An introduction to physics in the context of everyday objects.

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来自 弗吉尼亚大学 的课程

生活中的物理

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An introduction to physics in the context of everyday objects.

从本节课中

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' weights and positions affect the seesaw's motion?

The short answer to that question is that they affect the net torque on the seesaw,

and therefore the seesaw's angular acceleration.

In most cases, the riders of a seesaw position themselves so the net torque on

the seesaw is 0 or very nearly 0. As a result, the angular acceleration of

the seesaw is either 0. That is, its coasting rotationally.

Or it's just got the smallest amount of angular acceleration.

Well, that then requires a longer explanation.

How does that come about? You put riders on the seesaw; why don't

they produce enormous net torques? So we know that if we put one rider on the

seesaw, that rider's, because of the rider's weight, the rider pushes down on

the seesaw over here, on the, on your left...

[unknown] That's to the lever arm from the pivot.

It produces a torque and boom, the seesaw undergoes rapid angular acceleration such

that, that rider drops to the ground. But what if we put two riders on the, on

the seesaw simultaneously? And what I'm going to do is I'm going to

position them very carefully. And look.

The seesaw is experiencing very little angular acceleration, so the net torque on

it is either zero or very near zero. How did that happen?

Aren't these riders producing big torques on the seesaw?

There are two of them. >> Glad you asked that question.

Here's the story, this is now the longer explanation to the question that's

prompted this video. That rider because the rider's weight is

pushing down on the board, over here to your left, the lever arm that, that riders

force is using to produce a torque, points towards your left.

Here it is, and using the right hand rule now, we can see the direction of the

torque produced by that rider. The torque, we follow the lever arm and we

roll, I roll my finger down in the direction of the force and my thumb is

pointing toward you. That is the direction of a torque,

produced by this rep, a, this seesaw rider.

Let's come over to this seesaw rider. I need my right hand again.

I can't swap hands or I'll get the wrong answer.

So, that rider by virtue of his or her weight, is pushing down on the seesaw

board. The lever arm with which that rider is

producing a torque now, points to your right.

So, there it goes. And now I turn my, my index finger in the

direction of the force. And lo and behold, the torque produced by

that rider is away from you. So these two torques are in opposite

directions. This rider is producing a torque toward

you. This rider is producing a torque away from

you. When we add those two torques, and they

are the two torques acting on this seesaw, they sum to zero or very nearly zero.

And that's how it is that when I let go of this board and allow it to show you its

angular acceleration There's almost zero. If there is a little bit, and there is, I

can adjust the distance of one of the riders from the pivot.

This riders producing a little too much torque.

And now I, I move it toward the pivot, still a little too much torque.

So I move it a little closer to the pivot, and now That rider's producing almost jjst

the right torque. Let me move the rider in a little closer

and now this rider's producing too little torque.

I have been, I have adjusted the rider's positions, that is the lever arms they're

using, to show you that we can go all the way from.

Almost perfect balance, and I'll talk about balance in a minute, with that rider

dominating a little bit, to almost perfect balance with that rider dominating a

little bit. And everything in between, including in

principle, perfect balance where there's zero net torque on the seesaw.

>> Actually, balance is an interesting concept.

The balance that we talk about in the context of a seesaw, and many other

objects that teeter back and forth like a seesaw, is a situation where gravity

produces no torque on the object. So, when this seesaw is balanced It's

experiencing 0 torque due to gravity. I can come in and, and change things.

I'm, I'm here, very carefully adjusting positions in order to try get this

situation. This seesaw is almost perfectly balanced.

Meaning it's experiencing almost 0 torque due to gravity.

And that is the normal situation for a seesaw, and riders.

They like that situation because a balanced seesaw is free of torque, this

assumes nothing else is exerting torques on it, and it will turn at constant

angular velocity. It is an object that obeys Newton's First

Law of Rotational Motion. And it's not wobbling, it's rigid,

assuming the riders don't change their positions.

And therefore, in the absence of any torque, and there's no gravitational

torque on a balanced seesaw, it turns a constant negative velocity.

So, the, the reason the riders have to adjust their positions very carefully And

it, and their weights are important as well, is because they are trying to sum

their torques to zero, and how they place themselves mat, matters.

If, for example, the riders are, have, essentially identical weights, and these

two riders do They need to sit at equal distances from the pivot because the

torque they produce, after all, is the product of the force they exert on the

seesaw times the lever arm they have to work with.

There's some subtleties in here with, with regard to the angles involved between the

lever arm and the force but In this situation we can really ignore those.

The forces and lever arms are essentially at right angles to each other, and our

lives are simple. So these two identical riders, seated at

identical distances from the pivot, produce identical but oppositely directed

torques, and the seesaw balances. What if we have a heavier rider around,

though? So instead of this rider, we bring up one.

And this is made of steel. This is heavy stuff.

So I'm going to put this rider in. If I put this rider out at the same

distance as the rider on your right It completely dominates, and I run the risk

of tossing this rider. This is one of the flaws with seesaws, is

it's easy for one of the riders to become an astronaut, when a very heavy rider gets

on the seesaw[NOISE] and does that to it. But this rider cannot sit that far out

from the pivot. Too much lever arm for a large force, and

therefore this rider dominates it, and produces a torque that, that one cannot

compensate With. Comp-, compensate for.

So I have to bring the heavier rider in close.

How close? Pretty close.

I'm almost at balance. There we are, this is balance.

Alright? It's as close as I'm going to get.

And, again, the net torque on the seesaw is zero, or pretty close to zero.

>> And, you'll notice that, that now the lever arm with which this rider is

producing the torque, is quite short because this one weighs a lot, so big

downward force, short lever arm. And that is balancing, or cancelling out

the torque due to this one, which is in the opposite direction, but it's produced

by a by a smaller force acting at a larger lever arm.

So this is common in, in playing on a seesaw when you have two children of, of

significantly different weights. They have to sit at different distances

from the pivot. The heavier child sits close to produce a

certain torque, and the lighter child sits far from the pivot to produce an equal

amount of torque but in the opposite direction.

Well, that brings us to a question. And the question is this.

Can two riders, and we can adjust their weights as you like.

Ever sit on the same side of the seesaw, and still balance the seesaw?

Two riders cannot sit on the same side of the seesaw, and expect the seesaw to

balance. That's because those two riders.

Produce torques in the same direction about the pivot.

Their forces are in the same direction, their lever arms are in the same

direction, so their torques are in the same direction.

And when you add those torques, they sum to something larger than each one

individually. So you get a lot of torque on the see-saw,

and its Terribly unbalanced. In order to balance the seesaw, the two

riders, or however many you want to put on the seesaw, have to distribute themselves

on opposite sides of the pivot so that their, their torques cancel one another

and eventually, if you do it all right, they sum to zero and the seesaw is

rotationally inertial. It has zero net torque on it and no angular

acceleration. It coasts rotationally.

There are two seemingly different ways to think about the balanced see saw

situation. The first way is the way we've been doing.

Where this rider produces a torque, that rider produces a torque, the two torques

sum to zero and as a result the seesaw experience zero torque due to gravity.

It's balanced. The second way to think about this

situation is in terms of a concept known as the center of gravity.

Now center of gravity is the effective location of an object's weight.

I have one. You have one.

These riders have one. Even the seesaw board has one.

This rider's center of gravity, that is where its effective weight is located.

Is pretty much at its center. Same with that rider.

The seesaw board's center of gravity, the location, the effective location of its

weight, is at its middle. Right there.

And that might make you think that center of gravity, which is here, and center of

mass, which is here, are the same idea. Center of mass, center of gravity, aren't

they the same? They're not.

They happen to coincide for all objects here near the earth's surface.

Celestial objects violate this concepts for complicated reasons that I'll leave

for another day. But small objects do have their centers of

gravity at the same locations as their center of mass, but they're different

concepts. Center of mass is the effective location

of an object's mass. It's natural pivot.

We watch centers of mass in action when I threw various wobbling objects or sticks

and so on through the air and you'd watch. The center of mass was f, traveling in the

arc of a falling object. That's the mass moving and the inertial

properties of the object In play. So, center of mass is all about intertia

in motion. Center of gravity is about forces and it's

forces of gravity. It's got to do with gravity.

If there's no gravity around, center of gravity means nothing.

So it's the effected location object's weight The fact that weight is

proportional to mass here near the earth's surface, means that center of gravity and

center of mass share the same location. But they're different concepts and so if

you're dealing with the inertial aspects of an object, you're probably paying

attention to the center of mass. Mass.

If you're dealing with the gravitational or weight aspect of an object, you're

probably dealing with center of gravity. So, back to the situation here.

We have objects with various centers of gravity and that brinks us to an

observation that this entire structure Two riders in a seesaw is, we can consider it

as a single object. Where is its center of gravity?

That composite object. And it turns out that this overall

object's center of gravity is located right above that Pivot.

And it's being pulled straight down, like the centers of gravity are pulled straight

down. They're gravity after all, right?

The forces of gravity are toward the center of the earth.

It's being pulled straight down right towards the pivot, the center of rotation.

And as we've seen before, forces that act toward the center of rotation produce no

torque about the center of rotation. So, this seesaw is balanced for two, you

know, in two ways you can think of it. One is in terms of the individual riders

producing torques that sum to zero. The other way, which is kind of cool, is

that the riders and seesaw together have a center of gravity located vertically above

the pivot. And therefore, the, the force of gravity

acting on this entire structure acts right toward the pivot, and produces no torque.

It's along the, the lever arm and produces no torque.

[LAUGH]So Annie and Megan here are riding a real seesaw, not one of the little

things I have in my lab. And They're balanced right now.

Can you show us this? It takes delicate adjustment, but Megan's

distance is just right from the pivot, the pivot's right here.

Andy's distance is just right from the pivot They've adjusted it, so the net

torque on this thing is, is as close to 0 basically, as they can get it.

But this is a boring way to ride seesaws, if you just sit here balancing.

I guess it's not too boring. It's kind of exciting, trying to keep it

balanced. But they can unbalance it In order to rock

back and forth in one of two ways. They can either push on the ground with

their feet. So, so Meagan, why don't you push on the

ground. Okay, and that extra force produces

another torque, which causes Annie to rotate down.

Now Annie can push down on the ground and cause Meagan to rotate down.

So, they're causing angular accelerations back and forth by exerting new torques on

it. The other way they can unbalance this Is

by leaning. So the, each, each one of them has a

center of gravity that's located somewhere sort of mid-body.

But if they lean, they can shift the location of their center of gravity and

therefore. Exactly where they're exerting the forces

on the seesaw board, and cause it again to experience a net torque so it undergoes

angular acceleration. So, if you both lean towards Annie, what

happens? >> It goes down[LAUGH] >> Annie goes down,

because basically the lever arm With which she's working gets longer.

And the one that Megan's working with gets shorter.

So the torque is this way, toward me. But how if we lean, everybody lean towards

Megan. >> [laugh] Now the lever arms get longer

and shorter in the opposite direction. Speaker:and then that torque is toward

you. So, they can rock back and forth, so, this

is how a seesaw works. Okay, you guys can go at it.

>> All right, here we go. Speaker:[unknown] Speaker:[LAUGH] Either

way. Speaker:[laugh] >> And this is what makes

seesaw fun right, is all the adjustments of the torque so that you[UNKNOWN] Undergo

angular acceleration in opposite directions, back and forth.

>> This is so funny! >> Seesaws are not the only structures in

our world that need to balance. Mobile sculptures do as well.

This mobile sculpture is entitled, happy hanging hardware.

And I built it out of a torque wrench, a ball peen hammer, and a metal file.

Amazingly enough, each of these components is rotationally inertial.

You don't see any of the them undergoing angular acceleration, after all.

And that brings us to a question. For all of the components of a mobile

structure, to be rotationally inertial, how must those components be arranged?

Each component of this mobile structure has its center of gravity at or below the

point at which that component is supported.

In effect, the pivot, about which that component could rotate.

This is actually a relatively complicated concept though.

Because there are three components here which aren't the individual tools.

First component, the, the simplest, is the, is the file That file has its center

or gravity at or below this support point. Which is the loop of string going around

it. That's the pivot about which the file can

rotate. And so, the file has its center or gravity

at or below that, that pivot. And therefore.

Gravity produces no torque on the file, it's rotational inertial.

So far, so good. The ball peen hammer isn't an object by

itself, it's not the component by itself. Rather, the ball peen hammer and the file

together are the next component of this mobile.

And that combined object. Ball peen hammer and file has its overall

center of gravity at or below its support point.

This loop of string. And lastly, the torque wrench and

everything below it has its combined center of gravity at or below this.

Support point. The support point that, that is acting on

the torque.wrench. So, each of these components, the file and

the hammer and file and the wrench, hammer and file, each of those components has

it's center of gravity directly below it's support.

And therefor, gravity and pulling down on the center of gravity produces no torque

on that component about it's pivot. It doesn't undergo any angular

acceleration then due to gravity, it's balanced.

And so the file is balanced. The hammer and file are balanced.

The wrench, hammer, and file are balanced. The entire mobile then, is balanced, and

it's all rotationally inertial. So we see that objects that can rock or

tip are only rotationally inertial if you balance them carefully.

Sometimes that's what you want, like with a mobile.

Sometimes that's almost what you want, like with a seesaw, where getting it

perfectly balanced is interesting, but kind of unexciting in the long run and you

want to unbalance it a little bit to get some action happening.

We'll talk more about balance in the episode on bicycles, but for now.

It's clear that in the context of seesaws, balance and near balance are the name of

the game.