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13. Rotational Inertia & Energy

# Conservation of Energy with Rotation

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concept

## Conservation of Energy with Rotation 13m
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example

## Work to accelerate cylinder 3m
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Hey, guys, let's check out this example Here we have a solid cylinder and we want to know how much work is needed to accelerate that cylinder. So solid cylinder means that I is going to be half m r squared because that's the equation for the moment of inertia. Of a cylinder masses 10 in radius is two. I'm gonna put these here if you want to. You could already calculate the moment of inertia. Right. So I is half 10. Our square in the moment of inertia is 20. Okay, so we can already get that 20 kg meters square. It says it is mounted and free to rotate, um, on a perpendicular access through its center. Again. You have a cylinder, which is the same thing as a disk, and it has an axis. It's mounted on an axis that is perpendicular to it. So it looks like this right, and it's free to rotate about. That axis just doesn't wobble like that, right? It rotates like this. Now, most of the time, you actually have this where the access is horizontal, so it's on a wall, right? So it's something that's on a wall, and you have the disks spinning like this. All right, so it says that cylinders initially at rest. So the cylinder spins around itself, but it's initially at rest. Omega initially equals zero. And we want to know what is the work done to accelerate it from rest to 120 our PM. Okay, remember, most of the time when you have rpm, you're supposed to change that into W so that you could be using an equation. So let's do that real quick. Just to get that out of the way or make a final is two pi f or two pi our PM over 60. If you plug 1 20 here, you end up with 1 20 but by sixties to you end up with four pi radiance per second. Okay, so I'm going from 0 to 4 pi and I want to know how much work does that take. So work is energy. So hopefully you thought of using the conservation of energy equation K initial plus you initial plus work non conservative equals K final plus you final. In the beginning, there is no kinetic energy because it's not spinning, it's not moving sideways. Um, the potential energy is canceled because the height of the cylinder doesn't change, it stays in place. Right? Work, Non conservative is the work done by U plus the work done by friction. There is no work done by friction Just the work done by you. Which is exactly what we're looking for. And the kinetic energy which is Onley, kinetic, rotational, right? There's no linear. It's not moving sideways. The center of massive disc stays in place. So the equal zero. So the only type of kinetic energy we have is rotational, which is half Oh, May I Omega squared. We're looking for this. So all we gotta do is plug in this number. Work is going to be half. I we already calculated I over here it was 20. And Omega is four pi square. Okay, so if you multiply all of this, you get that it is 15 80 you get 80 jewels of energy, and that's how much energy is needed to get this solid cylinder from rest all the way to a speed of four pi or 120 rpm. Cool. Very straightforward. Plug it into the energy equation because we were asked for work. All right, hold makes sense. Let me know if you guys have any questions and let's keep going
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Problem

How much work is needed to stop a hollow sphere of mass 2 kg and radius 3 m that spins at 40 rad/s around an axis through its center?

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example

## Which shape reaches bottom first? 4m
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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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example

## Cylinders racing down:rolling vs. sliding 6m
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