13. Rotational Inertia & Energy

Conservation of Energy with Rotation

# Which shape reaches bottom first?

Patrick Ford

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Hey, guys, let's check out this conservation of energy example. Um, here we have three objects of equal mass and equal ratings, but they have different shapes. Remember, your shape is what determines what equation your moment of inertia has. And it's usually something like a fraction m r squared. But the number in here depends on the shape. So if you have different shapes, you can have different I equations. They're all released from rest at the same time from the top oven inclined plane. So I'm gonna have Here's a solid cylinder. Um, here is a hollow cylinder and here is a solid sphere. They're all from rest. They all have the same mass, right? So they have the same mass, the same radius. They all start from rest, and they all starts from the top of the inclined plane. They're gonna start from the same height as well. Everything is the same except the shapes. And I wanna know who reaches the bottom first if they're released at the same time. And this question will depend on your moments of inertia. What I want to remind you, that moment of inertia is a measurement of angular resistance of rotational resistance. So you can think you can think that the greater my eye, the heavier I am, the more I resist rotation. Therefore, I will get to the bottom last because I'm slower. Okay, So, um or I you can think of this as being heavier now. It doesn't mean that I have more mass, right? That's why I have heavier. I have mawr resistance. Therefore, I am slower. Okay. Now, a solid cylinder has a moment of inertia of half m r square, a hollow cylinder at the moment of inertia of m r squared. So you can think that there's a one in the front in a solid sphere, has a moment of inertia of 2/5 and Mars Square. So in this question, all we're doing is comparing these numbers because the m and they are the same. Now, this is a little bit easier if you use decimals. So this is 0.5. This is 1.0 and to over 50 point four. Okay. And you can see from here that this one is the lightest one. Okay, because the coefficient number in front of the M R is the lowest. It's the lightest one. Therefore, it is the fastest one. Therefore, it gets to the bottom first. Okay, It gets to the bottom first. So the sequence is that V that the solid sphere his first. Um, I'm gonna write it like this gets to the bottom first. The second one is going to be The solid cylinder is second, and the third one is going to be the hollow cylinder. Okay, now there is a pattern here. There's a reason why the hostile cylinder is slower than the solid cylinder. Solid cylinder has very good mass distribution. The masses very evenly distributed. And remember, the more evenly distributed the mass. The lighter you are, the less I so better mass distribution means lower I which means you are lighter. Okay. The whole cylinder has all of its mass concentrated on the edge. It has worse mass distribution, which means it has a higher I, which means it is heavier. So it has a worse mass distribution. Therefore, it is heavier now. A solid sphere is even mawr well distributed than a solid disk. A solid disk has all the mass on a thin layer like this. A sphere has basically the most perfect mass distribution you can have. That's why it has the most symmetrical one. That's why it has the lowest of them all. So the sphere is always fastest. Okay, so that's it for this one. Let's keep going.

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