13. Rotational Inertia & Energy

Conservation of Energy with Rotation

# Work to accelerate cylinder

Patrick Ford

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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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