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

Seesaws

Professor Bloomfield illustrates the physics concepts of rotational versus translational motion, Newton's law of rotation, and 5 physical quantities: angular position, angular velocity, angular acceleration, torque, and rotational mass using seesaws.

- Louis A. BloomfieldProfessor of Physics

Why do the riders' distances from the pivot affect the seesaw's responsiveness?

The answer to that question is that the farther the riders masses are from the

pivot, the greater the seesaw's overall rotational mass and the slower its angular

accelerations. Two riders can balance the seesaw in a

variety of ways. To begin with, they can go to the ends of

the board and adjust their distances from the pivot carefully until it balances.

That is, until it experiences zero overall torque due to gravity.

I mean I'm pretty much there. Balanced seesaw.

But they can also come in close to the pivot and sit like this, with much smaller

lever arms to work with now. So that they're producing much small

torques as individuals. But once again, those two torques sum to

zero. And there's zero overall torque due to

gravity on this seesaw. So, there are a variety of ways to balance

the seesaw. And you might think that there's no

significant difference between those choices, but that's not true, there is a

significant difference. The farther these two riders sit from that

central pivot, and therefore the axis of rotation, the greater the seesaw's overall

rotational mass. Now, rotational mass is not something

completely independent of ordinary mass. They're related, just as forces and

torques are related. Every portion of the seesaw's ordinary

mass contributes to the seesaw's rotational mass.

And the amount of that contribution depends on where the portion of ordinary

mass is. More specifically, on how far that portion

of ordinary mass is from the axis of rotation.

And that dependence on axis of rotation is, is a strong one.

Small, modest changes in distance from the axis of rotation can lead to large changes

in the rotational mass contribution. The amount of rotational mass contributed

by a portion of ordinary mass. Is equal to the ordinary mass itself times

the square of the distance. Between that portion of ordinary mass and

the axis of rotation. So, taking a small portion here and

doubling its distance from the axis of rotation doesn't just double its

contribution to the rotational mass, it quadruples it.

That means that for the riders, when they sit in close, and their distance from that

central pivot and center of rotation, axis of rotation, is small, they contribute

very little to its rotational mass, the overall rotational mass of the seesaw and

riders. Even though these, these riders have large

masses, they're too close to the axis of rotation to contribute very strongly to

its rotational mass. But if they go out like this, to a large

distance, well then their contribution to rotational mass is huge.

They might be only ten times as far away from the pivot as before.

From that axis of rotation. But an increase of distance by a factor of

ten is an increase in contribution to rotational mass of ten times ten, or ten

squared, which is 100. So the rotational mass contribution of

these riders could easily be 100 times. That of these riders.

It's a big effect. To see how big, I've made these two rods

that look the same, and have the same the masses.

They contain the same materials actually. But the difference is that in one of these

rods, the mass is all in the middle, near my hand, under my hand, hidden from view.

And in the other bar, all of the mass is far from my hand, at the ends of the bar.

So, same mass, but in, in this case it's moved way out far from the center of

rotation, which will be here in the middle bar.

And the rotational mass of this bar is something like 30, 40 times that of this

one. Big difference.

To see that difference, since you can't hold the bars, let's go get some help.

Annie and Megan are going to help me here with these two bars.

Now, these bars have the same masses and the same weights.

So you could check that out. Just, just compare the weights, do they

feel the same? >> Yeah, definitely even, uh-hm.

>> Yeah, definitely. >> So, just by weighing them in your hands

you can't tell a difference between these two bars.

But that doesn't mean they're identical. So the difference is going to be subtle,

and this difference will show up maybe when you begin to try to rock them back

and forth. So I want, Annie grab it in the middle of

the bar, Megan same thing. And I'm going to count to three, and on

the number three. I want the two of you to rock it back and

forth as fast as you can. That is, make it undergo angular

acceleration first one way and then the other, back and forth as fast as you can.

>> Uh-hm. >> 'Kay.

>> One. Two.

Three. >> This is really difficult.

>> [laugh] So Annie's having no trouble here, and Meagan's really lagging behind.

Must be weak today, right? Forgot to eat your breakfast.

Okay, now swap bars. >> Okay.

>> Thank you. >> Now, I'll count again.

One, two, three. >> Oh.

[laugh]. >> Miraculous change.

>> What? >> Something's different about these two

bars. What do you think's different about the

bars? They have the same mass, what's different?

Megan? >> Maybe the distribution of the mass

within the bar? >> So the distribution of the mass in,

within the bar is different. Where is the mass in your bar, right now?

>> In the center. >> So your bar has almost all of the mass

in your hand. >> Yeah.

>> As a result, the moment of inertia, or the rotational mass of this bar is very

small. >> Yeah.

>> It's very easy to make this bar undergo angular salvation.

How about yours, Annie. Where do you think the mass is located?

>> I mean they must be at the end, right? >> Yeah, so all the mass in this bar are,

is at the ends where it contributes enormously to ro-, rotational mass.

So they have the same mass, it's just distributed differently.

In Annie's bar it's at the ends, in Megan's bar it's in the middle.

And they behave totally differently when you try to rock them back and forth.

>> Uh-hm. >> Show us a little more time.

>> Sure. [laugh].

>> I should make a face. >> Yeah, it's, it's a pretty dramatic

difference. Unfortunately you guys can't try it out.

>> [laugh]. >> But, but if you ever come by, test out

these bars and see how different they are. >> Okay.

>> Thanks. >> Sure.

>> So you see, if you place an object's ordinary mass far from its axis of

rotation, that object can have a surprisingly large rotational mass.

Now, the distance involved here is between ordinary mass and the rotational axis.

And you're often free to choose an object's rotational axis.

If you do that, and if you change your choice, you may well change the object's

rotational mass. That ability to change an object's

rotational mass is why rotational motion is so complicated to calculate

quantitatively. And that's a fact the keeps first year

physics graduate students rather busy. It's hard work.

I want to give you a, a taste of the issues without trying to overwhelm you

with them. But look at this rod.

This is the rod that's very hard to wobble back and forth like this because it has an

enormous rotational mass when you twist it back and forth about this axis.

The one pointing toward you through my hand or away from you through my hand.

About that axis, gigantic rotational mass. But what about this axis in which I'm

twisting it back and forth like, like a drill or a screwdriver?

It's easy. This, this direction has almost no

rotational mass. That's because all the portions of

ordinary mass are very close to this spindle-like axis about which I'm twisting

it. So, this, this rod here, has two very

different rotational masses, a huge one when you do this motion and a tiny one

when you do this motion. Now this is a you know, fun and games rod.

But something you're more familiar with is perhaps a tennis racket.

The tennis racket is a classic example of something that has three particularly

important rotational masses. It's smallest rotational mass is for this

rotation about it's There, the, the top bottom axis right now.

This motion, most of the mass is pretty close to that axis, the spindle about

which I'm twisting it, and therefore it has a relatively small rotational mass.

The next larger rotational mass is for this motion.

Sometimes referred to as the frying pan motion, when you're, you're flipping

pancakes. So this is the intermediate rotational

mass. And the biggest rotational mass is for

this rotation. In between these motions, life is

extremely complicated. And it's beyond the scope of this class as

something I don't like to deal with anymore.

I've done it. Been there, done that, I'll leave it.

But these three distinct rotational masses, this small one, bigger one,

biggest one, give the motion of a tennis racket or anything shaped like a tennis

racket the rotational motion's quite complicated.

To make things even worse, these are all motions about the center of mass.

These are all rotations in which the center of mass stays put.

What if you shift the rotation, so that you don't care about the center of mass of

the tennis racket. For example, when you're swinging a tennis

racket about your shoulder. In that case, you're shifting the mass of

the tennis racket even farther from the center of rotation, the axis about which

you're spinning it, and creating an even larger rotational mass for the tennis

racket. So the bottom line with all of this is

rotational mass depends on your choice of axis of rotation.

We're finally ready for the question I asked you to think about in the

introduction of this episode. To remind you, that question asked if you

and a child half your height lean out over a swimming pool at the same angle and let

go at the same moment, which of the two of you will hit the water first?

Despite a fair amount of rotational physics under our belts, that remains a

challenging question. So before I ask it, and leave you free to

answer it, I want to give you a little more background.

Get you all prepped for this question. First, what's the big picture issue?

What is going to determine who hits first? It's going to be angular acceleration.

The one of you that undergoes the fastest angular acceleration will tip, will, will

de-, develop the fastest angular velocity, will tip over the fastest and will hit the

water first. So look for big angular acceleration.

Second, what is the axis of rotation about which the two of you are going to be

rotating? It's not your centers of mass, it's going

to be here at your feet. That leaves two more issues.

One is the cause of angular acceleration and the other is the resistance to angular

acceleration. The cause is the neck torque on you.

The resistance is your rotational mass. And let's look at each one individually.

First Torque. The torque is due to gravity, and to make

our lives simple, let's compare the torques about this, about your feet,

that's the ax-, that's the axis rotation here, for, for you and the child.

Now, I've made life very simple by using exactly the same board material.

One is just half as long as the other, and this makes, you know.

This has all the physics in it, but no, none of the details.

Life is easier. So, you have twice the weight of the

child, that's no suprise. And that weight effectively acts at your

center of gravity, which is twice as far, it's right here in the middle.

It twice as far from the axis of rotation as for the child.

So, you're experiencing four times the gravitational torque of the ch-, of the

child. You have twice the weight acting at twice

the lever arm. 2 times 2 is 4.

Right? 4 times the torque.

That's the cause of angular acceleration. How about the resistance to angular

acceleration, the rotational mass? Well, as compared to the child who has

half the mass here, distributed around here with the center of mass being about

there. You have twice the mass distributed here

and there, there's a center of mass here. The center of mass has moved out by a

factor of two. You have twice as much mass, that is on

average at twice the distance from the axis of rotation, namely your feet.

Well, remember that the distance involved here In calculating, we-, it, the

contribution of mass depends, not on distance, but on distance squared.

So the rotational mass that you have is eight times that of the child.

Twice as much mass, at twice the distance and you, you square the distance.

So it's two times two times two, that's eight.

This has, this, you have eight times the rotational mass of the child.

With that as background now, answer the question.

Which of the two of you tips over fastest and hits the water first?

The child undergoes greater angular acceleration, develops a bigger angular

velocity and hits the water first. If you haven't already tried this

experimentally, give it a go. All you need is two sticks, one twice as

long as the other. I can show you what you'll see when you

try it. Here's you, here's the child, and we'll

put you both on your feet and tip you to the same angle, and then let go.

3, 2, 1. No question, the child reached the water first.

It's a battle between torque and rotational mass.

In both cases about your feet. You have four times the gravitational

torque acting on you as, as the child has. So there's four times as much twist trying

to propel angular acceleration but you have eight times the rotational mass as

the child, resisting that angular acceleration.

Four times more impetus. Eight times more resistance, you get only

half the angular acceleration of the child.

So, the child undergoes twice you angular acceleration and just goes through the

whole rotational motion faster. And wins the race to the water.

Rotational motion clearly has some subtle complications, like a single object having

more than one rotational mass depending on your choice of axis of rotation.

But let's leave all those complications for the experts.

I chose seesaws as the topic for this episode because seesaws are comparatively

simple. That pivot fixes the axis' rotation so

that the seesaw can only rotate in one fashion.

And in general, it only has one rotational mass.

Things are simple. Nonetheless, the seesaw exhibits most of

the issues of rotational motion. Or at least the ones that I want to talk

about and try to convey to you. I've already done that now.

I've, I've shown you most of what happens in a rotating system like a seesaw with

one important exception. Energy.

As the seesaw rotates, the riders are exchanging energy.

And that, is the topic for the next video.