> 01:05:15.960 OK. 01:05:51.590 --> 01:05:55.950 So you have a homework problem that has just about this. 01:05:55.950 --> 01:05:58.840 You have a bunch of particles. 01:05:58.840 --> 01:06:01.230 m 1. 01:06:01.230 --> 01:06:03.128 m i. 01:06:03.128 --> 01:06:10.830 r 1 with respect to O. r i with respect to O. 01:06:10.830 --> 01:06:13.779 And you have all these particles out there. 01:06:13.779 --> 01:06:15.320 And I want to find the center of mass 01:06:15.320 --> 01:06:16.445 of this group of particles. 01:06:20.170 --> 01:06:22.340 So the center of mass is a vector. 01:06:22.340 --> 01:06:24.900 And let's just say it's here. 01:06:24.900 --> 01:06:31.000 So I'm looking for r G with respect to O. 01:06:31.000 --> 01:06:32.910 So it's a quantity that I defined-- 01:06:32.910 --> 01:06:40.566 r G with respect to O-- times the summation of the m i's. 01:06:40.566 --> 01:06:44.560 So I'm postulating there's a place out there that if I 01:06:44.560 --> 01:06:49.120 multiply it by the sum of the m i's, I get the same answer 01:06:49.120 --> 01:06:54.520 as if I summed the m i r i/O's. 01:06:57.190 --> 01:06:59.930 And I'm going to define the center of mass 01:06:59.930 --> 01:07:07.450 as r G with respect to O is the summation of each particle 01:07:07.450 --> 01:07:12.140 times its position vector divided 01:07:12.140 --> 01:07:13.920 by the sum of the masses, which is just 01:07:13.920 --> 01:07:16.398 the total mass of the system. 01:07:16.398 --> 01:07:17.210 OK? 01:07:17.210 --> 01:07:19.910 And that's a definition of center of mass. 01:07:19.910 --> 01:07:21.490 And this is a vector. 01:07:21.490 --> 01:07:23.470 And these are vectors. 01:07:23.470 --> 01:07:24.020 OK? 01:07:24.020 --> 01:07:26.380 So that's all there is to the center of mass. 01:07:26.380 --> 01:07:45.240 And if I take a time derivative of r G/O, 01:07:45.240 --> 01:07:52.025 then it's equal to the summation of the m i times the r i/O's. 01:07:52.025 --> 01:07:53.650 But they're timed derivatives-- and I'm 01:07:53.650 --> 01:07:55.608 going to put a dot right there, so I don't have 01:07:55.608 --> 01:07:59.010 to write out d by dt-- over MT. 01:07:59.010 --> 01:08:05.800 But that's just the summation of the individual momenta of each 01:08:05.800 --> 01:08:07.470 of the particles over MT. 01:08:11.760 --> 01:08:14.235 Very handy little formula. 01:08:16.810 --> 01:08:20.600 And if I take another time derivative, 01:08:20.600 --> 01:08:25.729 so I get an r G/O double dot, then it's 01:08:25.729 --> 01:08:32.029 just the summation of the time derivatives 01:08:32.029 --> 01:08:35.160 of the individual momenta again over this. 01:08:35.160 --> 01:08:36.189 So this is a statement. 01:08:36.189 --> 01:08:37.569 We move this to the other side. 01:08:37.569 --> 01:08:44.720 M T r G/O double dot equals the summation 01:08:44.720 --> 01:08:49.010 of the timed derivatives of the individual momenta. 01:08:49.010 --> 01:08:53.520 And that's the total momentum of the system 01:08:53.520 --> 01:08:55.189 times its timed derivative. 01:08:55.189 --> 01:09:02.420 So just from third law, you can come up 01:09:02.420 --> 01:09:06.740 with all of the linear momentum formulas. 01:09:06.740 --> 01:09:09.229 This is a statement for rigid body. 01:09:09.229 --> 01:09:13.399 The mass times the acceleration of the center of mass. 01:09:13.399 --> 01:09:16.899 The total mass of a rigid body times the acceleration 01:09:16.899 --> 01:09:20.069 of the center of mass is equal to the time 01:09:20.069 --> 01:09:23.481 rate of change of the total momentum of that object. 01:09:23.481 --> 01:09:24.439 Very important formula. 01:09:24.439 --> 01:09:26.020 You've used it a lot, right? 01:09:26.020 --> 01:09:28.990 So I'll tell you a quick story, and then we'll knock off. 01:09:28.990 --> 01:09:33.270 So two, three years ago, we have doctoral exams 01:09:33.270 --> 01:09:34.899 in mechanical engineering. 01:09:34.899 --> 01:09:37.479 And in the dynamics oral exam, we had eight students 01:09:37.479 --> 01:09:39.229 a few years ago. 01:09:39.229 --> 01:09:42.120 And they were asked to find the center of mass of an object. 01:09:42.120 --> 01:09:44.910 It was just a step in a harder problem. 01:09:44.910 --> 01:09:46.810 And seven out of the eight students 01:09:46.810 --> 01:09:49.260 could not remember the definition 01:09:49.260 --> 01:09:50.840 of the center of mass. 01:09:50.840 --> 01:09:54.037 Now they could do Lagrange equations and nasty dynamics 01:09:54.037 --> 01:09:55.620 problems, but they'd kind of forgotten 01:09:55.620 --> 01:09:58.290 some of these really, really basic things. 01:09:58.290 --> 01:10:01.910 So I have one demo to show you, just 01:10:01.910 --> 01:10:03.520 to illustrate center of mass. 01:10:03.520 --> 01:10:08.490 So center of mass, we know how to calculate it. 01:10:08.490 --> 01:10:10.570 And this is a rod. 01:10:10.570 --> 01:10:13.657 How can I simply find the center of mass of this thing? 01:10:13.657 --> 01:10:14.790 AUDIENCE: Balance it. 01:10:14.790 --> 01:10:16.861 PROFESSOR: Balance it. 01:10:16.861 --> 01:10:17.360 OK. 01:10:17.360 --> 01:10:19.980 Young lady here says move my fingers. 01:10:19.980 --> 01:10:22.210 Where do you think the center of mass is? 01:10:22.210 --> 01:10:23.130 Right in the middle. 01:10:23.130 --> 01:10:23.410 OK. 01:10:23.410 --> 01:10:23.993 Let's find it. 01:10:29.780 --> 01:10:31.330 Should this work, by the way? 01:10:31.330 --> 01:10:32.330 You ever done this? 01:10:35.762 --> 01:10:37.345 All right, the center of mass, I ought 01:10:37.345 --> 01:10:39.435 to be able to just about balance this thing there. 01:10:39.435 --> 01:10:41.060 So the center of mass is in the middle. 01:10:41.060 --> 01:10:41.960 That's because I cheated. 01:10:41.960 --> 01:10:43.418 I put a piece of steel in this end. 01:10:43.418 --> 01:10:44.709 OK? 01:10:44.709 --> 01:10:46.010 All right. 01:10:46.010 --> 01:10:49.729 But indeed, you might figure out why is it that I can actually 01:10:49.729 --> 01:10:50.770 do this and make it work. 01:10:50.770 --> 01:10:51.530 You can do it with a broomstick. 01:10:51.530 --> 01:10:53.340 You can do it with any object. 01:10:53.340 --> 01:10:56.330 The key is it'll work so long as the friction coefficients are 01:10:56.330 --> 01:10:58.550 the same on both fingers. 01:10:58.550 --> 01:11:01.050 It's really easy to find center of mass that way. 01:11:01.050 --> 01:11:03.230 So this got a little steel weight in the end. 01:11:03.230 --> 01:11:03.730 OK. 01:11:03.730 --> 01:11:05.220 We're done for today. 01:11:05.220 --> 01:11:09.150 And see you on Thursday.