### 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, letβs 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 π is equal to one-half π, the mass of the object, times π£ squared, where
π£ 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.

Letβs now compare this to the
rotational kinetic energy of an object. The rotational kinetic energy π,
in this video weβve also added a subscript capital R to remind ourselves that itβs
rotational, is equal to one-half πΌ, the moment of inertia of the object, π
squared, where π 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, itβs
also based on the velocity, but in this case, our object is spinning, so we care
about the π 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 thatβs
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.

Letβs 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, π,
of an object is equal to one-half πΌ, the moment of inertia of the object, times π
squared, where π 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 π. 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 π equals
one-half πΌπ squared, where π is the rotational kinetic energy, πΌ is the moment
of inertia, and π 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 πΌ and 28 radians per second for
π. 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 π, the rotational
kinetic energy, is equal to one-half πΌ, the moment of inertia of an object, times
π squared, where π 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 π squared, we must
multiply both sides of the equation by two and one over πΌ. This will cancel out the one-half
on the right-hand side of the equation as well as the πΌ 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 π squared into just π, 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
π over πΌ equals π, and we are ready to plug in our values. We replace π with 248 joules and
πΌ 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 π equals one-half πΌπ squared
to calculate rotational kinetic energy of an object. π equals one-half πΌπ squared is
the rotational analogue of π equals one-half ππ£ squared. This is a summary of rotational
kinetic energy.