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, a solid cylinder means that \( I \) is going to be half \( m r^2 \) because that's the equation for the moment of inertia of a cylinder. Mass is 10 and radius is 2. I'm going to put these here. If you want to, you could already calculate the moment of inertia, right? So \( I \) is \( \frac{1}{2} \times 10 \times 2^2 \), and the moment of inertia is 20. Okay? So, we can already get that 20 kilograms meters squared.
It says it is mounted and free to rotate, on a perpendicular axis 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. It just doesn't wobble like that, right? It rotates like this. Now, most of the time you actually have this where the axis is horizontal. So it's on a wall, right? So it's something that's on a wall, and you have the disc spinning like this, all right?
So it says that the cylinder is initially at rest, so the cylinder spins around itself, but it's initially at rest. \( \omega \) initial equals 0. And we want to know what the work done is to accelerate it from rest to a 120 RPM. Okay? Remember, most of the time when you have RPM, you're supposed to change that into \( \omega \) so that you can use it in the equation. So let's do that real quick just to get that out of the way. \( \omega \) final is \( 2 \pi f \) or \( \frac{2 \pi \times \text{RPM}}{60} \). If you plug 120 here, you end up with 120 divided by 60 is 2. You end up with \( 4 \pi \) radians per second.
Okay? So, I am going from 0 to \( 4 \pi \), and I want to know how much work that takes. So work is energy. Hopefully, you thought of using the conservation of energy equation, \( k_{\text{initial}} + u_{\text{initial}} + W_{\text{nonconservative}} = k_{\text{final}} + u_{\text{final}} \). In the beginning, there's no kinetic energy because it's not spinning, it's not moving sideways. The potential energies cancel because the height of the cylinder doesn't change. It stays in place. Right? \( W_{\text{nonconservative}} \) is the work done by you 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 kinetic energy, which is only kinetic rotational. Right? There's no linear. It's not moving sideways. The center of mass of the disk stays in place. So \( v \) equals 0. So, the only type of kinetic energy we have is rotational, which is \( \frac{1}{2} I \omega^2 \). We're looking for this, so all we got to do is plug in this number. Work is going to be \( \frac{1}{2} \times 20 \times (4 \pi)^2 \).
Okay? So, if you multiply all of this, you get that it is 1580 joules of energy, and that's how much energy is needed to get the solid cylinder from rest all the way to a speed of \( 4 \pi \) or a 120 RPM. Cool? Very straightforward. Plug it into the energy equation because we were asked for work. Alright? Hope it makes sense. Let me know if you guys have any questions and let's keep going.