Mr.p: Good morning. Bobby, what is the equation for kinetic energy?
"Flipping Physics" Intro
Bobby: Kinetic energy equals one-half mass times velocity squared.
Mr.p: Now, let's say we have an object that is rotating around its center of mass but,
its center of mass is not moving like a bike wheel on an upside-down bicycle.
The center of mass of the bike wheel is not moving.
So the velocity of the center of mass of the wheel is zero. So the wheel has zero kinetic energy, right?
Billy: Right!
Bobby: That doesn't make sense to me.
Mr.p: I agree Bobby. Kinetic energy is the energy of motion,
so if an object is in motion, it should have kinetic energy.
It is just that this equation does not describe the type of kinetic energy this wheel has.
Rather than using center of mass to look at the wheel as a whole,
we need to look at the kinetic energy of all the individual small pieces that make up the wheel.
The total kinetic energy of the wheel
equals the sum of the kinetic energies of all the small pieces that make up the wheel.
The letter "i" represents that the number goes from 1 to "i",
the total number of small pieces which make up the bike wheel.
Substituting in the equation for kinetic energy gives us
The sum of one half mass times velocity squared for every small piece, which makes up the object.
Notice the velocity in this equation is the velocity of every small piece which makes up the object.
This velocity must be a tangential velocity because the object is rotating
and therefore every part of the object is moving in a circle.
Bo, what is the equation that relates tangential velocity to angular velocity?
Bo: Tangential velocity equals radius times angular velocity.
Which we can then substitute into our equation. A couple of things to point out:
First off, realize "r" in this equation is the distance each small piece is from the axis of rotation.
Second, because the wheel is a "rigid object with shape", meaning it does not change shape as it rotates;
the angular velocity of every piece is the same.
This means we do not need these subscript "i" for each piece's angular velocity,
we can simply write omega without the "i" subscript.
Now we have parenthetically isolated the sum of the mass of each piece
times the square of the distance each piece is from the axis of rotation.
This is defined as the Moment of Inertia of the object.
The moment of inertia of an object, the symbol for which is a capital "I",
is defined as the sum of, for every small piece which makes up the object,
the quantity mass times the square of the distance each small piece is from the object's axis of rotation.
When we substitute that back into the equation for the total kinetic energy of the rotating object, we get
the total kinetic energy of the rotating object, or what is called the object's rotational kinetic energy,
equals one half times the moment of inertia of the object
times the square of the angular velocity of the object.
Now that we have the rotational kinetic energy defined,
we need to more specifically identify the original kinetic energy equation.
The kinetic energy we have been using up to this point is called Translational Kinetic Energy.
It is the energy associated with the motion of the center of mass of an object
moving from one point in space to another point in space.
Billy: Mr.p?
Mr.p: Yes, Billy?
Billy: Those two equations look very similar.
They both have one-half times a quantity times the square of a velocity. That's kind of weird.
Bobby: Also, I get that rotational kinetic energy is the kinetic energy of a rotating object and
translational kinetic energy is the kinetic energy of the objects moving from one place to another.
But, what is moment of inertia?
Bo: Yeah, what is moment of inertia?
Mr.p: Sure.
Let's discuss what Billy pointed out about the similarities between the two kinetic energy equations.
Billy, what is the difference between the "v" in the translational kinetic energy equation
and the "omega" in the rotational kinetic energy equation?
Billy: Well, both v and omega are velocities. Omega is the angular velocity of the object,
so then the v is the linear velocity of the object, right?
Mr.p: Correct. Notice then that in the rotational kinetic energy equation, the capital I, the moment of inertia,
takes the place of the inertial mass in the translational kinetic energy equation.
That is why I also like to refer to the moment of inertia as "rotational mass".
In order to understand what I mean when I say moment of inertia is rotational mass,
Bo, remind me, what is inertial mass?
Bo: Inertial mass is a measure of inertia. Inertia is a tendency to resist acceleration.
So, inertial mass is the measure of an object's resistance to acceleration.
Mr.P: Correct.
Inertial mass is a measure of the tendency of an object to resist acceleration.
The more mass something has the more it resists acceleration.
This means that moment of inertia,
or "rotational mass", is a measure of the tendency of an object to resist angular acceleration.
The more moment of inertia, or "rotational mass " something has, the more it resists angular acceleration.
Bobby: Uh, can you show us what that means?
Mr.p: Sure.
Let's start with two eggs in an egg carton with both of those eggs near the middle of the egg carton.
For this example, we are going to assume
the mass of the egg carton itself is very small relative to the mass of the two eggs.
Therefore, we can assume the mass of the egg carton is negligible. We can ignore the mass of the egg carton.
Because there are two objects in the system, the two eggs, the moment of inertia of the system will be
the sum of two expressions of mass times distance from axis of rotation squared, one for each egg.
When I hold the egg carton in the middle and rotate it, it is relatively easy to rotate the system,
In other words, because the eggs are close to the axis of rotation, the "r" value of each egg is small,
so the moment of inertia is small, and it is relatively easy to cause the system to angularly accelerate
BBB: Okay. Sure. Yeah.
Mr.p: Now let's move the eggs so there are on opposite ends of the egg carton.
Bobby, assuming the same axis of rotation, the middle of the egg carton, how does
moving the two eggs to the opposite ends of the egg carton affect the moment of inertia of the system?
Bobby: Well, okay. Moving the eggs farther from the axis of rotation increases the value of "r",
the distance from the axis of rotation, so the moment of inertia of the system should be increased.
And increasing the moment of inertia should make it more difficult to rotate the eggs.
Right because we increased the distance the eggs are from the axis of rotation
we increased the moment of inertia, or "rotational mass",
and therefore it is more difficult to cause the eggs to angularly accelerate.
Mr.p: It is important to realize we have not changed the mass of the system;
we have only changed the locations of the masses in the system.
Increasing the distance the eggs are from the axis of rotation increases the moment of inertia,
or "rotational mass", of the system.
Which makes it more difficult to angularly accelerate the system.
However, the inertial mass of the system remains the same.
Also notice how "r", the distance from the axis of rotation of each particle,
is squared in the moment of inertia equation.
This means the distance each particle is from the axis of rotation of the system
has a much larger influence over the moment of inertia than the mass of each particle.
Bobby: Right. In other words,
if the distance from the axis of rotation is tripled, the moment of inertia has increased nine times
because three squared is nine.
But tripling the mass only triples the moment of inertia.
Billy: Yeah.
Bo: Right.
Mr.P: Thanks Bobby. And thank you for learning with me today, I enjoyed learning with you.