Mr.p: Good morning.
Today we are going to use an item from Arbor Scientific called a "Rotational Inertia Demonstrator"
to demonstrate the concept of rotational inertia, or what is often called moment of inertia.
"Flipping Physics" Intro
The basic idea is that we have three different pulley sizes all mounted to the same axle.
The axle has very little friction and, for our demonstration purposes,
we are going to assume that it actually has zero friction.
Notice the axle is the axis of rotation of the system and attached to the pulleys we have these four spokes.
And on those four spokes, we have four masses which we can adjust the locations of.
Remind me Bobby, what is the equation for the rotational inertia of a system of particles?
Bobby: The moment of inertia, or rotational inertia, of a system of particles
equals the sum of the quantity of the
mass of each particle times the square of the distance each particle is from the axis of rotation.
But, that thing is not a system of particles, right?
Mr.p: Bobby, the basic idea is that we can use the equation
you just gave us for the rotational inertia of a system of particles to understand
what happens to the rotational inertia of the rotational inertia demonstrator when we make changes to it.
Billy, what is the rotational form of Newton's second law?
Billy: Net torque equals rotational inertia times angular acceleration
where both torque and angular acceleration are vectors.
Mr.p: Okay, let's start with the four adjustable masses as close to the axis of rotation as possible
and let's suspend a 100 gram mass from the largest sized pulley.
Observe what happens when I let go of the system.
The torque caused by the hanging mass causes an angular acceleration of the system. Now, what will
happen when I adjust the locations of the four masses such that they are farthest from the axis of rotation?
Bo, will the angular acceleration of the new setup be increased, decreased, or will it remain the same?
Bo: Moving the masses so they are farther from the axis of rotation will
increase the "r" in the rotational inertia equation. And therefore increase the rotational inertia of the system.
Assuming you leave the mass hanging the same, the torque should be about the same, then the
rotational form of Newton's second law shows
that when you increase the rotational inertia, the angular acceleration must decrease.
Mr.p: Okay, let's see if that is correct.
As you can see, moving the masses farther for the axis of rotation
increases the rotational inertia of the system, which decreases the angular acceleration of the system.
Bo: mr.p.?
Mr.p: Yes, Bo?
Bo: Why are you making it so the four masses are always the same distance from the center?
Mr.p: Bo, that is a fair question. You can see I have rearranged the rotational inertia demonstrators
and both of these examples actually have no hanging mass from the pulleys;
one has the masses equally spaced from the axis of rotation
and one has a single mass farther from the axis of rotation.
Bo, how does this demonstrate your answer?
Bo: Well, when the masses are equally spaced from the axis of rotation,
it continues to rotate at what looks like a constant angular velocity, the other one does not.
Bobby: Oh, it's the center of mass.
Billy: Right. When the masses are equally spaced from the axis of rotation
the center of mass of the system is at the axis of rotation.
So because the r value in the torque equation for the force of gravity equals zero,
the force of gravity does not cause a torque on a system.
Sure. That means when the masses are not equally spaced from the axis of rotation,
the center of mass of the system is displaced from the axis of rotation and the
force of gravity causes a torque on the system which is always angularly accelerating the center of mass
of the system toward a point below the axis of rotation. That really complicates things.
Mr.p: That is correct, keeping the center of mass of the system at the axis of rotation of the system
makes it so much easier situation to analyze and understand.
Now, let's get back to the original demonstrations.
We had just done a demonstration with the 100 gram hanging mass on the largest sized pulley and
all four adjustable masses far from the axis of rotation.
Billy, what if we now change the location of the string so instead of being on the largest sized pulley,
the 100 gram mass is hanging from the smallest sized pulley.
How does that change the angular acceleration of the system?
Billy: Well, we know torque equals the r vector times the force times the angle between those two vectors.
Therefore, if you decrease the distance from the axis of rotation to where the force acts,
you are decreasing the "r" value, therefore you decrease the net torque.
Because the rotational inertia stays the same,
according to the rotational form of Newton's second law,
the angular acceleration of the system must decrease.
Mr.p: Again, let's see if that is correct.
You can see that decreasing the distance the force is from the axis of rotation
did decreas the net torque, and therefore decreased the angular acceleration of the system.
Okay. Now, what if I add 200 grams of mass
to the same pulley only wrapped around the other direction,
such that it hangs over the other side of the pulley.
Bobby, what happens to the angular acceleration of the system now?
Bobby: Wait, now there are two masses hanging from it.
I mean ... where are you taking this thing?
Bo: Yeah.
Billy: Oh! Remember the rotational form of Newton's second law includes the net torque, not just the torque.
You need to add those two torques together.
Bobby: Oh, right. One of the masses is 100 grams and the other is 200 grams.
But the two torques act in opposite directions and therefore, the
net torque actually ends up having the same magnitude as it did before, only now in the opposite direction.
Therefore the angular acceleration should have the same magnitude, only opposite in direction.
Mr.p: Very nice. Let's see if you are correct.
Will you look at that,
they do have roughly the same magnitude angular acceleration only opposite directions.
Please everybody remember it is the net torque not just a single torque.
It is the sum of all torques acting on the object,
which equals the rotational inertia of the object times the angular acceleration of the object.
Thank you very much for learning with me today, I enjoyed learning with you.