Mr.p: Good morning. Let's learn about conservation of angular momentum.
But before we do, let's review the equation for conservation of linear momentum.
Bo, what is that equation?
"Flipping Physics" Intro
Bo: The sum of the initial linear momentum of a system
equals the sum of the final linear momentum of a system.
Billy: And remember... Bo: And remember, linear momentum is a vector,
Mr.p: Thanks Bo. Bobby, what then do you think is the equation for conservation of angular momentum?
Bobby: The sum of the initial angular momentum of a system
equals the sum of the final angular momentum of a system.
Billy: But, remember... Bobby: And remember angular momentum is a vector.
Mr.p: Great. In a bit we will talk about when angular momentum is conserved,
however, right now let's start with a simple demonstration. Alright,
This is me sitting on a stool which is free to rotate about a vertical axis through the center of the stool
and I am holding two masses at arm's length straight out from my body.
If I give myself an external torque to cause the stool and me to rotate,
I am now spinning with roughly a constant angular velocity.
Class, what will happen if I bring my arms in closer to my body?
Billy: I feel like I've seen something like this before.
Bo: Yeah. It's just like when a figure skater brings their arms in while spinning.
Bobby: Or a gymnast. Billy: Or a diver.
Bo: Or a dancer. Mr.p: Sure, but what happens?
Billy: Bringing your arms in will increase your angular velocity, right?
Mr.p: That is correct,
when I bring my arms in closer to my body, the angular velocity of the stool and me increases.
But the question is why?
Why does bringing my arms and the two masses in closer to my body
increase the angular velocity of the system?
Bo: Can we assume the angular momentum of the system is conserved?
Mr.p: Yes, we may assume angular momentum is conserved in this example.
Bo: Well then, the initial angular momentum of the system equals the final angular moment of the system.
Billy: And angular momentum of a rigid object with shape equals rotational inertia times angular velocity.
Bobby: So, rotational inertia initial times angular velocity initial
equals rotational inertia final times angular velocity final.
Billy: Oh, yeah so rotational inertia of a system of particles equals
the sum of the mass of each particle times the square of the distance each particle is from the axis of rotation,
or r squared.
Bo: Bringing your arms decreases the average distance
everything is from the axis of rotation,
which decreases the rotational inertia.
Bobby: And if the rotational inertia is decreased,
the angular velocity must increase to keep the angular momentum of the system constant.
Billy: You know, I just want to specifically point out
that the axis of rotation is the vertical axis which passes through the center of the stool.
We always have to define the axis of rotation when talking about rotational inertia and angular momentum.
Mr.p: Very nice. And notice the reverse is also true.
Moving the masses from close to the axis of rotation to farther from the axis of rotation
increases the average distance each object is from the axis of rotation,
increasing the rotational inertia of the system
and the angular velocity of the system must decrease
to maintain a constant angular momentum of the system.
As was pointed out earlier, conservation of angular momentum occurs in many sports,
for example, this is me diving in high school a couple of years ago. I am doing an inward 1 and a half tuck.
Bo: Wait a second, that's you in high school?
Mr.p: Yes. That is me during my 10th grade year of high school.
Billy: Oh wow, look at that.
Mr.p: Let's get back to physics.
As I leave the board and go into the air, I pull my legs and arms in to form the "tuck" position.
This decreases the rotational inertia of my body considerably
and therefore increases my body's angular velocity.
And while I'm in the air, angular momentum is conserved.
When I am almost done with 1 and a half rotations,
I straighten my body to increase my rotational inertia and decrease my angular velocity.
This is similar to moving my arms on the stool like we just demonstrated.
Now let's talk about when angular momentum is conserved.
But first, remind me Billy, when is linear momentum conserved?
Billy: Uh...
Mr.p: Okay, so remember you start with Newton's second law.
Billy: Net force equals mass times acceleration.
Bo: No, actually it starts with the version of Newton's second law that has linear momentum in it.
Billy: Okay. So the net external force on the system equals
change in linear momentum over change in time.
And linear momentum is conserved when the net force acting on the system equals zero.
So, multiply both sides by change in time.
Change on time cancels out and we are left with zero equals the change in linear momentum of the system.
Change in linear momentum equals the final linear momentum minus the initial linear momentum
Add initial linear momentum of the system to both sides
and we get the initial linear momentum of the system equals the final linear momentum of the system.
Bobby: Then conservation of angular momentum should use the rotational form of Newton's second law;
the net external torque acting on the system equals change in angular momentum over change in time
and angular momentum should be conserved when the net external torque acting on the system equals zero.
And we need to identify the axis of rotation. We got to make sure we do that
Then we can multiply both sides by change in time.
Change in time cancels out and we are left with zero equals the change in angular momentum of the system.
Chang in angular momentum equals final angular momentum minus initial angular momentum.
Add angular momentum of the system to both sides
and we get the initial angular momentum of the system equals the final angular momentum of the system.
Billy: All you did was pretty much say what I said only with "angular" instead of "linear".
Bobby: Yup. That's what I did.
Mr.p: Bobby you are correct
that angular momentum is conserved when the net external torque acting on the system equals zero.
To demonstrate this let's return back to me on the stool.
While I am just sitting there, class, what is the angular momentum of the mr.p and stool system?
BBB: Zero. Billy: because you have zero angular velocity.
And if I do not apply a force on anything,
do you see how the net external torque acting on the stool and me will be zero?
Billy: Yes. Bobby: Uh huh.
Bo: Right, so the angular momentum of you and the stool will stay equal to zero.
Mr.p: Exactly Bo. Which is why no matter how much I wave my arms and legs about,
I cannot change the angular momentum of the system
and the angular velocity of the system remains equal to zero.
However, if I simply push off of something,
the object, which is external to the system, applies a net external torque to the system
and the stool and I begin to rotate because angular momentum is no longer conserved.
Booby: Okay. Billy: That's cool. Bo: Right.
Mr.p: Okay, I want to talk for a bit about the fact that angular momentum is a vector.
Which means class, angular momentum has both BBB: Magnitude and direction.
Mr.p: Therefore, when we look at this top which is rotating clockwise when viewed from above,
Bo, what is the direction of the angular momentum of the spinning top?
Bo: We still use the right hand rule to find the direction of the angular momentum, right?
Mr.p: Yes, absolutely we use the right hand rule.
Bo: Okay. So I take my right hand, curl my fingers in the direction the top is spinning and stick out my thumb.
My thumb points down, so the angular momentum of the spinning top is down.
Mr.p: That is correct.
And just like an object will tend to maintain its linear momentum,
an object will also tend to maintain its angular momentum.
The direction of the spinning top is down and, as long as the top is spinning,
the angular momentum will tend to remain in that downward direction.
But notice when a top is not spinning,
it will fall over because zero angular momentum does not have a direction.
This is actually the same thing which makes it easier to maintain your balance when a bicycle is moving
than when it is not moving.
If you are moving forward on a bicycle,
the angular momentum of the wheels, according to the right hand rule, is to the left of the bicycle rider.
The angular momentum tends to maintain that direction, and therefore helps hold the bicycle upright.
But when you are not moving,
there is no angular momentum and it is much more difficult to keep your balance.
Billy: That makes sense. Bo: Sure. Bobby: Cool.
Mr.p: Well, there you go. Thank you very much for learning with me today. I enjoyed learning with you.
Billy: Mr.p, can we please see more of you in high school?
Bo: Yeah. That was pretty funny. Bobby: Yes, please?
Mr.p: Sure, here's a video from right after that diving meet my sophomore year.