Mr.p: Good morning. We have already worked with the angular momentum of rotating rigid objects with shape
like records, spinning tops, and merry-go-rounds.
Because those objects are rotating, it is probably clear they have angular momentum.
Today we are going to show that
an object which is moving in a straight line can also have angular momentum.
"Flipping Physics" Intro
Bobby: How can an object moving in a straight line have angular momentum?
Bo: Yeah, if it's not rotating, how does it have angular momentum?
Billy: But the ball is rotating as it rolls, right?
Mr.p: Oh, yeah, sorry. So, yeah, the ball is rotating as it rolls.
The ideal examples of this typically have a point particle moving at a constant velocity,
which is difficult to do in the real world without having the object be a rolling sphere.
So realize, I am talking about the angular momentum of the ball as it moves in a straight line.
Bo: Okay, but I still don't get how anything moving in a straight line could have angular momentum.
Mr.p: Right. To help you understand that objects moving in a straight line can have angular momentum,
I mounted a wooden board on a Lazy Susan, which allows the board to rotate.
Ideally, there would be no friction,
however, unfortunately, friction does slow down the rotation of the board.
Then I put an incline right in front of the board
so I can give a ball a consistent linear velocity right before it runs into the rotating board.
Just so you know, the lighter colored wooden board at the end of the track
is there simply to keep the ball at roughly the same height as the rotating wooden board.
Now that we understand the setup, I'm going to change our viewpoint
so we are directly above where the ball collides with the rotating wooden board.
Now I have slowed the video down to 25% speed to make it easier to see what is happening.
Let's discuss how this shows that this ball which is moving in a straight line has angular momentum.
Class, before the ball runs into the board, does the board have angular momentum?
BBB: No.
Billy: Angular momentum of a rigid object with shape equals rotational inertia times angular velocity.
The angular velocity of the board, which is a rigid object with shape, before the ball strikes, it is zero,
so the angular momentum of the board before the ball strikes it is also zero.
Mr.p: Correct. Now class, after the ball strikes the board, does the board have angular momentum?
Bobby and Bo: Yes.
Billy: Yes, the board has an angular velocity after the ball runs in to it,
so the board must have an angular momentum while it is rotating.
Mr.p: So if before the collision with the ball, the board does not have angular momentum
and after the collision, the board does have angular momentum,
where did the angular momentum of the board come from?
Bobby: The ball must have transferred angular momentum to the board.
Billy: So, in order to transfer angular momentum to the board,
the ball had to have angular momentum before the collision.
Bo: Okay, that makes sense. But it still seems weird that an object moving linearly can have angular momentum.
Billy: It is weird. Mr.p: I agree it is a bit strange,
but we did just show that an object moving linearly can have angular momentum.
And remember, angular momentum is about an axis of rotation,
so realize the ball has an angular momentum, in this case, about the location the board is rotating around.
Now, I want to construct the equation for the angular momentum of a point particle.
Let's look at a lacrosse ball striking the board a short distance from the axis of rotation of the board.
Now let's repeat the experiments only using a metal ball, which has a larger mass than the lacrosse ball.
Again, this is a more massive ball moving at the same speed and striking the board at the same location.
When we compare those two collisions, do y'all see how
the more massive metal ball gave the board a larger angular momentum after the collision
than the less massive lacrosse ball?
BBB: Yes. Yep. Sure.
Bo: The board which was hit by the metal ball has a larger angular velocity,
so that board should also have a larger angular momentum.
Billy: That must mean, before the collision, the angular momentum of the more massive metal ball
is larger than the angular momentum of the less massive lacrosse ball, right?
Mr.p: Correct.
In fact, the angular momentum of a point particle is linearly proportional to the mass of the point particle.
Bo: That makes sense. More mass means more angular momentum.
Mr.p: Now, let's give the lacrosse ball a larger velocity and see what happens.
Do y'all see that, at the same distance from the axis of rotation and with the same mass,
a larger velocity of the point particle causes a larger angular momentum of the board?
BBB: Yes. Yep. Yea
Bobby: Is the angular momentum of a point particle
also linearly proportional to the linear velocity of the point particle?
Mr.p: That is correct, Bobby.
So far we have shown that the angular momentum of a point particle is linearly proportional
to both the mass and linear velocity of the point particle.
Billy: Uh, mr.p? Mr.p: Yes Billy?
Billy: These videos do not actually prove that the angular momentum of a point particle
is linearly proportional to both the mass and linear velocity of the point particle.
I mean, we have shown that it could be, but we have not actually shown that it is.
Bo: That is a good point. Bobby: Yeah.
Mr.p: Sorry. Yes, that is absolutely correct Billy. My apologies.
Okay, what we were doing here is clearly not a rigid proof;
that would take a lot more experimenting and a lot more time.
What we're doing here is just meant to show that the equation could be true.
Thanks for pointing that out Billy.
Billy: You are welcome, mr.p!
Now, watch what happens if we move the point particle farther from the axis of rotation.
Again, this is with the same velocity and same particle mass,
however, the distance from the axis of rotation is increased. Bo, what happens here?
Bo: Well, the larger distance from the axis of rotation caused a larger angular momentum of the board.
So, I bet angular momentum is also
linearly proportional to the distance the point particle is from the axis of rotation.
I'm guessing this symbol for that is "r" because it looks a lot like the "r" value for torque,
because the "r" value for torque is also a distance from the axis of rotation. Am I right?
Mr.p: That is correct Bo. The angular momentum of the point particle equals r,
the distance from the axis of rotation to the point particle,
times the object's mass times the object's linear velocity times.... well, there is one more thing.
Look at what happens when the point particle collides with the board
while moving at an angle which is not 90 degrees relative to the board.
Billy: Okay, the angular momentum delivered to the board is decreased when the angle is not 90 degrees.
Angular momentum has the same r value as torque,
does that mean angular momentum also has the same sine of theta as the torque equation?
Mr.p: That is absolutely correct Billy. The magnitude of the angular momentum of a point particle equals
the r vector times the mass of the object times the linear velocity of the object times the sine of theta,
where theta is the angle between the direction of the r vector and the direction of the object's velocity.
Bobby: The r vector goes from the axis of rotation to the center of mass of the point particle, right?
Mr.p: Yes, Bobby, that is correct.
The r vector goes from the axis of rotation to the center of mass of the point particle
and the angle theta in the angular momentum equation
is between the direction of the r vector and the direction of the linear velocity of the point particle.
Billy: Wow. That really is similar to the way the torque equation works.
I mean torque equals the r vector times force times the sine of the angle between those two vectors.
Replace force with mass times velocity, and you have the angular momentum of a point particle equation.
Mr.p: Very nice observation, Billy.
Now class, watch what happens when the point particle is moving directly toward the axis of rotation.
Why do you think the point particle gave the board zero angular momentum?
Well, because before the collision
the r vector goes from the axis of rotation to the center of mass of the point particle,
the r vector and linear velocity of the point particle are in opposite directions.
Bobby: So the angle, theta, between those two directions is 180 degrees.
Bo: And the sine of 180 degrees equals zero.
So a point particle moving directly toward an axis of rotation has zero angular momentum.
Mr.p: Very nice everybody. One last piece.
Notice what happens if the point particle collides with the board on the other side of the axis of rotation.
Bo: Yeah. The board will rotate in the opposite direction. That totally makes sense.
Angular momentum is a vector, so it has direction.
I bet we still use the right hand rule to determine the direction of angular momentum, right?
Mr.p: That is correct.
If you point to the fingers of your right hand in the direction of the r vector,
curl your fingers in the direction of the velocity, and stick out your thumb,
you can see the ball on the left has angular momentum which is out of the screen before the collision
and the ball on the right has angular momentum which is into the screen before the collision.
Notice the angular momentum of each of the rotating boards after the collision
matches the direction of the angular momentum of the ball before the collision.
Bo: Yeah, that makes sense.
The angular momentum of the rotating board on the left is out of the screen
the angular momentum of the rotating board on the right is into the screen.
Mr.p: That's right, Bo, thanks.
And thank you for learning with me today, I enjoyed learning with you.