WEBVTT 00:00:03.770 --> 00:00:08.020 Here are two people. They're both standing, but they're standing in two completely different 00:00:08.020 --> 00:00:13.530 ways. Which person would it be easier to push over? If you wanted to push this person [left] 00:00:13.530 --> 00:00:18.199 over, where would you apply a force? What about this [right] person? The answers to 00:00:18.199 --> 00:00:22.380 all these questions can be explained using the concept of torque. 00:00:22.380 --> 00:00:26.510 This video is part of the Representations video series. Information can be represented 00:00:26.510 --> 00:00:32.590 in words, through mathematical symbols, graphically, or in 3-D models. Representations are used 00:00:32.590 --> 00:00:38.000 to develop a deeper and more flexible understanding of objects, systems, and processes. 00:00:38.000 --> 00:00:44.060 Hi. I'm Sanjay Sarma. Professor of mechanical engineering at MIT, In this video, we'll be 00:00:44.060 --> 00:00:46.820 talking torque and balance. 00:00:46.820 --> 00:00:51.510 In order to understand these core concepts, you'll need a working knowledge of vectors 00:00:51.510 --> 00:00:57.390 and their uses. Specifically, you must be familiar with force, displacement, and torque. 00:00:57.390 --> 00:01:02.760 We will also assume that you know how to compute a cross product, and how to use the Right-Hand 00:01:02.760 --> 00:01:08.060 Rule, and that you have done problems involving the center of mass of an object. 00:01:08.060 --> 00:01:13.109 Our objective is to improve your ability to draw torque diagrams, and give you some practice 00:01:13.109 --> 00:01:18.959 with setting them up. By the end you should also understand what is needed for human beings 00:01:18.959 --> 00:01:21.310 to balance. 00:01:21.310 --> 00:01:25.959 We'll start with an activity. Everyone stand up and spread out across the room. You'll 00:01:25.959 --> 00:01:28.630 need a partner for this activity. 00:01:28.630 --> 00:01:33.908 When I say "go," your goal is to carefully push your partner over. Use the smallest amount 00:01:33.908 --> 00:01:38.380 of force you can. You will switch partners halfway through, so be gentle. 00:01:38.380 --> 00:01:43.899 When you push, consider where you should push, and in what direction. Try many different 00:01:43.899 --> 00:01:45.889 approaches. 00:01:45.889 --> 00:01:50.469 Here are some questions that may help you think about this in a scientific manner. When 00:01:50.469 --> 00:01:53.848 it comes to your push, where will you push? 00:01:53.848 --> 00:01:58.908 What direction will you push? How hard will you push? Consider your partner as well: how 00:01:58.908 --> 00:02:05.090 is your partner standing? What is the floor like under your partner's feet? Can your partner 00:02:05.090 --> 00:02:07.149 balance well? 00:02:07.149 --> 00:02:39.220 Are you ready? Go! 00:02:39.220 --> 00:03:11.390 Switch partners! 00:03:11.390 --> 00:03:16.170 Your teacher will now lead you in a short discussion about this activity. Pause the 00:03:16.170 --> 00:03:24.120 video here. 00:03:24.120 --> 00:03:28.630 Let's continue our investigation of torque and balance in the human body. Take a look 00:03:28.630 --> 00:03:35.270 at this next video clip, in which MIT researchers Colin Fredericks and Jennifer French demonstrate 00:03:35.270 --> 00:03:38.930 the effectiveness of properly applied torque. 00:03:38.930 --> 00:03:44.980 [Colin speaking] This is called "Sai son" stance. This is a basic stance in martial 00:03:44.980 --> 00:03:48.870 arts. Known for it's stability along a particular direction. 00:03:48.870 --> 00:03:52.210 This is my colleague, mathematician Jennifer French. 00:03:52.210 --> 00:03:56.400 Jen knows the stance is strong along certain directions–such as the one she is pulling 00:03:56.400 --> 00:03:58.120 in now. 00:03:58.120 --> 00:03:59.850 [laughing] Oops. 00:03:59.850 --> 00:04:03.370 It can also withstand a great deal of force in the opposite direction. 00:04:03.370 --> 00:04:09.980 I'm not going to be knocked over if she is pushing or pulling along that particular line. 00:04:09.980 --> 00:04:14.940 However, Jen knows the stance's weak point. 00:04:14.940 --> 00:04:22.199 If she pushes along a different line, it'll knock me right down to the ground. 00:04:22.199 --> 00:04:24.879 This isn't just martial arts. This is science. 00:04:24.879 --> 00:04:28.659 Torque controls my ability to balance and we're going to show you how today. 00:04:28.659 --> 00:04:34.610 [Sanjay speaking] To analyze the situation, let's look at what physical properties are 00:04:34.610 --> 00:04:36.550 important here. 00:04:36.550 --> 00:04:42.030 What forces do you think are involved? Pause the video to discuss. 00:04:46.340 --> 00:04:53.199 Next, draw a simple diagram that you can use to find the net torque on this man. Pause 00:04:53.199 --> 00:04:56.399 the video while you do this. 00:05:01.069 --> 00:05:05.919 The simplest way to represent this man is with a rectangle. You should remove any other 00:05:05.920 --> 00:05:12.479 details, and draw our forces so that we can tell exactly where they are applied to his body. 00:05:12.479 --> 00:05:16.900 Finally, if you were to draw a diagram showing someone resisting a push, how would you do 00:05:16.900 --> 00:05:21.129 it? Which of these three is most appropriate? 00:05:21.129 --> 00:05:24.329 Pause the video to discuss. 00:05:28.090 --> 00:05:34.120 You may be wondering why we can use a two-dimensional diagram to discuss a three-dimensional situation. 00:05:34.129 --> 00:05:40.509 The reason is that all of our forces are applied in the same plane, simplifying the problem. 00:05:40.509 --> 00:05:45.058 While our torque vectors point in and out of the screen on this diagram, we can represent 00:05:45.058 --> 00:05:51.889 that fairly easily. In a more complex situation, we may need to draw something more fully three-dimensional, 00:05:51.889 --> 00:05:56.389 as our torques might point in other directions. 00:05:56.389 --> 00:06:02.129 This next video clip will walk you through a partial analysis of this situation. You 00:06:02.129 --> 00:06:07.089 will need a way to take notes and draw diagrams while you watch. 00:06:10.820 --> 00:06:15.379 [Jen speaking] I couldn't push Colin over by applying force in the stable direction. 00:06:15.379 --> 00:06:19.739 Yet when I applies the same force in a different direction, I could push him over. 00:06:19.740 --> 00:06:21.340 How did I know this? 00:06:21.340 --> 00:06:22.360 Torque. 00:06:22.360 --> 00:06:26.759 Here you see two 2-dimensional views representing Colin. 00:06:26.759 --> 00:06:29.740 The width of the base represents the distance between his feet. 00:06:29.740 --> 00:06:32.939 The wide square is the view of Colin from the side and stable states. 00:06:32.939 --> 00:06:36.839 The narrow rectangle is the view of Colin from straight on. 00:06:36.839 --> 00:06:41.319 Lets draw in the forces acting on Colin as I push him in the stable directions. 00:06:41.319 --> 00:06:43.219 Start by drawing the center of mass. 00:06:43.219 --> 00:06:44.999 Which we assume is at the center of the square. 00:06:44.999 --> 00:06:48.529 The force of gravity pulls straight down with magnitude three halfs. 00:06:48.529 --> 00:06:52.990 There is also the force of my push of length one. Which we place in the location where 00:06:52.990 --> 00:06:55.360 the force is applied. 00:06:55.360 --> 00:07:00.759 This push shifts Colin's weight entirely to his back foot. So the normal force is applied 00:07:00.759 --> 00:07:04.210 there with a magnitude equal and opposite that of force p. 00:07:04.210 --> 00:07:08.849 The force of friction is also at this foot. Equal and opposite the magnitude and direction 00:07:08.849 --> 00:07:10.319 of the push force. 00:07:11.240 --> 00:07:14.339 Recall that torque is r cross F. 00:07:14.339 --> 00:07:18.210 That is, torque occurs when a force is applied some displacement distance from a reference 00:07:18.210 --> 00:07:19.080 point. 00:07:19.080 --> 00:07:22.729 We can choose any reference point we like, and the net torque we compute will be the 00:07:22.729 --> 00:07:23.439 same. 00:07:23.439 --> 00:07:26.599 For this problem I will choose the reference point to be the pivot point. 00:07:26.599 --> 00:07:30.469 Why don't you try this problem but choose the center of mass as the reference point. 00:07:30.469 --> 00:07:32.550 Lets get back to the problem. 00:07:32.550 --> 00:07:36.569 The push force and the gravitation force are applied some distance from the pivot point 00:07:36.569 --> 00:07:37.360 at the foot. 00:07:37.360 --> 00:07:38.869 So they will create torque. 00:07:38.869 --> 00:07:43.150 However, the friction force and normal force are applied at the pivot point. 00:07:43.150 --> 00:07:47.210 So R is zero and we can ignore that when we compute the net torque. 00:07:47.210 --> 00:07:49.479 Lets compute the torque due to the push force. 00:07:49.479 --> 00:07:53.279 The vector r is the displacement from the pivot point to where the force is applied. 00:07:53.279 --> 00:07:56.580 We decompose this vector into its x and y components. 00:07:56.580 --> 00:08:01.520 We use a coordinate system with origin at the pivot point to determine the r vector 00:08:01.520 --> 00:08:05.900 to be 4i + 3j. 00:08:05.900 --> 00:08:12.499 The magnitude of the push force is negative one i. 00:08:12.499 --> 00:08:17.469 The magnitude of torque can be found as the magnitude of r times the magnitude of F times 00:08:17.469 --> 00:08:18.219 sin theta. 00:08:18.219 --> 00:08:20.849 Where theta is the angle between the two vectors. 00:08:20.849 --> 00:08:27.619 Thus the magnitude is the area of the parallelogram formed by r and F. 00:08:27.619 --> 00:08:33.750 Since the area of this parallelogram is also the area of the rectangle formed by F and 00:08:33.750 --> 00:08:37.890 r sin theta or the component of r that is perpendicular to F. 00:08:37.890 --> 00:08:41.289 We can see this magnitude visually as the area of this rectangle. 00:08:41.289 --> 00:08:46.130 We will show the magnitudes in this way because it is very easy to see the relative magnitudes 00:08:46.130 --> 00:08:47.790 of the various torques. 00:08:47.790 --> 00:08:49.200 So... 00:08:49.200 --> 00:08:58.280 For the push force the area is 3, which mean the magnitude of the torque is 3. 00:08:58.290 --> 00:09:03.020 The direction of the torque vector is along the axis of rotation caused by this force. 00:09:03.020 --> 00:09:05.290 We find the direction using the right hand wall. 00:09:05.290 --> 00:09:09.690 Point our fingers along r and curl along F. 00:09:09.720 --> 00:09:15.430 In this case we find the direction to be out of the board or positive k. 00:09:15.430 --> 00:09:23.050 So this tells us that the torque due to the push force is equal to 3k. 00:09:23.050 --> 00:09:28.200 We can also find the torque vector by taking the determinate of the following three by 00:09:28.200 --> 00:09:29.990 three matrix. 00:09:29.990 --> 00:09:33.530 The first row is i, j, and k. 00:09:33.530 --> 00:09:38.390 The second row are the x, y, and z components of the r vector. 00:09:38.390 --> 00:09:43.060 And the third row are the x, y , and z components of the force vector. 00:09:43.060 --> 00:09:50.060 Because r and F lie in the same plane, the z components are zero. 00:09:50.060 --> 00:09:56.180 In our example, the force is pointing in the negative i direction and r we found to be 00:09:56.180 --> 00:09:57.980 4 i plus 3 j. 00:09:57.980 --> 00:10:06.090 Computing it this way, we also find that the torque is 3k. 00:10:06.090 --> 00:10:08.490 Now let's find the torque due to gravity. 00:10:08.490 --> 00:10:12.080 The vector r is found by connecting the pivot point to the center of mass. 00:10:12.080 --> 00:10:17.220 We leave to you to pause the video and to determine the components of r using this coordinate 00:10:17.220 --> 00:10:20.830 system. 00:10:20.830 --> 00:10:26.540 We find the magnitude of the torque by taking the component of r that is perpendicular to 00:10:26.540 --> 00:10:32.600 F pointing in the x direction, and the magnitude is the area of the rectangle between these 00:10:32.600 --> 00:10:33.850 two vectors. 00:10:33.850 --> 00:10:36.880 Again, we find the direction using the right hand rule. 00:10:36.880 --> 00:10:42.530 We find the torque to be pointing into the board at the pivot point. 00:10:42.530 --> 00:10:47.940 Since the x component of the r vector is 2, and the force due to gravity is 3 halves, 00:10:47.940 --> 00:10:49.830 pointing in the negative j direction. 00:10:49.830 --> 00:10:55.020 The magnitude of the torque due to gravity is 3 and it points in the negative k direction. 00:10:55.020 --> 00:10:58.600 So the torque is negative 3 k at the pivot point. 00:10:58.600 --> 00:11:04.210 Adding the push torque and the gravity torque together we find that we get zero. 00:11:04.210 --> 00:11:08.950 This means that there is no rotation about the pivot point. 00:11:08.950 --> 00:11:10.890 Now let's look in the unstable direction. 00:11:10.890 --> 00:11:16.450 We have all the same forces as before- force of gravity at the center of mass, the push 00:11:16.450 --> 00:11:25.080 force, a normal force at the back foot, as well as a friction force. 00:11:25.080 --> 00:11:29.900 The magnitudes of these forces are all the same as in the other diagram. 00:11:29.910 --> 00:11:35.370 Let's compute the torque due to the push force. The vector r connects the pivot point to the 00:11:35.370 --> 00:11:39.110 place where the force is applied. 00:11:39.110 --> 00:11:44.050 Using our coordinate axes with the origin placed at the pivot point, we find this vector 00:11:44.050 --> 00:11:51.300 to be i plus 3 j. 00:11:51.300 --> 00:11:55.390 We find the magnitude of the torque vector is the area of the rectangle formed by force 00:11:55.390 --> 00:11:59.440 vector and the component of r, perpendicular to F. 00:11:59.440 --> 00:12:06.070 Thus the area is one times three, or three. 00:12:06.070 --> 00:12:08.830 In order to find the direction, we use the right hand rule. Point our fingers along r, 00:12:08.830 --> 00:12:14.090 curl in the direction of F, and our thumb points out of the board, so the direction 00:12:14.090 --> 00:12:16.090 of our torque is k. 00:12:16.090 --> 00:12:21.300 Since the magnitude is 3, that tells us that torque due to the push is 3 k at the pivot 00:12:21.300 --> 00:12:22.500 point. 00:12:22.500 --> 00:12:25.670 Finally, let's find the torque due to gravity. 00:12:25.670 --> 00:12:29.620 Start by drawing the r vector from the pivot point to the center of mass. 00:12:29.620 --> 00:12:33.320 Now let's decompose this vector into it's x and y components. 00:12:33.320 --> 00:12:42.620 We leave this as an exercise to you, using the coordinate system shown here. 00:12:42.620 --> 00:12:46.510 The magnitude of the torque due to gravity at the pivot point is found by the area of 00:12:46.510 --> 00:12:52.570 the rectangle form by the force vector and the x component of the r vector. 00:12:52.570 --> 00:12:56.500 The direction is found using the right hand rule, pointing our fingers along r and curling 00:12:56.500 --> 00:12:57.800 around F. 00:12:57.800 --> 00:13:02.260 In other words, into the board or negative k. 00:13:02.260 --> 00:13:11.680 So the torque due to gravity is one-half times three-halves or three-fourths pointing in 00:13:11.680 --> 00:13:14.310 the negative k direction. 00:13:14.310 --> 00:13:17.170 The net torque about the pivot point is the sum of these two. 00:13:17.170 --> 00:13:22.200 In other words, the torque push plus the torque due to gravity is the net torque around the 00:13:22.200 --> 00:13:26.550 pivot point, which is two and a quarter the positive k direction. 00:13:26.550 --> 00:13:36.450 That means there is a total net rotation about the pivot point. Causing Colin to fall over. 00:13:36.450 --> 00:13:41.550 We leave it as an exercise for you to use the determinate in order to compute the torque 00:13:41.550 --> 00:13:42.650 vectors. 00:13:42.650 --> 00:13:47.090 If you computed the net torque using the center of mass as the reference point, you should 00:13:47.090 --> 00:13:51.680 notice that the net torques that you found are equal to the net torques that we computed 00:13:51.680 --> 00:13:53.020 here. 00:13:54.840 --> 00:13:59.940 [Sanjay speaking] Now that you've seen torque and balancing in detail, let's consider a 00:13:59.940 --> 00:14:04.600 more complex problem. This one is a bit tricky. I'll show you how it works. 00:14:04.600 --> 00:14:07.380 I'm going to take a chair and place it next to the wall. 00:14:07.380 --> 00:14:13.760 I'll put my toes up to the wall, and step back, toe-to-heel, twice. 00:14:13.760 --> 00:14:17.680 I step sideways until I'm over the chair. 00:14:17.680 --> 00:14:23.520 Then I bend forward until my head touches the wall, pick up the chair, and stand up... 00:14:23.520 --> 00:14:25.690 er... or not. 00:14:25.690 --> 00:14:30.430 This is something that most women can do, but men cannot. 00:14:30.430 --> 00:14:37.810 Maybe you think it's a trick? Here's a video of some of your professors trying to lift 00:14:37.810 --> 00:14:39.350 [Jen speaking]Okay, walk up to the wall. 00:14:39.350 --> 00:14:39.850 Yes. 00:14:39.850 --> 00:14:41.810 Then heel toe heel toe. 00:14:41.810 --> 00:14:43.620 That's it. One more time. 00:14:43.620 --> 00:14:46.860 And now translate over but don't come in or out. 00:14:46.860 --> 00:14:47.680 Okay. 00:14:47.680 --> 00:14:50.740 Now bend over to the the wall. Bend over. Put your head on the wall. 00:14:50.740 --> 00:14:51.240 Mmm-hmm. 00:14:51.240 --> 00:14:53.560 Pick the chair up. 00:14:53.560 --> 00:14:54.760 Now stand up. 00:14:54.770 --> 00:14:55.300 Hah! 00:14:55.300 --> 00:15:01.240 [laughing] 00:15:01.240 --> 00:15:03.180 That's fair enough. Don't hurt yourself. 00:15:03.180 --> 00:15:06.140 No, no. There was actually no way. [laughing] 00:15:07.320 --> 00:15:09.200 Toes against the wall. 00:15:10.980 --> 00:15:11.600 One. 00:15:12.260 --> 00:15:16.400 Two. Okay, translate over. You might be able to do it because his feet are proportionately 00:15:16.400 --> 00:15:17.740 small. 00:15:17.740 --> 00:15:19.860 Okay, down. 00:15:19.860 --> 00:15:21.260 Pick it up. 00:15:21.260 --> 00:15:23.080 [giggle] 00:15:23.080 --> 00:15:26.200 Pick up the chair first. Please pick up of the chair please. 00:15:26.200 --> 00:15:27.720 Now stand up. 00:15:27.720 --> 00:15:29.400 [laughing] 00:15:31.380 --> 00:15:33.600 Okay. One over. 00:15:33.600 --> 00:15:35.540 Bend over. 00:15:37.380 --> 00:15:39.880 Ohhh, okay. 00:15:39.880 --> 00:15:41.280 Alright. 00:15:41.960 --> 00:15:45.960 Okay, then do it one more time with the other foot. 00:15:45.960 --> 00:15:52.060 Okay, now can translate over a little bit? Just move over but don't go in or out. 00:15:53.500 --> 00:15:58.320 Well, no. You want a line there. Actually... be in a little. 00:15:58.320 --> 00:15:59.880 There you go. 00:15:59.880 --> 00:16:02.000 No no no- you're fine! Just even up your feet. 00:16:02.000 --> 00:16:02.500 Okay. 00:16:02.500 --> 00:16:05.960 Bend over. Put your hands on the wall, just so you don't hit your head. 00:16:05.960 --> 00:16:07.560 Okay... 00:16:07.560 --> 00:16:10.320 now pick up the chair. 00:16:10.320 --> 00:16:12.280 Now stand up. 00:16:12.280 --> 00:16:17.100 No no. Don't do it again. You have to pick up the chair. Clean pick up. 00:16:17.100 --> 00:16:19.080 Now stand up. 00:16:21.200 --> 00:16:26.120 Alright! [clapping] 00:16:26.880 --> 00:16:28.520 [Sanjay speaking] Now it's your turn to try. 00:16:28.520 --> 00:16:32.480 Get together with your partner again and bring a chair over to the wall. 00:16:32.480 --> 00:16:42.020 Why don't you pause the video and give it a try? 00:16:42.020 --> 00:16:45.900 Now that you've attempted the chair lift, let's return to our seats and discuss what 00:16:45.900 --> 00:16:47.550 happened. 00:16:47.550 --> 00:16:51.990 If you wanted to figure out why this happens, what information would you need? 00:16:51.990 --> 00:16:54.690 What assumptions would you have to make? 00:16:54.690 --> 00:16:59.600 Discuss the matter with your partner. Try to draw a diagram of the situation. 00:16:59.600 --> 00:17:03.040 Pause the video here while you work this out. 00:17:07.060 --> 00:17:11.359 Many of you should be wondering about the center of mass for this situation. 00:17:11.369 --> 00:17:17.099 This diagram shows typical locations for the center of mass in men (on the left) and women 00:17:17.099 --> 00:17:18.809 (on the right). 00:17:18.809 --> 00:17:22.880 Not only are men typically taller, but their center of mass is usually higher in their 00:17:22.880 --> 00:17:23.970 bodies. 00:17:23.970 --> 00:17:28.309 This is not always true, but it is fairly typical. 00:17:28.309 --> 00:17:33.230 Using this information, work with your partner to try to explain what is happening. 00:17:33.230 --> 00:17:37.230 Why can women lift the chair when men cannot? 00:17:37.230 --> 00:17:40.280 You do not need to obtain a numerical solution. 00:17:40.280 --> 00:17:45.530 Instead, use reasoned arguments, diagrams, and well-supported assumptions to prove your 00:17:45.530 --> 00:17:46.910 answer. 00:17:46.910 --> 00:17:52.690 Only use calculations if you cannot support your answer in any other way. 00:17:52.690 --> 00:17:56.210 Pause the video and give it a try. 00:17:59.770 --> 00:18:01.980 Are you ready to see the solution? 00:18:01.980 --> 00:18:04.780 Let's take a look. 00:18:04.780 --> 00:18:09.030 For this section we will measure torques around the center of mass. 00:18:09.030 --> 00:18:13.280 This will simplify our work so that we don't have to worry about the person's mass and 00:18:13.280 --> 00:18:15.809 the pull of gravity. 00:18:15.809 --> 00:18:18.590 This diagram shows the man lifting the chair. 00:18:18.590 --> 00:18:23.050 You can see that the center of mass is outside his body, and has been moved farther forward 00:18:23.050 --> 00:18:25.580 and down by the chair. 00:18:25.580 --> 00:18:27.660 This diagram shows the same for the woman. 00:18:27.660 --> 00:18:34.100 Her center of mass is also outside her body, but is much closer to her legs. 00:18:34.100 --> 00:18:39.490 If we draw a line for the man to indicate where his toes are, you see that the center 00:18:39.490 --> 00:18:41.820 of mass is beyond the edge of his foot! 00:18:41.820 --> 00:18:47.420 No matter how hard he tries, he simply cannot apply force in the right place to lift that 00:18:47.420 --> 00:18:49.290 chair. 00:18:49.290 --> 00:18:54.180 On our diagram, we can see that the torque will always point in the direction out of 00:18:54.180 --> 00:18:55.170 the screen. 00:18:55.170 --> 00:19:00.660 It will rotate the man counterclockwise, pushing his head harder into the wall. 00:19:00.660 --> 00:19:04.490 The harder he pushes, the worse it will be. 00:19:04.490 --> 00:19:08.430 The woman, however, has her center of mass above her feet. 00:19:08.430 --> 00:19:13.830 She can stand up because her feet are able to apply force in the proper location and 00:19:13.830 --> 00:19:14.680 direction. 00:19:14.680 --> 00:19:22.380 If we draw a torque diagram, we can see that the direction of torque applied by her feet 00:19:22.380 --> 00:19:24.490 will be into the screen. 00:19:24.490 --> 00:19:29.790 By applying more force, she can rotate her upper body clockwise and stand up, whereas 00:19:29.790 --> 00:19:32.920 the man cannot. 00:19:32.920 --> 00:19:37.610 Today we hope that you have improved your ability to draw torque diagrams, and to analyze 00:19:37.610 --> 00:19:41.050 torque problems that occur in the real world. 00:19:41.050 --> 00:19:45.710 Torque is an important quantity that comes into play in countless situations around us, 00:19:45.710 --> 00:19:50.660 from machinery to buildings to the simple act of walking. I hope you enjoyed this look 00:19:50.660 --> 00:19:55.320 at one of its fascinating applications.