Mr.p: Good morning. In our previous lesson we used a wooden board mounted on a Lazy Susan
to introduce the equation for the angular momentum of a point particle.
"Flipping Physics" Intro
Billy, please give us the equation for the angular momentum of a point particle.
Billy: The angular momentum of a point particle equals r m v sine theta,
where r is the vector pointing from the axis of rotation to the center of mass of the point particle,
m is the mass of the point particle, v is the linear velocity of the point particle,
and theta is the angle between the direction of the r vector and the velocity of the point particle.
Mr.p: Correct. Now, what I want to do today is
to introduce a common triangle used when a point particle is moving toward a rigid object with shape.
In this example, the point particle is moving toward the wooden board and will collide with the board.
During the collision, angular momentum is conserved. And an important concept to understand
is the value of the angle momentum of the point particle before it strikes the wooden board.
First off, we are going to assume the point particle is moving at a constant linear velocity,
so that is the v in the angular momentum equation.
The r vector goes from the axis of rotation to the center of mass of the point particle.
Please notice
the magnitude and direction of the r vector change as a function of position of the point particle.
Now, the angle in our angular momentum equation for the point particle,
should that be theta 1 or theta 2 theta?
Bo Theta 1. Billy: Theta 2.
Bobby: Um. Actually, it's theta 1.
The direction of the r vector is up and to the right, and the velocity vector is to the left,
so the angle between those two directions is theta 1.
Mr.p: That is correct Bobby. But notice that theta 1 and theta 2 are supplementary angles.
Billy: Oh right. Theta 1 and theta 2 add up to 180 degrees, so they are supplementary angles,
so the sine of theta 1 equals the sine of theta 2,
so it does not really matter which angle we use as far as the angular momentum of the particle is concerned,
right mr.p? Mr.p: Sure Billy, that's fine.
Bo: But I was right, theta 1 is the correct angle. Billy: You are only correct when it does not matter.
Bo: But still, I was right. Mr.p: Okay, Bo, you were right. Moving on.
Looking at the equation
for the angular momentum of the point particle before it collides with the wooden board:
m, the mass of the point particle, is constant, and v, its linear velocity, is also constant.
But do y'all see how both the r vector and theta 1 are changing as a function of position of the point particle?
BBB: Yes. Yeah. Yep. Billy: That complicates things, does it not?
Mr.p: Yes, Billy, it does, but the whole point of this lesson is to actually make this less complicated.
So, let's add a line here to create a right triangle and label the new side of the triangle "y".
Bo, please take the sine of theta 2 and let's see what happens.
Bo: Sine of theta 2 equals....
Billy: Actually, the sine of theta 1and the sine of theta 2 are the same, so can we just call it the sine of theta?
Mr.p: Sure, just call the angle theta. Bo: Sine of theta equals opposite over hypotenuse.
Opposite theta 2 is side y and the hypotenuse is the r vector.
Oh, we can solve for y which equals r times sine theta.
Bobby: Wait a minute.
y has a constant value. That means that r times sine theta is also constant and it equals y.
Billy: That means that, even though r and theta change as the point particle moves toward the wooden board,
r times sine theta, which equals y, does not change.
Bo: Actually, we can plug that back in to our angular momentum equation and show that
the angular momentum of the point particle as it moves toward the wooden board at a constant velocity
also has a constant angular momentum.
Billy: And it equals mass times velocity times y.
Knowing the angular momentum of the point particles constant makes this much easier.
Thank you mr.p. Mr.p: You are welcome Billy.
Please, everybody, remember this when you are solving conservation of angular momentum problems
involving a point particle and a rigid object with shape.
Thank you very much for learning with me today, I enjoyed learning with you.