An introduction to physics in the context of everyday objects.

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來自 University of Virginia 的課程

生活中的物理

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An introduction to physics in the context of everyday objects.

從本節課中

Bumper Cars

Professor Bloomfield examines the physics concepts of momentum, impulse, angular momentum, angular impulse, and the relationship between potential energy and force using bumper cars.

- Louis A. BloomfieldProfessor of Physics

Can a spinning bumper car carry a torque? The answer to that question is no.

The spinning bumper car carries angular momentum, but it can't carry a torque.

As you've probably guessed, the story I'm about to tell you is a rotational

equivalent of the translational story I told in this episode's first video.

Like a force, a torque is exerted by 1 object on another object.

So a single object can't carry a torque. But there is a physical quality of

rotational motion that a spinning bumper car does carry.

A capacity to make other things turn in the direction the bumper car is turning.

That physical quantity of rotational motion is called angular momentum.

And angular momentum is the conserved quantity of turning.

As required of any conserved quantity angular momentum can't be created or

destroyed. It can only be transferred between

objects. So, one bumper car can transfer angular

momentum to a second bumper car. Fortunately, for us angular momentum has

many similarities to ordinary or what is called linear momentum.

Still, those two are separate quantities and they're conserved separately.

And a bumper car carries both of the simultaneously, so I can send this guy

along with momentum, angular momentum, and both.

[LAUGH] Out of control, there we go. Together with energy, a bumper car can

sit, can carry Three conserved quantities altogether.

And those three conserved quantities determine much of how the bumper car

behaves. Like linear momentum, angular momentum

involves motion. There is no such thing as potential

angular momentum. A bumper car that's turning around some

center. Has angular momentum, and one that's not

turning about that center has zero angular momentum.

Angular momentum is also a vector quantity.

It's about rotation, and rotation is a three-dimensional activity that requires

direction. I can rotate like this.

I can rotate like this. Not very far.

But you get the idea that rotation is complicated in 3 dimensions.

And so you have to specify it with direction.

And just as anger velocity has a direction, so anger momentum has a

direction. And you use the same rules.

Now, I can show you the direction of angular momentum in a single bumper car

by choosing as the center of rotation the bumper car center of mass that's sort of

the natural center of rotation for the bumper car and I'm going to give it first

angular momentum upward like this here it is.

It has angular momentum upward. And now, I'm going to get angular

momentum downward, like that. And it's a full, it's got all the variety

of a vector quantity, upward, downward. I could, in principle, I can, I can do, I

can do toward me, there it is toward you. Not very good for a bumper car.

And finally. It has an amount.

This is a little bit of angular momentum upward, and this is a lot of angular

momentum upward, always about that center of rotation which is a center of mass of

my little lone bumper car. Well, to show you more about angular

momentum And in particular, that it's a conserved quantity.

I'm going to return to the classroom, and go for a spin myself.

So here I am in a classroom. And I'm going to sit on this swivel

chair, which is the rotational equipment of the wheelie cart that I used for

linear momentum. This swivel chair spins so freely, or so

nearly freely, that I can't exchange any angular momentum about my center of mass

with the earth and it allows me to kind of coast in terms of angular momentum.

The angular momentum that's in me won't leak out very quickly into the ground.

There are issues of air resistance too, but basically I can, I can show you that,

that my angle management's conserved. Now I'm having trouble trying to, to look

at the camera because thing swivels so easily, and I can't get started turning,

without something investing angle management in me because I can't create

any momentum. Whatever I've got, I've got.

Now I've got some slight rotations going on here that have to do with the

imperfections of this system, but basically, my angular momentum is 0 right

now. And to get some angular momentum, I need

to obtain some from the ground. So I'm going to put my foot down And I'm

going to have the ground twist me. It's going to exert torque on me, and

I'll get some angular momentum. Here we go, ready?

I'm going to get some angular momentum. I got it.

And now that I got it, I can't get rid of it.

I gotta keep going. this is a little less pleasant than

coasting across a room. I'm going to get awfully dizzy So I have

to get rid of my angular momentum and the way I gotta do that is by giving it to

something else. So I'm going to give it to the ground

with my foot again. Ready, there we go, gave it to the ground

and I wish you guys would stop moving. Alright so that was angular momentum

upward. Now, I can get some manual momentum

downward by twisting the ground the other way having it twist me back and woof ,

I've got angular momentum it's now downward.

I've got downward angular momentum. And I can't stop, I can't stop, I can't

stop until I give it away. There we go.

And I mean its got a direction, and the same kind of ideas apply as with linear

momentum: that if I want to reverse directions, I need a huge transfer of

angular momentum, because not only do I have to come to a stop and get rid of my

initial angular momentum I have to reverse, go the other direction around,

which requires still more exchange of momentum with the ground.

To show you that more concretely, let me get some angular momentum upward.

Here we go, okay, I've got it now. A little less, so I'm not quite so dizzy.

And now in order to reverse my direction, I'm going to have to, to give all my

upward angular momentum to the ground. Here we go.

And, I have to give more than I had in reverse directions.

So I gave the ground more upward momentum than I actually had and I ended up with a

deficit of upward momentum which is to say, downward momentum.

And here I am, rotating downward. So angular momentum is a conserved

quantity. Once you're isolated and you can't

exchange it with anything, whatever you've got, you've got.

It's trapped in you. And to exchange it, well, there's a

mechanism for exchanging it, and that will be the subject of the next video.

My angular momentum depends on two things, my rotational mass and my angular

velocity. And it turns out to be proportional to

each of those quantities. That is, the more rotational mass I have,

the more angular momentum I carry, or the more angular velocity I have, the more

angular momentum I carry. Actually, angular momentum is equal to

Rotational mass times angular velocity. So if I turn, if I, if I bring myself in

close to the center of rotation and shrink my rotational mass, then I'm have

a small rotational mass. And if I rotate very slowly with a small

angular velocity, then the product of a small rotational mass times a small

angular velocity Isn't quite as small angular momentum.

On the other hand, if I increase my rotational mass by spreading myself out

far away from the center of rotation, now even at the same angular velocity as

before I'm carrying more angular momentum or if I go back to my original rotational

mass and spin faster. Again, I'm carrying more angular

momentum. Where things get interesting is, if I

have a ceratin angular momentum invested in me by the ground, and I change my

shape, I can change my rotational mass. I'm not a rigid object.

If I change my rotational mass But I don't exchange angular momentum with

anything around me, my angular momentum has to say the same.

So if my rotational mass changes, then my angular velocity has to change to

compensate in order to keep my angular momentum constant.

This is actually a fun demonstration, one you've seen before.

I'm going to do it in a physicist version, but you've seen it as the ice

skater trick if you've watched ice skaters spinning.

If I start spinning with my arms out and these massive dumbbells in them I have a

huge rotational mass, and even when I'm turning relatively slowly, small angular

velocity, I have a lot of angular momentum.

If I then, shrink my rotational mass, by pulling the dumb bells in, I still have a

lot of angular momentum, my angular velocity has to increase to compensate.

Because the product of my now small rotational mass times what will have to

be a large angular velocity has to come out equal to my angular momentum which is

large. So here we go.

I'm going to, I have a large rotational mass and so even though I'm turning

slowly I've got lots of angular momentum in me.

It's upward and now as I pull these in, these masses in actually, I spin faster.

I have to because my Angular momentum is the thing that's constant.

Everything else can change. But not my angular momentum.

So skaters do this trick that they start spinning with their arms out and then

they pull in tight and they spin faster as their angular momentum stays constant,

but their rotational mass shrinks and their angular velocity increases.

Well, with that then, I'm going to go back to the lab.

because I'm getting pretty dizzy. It's time to stop.

Wait a second. This isn't my laboratory.

What am I doing here? Is this some sort of big picture moment

or something? Actually it is.

I'm going to revisit Newton's first law of translation motion and rotation motion

with some new insights gathered from our experiences with momentum and angular

momentum. Think about first, Newton's first law of

translational motion. It says that an object that's free of

external forces moves at constant velocity, and I'm going to make myself an

object that's free of external forces so I move at constant velocity.

What underlies that law, Newton's first law of translation motion, is actually

conservation momentum. Once you're isolated, free of external

forces, your momentum is constant. And because your mass can't change your

velocity can't, change either. After all, your fixed momentum.

The momentum of an isolated object, is equal to your mass, which can't change.

Times your velocity, which therefore, can't change.

So once I start moving along, I have a fixed velocity because my momentum is

constant. And because my mass is constant.

So that's the real origin of Newton's first law of translational motion,

conservation of momentum. What about Newton's first law of

rotational motion? things are a little different.

Newton's first law of rotational motion depends, has underlying it Conservation

of angular momentum. So, once you're free of external torques

and I'll add the word rigid and I'll come back to it.

And you're rigid, once you're free of external torques, your angular momentum

can't change. So, be if, if you're rigid your

rotational mass can't change and therefore.

Your angular, your angular velocity can't change.

So the angular velocity's constant because your angular momentum is constant

and your rotational mass is constant. So, your angular velocity has to come

along for the ride. It's got no choice.

But, what if you're not rigid. Remember that word rigid in Newton's

first law of rotational motion. Why was the word rigid in there?

It's because if you're not rigid, you can change your rotational mass.

You can't change your ordinary mass. That's really, really, really fixed, but

you can change your rotational mass by changing your shape.

And if you do that You, you, first off you, Newton's first law of rotational

motion no longer applies to you. And second off, you do experience changes

in rot, in angular velocity, for a good reason.

So I'll isolate myself again into some separate from the whole world and now my

angular momentum is, is, is constant because it's conserved and I can't

exchange it with anything. And my rotational mass is constant

because I'm playing at being rigid. So therefore my angular velocity is

constant. It's coming along for the ride.

But if I change my shape all bets are off.

My angular momentum still is constant. But because I changed my rotational mass,

I increased it My angular velocity decreased.

That is a remarkable thing that you can do in the world of rotation because you

can change your rotational mass. In the world of translation, you can't

change your mass or you can't change your velocity all by yourself.

Again, the world of rotation Because you can change your rotation mass, you can

actually change your angular velocity all by yourself.

Just change your shape. And with that, I'm dizzy again and it

really is time to go back to the laboratory.

It's nice to be back. And before I forget.

It's time to pose the question that I asked you to think about in the

introduction to this episode. That question was this.

It says, first off that. Suppose your on a playground marry go

round. Or one of these spinning platforms.

And your on the outside And if you pull yourself to the center of that spinning

playground merry-go-round, what will happen to its rotation?

As you climb to the center of this rotating disk, it's angular momentum

won't change because that's, that's fixed.

It's an isolated object. Spinning about it's center of mass, but

you will be shrinking it's rotational mass.

And consequently, it's angular velocity has to increase to compensate in order to

keep the total angular momentum of this spinning object constant.

By now, it should be pretty clear. That once a bumper car is spinning it has

angular momentum. And it can't stop spinning until it

transfers that angular momentum to something else.

The riders in the car do matter because they contribute to the bumper car's

rotational mass. And the more mass of those riders are,

and the farther they are from the center of mass of the car that is, the center of

rotation we have in mind for our angular momentum, the bigger the rotational mass

of the car and therefore the more angular momentum it carries for a given angle of

velocity. So, a car filled with very massive people

sitting far from the center of mass the car is spread out widely within the car

can allow that car to carry awful lot of angular momentum.

On the other hand a single child seated right in the middle of the car doesn't

contribute much of the rotational mass. So that car then, can turn fairly fast.

And still not be carrying very much angular momentum.

If people move around, they can change the rotational mass of the car.

And if it's already spinning, they'll affect the rotation, the angular velocity

of the car. According to this, this.

All this stuff I've been talking about. Where the angular momentum stays the

same, the same. But As the rotational mass for example

decreases the angular velocity has to increase to compensate.

Well most people don't set their own car's spinning deliberately.

That is they don't obtain angular momentum out of the ground by driving

funny. Most of the time they wait for collisions

to occur and it's those collisions That transfer angular momentum and start the

cars spinning, at least when the cars are spinning wildly.

So, we need to look at those transfers of angular momentum.

And that is the job for the next video.