WEBVTT 00:00:00.415 --> 00:00:02.103 - [Voiceover] When a major league baseball player throws 00:00:02.103 --> 00:00:05.637 a fast ball, that ball's definitely got kinetic energy. 00:00:05.637 --> 00:00:07.581 We know that cause if you get in the way, 00:00:07.581 --> 00:00:09.429 it could do work on you, that's gonna hurt. 00:00:09.429 --> 00:00:10.583 You gotta watch out. 00:00:10.583 --> 00:00:14.212 But here's my question: does the fact that most pitches, 00:00:14.212 --> 00:00:15.855 unless you're throwing a knuckle ball, 00:00:15.855 --> 00:00:19.157 does the fact that most pitches head toward home plate 00:00:19.157 --> 00:00:21.805 with the baseball spinning mean that that ball 00:00:21.805 --> 00:00:24.197 has extra kinetic energy? 00:00:24.197 --> 00:00:26.571 Well it does, and how do we figure that out, 00:00:26.571 --> 00:00:28.986 that's the goal for this video. 00:00:28.986 --> 00:00:31.267 How do we determine what the rotational 00:00:31.267 --> 00:00:33.647 kinetic energy is of an object? 00:00:33.647 --> 00:00:35.770 Well if I was coming at this for the first time, 00:00:35.770 --> 00:00:37.803 my first guest I'd say okay, 00:00:37.803 --> 00:00:40.561 I'd say I know what regular kinetic energy looks like. 00:00:40.561 --> 00:00:42.875 The formula for regular kinetic energy is 00:00:42.875 --> 00:00:45.954 just one half m v squared. 00:00:45.954 --> 00:00:48.567 So let's say alright, I want rotational kinetic energy. 00:00:48.567 --> 00:00:50.963 Let me just call that k rotational 00:00:50.963 --> 00:00:52.497 and what is that gonna be? 00:00:52.497 --> 00:00:54.964 Well I know for objects that are rotating, 00:00:54.964 --> 00:00:58.764 the rotational equivalent of mass is moment of inertia. 00:00:58.764 --> 00:01:01.461 So I might guess alright instead of mass, 00:01:01.461 --> 00:01:04.354 I'd have moment of inertia cause in Newton's second law 00:01:04.354 --> 00:01:06.813 for rotation I know that instead of mass there's 00:01:06.813 --> 00:01:09.091 moment of inertia so maybe I replace that. 00:01:09.091 --> 00:01:12.188 And instead of speed squared, maybe since I have 00:01:12.188 --> 00:01:15.284 something rotating I'd have angular speed squared. 00:01:15.284 --> 00:01:16.898 It turns out this works. 00:01:16.898 --> 00:01:20.014 You can often derive, it's not really a derivation, 00:01:20.014 --> 00:01:22.516 you're just kind of guessing educatedly but you could 00:01:22.516 --> 00:01:25.797 often get a formula for the rotational analog of some 00:01:25.797 --> 00:01:29.780 linear formula by just substituting the rotational analog 00:01:29.780 --> 00:01:32.415 for each of the variables, so if I replaced mass with 00:01:32.415 --> 00:01:35.247 rotational mass, I get the moment of inertia. 00:01:35.247 --> 00:01:37.721 If I replace speed with rotational speed, 00:01:37.721 --> 00:01:40.743 I get the angular speed and this is the correct formula. 00:01:40.743 --> 00:01:43.303 So in this video we needed to ride this cause that 00:01:43.303 --> 00:01:44.913 is not really a derivation, we didn't really 00:01:44.913 --> 00:01:47.720 prove this, we just showed that it's plausible. 00:01:47.720 --> 00:01:50.111 How do we prove that this is the rotational 00:01:50.111 --> 00:01:52.991 kinetic energy of an object that's rotating 00:01:52.991 --> 00:01:54.349 like a baseball. 00:01:54.349 --> 00:01:56.997 The first thing to recognize is that this rotational 00:01:56.997 --> 00:01:59.684 kinetic energy isn't really a new kind of 00:01:59.684 --> 00:02:02.301 kinetic energy, it's still just the same old 00:02:02.301 --> 00:02:05.721 regular kinetic energy for something that's rotating. 00:02:05.721 --> 00:02:07.051 What I mean by that is this. 00:02:07.051 --> 00:02:09.820 Imagine this baseball is rotating in a circle. 00:02:09.820 --> 00:02:13.320 Every point on the baseball is moving with some speed, 00:02:13.320 --> 00:02:15.462 so what I mean by that is this, so this point at the top 00:02:15.462 --> 00:02:18.613 here imagine the little piece of leather right here, 00:02:18.613 --> 00:02:20.367 it's gonna have some speed forward. 00:02:20.367 --> 00:02:23.485 I'm gonna call this mass M one, that little piece 00:02:23.485 --> 00:02:27.188 of mass right now and I'll call the speed of it V one. 00:02:27.188 --> 00:02:29.744 Similarly, this point on the leather right there, 00:02:29.744 --> 00:02:32.288 I'm gonna call that M two, it's gonna be moving down 00:02:32.288 --> 00:02:35.713 cause it's a rotating circle, so I'll call that V two 00:02:35.713 --> 00:02:38.370 and points closer to the axis are gonna be moving 00:02:38.370 --> 00:02:41.027 with smaller speed so this point right here, 00:02:41.027 --> 00:02:43.779 we'll call it M three, moving down with a speed V three, 00:02:43.779 --> 00:02:46.771 that is not as big as V two or V one. 00:02:46.771 --> 00:02:48.082 You can't see that very well, 00:02:48.082 --> 00:02:51.587 I'll use a darker green so this M three right here 00:02:51.587 --> 00:02:54.921 closer to the axis, axis being right at this point 00:02:54.921 --> 00:02:58.781 in the center, closer to the axis so it's speed is smaller 00:02:58.781 --> 00:03:01.411 than points that are farther away from this axis, 00:03:01.411 --> 00:03:03.373 so you can see this is kinda complicated. 00:03:03.373 --> 00:03:05.539 All points on this baseball are gonna be moving with 00:03:05.539 --> 00:03:08.126 different speeds so points over here that are really 00:03:08.126 --> 00:03:10.796 close to the axis, barely moving at all. 00:03:10.796 --> 00:03:12.946 I'll call this M four and it would be 00:03:12.946 --> 00:03:15.093 moving at speed V four. 00:03:15.093 --> 00:03:17.676 What we mean by the rotational kinetic energy is 00:03:17.676 --> 00:03:19.892 really just all the regular kinetic energy these 00:03:19.892 --> 00:03:23.846 masses have about the center of mass of the baseball. 00:03:23.846 --> 00:03:26.681 So in other words, what we mean by K rotational, 00:03:26.681 --> 00:03:29.455 is you just add up all of these energies. 00:03:29.455 --> 00:03:32.021 You have one half, this little piece of leather 00:03:32.021 --> 00:03:33.737 up here would have some kinetic energy 00:03:33.737 --> 00:03:37.487 so you do one half M one, V one squared plus. 00:03:38.415 --> 00:03:41.050 And this M two has some kinetic energy, 00:03:41.050 --> 00:03:43.152 don't worry that it points downward, 00:03:43.152 --> 00:03:45.954 downward doesn't matter for things that aren't vectors, 00:03:45.954 --> 00:03:49.258 this V gets squared so kinetic energy's not a vector 00:03:49.258 --> 00:03:51.779 so it doesn't matter that one velocity points down 00:03:51.779 --> 00:03:54.443 cause this is just the speed and similarly, 00:03:54.443 --> 00:03:58.950 you'd add up one half M three, V three squared, 00:03:58.950 --> 00:04:00.517 but you might be like this is impossible, 00:04:00.517 --> 00:04:02.926 there's infinitely many points in this baseball, 00:04:02.926 --> 00:04:05.388 how am I ever going to do this. 00:04:05.388 --> 00:04:07.379 Well something magical is about to happen, 00:04:07.379 --> 00:04:09.526 this is one of my favorite little derivations, 00:04:09.526 --> 00:04:12.133 short and sweet, watch what happens. 00:04:12.133 --> 00:04:15.067 K E rotational is really just the sum, 00:04:15.067 --> 00:04:17.661 if I add all these up I can write is as a sum 00:04:17.661 --> 00:04:21.494 of all the one half M V squares of every point 00:04:22.457 --> 00:04:25.416 on this baseball so imagine breaking this baseball 00:04:25.416 --> 00:04:27.756 up into very, very small pieces. 00:04:27.756 --> 00:04:30.072 Don't do it physically but just think about it mentally, 00:04:30.072 --> 00:04:33.039 just visualize considering very small pieces, 00:04:33.039 --> 00:04:35.919 particles of this baseball and how fast they're going. 00:04:35.919 --> 00:04:38.938 What I'm saying is that if you add all of that up, 00:04:38.938 --> 00:04:41.359 you get the total rotational kinetic energy, 00:04:41.359 --> 00:04:42.967 this looks impossible to do. 00:04:42.967 --> 00:04:44.552 But something magical is about to happen, 00:04:44.552 --> 00:04:45.766 here's what we can do. 00:04:45.766 --> 00:04:48.352 We can rewrite, see the problem here is V. 00:04:48.352 --> 00:04:50.755 All these points have a different speed V, 00:04:50.755 --> 00:04:52.711 but we can use a trick, a trick that we love 00:04:52.711 --> 00:04:55.125 to use in physics, instead of writing this as V, 00:04:55.125 --> 00:04:57.773 we're gonna write V as, so remember that for things 00:04:57.773 --> 00:05:01.564 that are rotating, V is just R times omega. 00:05:01.564 --> 00:05:04.133 The radius, how far from the axis you are, 00:05:04.133 --> 00:05:06.885 times the angular velocity, or the angular speed 00:05:06.885 --> 00:05:09.358 gives you the regular speed. 00:05:09.358 --> 00:05:12.145 This formula is really handy, so we're gonna replace 00:05:12.145 --> 00:05:16.185 V with R omega, and this is gonna give us R omega 00:05:16.185 --> 00:05:18.352 and you still have to square it and at this point 00:05:18.352 --> 00:05:19.993 you're probably thinking like this is even worse, 00:05:19.993 --> 00:05:21.079 what do we do this for. 00:05:21.079 --> 00:05:24.023 Well watch, if we add this is up I'll have one half M. 00:05:24.023 --> 00:05:26.848 I'm gonna get an R squared and an omega squared, 00:05:26.848 --> 00:05:28.958 and the reason this is better is that even though 00:05:28.958 --> 00:05:32.626 every point on this baseball has a different speed V, 00:05:32.626 --> 00:05:35.491 they all have the same angular speed omega, 00:05:35.491 --> 00:05:38.315 that was what was good about these angular quantities 00:05:38.315 --> 00:05:41.618 is that they're the same for every point on the baseball 00:05:41.618 --> 00:05:43.870 no matter how far away you are from the axis, 00:05:43.870 --> 00:05:46.042 and since they're the same for every point I can 00:05:46.042 --> 00:05:48.634 bring that out of the summation so I can rewrite 00:05:48.634 --> 00:05:51.609 this summation and bring everything that's constant 00:05:51.609 --> 00:05:54.818 for all of the masses out of the summation so I can 00:05:54.818 --> 00:05:58.220 write this as one half times the summation 00:05:58.220 --> 00:06:01.803 of M times R squared and end that quantity, 00:06:02.782 --> 00:06:06.597 end that summation and just pull the omega squared out 00:06:06.597 --> 00:06:08.565 because it's the same for each term. 00:06:08.565 --> 00:06:11.444 I'm basically factoring this out of all of these 00:06:11.444 --> 00:06:13.857 terms in the summation, it's like up here, 00:06:13.857 --> 00:06:15.548 all of these have a one half. 00:06:15.548 --> 00:06:17.487 You could imagine factoring out a one half 00:06:17.487 --> 00:06:18.985 and just writing this whole quantity as 00:06:18.985 --> 00:06:22.135 one half times M one V one squared plus 00:06:22.135 --> 00:06:24.167 M two V two squared and so on. 00:06:24.167 --> 00:06:26.055 That's what I'm doing down here for the one half 00:06:26.055 --> 00:06:28.615 and for the omega squared, so that's what was good 00:06:28.615 --> 00:06:31.077 about replacing V with R omega. 00:06:31.077 --> 00:06:32.540 The omega's the same for all of them, 00:06:32.540 --> 00:06:33.816 you can bring that out. 00:06:33.816 --> 00:06:35.514 You might still be concerned, you might be like, 00:06:35.514 --> 00:06:37.993 we're still stuck with the M in here cause you've 00:06:37.993 --> 00:06:39.990 got different Ms at different points. 00:06:39.990 --> 00:06:42.160 We're stuck with all these R squareds in here, 00:06:42.160 --> 00:06:44.628 all these points at the baseball are different Rs, 00:06:44.628 --> 00:06:46.328 they're all different points from the axis, 00:06:46.328 --> 00:06:48.558 different distances from the axis, we can't bring 00:06:48.558 --> 00:06:51.449 those out so now what do we do, well if you're clever 00:06:51.449 --> 00:06:53.792 you recognize this term. 00:06:53.792 --> 00:06:56.615 This summation term is nothing but the total moment 00:06:56.615 --> 00:06:59.296 of inertia of the object. 00:06:59.296 --> 00:07:01.628 Remember that the moment of inertia of an object, 00:07:01.628 --> 00:07:04.394 we learned previously, is just M R squared, 00:07:04.394 --> 00:07:06.410 so the moment of inertia of a point mass 00:07:06.410 --> 00:07:09.122 is M R squared and the moment of inertia 00:07:09.122 --> 00:07:12.483 of a bunch of point masses is the sum of all 00:07:12.483 --> 00:07:15.402 the M R squareds and that's what we've got right here, 00:07:15.402 --> 00:07:19.514 this is just the moment of inertia of this baseball 00:07:19.514 --> 00:07:22.115 or whatever the object is, it doesn't even have to be 00:07:22.115 --> 00:07:24.287 of a particular shape, we're gonna add all 00:07:24.287 --> 00:07:26.998 the M R squareds, that's always going to be 00:07:26.998 --> 00:07:28.637 the total moment of inertia. 00:07:28.637 --> 00:07:30.925 So what we've found is that the K rotational 00:07:30.925 --> 00:07:34.066 is equal to one half times this quantity, 00:07:34.066 --> 00:07:35.947 which is I, the moment of inertia, 00:07:35.947 --> 00:07:38.284 times omega squared and that's the formula 00:07:38.284 --> 00:07:40.004 we got up here just by guessing. 00:07:40.004 --> 00:07:41.850 But it actually works and this is why it works, 00:07:41.850 --> 00:07:43.859 because you always get this quantity down here, 00:07:43.859 --> 00:07:46.204 which is one half I omega squared, no matter what 00:07:46.204 --> 00:07:47.676 the shape of the object is. 00:07:47.676 --> 00:07:49.420 So what this is telling you, what this quantity 00:07:49.420 --> 00:07:52.346 gives us is the total rotational kinetic energy 00:07:52.346 --> 00:07:55.666 of all the points on that mass about the center 00:07:55.666 --> 00:07:58.591 of the mass but here's what it doesn't give you. 00:07:58.591 --> 00:08:01.036 This term right here does not include 00:08:01.036 --> 00:08:03.451 the translational kinetic energy so the fact that 00:08:03.451 --> 00:08:06.292 this baseball was flying through the air does not 00:08:06.292 --> 00:08:08.142 get incorporated by this formula. 00:08:08.142 --> 00:08:10.264 We didn't take into account the fact that 00:08:10.264 --> 00:08:12.391 the baseball was moving through the air, 00:08:12.391 --> 00:08:13.976 in other words, we didn't take into account 00:08:13.976 --> 00:08:16.791 that the actual center of mass in this baseball 00:08:16.791 --> 00:08:19.200 was translating through the air. 00:08:19.200 --> 00:08:21.365 But we can do that easily with this formula here. 00:08:21.365 --> 00:08:24.279 This is the translational kinetic energy. 00:08:24.279 --> 00:08:26.930 Sometimes instead of writing regular kinetic energy, 00:08:26.930 --> 00:08:29.841 now that we've got two we should specify this is really 00:08:29.841 --> 00:08:31.791 translational kinetic energy. 00:08:31.791 --> 00:08:34.361 We've got a formula for translational kinetic energy, 00:08:34.361 --> 00:08:37.701 the energy something has due to the fact that the center 00:08:37.701 --> 00:08:40.522 of mass of that object is moving and we have a formula 00:08:40.522 --> 00:08:42.972 that takes into account the fact that something can have 00:08:42.972 --> 00:08:45.494 kinetic energy due to its rotation. 00:08:45.494 --> 00:08:48.316 That's this K rotational, so if an object's rotating, 00:08:48.316 --> 00:08:50.483 it has rotational kinetic energy. 00:08:50.483 --> 00:08:52.718 If an object is translating it has 00:08:52.718 --> 00:08:54.500 translational kinetic energy, 00:08:54.500 --> 00:08:56.515 i.e. if the center of mass is moving, 00:08:56.515 --> 00:08:59.986 and if the object is translating and it's rotating 00:08:59.986 --> 00:09:02.435 then it would have those both of these kinetic energies, 00:09:02.435 --> 00:09:04.948 both at the same time and this is the beautiful thing. 00:09:04.948 --> 00:09:08.430 If an object is translating and rotating and you want 00:09:08.430 --> 00:09:11.390 to find the total kinetic energy of the entire thing, 00:09:11.390 --> 00:09:14.004 you can just add these two terms up. 00:09:14.004 --> 00:09:17.147 If I just take the translational one half M V squared, 00:09:17.147 --> 00:09:20.573 and this would then be the velocity of the center of mass. 00:09:20.573 --> 00:09:22.157 So you have to be careful. 00:09:22.157 --> 00:09:23.749 Let me make some room here, so let me get rid 00:09:23.749 --> 00:09:25.130 of all this stuff here. 00:09:25.130 --> 00:09:28.741 If you take one half M, times the speed of the center 00:09:28.741 --> 00:09:31.655 of mass squared, you'll get the total translational 00:09:31.655 --> 00:09:33.239 kinetic energy of the baseball. 00:09:33.239 --> 00:09:36.386 And if we add to that the one half I omega squared, 00:09:36.386 --> 00:09:39.184 so the omega about the center of mass you'll get 00:09:39.184 --> 00:09:43.688 the total kinetic energy, both translational and rotational, 00:09:43.688 --> 00:09:46.624 so this is great, we can determine the total kinetic energy 00:09:46.624 --> 00:09:49.889 altogether, rotational motion, translational motion, 00:09:49.889 --> 00:09:52.580 from just taking these two terms added up. 00:09:52.580 --> 00:09:54.049 So what would an example of this be, 00:09:54.049 --> 00:09:55.796 let's just get rid of all this. 00:09:55.796 --> 00:09:59.180 Let's say this baseball, someone pitched this thing, 00:09:59.180 --> 00:10:02.582 and the radar gun shows that this baseball was hurled 00:10:02.582 --> 00:10:04.799 through the air at 40 meters per second. 00:10:04.799 --> 00:10:07.452 So it's heading toward home plate at 40 meters per second. 00:10:07.452 --> 00:10:09.858 The center of mass of this baseball is going 00:10:09.858 --> 00:10:12.551 40 meters per second toward home plate. 00:10:12.551 --> 00:10:15.094 Let's say it's also, someone really threw the fastball. 00:10:15.094 --> 00:10:18.107 This thing's rotating with an angular velocity 00:10:18.107 --> 00:10:20.190 of 50 radians per second. 00:10:22.264 --> 00:10:24.376 We know the mass of a baseball, I've looked it up. 00:10:24.376 --> 00:10:28.781 The mass of a baseball is about 0.145 kilograms 00:10:28.781 --> 00:10:31.795 and the radius of the baseball, so a radius of a baseball 00:10:31.795 --> 00:10:35.388 is around seven centimeters, so in terms of meters that 00:10:35.388 --> 00:10:38.865 would be 0.07 meters, so we can figure out what's 00:10:38.865 --> 00:10:41.240 the total kinetic energy, well there's gonna be 00:10:41.240 --> 00:10:43.202 a rotational kinetic energy and there's gonna be 00:10:43.202 --> 00:10:45.048 a translational kinetic energy. 00:10:45.048 --> 00:10:47.875 The translational kinetic energy, gonna be one half 00:10:47.875 --> 00:10:50.835 the mass of the baseball times the center of mass speed 00:10:50.835 --> 00:10:53.993 of the baseball squared which is gonna give us one half. 00:10:53.993 --> 00:10:57.626 The mass of the baseball was 0.145 and the center of mass 00:10:57.626 --> 00:11:00.650 speed of the baseball is 40, that's how fast the center 00:11:00.650 --> 00:11:02.712 of mass of this baseball is traveling. 00:11:02.712 --> 00:11:06.712 If we add all that up we get 116 Jules of regular 00:11:06.712 --> 00:11:08.894 translational kinetic energy. 00:11:08.894 --> 00:11:11.246 How much rotational kinetic energy is there, 00:11:11.246 --> 00:11:13.281 so we're gonna have rotational kinetic energy 00:11:13.281 --> 00:11:16.088 due to the fact that the baseball is also rotating. 00:11:16.088 --> 00:11:19.587 How much, well we're gonna use one half I omega squared. 00:11:19.587 --> 00:11:22.484 I'm gonna have one half, what's the I, well the baseball is 00:11:22.484 --> 00:11:26.328 a sphere, if you look up the moment of inertia of a sphere 00:11:26.328 --> 00:11:29.665 cause I don't wanna have to do summation of all 00:11:29.665 --> 00:11:32.999 the M R squareds, if you do that using calculus, 00:11:32.999 --> 00:11:34.873 you get this formula. 00:11:34.873 --> 00:11:36.995 That means in an algebra based physics class 00:11:36.995 --> 00:11:38.900 you just have to look this up, it's either in your book 00:11:38.900 --> 00:11:41.635 in a chart or a table or you could always look it up online. 00:11:41.635 --> 00:11:45.763 For a sphere the moment of inertia is two fifths M R squared 00:11:45.763 --> 00:11:48.619 in other words two fifths the mass of a baseball 00:11:48.619 --> 00:11:50.459 times the raise of the baseball squared. 00:11:50.459 --> 00:11:53.627 That's just I, that's the moment of inertia of a sphere. 00:11:53.627 --> 00:11:56.235 So we're assuming this baseball is a perfect sphere. 00:11:56.235 --> 00:11:59.358 It's got uniform density, that's not completely true. 00:11:59.358 --> 00:12:00.954 But it's a pretty good approximation. 00:12:00.954 --> 00:12:03.019 Then we multiply by this omega squared, 00:12:03.019 --> 00:12:04.754 the angular speed squared. 00:12:04.754 --> 00:12:07.137 So what do we get, we're gonna get one half times 00:12:07.137 --> 00:12:11.222 two fifths, the mass of a baseball was 0.145. 00:12:11.222 --> 00:12:13.284 The radius of the baseball was about, what did we say, 00:12:13.284 --> 00:12:18.027 .07 meters so that's .07 meters squared and then finally 00:12:18.027 --> 00:12:20.494 we multiply by omega squared and this would make it 00:12:20.494 --> 00:12:23.238 50 radians per second and we square 00:12:23.238 --> 00:12:25.821 it which adds up to 0.355 Jules 00:12:28.705 --> 00:12:31.416 so hardly any of the energy of this baseball 00:12:31.416 --> 00:12:33.034 is in its rotation. 00:12:33.034 --> 00:12:36.449 Almost all of the energy is in the form of translational 00:12:36.449 --> 00:12:38.521 energy, that kinda makes sense. 00:12:38.521 --> 00:12:40.895 It's the fact that this baseball is hurling toward 00:12:40.895 --> 00:12:43.901 home plate that's gonna make it hurt if it hits you 00:12:43.901 --> 00:12:46.049 as opposed to the fact that it was spinning when 00:12:46.049 --> 00:12:48.545 it hits you, that doesn't actually cause as much damage 00:12:48.545 --> 00:12:50.705 as the fact that this baseball's kinetic energy 00:12:50.705 --> 00:12:54.466 is mostly in the form of translational kinetic energy. 00:12:54.466 --> 00:12:57.154 But if you wanted the total kinetic energy of the baseball, 00:12:57.154 --> 00:12:59.135 you would add both of these terms up. 00:12:59.135 --> 00:13:02.641 K total would be the translational kinetic energy 00:13:02.641 --> 00:13:04.937 plus the rotational kinetic energy. 00:13:04.937 --> 00:13:09.104 That means the total kinetic energy which is the 116 Jules 00:13:10.046 --> 00:13:12.546 plus 0.355 Jules which give us 00:13:14.425 --> 00:13:15.592 116.355 Jules. 00:13:18.343 --> 00:13:20.590 So recapping if an object is both rotating 00:13:20.590 --> 00:13:23.156 and translating you can find the translational 00:13:23.156 --> 00:13:26.787 kinetic energy using one half M the speed of the 00:13:26.787 --> 00:13:29.564 center of mass of that object squared and you can 00:13:29.564 --> 00:13:32.071 find the rotational kinetic energy by using 00:13:32.071 --> 00:13:34.552 one half I, the moment of inertia. 00:13:34.552 --> 00:13:36.161 We'll infer whatever shape it is, 00:13:36.161 --> 00:13:38.640 if it's a point mass going in a huge circle 00:13:38.640 --> 00:13:41.035 you could use M R squared, if it's a sphere 00:13:41.035 --> 00:13:43.635 rotating about its center you could use two fifths 00:13:43.635 --> 00:13:46.209 M R squared, cylinders are one half M R squared, 00:13:46.209 --> 00:13:49.007 you can look these up in tables to figure out 00:13:49.007 --> 00:13:52.032 whatever the I is that you need times the angular 00:13:52.032 --> 00:13:56.319 speed squared of the object about that center of mass. 00:13:56.319 --> 00:13:58.423 And if you add these two terms up you get the 00:13:58.423 --> 00:14:01.423 total kinetic energy of that object.