# Lesson Video: Rotational Kinetic Energy Physics • 9th Grade

In this video, we will learn how to calculate the rotational kinetic energy of a rotating object from its moment of inertia and its angular velocity.

07:26

### Video Transcript

In this video, we will learn how to calculate the rotational kinetic energy of a rotating object from its moment of inertia and its angular velocity. Before we go over rotational kinetic energy, let’s refresh our memory on translational kinetic energy, as we will see a lot of similarities between these two equations.

We should remember that the kinetic energy 𝑘 is equal to one-half 𝑚, the mass of the object, times 𝑣 squared, where 𝑣 is the speed of the object. We can write in a subscript T to remind ourselves that this is the translational kinetic energy. The kinetic energy is the energy associated with the motion of the object and, as we just stated, is based on the mass of the object and how fast the object is going through its velocity.

Let’s now compare this to the rotational kinetic energy of an object. The rotational kinetic energy 𝑘, in this video we’ve also added a subscript capital R to remind ourselves that it’s rotational, is equal to one-half 𝐼, the moment of inertia of the object, 𝜔 squared, where 𝜔 is the angular velocity. Just as we saw the translational kinetic energy was based on the mass of the object, the rotational kinetic energy is based on the moment of inertia, which is essentially how the mass is distributed throughout the object. We also saw that the translational kinetic energy was based on the velocity of the object. In rotational kinetic energy, it’s also based on the velocity, but in this case, our object is spinning, so we care about the 𝜔 or angular velocity.

Looking back at our formula for the rotational kinetic energy, we can see that the kinetic energy is directly related to the moment of inertia. This means that if the moment of inertia goes up, so does the kinetic energy by the same amount. In fact, if we were to double the moment of inertia of the object, the rotational kinetic energy would double as well. With respect to angular velocity, the rotational kinetic energy is directly related to the square of the angular velocity. If we increase the angular velocity, the rotational kinetic energy would also increase, but by a factor that’s related to the square of the angular velocity. For example, if we were to double the angular velocity, then the rotational kinetic energy would actually increase by a factor of four.

Let’s do a few example problems using this equation.

A solid metal disk is rotating with an angular velocity of 15 radians per second. The disk has a moment of inertia of 4.0 kilogram meter squared around its axis of rotation. What is the rotational kinetic energy of the disk?

In the problem, we are given angular velocity, moment of inertia, and asked to find the rotational kinetic energy. To solve the problem, we therefore need an equation that relates these three variables together. The rotational kinetic energy, 𝑘, of an object is equal to one-half 𝐼, the moment of inertia of the object, times 𝜔 squared, where 𝜔 is the angular velocity of the object. The question asked us to solve for the rotational kinetic energy of the disk. We therefore do not need to rearrange our formula to solve for our unknown variable.

Plugging in the values given to us in our problem, we have 4.0 kilogram meter squared for our moment of inertia and 15 radians per second for our 𝜔. When multiplying out our values, we have to be careful to make sure that we square our 15 radians per second before multiplying it by 4.0 kilograms meter squared and one-half. Multiplying out our values, we get a rotational kinetic energy of 450 joules. The rotational kinetic energy of the disk is 450 joules.

A football is rotating with an angular velocity of 28 radians per second. The football has a moment of inertia of 1.9 kilogram meter squared around its axis of rotation. What is the rotational kinetic energy of the football? Give your answer to two significant figures.

In the problem, we are given angular velocity, moment of inertia, and asked to solve for the rotational kinetic energy. To begin to solve the problem, we need to start with an equation that has these three variables in it. We can use the equation 𝑘 equals one-half 𝐼𝜔 squared, where 𝑘 is the rotational kinetic energy, 𝐼 is the moment of inertia, and 𝜔 is the angular velocity. In our problem, we are asked to solve for the rotational kinetic energy of the football, which means we do not need to rearrange our formula to solve for our unknown variable.

Plugging in the values given from the problem, we have 1.9 kilogram meter squared for 𝐼 and 28 radians per second for 𝜔. Before we multiply out our values, we must make sure that we square our angular velocity. When we multiply out our values, we get a kinetic energy of 744.8 joules. This is not our final answer as the problem asked us to give our answer to two significant figures. When we round 744.8 joules to two significant figures, we get 740 joules. The rotational kinetic energy of the football is 740 joules.

A tennis ball has a rotational kinetic energy of 248 joules. Its moment of inertia around its axis of rotation is 0.22 kilogram meter squared. What is the angular velocity of the tennis ball? Give your answer to two significant figures.

In our problem, we are given rotational kinetic energy, moment of inertia, and we are asked to solve for the angular velocity. This means that we need an equation that relates these three variables together. We know that 𝑘, the rotational kinetic energy, is equal to one-half 𝐼, the moment of inertia of an object, times 𝜔 squared, where 𝜔 is the angular velocity of the object.

Our problem asked us to solve for the angular velocity of our tennis ball. Therefore, we must rearrange our formula to solve for our unknown variable. To isolate 𝜔 squared, we must multiply both sides of the equation by two and one over 𝐼. This will cancel out the one-half on the right-hand side of the equation as well as the 𝐼 on the right-hand side of the equation, leaving us with two times the rotational kinetic energy divided by the moment of inertia is equal to the angular velocity squared. To turn 𝜔 squared into just 𝜔, we must square root both sides of the equation. This will cancel out the square on the right-hand side.

Now we have the square root of two 𝑘 over 𝐼 equals 𝜔, and we are ready to plug in our values. We replace 𝑘 with 248 joules and 𝐼 with 0.22 kilogram meter squared. When we calculate our angular velocity, we get 47.48 radians per second. This, however, is not our final answer as we are asked to give our answer to two significant figures. When we round 47.48 radians per second to two significant figures, we get 47 radians per second. The angular velocity of the tennis ball is 47 radians per second.

Key Points

Use 𝑘 equals one-half 𝐼𝜔 squared to calculate rotational kinetic energy of an object. 𝑘 equals one-half 𝐼𝜔 squared is the rotational analogue of 𝑘 equals one-half 𝑚𝑣 squared. This is a summary of rotational kinetic energy.