Intro to Rotational Kinetic Energy: Study with Video Lessons, Practice Problems & Examples

Rotational kinetic energy (K_r) is the energy associated with spinning motion, represented by the equation ${\frac{1}{2}}^I\omega \omega$. In rolling motion, both linear (K_l) and rotational kinetic energy contribute to total kinetic energy (K_total = K_l + K_r). The moment of inertia (I) varies based on the object's shape, with point masses calculated as I = m r² and hollow spheres using I = (2/3) m r². Understanding these concepts is crucial for analyzing motion in physics.

Intro to Rotational Kinetic Energy

Hey, guys. In this video, we're going to talk about rotational kinetic energy, which is the energy associated with the motion of spinning. Let's check it out. Alright. So if you remember, if you had linear speed, which is v, you had kinetic energy. Now there's going to be 2 types of kinetic energy. So we're going to specify that this is linear kinetic energy and you're used to this equation k=12mv2. I put a little l there 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 ω or omega, you have rotational kinetic energy. And instead of k_l, we call it k_r. Now the equation is very similar. It's 12. Now instead of using m, we're going to use the rotation equivalent of m, which is I moment of inertia. And instead of V, we're going to use the rotation equivalent of V, which is Omega. So I get this. Right? So if you remember the first equation, it 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 and a ω. 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 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 going to say that the k_total is k_l+k_r. Cool? And the last thing I want to remind you, we'll do a quick example, is that for point masses, point masses are tiny objects that don't have a shape, that have negligible size and radius. They have no volume. The moment of inertia I is mr2, where r is a distance between the objects and the axis. Okay? Remember also that if you have a shape or a rigid body, an object with non-negligible radius and volume, we're going to get the moment of inertia from a table lookup. For example, if you have a solid cylinder or a solid disk, same thing, the equation for that is 12mr2. So point mass is always this and some sort of shape will have a different equation each time. Cool? Awesome. So let's do a very quick example here. I have a basketball player that spins a basketball around itself on top of his finger. K. So I'm going to try to draw this. It's going to come out terrible. So here's a basketball player. Here's his finger, 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? And it says here the ball has a mass of 0.62, a diameter of 24 centimeters, so 0.24 meters, and it spins at 15 radians per second. Radians per second is angular speed, angular velocity, omega. K. 15. And we want to know the ball's linear, rotational, and total kinetic energy. In other words, we want to know what is k_l, what is k_r, and what is k_total. Alright. So first things first, you may already have caught this. In physics, we never use diameter. We always use radius. So when you see diameter, you immediately convert it to radius. Radius is half, so it's 0.12. Now we're going to plug into the equation here. Kinetic energy is 12mv2 and this ball has no kinetic energy. No linear kinetic energy I should say and that's because it spins in place. It's rotating, but it's not actually moving. Right? It doesn't have it has rotational motion, but it doesn't have linear motion. It doesn't have translational motion. It just stays in place spinning around itself. So we're going to 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 12Iω2. 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 shell and then there's air inside. So it is a hollow sphere. So I for a hollow sphere, you would look it up or it would be given to you, is 23mr2. So what I'm going to do is I'm going to plug that in here. 23mr2 and then omega squared, which I have. Okay. So now we can just plug in numbers. The 2 cancels with the 2 and I'm left with 13. The mass is 0.62. The radius is 0.12 squared and omega, we have it right here, 15². And if you multiply all of this, I got it here, you get 0.67 joules. 0.67 joules. And so that's it. For the last part, we want to do the total kinetic energy. Remember the total kinetic energy is just an addition of the 2 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 made sense. Let me know if you have any questions and let's keep going.

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?

• A. 4.83 × 10⁶ J
• B. 9.87 × 10⁶ J
• C. 1.97 × 10⁷ J
• D. 3.95 × 10⁷ J

Mass of re-designed flywheel

Hey, guys. So here we have a rotational kinetic energy problem of 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, and you want to decrease the radius by half. So first things first, solid disc means that the moment of inertia is half *m*r². That's the equation for a solid disc or solid cylinder. And you want the new radius, I'm going to call this *r₂*, to be half of *r₁*. And I want to know by how much mass or how much mass must the new flywheel have, so what'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 stored to be the same. The amount of energy you stored is given by *k*r, that's energy stored, right, which is given by half *I*ω². This is energy stored as rotational kinetic energy in a flywheel. You want this number not to change. You want this number to be constant, constant. Okay? So how do you do this? Well, if *r* changes if *r* changes right here, then *I* is going to change. And if *I* changes, *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, 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 *r* with changing *m*. Okay? So what I'm gonna do here is I'm gonna expand this equation, half, *I* is half *m*r²ω². 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 2. Okay. So but the the *r* is squared, which means that when I reduce *r* by a factor of 2, I also have to square this. And *r* is becoming half as large, but then the whole thing, *r* squared, is becoming 4 times smaller. Okay? 4 times smaller. What that means is that if you wanna keep everything constant, my mass has to grow by a factor of 4×. Okay? So my new mass has to be 4 times my old mass, and that's the answer. Cool? So again, *r* decreases by the factor of 2, but then you have to square because there's a square here, you get a 4. If one variable decreases by 4, the other one has to increase by 4. Notice there are no squares in the *m*. So it's just a 4, not a 2. Nothing crazy like that. Cool? That's it for this one. Let me know if you have any questions.

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.

• A. 79 kg
• B. 317 kg
• C. 493 kg
• D. 1.97×10⁶ kg

Here’s what students ask on this topic:

What is the formula for rotational kinetic energy?

The formula for rotational kinetic energy (K_r) is given by:

$K_r=\frac{1}{2}I{\omega }^2$

Here, $I$ is the moment of inertia, and $\omega$ (omega) is the angular velocity. This equation is analogous to the linear kinetic energy formula $K_l=\frac{1}{2}m{v}^2$, where mass (m) and linear velocity (v) are replaced by their rotational counterparts.

How do you calculate the moment of inertia for different shapes?

The moment of inertia (I) depends on the shape and mass distribution of the object. For point masses, it is calculated as:

$I=m{r}^2$

where m is the mass and r is the distance from the axis of rotation. For rigid bodies, the moment of inertia varies:

• Solid cylinder or disk: $I=\frac{1}{2}m{r}^2$
• Hollow sphere: $I=\frac{2}{3}m{r}^2$

These values can be found in tables or derived based on the geometry of the object.

What is the difference between linear and rotational kinetic energy?

Linear kinetic energy (K_l) is associated with the motion of an object moving in a straight line and is given by:

$K_l=\frac{1}{2}m{v}^2$

where m is the mass and v is the linear velocity. Rotational kinetic energy (K_r) is associated with the spinning motion of an object and is given by:

$K_r=\frac{1}{2}I{\omega }^2$

where I is the moment of inertia and ω (omega) is the angular velocity. In rolling motion, both types of kinetic energy contribute to the total kinetic energy.

How do you calculate the total kinetic energy in rolling motion?

In rolling motion, an object has both linear and rotational kinetic energy. The total kinetic energy (K_total) is the sum of these two components:

$K_{\mathrm{total}}=K_l+K_r$

where:

$K_l=\frac{1}{2}m{v}^2$

and

$K_r=\frac{1}{2}I{\omega }^2$

Here, m is the mass, v is the linear velocity, I is the moment of inertia, and ω (omega) is the angular velocity. By calculating both components and adding them, you get the total kinetic energy.

What is the moment of inertia for a hollow sphere?

The moment of inertia (I) for a hollow sphere is given by:

$I=\frac{2}{3}m{r}^2$

where m is the mass of the sphere and r is the radius. This formula accounts for the mass being distributed on the surface of the sphere, with the interior being hollow. This is different from a solid sphere, which has a different moment of inertia due to its mass distribution.