How is angular momentum transferred from one bumper car to another?
The answer to that question, is that the first bumper car does an angular impulse
on the second bumper car. Just as you can transfer energy by doing
work and linear momentum by doing a impulse, you can transfer angular
momentum by doing a angular impulse. So, what exactly is an angular impulse?
An angular impulse is a torque exhorted for time, for example, I can do an
angular impulse on this bumper car by twisting it, that's the torque, as time
passes, here we go. Okay, I did an angular impulse on that
bumper car and I transfer a momentum to it.
So, that product, my torque times the time over which it acted i the angular
impulse, and therefore, the amount of angular momentum that I transfer to the
bumper car. And angular impulse transfers a vector
quantity, angular momentum, so that angular impulse itself is a vector
quantity. It points in the direction of the torque
that's responsible for it. So, for example, if I twist the bumper
car, downward, I transfer downward angular momentum to the, to the bumper
car. So I did downward angular impulse.
On the other hand, if I twist it upward, that's upward torque produced an upward
angular impulse that produce that resulted in an upward transfer of angular
momentum to the bumper car. And also like impulses, I have some
flexibility in the angular impulse I'm doing.
For example, can do the same angular impulse with different combinations of
Torque and time or, or rather duration. For example, here is an angular impulse
that involves a medium torque exerted for a medium amount of time.
Here it goes,[NOISE]. Okay.
I transferred a certain amount of angular momentum, in this case downward, into the
bumper car. And I did it with a medium torque exerted
for a medium time. I could also use a much larger torque,
exerted for a very short time. And that was the same angular impulse,
but with a bigger torque for a shorter time.
And finally, I can do the same angular impulse with a smaller torque exerted for
a longer time. [NOISE] It's a little hard, 'kay?
But you get the idea. And, in bumper cars, many of the twisting
jolts that you experience you feel during the, during the motion around the arena,
occurred during these collisions that transfer a lot of angular momentum in a
very short period of time. And so they are high torque, short
duration, and your impulses that really,[LAUGH] spin you around, hard.
Also, like anger, like ordinary impulses, like the impulses that transfer momentum,
anger impulses come in equal but opposite pairs.
When I twist this bumper car downward, that's a downward twist.
I did, I, downward transfer of angular momentum by way of a downward angular
impulse. It does an angular impulse back on me in
the opposite direction. So, I transferred downward angular
momentum to it. It, on the other hand, twisted me
backwards, because it has to. That's, that's Newton's third law of
rotational motion, motion. It twisted me upward and did an angular
impulse upward on me and gave me upward angular momentum, which is the same as a
deficit. A negative amount of downward angular
momentum, it took away the downward angular momentum that I gave to it and
that makes the transfer complete. I gave the bumper car downward angular
momentum. At the same time, it took away downward
angular momentum from me. So that what I lost, it gained.
Angular momentum is conserved. And this has to happen.
Whatever I give it, I have to give up. All right.
Well, so far, I'm just doing all of this with my hands and a bumper car.
But in the bumper car, most of the angular impulses, and certainly the ones
that are most exciting involve collisions.
And the angular impulses in those contexts, in the collision context, are
kind of subtle and they often involve frictional forces.
So let me show you how an angular impulse occurs in bumper cars.
as I say, they often involve friction, and so, having these rubber bumpers,
which have a lot of friction, they, they really grip each other pretty well, is
important. Now, angular momentum, like all the
rotational quantities is defined around, about a center of rotation, so we have to
pick it and stick with it. The center of rotation I'm going to pick
is the center of mass of this car. This is the, this is our center of
attention, too. We're going to pay attention to this car.
This guy is just an interlope where it's going to come through and spin the first
car. At the start of the story, the angular
momentum will be present already. It will be in the form of this cars
moving around, it will effect orbiting. It's going around the center of rotation,
for our story. It won't actually be travelling in a
circle. It'll be travelling in a straight line
as, as objects do when they're free of external forces.
But the fact that it's moving like this, means that it is swinging around that
center rotation. It's actually, there's rotation in here
already, and therefore, angular momentum here already.
So this guy's going to come along and it's going to clip the edge of that
bumper car. It's going to catch it and twist it.
And the, the act of catching that edge and twisting that bumper car there.
A force, a frictional force, exerted at a lever arm from the center of rotation,
produces a torque, and will twist our main focus bumper car, and do the anger
impulse. So, as this guy comes along, heading
straight and true and it grabs the edge of the bumper car, we care about it's
going to exert the torque for the time and do the anger impulse on that bumper
car. So, let me see if I can pull this off.
The, what you should be looking for is this thing traveling straight, that one
not spinning, the collision. And after that, this guy should be
spinning, because the angular momentum will be transferred from this one to the,
to our main focus. Ready, get set.
There it is. Of course, I've got so much junk in the
way. Let me get this out of the way.
Come back guys. It's spinning.
Spinning. So, those kinds of collisions in bumper
cars break, I can try to, I'm going to hold this guy and really clip it hard.
[NOISE] It was spinning fast before[LAUGH] it flew off onto the floor.
But, those kinds of collisions occur in real bumper cars, and all of sudden, you
get clipped, our bumper car gets caught on the edge by a fast moving nearby
bumper car, and suddenly, whoa. You're spinning around.
That's where that comes from. It's time for a question.
This is a gyroscope, which consists of a wheel on an axle.
It's mounted in a frame that allows that wheel and axle to turn very freely,
almost without any frictional torques at all.
This particular gyroscope resides in a frame that's, that isolates it from the,
from the external world, so that it's very hard to exert any torques on this
wheel, this gyroscope wheel, about the gyroscope wheel's center of mass which is
the center rotation of this gyroscope. Now, there's a string wrapped around the
axle of this gyroscope. And the question is this.
If I pull that string while holding the frame so that axle can't move, except to
just rotate, all it can do is rotate. If I pull the string while the axle is is
held in in orientation, what will happen to the gyroscope?
When I pull that string, I will produce a torque on the gyroscope about it's center
of mass, center of rotation. And that torque will last for awhile,
good for actually a second. As a result, I will do a large angular
impulse on the gyroscope and transfer a lot of angular momentum to it.
And once I do, it'll be spinning fast. I'll show you.
Here we go. [NOISE].
It's now spinning quite fast and it has a lot of angular momentum.
And the interesting thing to show is that, as I try to move, the surrounding,
the, the frame that holds this, this gyroscope in place, it tends to keep
turning about the same axis. As long as I don't exert a torque on it,
I transfer no angular momentum about its own center of rotation and it keeps
turning as it was. Now, it's not perfect.
It's pretty close, but the frame really tries to prevent any angular impulses on
that spinning gyroscope. Nice.
[NOISE]. Now, in my bumper car arena which I'll
bring out in a moment, I have bumper cars that don't have rubber bumpers, they're
all plastic, they don't have very much friction between them.
And so, one, when one clips the other one, it doesn't transfer very much
angular momentum. So the one it hit, I can help that by
making my, my bumper cars not perfectly round, not circular like this, and many
bumper cars aren't quite circular. And when they're not circular, it isn't
just frictional forces that contribute to those torque.
It's also support forces, so I'm going to bring out my little bumper car arena, and
I'm going to put on some bumper cars that I've made not round.
So here, we have my miniature bumper car arena again and I've modified two of the
bumper cars by putting wires on them that stick out.
And, those wires will allow a passing bumper car to exert a large torque on
these modified bumper cars. So that makes them more similar to bumper
cars that have either rubber bumpers that, that give you a lot of grip or to
bumper cars that aren't circular, that, that extend outward in certain areas.
So that a passing bumper car can exert support forces on that bumper car and
produce torque in that manner. So first, to show you that that a passing
bumper car, let me pick a different color.
The green one passing the red one will catch its wire, exert the torque on the
red bumper car, about the red bumper car's center of mass, which will be our
center for this story. And it will set the red bumper car
spinning by doing an angular impulse on the red bumper car.
So I'll start with the red bumper car as motionless as I can get it.
Here it is, motionless. We're going to spinning.
There it goes, and I can do the same with the little red bumper car.
I only have little red bumper cars and I can set it spinning.
Where things get a little more interesting though, are when I have
bumper cars of different masses bumping into each other.
For example, this bumper car, the, this big red one, has a relatively large
rotational mass, and so, you have to pour a lot of angular momentum into that in
order to make its angular velocity significant.
So let me, let me smack a small, one of the little red guys into that big red
behemoth. Not much action.
The little red one carried the angular momentum at the start.
It has a small mass. And therefore, had a relatively small
rotational mass about the center of the big red one.
So, it didn't have much angular momentum to transfer.
It transferred a good fraction of what it had, but that wasn't enough to set the
big red one with its large rotational mass, spinning very fast.
On the other hand, if we go, use the little red one as a target, and, and take
the, well, now, I can use the big green one.
A big green one as the passing monster. It sets the little red one spinning
furiously, because, the big green one was carrying a lot of rotation, a lot of
angular momentum. It has a big mass, and it, and as it
moved around the center of rotation, which was the center of mass from our
little one, the green one was carrying a lot of angular momentum.
The little red one received a good fraction of that angular momentum from
the green one and that was enough to make the little red one with its tiny
rotational mass spin like crazy. So, you've seen these effects in bumper
cars if you've gone there or experienced them.
And that is, the big massive cars passing you and clipping the edge of your car can
really cause some serous anger response from your car.
Your car receives a big dose of anger by way of a big anger impulse, and suddenly,
you're whipped around in a hurry. On the other hand, passing cars that are
occupied by, say small children don't carry very much angular momentum about
your center of mass. And so, when they give you even a good
fraction of their angular momentum, doesn't have much effect on you and your
car. Your car receives a modest angular
momentum dose by way of a modest angular momentum anger impulse, and the result
is, is underwhelming. So, once again, the big massive cars
affect the little cars a lot more than the other way around.