Mr.p: Good morning. Bobby, please read the problem, and Bo, please translate.
"Flipping Physics" Intro
Bobby: As shown, a point particle strikes a wooden board
mounted such that it is free to rotate about its center.
During this collision, is
Part (a) linear momentum of the point particle and wooden board system conserved?
And Part (b) is angular momentum of the point particle and wooden board system conserved?
Bo: So there are no known values.
So it is just Part (a) conservation of linear momentum equals question mark
or the sum of the initial linear momenta equals the sum of the final linear momenta with question marks.
And Part (b) conservation of angular momentum equals question mark
or the sum of the initial angular momenta equals the sum of the final angular momenta with question marks.
Mr.p: Okay. Let's start with Part (a),
is the linear momentum of the point particle and wooden board system conserved?
Billy, please look at this from the standpoint of the directions of the linear momenta.
Billy: Sure. As the point particle moves toward the wooden board,
it is moving to the left and therefore has linear momentum, which is to the left.
After the collision, the point particle is moving to the right
and therefore has linear momentum which is to the right.
So, clearly the linear momentum of the point particle changes and is not conserved. However,
we are talking about the total linear momentum of the point particle and wooden board system.
So, we need to look at the linear momentum of the wooden board as well.
Before the collision the wooden board is not moving, so it has zero linear momentum.
After the collision the wooden board is rotating,
however, its center of mass is still not moving, so it still has zero linear momentum.
Therefore, the wooden board has zero change in its linear momentum.
If we sum the changes in linear momenta of the wooden board and the point particle,
there must have been a change in linear momentum of the system
because the linear momentum of the point particle changed,
and the linear momentum of the wooden board did not.
Therefore, linear momentum of the point particle and wooden board system is not conserved.
Mr.p: Thank you Billy.
Now, I'd like to prove it using equations. Bobby, when is linear momentum conserved?
Bobby: Linear momentum is conserved when the net external force acting on the system equals zero.
That is because net external force acting on the system equals
change in linear momentum over change in time.
When that equals zero, the change in time cancels out,
because you can multiply both sides by the change in time.
Change in linear momentum equals final linear momentum minus initial linear momentum.
Therefore, the initial linear momentum of the system equals the final linear momentum of the system
which is the equation for conservation of linear momentum.
Mr.p: That means the net force on the system in this example must not equal zero.
When the particle collides with the board, the point particle applies a force on the board
and, according to Newton's Third Law, the board applies an equal but opposite force on the point particle.
That means those two forces are internal to the system and cancel one another out.
So, what is another force which is causing the net force on the system to not equal zero?
Bo: I got nothin'. Billy: I do not know.
Mr.p: Okay. What about the fact that,
during the collision, the board is fixed in place at the axis of rotation?
Bobby: Oh right.
If the board were not held in place at its center of mass, the board would start to move translationally.
Billy: That totally makes sense. There must be an external force acting at the pivot point of the board
which makes it so the net external force acting on the board and point particle system does not equal zero
and linear momentum of the system is not conserved! Mr.p: Correct.
Because the board is attached to a Lazy Susan which allows the board to rotate but not change locations,
while the point particle strikes the board, the
pivot point applies an external force on the board
which prevents the board from entering into translational motion.
That means the net external force acting on the board and point particle system during the collision,
does not equal zero and linear momentum of the system is not conserved.
Please realize this force from the pivot point only acts on the board and point particle system
while the point particle is in contact with the board, which is only a very short time interval.
Also, I will point out that, if the board were not held in place at the pivot point
and were instead on a frictionless surface, for example,
there would be no net external force acting on the board and point particle system
and linear momentum of the system would be conserved.
After that collision, the board would enter into both rotational and translational motion.
Now let's talk about conservation of angular momentum. Again, let's start with directions. Billy?
Billy: Okay. As the point particle is moving toward the board, it has angular momentum, which is,
according to the right-hand rule... Well, when we start with our fingers as the axis of rotation of the board,
which is the point the board rotates around,
and point our fingers from the axis of rotation to the center of mass of the point particle.
Then we curl our fingers in the direction of the velocity of the point particle, and stick out our thumb.
Our thumb points out of the screen,
so that is the direction of the angular momentum of the point particle before the collision.
At this point the board is not moving and has zero angular momentum.
So, before the collision, the system has angular momentum, which is out of the screen.
After the collision, the point particle is moving in the opposite direction and has,
according to the right-hand rule, angular momentum, which is into the screen.
And after the collision, the board has an angular momentum, which is out of the screen.
Therefore, the angular momentum before the collision is out of the screen and after the collision
is the summation of the point particle's into the screen
and wooden board's out of the screen angular momenta.
I guess that means
we do not know if the angular momentum of the point particle and wooden board system is conserved.
I mean, it could be, but we do not know if it is just by looking at the directions of the angular momenta, right?
Mr.p: That is correct Billy.
Looking at the directions of the angular momenta of the point particle and wooden board
is inconclusive as far as proving conservation of angular momentum is concerned.
So, we are going to have to rely on the equations.
Bobby, could you please use equations to show when angular momentum is conserved?
I'm just gonna say exactly what I said for conservation of linear momentum,
only I'm going to replace the word linear with angular and the word force with torque.
Do you do still need me to say it?
That is correct Bobby, however, I would still like you to say it, please.
Bobby: Okay.
Angular momentum is conserved when the net external torque acting on the system equals zero.
That is because the net external torque acting on the system
equals change in angular momentum over change in time.
When that equals zero, the change of time cancels out,
because you can multiply both sides by the change in time.
Change in angular momentum equals final angular momentum minus initial angular momentum.
Therefore, the initial angular momentum of the system equals the final angular momentum of the system,
which is the equation for conservation of angular momentum.
Mr.p: Alright, we have already identified all the forces acting on the system during the collision,
now we just need to identify the net torque caused by all those forces.
Bo, could you please discuss
the torque caused by the two forces which comprise the Newton's Third Law Force Pair?
Bo: Newton's Third Law Force Pair?
Bobby: The force of the particle on the board
and the equal but opposite force the board causes on the particle.
We know they are equal but opposite because of Newton's Third Law.
That means that that makes them a Newton's Third Law Force Pair.
Bo: That's right. Thanks. So the two torques caused by
the forces of the particle on the board and the board on the particle have the same r vector.
The forces in those torque equations have the same magnitudes.
And then there are the angles. Well, the two angles are supplementary,
Billy: The two angles add up to 180 degrees. Bo: Right.
That means the two supplementary angles have the same value for sine.
Therefore, those two torques are equal in magnitude. And,
because the directions of the two forces are opposite, the directions of their torques are also opposite
and those two torques cancel one another out. Mr.p: Absolutely Bo.
And Billy, what about the torque caused by the force from the hinge on the board during the collision?
Billy: Okay, the torque caused by the force from the hinge on the board during the collision...
Oh, because the force from the hinge on the board acts directly on the axis of rotation,
the r vector for that torque equals zero,
which means the torque caused by the force from the hinge on the board during the collision also equals zero.
That means the net torque on the system during the collision equals zero
and the angular momentum of the point particle and wooden board system is conserved
about the axis of rotation of the wooden board.
Mr.p: That is absolutely correct.
We have just shown that, during a collision between a point particle and a rigid object with shape,
linear momentum is not conserved for the system,
and angular momentum is conserved about the axis of rotation of the rigid object.
Thank you very much for learning with me today, I enjoyed learning with you.