A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at t = 0, the wheel turns through 8.20 revolutions in 12.0 s. At t = 12.0 s the kinetic energy of the wheel is 36.0 J. For an axis through its center, what is the moment of inertia of the wheel?

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Hey, everyone. Welcome back. In this problem. CD player rotates a compact disc about its central access with constant angular acceleration starting from rest at T equals zero seconds. The C D completes five revolutions in five seconds. The rotational kinetic energy of the disc at T equals five seconds is jewels. And were asked to calculate the moment of inertia with respect to the disks, central axis. So we're given some information about the kinetic energy. We're asked to find the moment of inertia. Let's recall how we can relate the to the kinetic energy. We're gonna call it K E U for here Is going to be equal to 1/2 I omega squared because we're talking about angular motion. Okay. When we have linear motion, we have that the kinetic energy is one half M V squared here. Very similar because we're talking about angular motion. We have the kinetic energy is one half I omega squared. OK. So I is the moment of inertia omega is the angular velocity. All right. So what do we know? What do we need to find it? Well, we know that the kinetic energy is equal to 25 jewels we want to find. I, we don't know Omega squeaked. So we need to find Omega in order to find our moment of inertia. I, that we're looking for. How can we do that? We have to be careful here. Okay. We're given that it completes five revolutions in five seconds. So it can be very easy to go ahead and write five revolutions per five seconds. So one revolution a second as our speed. However, our speed is not constant throughout the five seconds, we start at rest at T equals zero seconds. We have a constant acceleration and we want to find the speed at A T equals five seconds because that's where we have information about the kinetic energy. So instead of just using the five revolutions in five seconds, we need to use some of our kingdom attic equations or are you a M equations for angular motion? Okay. Now, let's recall, we have the following. We have that theta minus theta not, Is equal to 1/2 Omega not plus Omega. T. What do we know here? Well, we know that data minus data not, is going to be five revolutions. Okay. The change in position or the displacement, the total displacement that we go is five revolutions. We know that Omega not is equal to zero because we're told that we start from rest, we have Omega which we're trying to find. And we know that we're looking at the time point of five seconds. So we have five revolutions is equal to one half times zero plus omega times five seconds. Now, we want to convert our revolutions into radiance. Okay. And let's recall that for every revolution we have two pi radiance, okay. One revolution is going around the circle one time and we know that a circle consists of two pi radiance. And so this is gonna be times two pi radiance per revolution. Okay. On the right hand side, we have one half times five seconds. It's gonna give us 2.5 seconds and then we have times omega. Okay. We want to isolate omega and so we divide by 2.5 seconds. We get that Omega is equal to five revolutions times two pi radiance per revolution, the unit of revolution will cancel, we're left with 10 pi radiance divided by 2.5 seconds which gives us four pi radiance per second. Okay. And that is gonna be our angular speed omega. We can use that in our kinetic energy equation now to find I and so if we go back up to the top here, we have one half times I which is what we're looking for. Times are omega squared for pie radiance per second, all squared. Alright. And if we work this out on the right hand side so that we can still see the rest of our work, We're gonna have 25 jewels is equal to eight pi squared. Okay because we have four pi squared. So we get 16 pi squared divided by two, gives us eight pi squared brady in squared per second squared times a moment of inertia. II. We wanna isolate for I. And so we divide, we get 25 jules Divided by eight pi squared radiant squared per second squared. Remember that a jewel is a kilogram meter squared per second squared. And so we end up with 0. kgm squared. And if we compare that to our answer choices, we see that that is going to correspond with answer choice. A okay. So a moment of inertia of our compact disc about that central axis is going to be 0.32 kg meter squared. That's it for this one. Thanks everyone for watching. See you in the next video.

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Key Concepts

Angular Acceleration

Kinetic Energy of Rotation

Moment of Inertia