13. Rotational Inertia & Energy

Types of Motion & Energy

# Kinetic Energy of a Point Mass

Patrick Ford

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Hey, guys, in this video, I'm gonna show you how there are two ways to calculate the kinetic energy of a point mass going around a circle. Let's check it out. All right, So remember, if you have a point mass around a circle under in a circular path looks kind of like this. Around a distance of little are from the axis of rotation. You have rotational speed, omega, and you also have a linear equivalent, which is your tangential velocity. Okay. All right. Um, but you only have one type of motion. All you're doing is this Okay? You're on Lee Motion is really rotational motion. Um, your only emotion is rotational motion. So you only have one type of kinetic energy. Okay, But you can calculate using K, l or KR. You can use thean question for linear or for KR, and that's because these two equations as I'm gonna show you now or equivalent, Okay, the most important thing to do here is to make sure you don't double counting. Okay? When I ask you for the total kinetic energy of an object, you can't, uh, point mass like this. You can't look at it and say, Well, it's got a V. So it has a linear kinetic energy, and it has a w. So it has a rotational kinetic energy. It's got to kind of energies. Let's add the two of them together. You can't do that because these guys are equivalents, right? The tangential velocity is basically a mirror of W. It doesn't mean there is to. It just means that one basically reflects the other. Alright, so you can't do is double count. So let me show you how this works. Um, a small 2 kg objects, so mass tickles. 2 kg is going around with a rate of going around the vertical axis. So what is the vertical axis? Remember? Access. You can think of it as a, uh, imaginary line that you spin around. The vertical axis would look like this. So it means the object is going around like this. Okay, like this. Cool. So it would actually I could draw it like this. And the object is doing this, okay? And it does this in a rate of three radiance per second, maintaining a constant distance of 4 m to the access this distance to the axis is what we call little. Our little are 4 m and I wanna know the objects kinetic energy. And I want I want to do this using the Kael equation the KR equation. And the purpose of this question is to show you how the answer ends up being the same. And I'm gonna simplify. I'm gonna summarize it at the end so we can do que el We could do Que el, which is going to be half M V square. Okay, remember that thes two V and are are related right V and are related by V equals R Omega. So what I'm gonna do is also right. K r equals half I omega, and I'm gonna rewrite one of these questions of one of these equations and you're gonna notice how it's gonna look exactly like the other. So let's rewrite this one here half, Remember, I for a point Mass is m r Square someone. I replaced this with m r Square and I can rewrite omega as well. V equals R omega so omega equals V over R. So instead of omega here, I'm gonna put V over R. Now look what happens. This r squared cancels with this are square and we're left with half M V squared, which is exactly this equation. Okay, so you can go from one to the other four. A point mass. You could do this. Which means I could have calculated them either way. All right, so if I go here, K l equals half M V squared. Let's get these numbers. W equals three v equals R w So w equals V over R the I'm sorry, I'm trying to get V so v equals R four W three v is 12. So this is half mass is too. They canceled 12 squared. So this is 1 40 for Jules. Cool. And if I wanted to do it using K R. I already showed you how the equations turned out to be the same. Now, I'm just gonna plug in numbers differently. So if I wanted to do it this way, I could have done half m r squared omega squared, half right, which is this Half the mass is too. And the distances four square and the W is three squared. So these two cancer, I have 16 times nine, which is 1 40 for Jules. Okay, so If you calculated using linear, it's 1 44. If you calculate using rotation is 1 44. And if I ask you what is K Total? Thea answer is 1 44. Okay. And I want you to please right here, not to 88. You do not add the tude. You could get the same answer using the two different equations. Now, to make this simpler for you, I have a convention. I always think of a knob checked going around the circle like this. It has one motion. I always think of this as linear motion. I always I'm sorry. Rotational motion, not linear. So I would always do it like this K l plus K r. And I would say there's no que el There's Onley kr, and this will guarantee that you don't double counting. Cool. So this is just a potentially tricky thing. But once you understand, you get it out of the way. It's never gonna bother you again. Cool. Let me know if you any questions

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