Skip to main content
Pearson+ LogoPearson+ Logo
Start typing, then use the up and down arrows to select an option from the list.

Physics

Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

13. Rotational Inertia & Energy

Intro to Rotational Kinetic Energy

1
concept

Intro to Rotational Kinetic Energy

clock
6m
Play a video:
Was this helpful?
Hey, guys. In this video, we're gonna talk about rotational kinetic energy, which is the energy associated with the motion of spinning. Let's check it out. All right, So if you remember, if you had linear speed, which is V, you had kinetic energy. Now there's going to be two types of kinetic energy, So we're going to specify that this is linear kinetic energy, and you're used to this equation K equals half MV squared. I put a little elder to indicate that this is the linear type of kinetic energy. And that's because now we have a new one, which is, if you have rotational speed instead of V, it's W or Omega. You have rotational, kinetic energy and instead of Kael, we call it K are now the equation is very similar. It's half now. Instead of using M, we're going to use the rotational equivalent of em, which is I moment of inertia. And instead of V, we're gonna use the rotation equivalent of E, which is omega. So I get this right. So if you remember, the first equation should be easy to remember the second one. Now, on a special case, there's a special situation when you're moving and rotating, So you have a V N a W. This is called rolling motion. And one example of this is if you have a toilet paper roll that is sort of moving this way while rolling rolling around itself. So it's a toilet paper that's rolling on the floor has both kinds of motion. Therefore, it has both kinds of kinetic energy. So I'm gonna say that the K total is K l plus k R cool. And the last thing I will remind you will do a quick example. Um, is that for point masses point masses or tiny objects that don't have a shape? Um, that's have negligible size and radius. They have no volume. The moment of inertia. I is m r squared, where r is a distance between the object in the axis. Okay, remember also that if you have a shape or a rigid body, an object with, um non negligible radius and volume, we're going to get the moment of inertia from a table. Look up, for example, if you have a solid cylinder or a solid disc, same thing the equation for that is half m r squared so a point mass is always this and a some sort of shape will have a different equation each time. Quote. Awesome. So let's go. Very quick example. Here I have a basketball player that spins a basketball around itself on top of his finger. Okay, so I'm gonna try to draw. This is gonna come out terrible. Um, So here's basketball player. Here's his finger, uh, exaggerating some stuff. And here's a basketball and he's rotating the basketball around itself. So it looks kind of like this. Basketball spinning around itself on top of your finger. Right? Um and it says here the ball has a massive 0.62 a diameter of 24 centimeters. So 240.24 m and it spins at 15 radiance per second. Radiance per second is angular speed, angular velocity omega. Okay, 15. And we want to know the balls linear, rotational and total kinetic energy. In other words, we wanna know what is K l What is K R and what is K total? All right, So first things first. You may already have caught this. Um, in physics, we never use diameter. We always use radius. So when you see a diameter you immediately converted to Radius. Radius is half, so it's points 12. Now we're gonna plug into the equation here. Kinetic energy is half M V squared. And this bow has no kinetic energy. No linear, kinetic energy should say, and that's because it spins in place. It's rotating, but it's not actually moving right. It doesn't have has rotational motion, but it doesn't have linear motion. It doesn't have translational motion. It's just stays in place, spinning around itself. So we're gonna say that it has no linear kinetic energy. It does have rotational, kinetic energy because it's spinning around itself, and that's given by half I Omega squared. Okay, now a basketball, A basketball has moments of inertia, the moment of inertia of a hollow sphere. Okay, I didn't give you the equation for that. I didn't explicitly say it was a hollow sphere, but you should know that a basketball is a show. And then there's air inside eso. It is a hollow sphere. So I for a hollow sphere, you would look it up or it would be given to you is two thirds m r square. So what I'm gonna do is I'm gonna plug that in here? Two thirds m r squared and then Omega squared, which I have. Okay, so now we can just plug in numbers, the two cancer with the to and I'm left with one third. The mass is 10.62. The radius is 12 square in omega. We have it right here. 15 15 squared. And if you multiply all of this, I gotta hear you get a 0.67 Jules, 67 jewels. And so that's it. For the last part, we want to do the total kinetic energy. Remember the total kinetic energy? It's just in addition of the two types kinetic, linear, plus kinetic rotational, there is no kinetic linear. So the total kinetic energy is just the 0.67 that's coming from the rotational kinetic energy. Cool. So that's how this stuff works. Hopefully, this makes sense. Let me know if you have any questions and let's keep going
2
Problem

A flywheel is a rotating disc used to store energy. What is the maximum energy you can store on a flywheel built as a solid disc with mass 8 × 104 kg and diameter 5.0 m, if it can spin at a max of 120 RPM?

3
example

Mass of re-designed flywheel

clock
3m
Play a video:
Was this helpful?
Hey, guys. So here we have a rotational, kinetic energy problem off the proportional reasoning type and what that means. It's one of those questions where I ask you, how does changing one variable affect another variable? It's one of those. Okay, so let's check it out. I'm gonna show you what I think is the easiest way to solve these. So it says you're tasked with redesigning a solid disc flywheel on. Do you want to decrease the radius by half? So first things first. Solid disk means that the moment of inertia is half M R square. That's the equation for a solid desk or solid cylinder. And you want the new radius I'm gonna call. This are to to be half of our one, and I want to know by how much mass or how much mass must the new fly we'll have eso. It's the new mass relative to the original mass so that you can store the same amount of energy you want. The amount of energy that you store to be the same theme amount of energy stored is given by K R. That's energy stored, right, um which is given by half. I Omega squared. This is energy stored as rotational kinetic energy in a flywheel. You want this number not to change. You want this number to be a Constance Constance. Okay, So how do you do this? Well, if our changes, if our changes right here, then eyes going to change, and if I change is K is going to change. And that's bad news. So how do we change something else so that the K doesn't change well for the k not to change. Um, for the k not to change. You have to make sure that the I doesn't change. And for the I not to change, you have to cancel out changing our with changing em. Okay, so what I'm gonna do here is I'm gonna expand this equation half. Um, I is half m r squared Omega Square. So now I see all the variables that affect my K. And again, the K has to remain constant. So if my radius is becoming half as large, it means that it is decreasing by a factor of two. Okay, so but the the R is squared, which means that when I reduce our by a factor of two I also have to square this and are is becoming half a xlat. But then the whole thing R squared is becoming four times smaller. Okay, four times smaller. What that means is that if you wanna keep everything constant, mass has to grow by a factor of four X. Okay, so my new mass has to be four times my old mass, and that's the answer. Cool. So again, our decreases by a factor of two. But then you have to square because there's a square here you get a four. If one variable decreases by four, the other one has to increase by four notice. There's no squares in the M, so it's just a four. Not to nothing crazy like that. Cool. That's it for this one. Let me know if have any questions.
4
Problem

When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 10,000 J in kinetic energy. Calculate the sphere’s mass.

Divider