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 does a lone seesaw ride plummet to the ground?

The answer to that question is that the lone rider produces a torque on the seesaw

and causes it to undergo angular acceleration.

The seesaw rotates such that the rider descends toward the ground.

And hits it. There are several ways of examining this

situation. So I'm going to follow the path that I

think is most straightforward. The rider's weight gives rise to a torque

on the seesaw. And since the seesaw is no longer

rotationally inertial, its angular velocity is no longer constant.

Instead, that angular velocity changes with time, and the rider soon plummets to

the ground. But those observations give rise to two

more questions. How does the seesaw respond to torques,

and what is the origin of this particular torque?

So let me start by looking at the seesaws response to torques.

When the seesaw is experiencing no outside torques it's covered by Newton's first law

of rotational motion. So it rotates at constant angular

velocity, like this. But once there is a torque acting on the

seesaw, the seesaw is no longer covered by angu-, by Newton's First Law of Rotational

Motion. And its angular velocity is no longer

constant. Instead, it's angular velocity begins to

change with time. The seesaw undergoes angular acceleration.

Angular acceleration is another vector physical quantity of rotational motion,

and it is the rate at which angular velocity is changing with time.

Like ordinary acceleration, translational acceleration, it's a subtle quantity.

It's hard to see. You have to look carefully.

It takes three glances. To see acceleration and it takes three

glances to see angle acceleration. So I'm going to illustrate angle

acceleration with my body and hope that you can see.

It happening. So here we go, let me start, motionless,

rotationally motionless. That is my angle velocity is 0.

If I change my angular velocity, during the time over which that angular

velocity's changing, I am undergoing angular acceleration.

So here we go. I'm going to g-, undergo angular

acceleration, and then I'm going to stop undergoing angular acceleration, and watch

what happens. Here goes the angular acceleration; it's

going to be... (End of transcription.) Up.

Right hand rule again. I'm going to rotate.

Here we go. Shoop.

Okay, I did it. It;s over.

I am now coasting rotationally, at constant angular velocity.

But when I first got started I was undergoing angular acceleration.

If I don't go, undergo angular acceleration again, I'm going to keep

spinning here forever, and this will make me very dizzy.

So I'm going to undergo angular acceleration downward in a moment.

Ready? Get set, whoop, there I did it.

So during those two moments when I changed, extended moments.

When I changed my angular velocity I did it by way of angular acceleration.

I'll show it to you again. I'm going to do A intersection upward for

about a quarter of a second and then, I'm going to do A intersection downward for

about a quarter of a second and come to stop.

Ready? There.

Now, I'm coasting and now. So, the angular acceleration portion of

that situation Was during the changes in my angular velocity.

Coming back to the seesaw then, the angular acceleration is absent now.

Ready, get set, there it is. There were a lot of angualar accelerations

there at the bottom, but they initially kicked in, the first angular accelerations

Kicked in when the rider got on the seesaw.

Right now. So we see, a torque causes a seesaw to

undergo an angular acceleration. But what if there's more than one torque,

acting on that seesaw at the same time? In that case, those torques add together.

To be come a net torque, and the net torque is what causes the angular

acceleration. So for example if I've got 2 riders

hopping on to the seesaw at once, the seesaw can't respond with several separate

angular accelerations at the same time, it, it only has 1.

Instead it responds to the net torque, produced by those 2 riders.

Well, if net torque causes angular acceleration, the question comes up is,

how much angular acceleration? It turns out that the seesaw's angular

acceleration is proportional to the net torque acting on it.

So if a gently net torque acts on a seesaw.

It undergoes a small angular acceleration. But if a large net torque acts on the

seesaw,[SOUND] it undergoes a large angular acceleration.

But there's a second factor involved in determining the seesaw's angular

acceleration, the seesaw's rotational mass.

Rotational mass is the measure of an object's rotational inertia, its

resistance to undergoing angular acceleration.

Now traditionally that physical quantity is called moment of inertia and it has

various complexities to it. They're beyond the, the scope of our

little discussion here. Not relevant really to seesaws.

So rather than trying to have you remember a name like moment of inertia, with its

complexities, I'll make our lives simpler by simply calling it.

Rotational mass. That conveys the characteristic that it's

a mass like thing, it's a resistance to acceleration of some form.

In this case, rotational acceleration. So, this seesaw has a certain rotational

mass, a certain resistance to angular acceleration.

So if I exert a certain torque on it. It responds with a specific angular

acceleration. And I go back to, to putting a rider.

So, my little rubber stopper rider is here.

If I put a certain rider on this seesaw and let it undergo angular acceleration,

well, it undergoes rather rapid angular acceleration, and down goes the rider.

But I can increase the rotational mass of this seesaw.

By adding a second board. When I do this, I'm increasing the

rotational inertia of the seesaw. Try to glue it and tape it in place.

And now, it's less responsive to the same torques as before.

In this case, if I put 1 rider on. It undergoes angular acceleration, but not

as much. Overall, the seesaw's angular acceleration

is proportional to the net torque acting on the seesaw.

And also inversely proportional to the seesaw's rotational mass.

Those two observations form the basis. For Newton's Second Law of Rotational

Motion, which states that an object's angular acceleration is equal to the net

torque acting on that object divided by that object's rotational mass.

I'm going to ask a question about angular acceleration, but I'm going to do it in

the context of a bicycle wheel that I can hold in my hands.

At present the bicycle wheel is motionless.

And I'm going to do three things to it, in sequence.

First, I'm going to start it spinning. Second, I'm going to turn the wheel all

the way around like this, so it's spinning in the opposite direction.

And now as a third thing, I'm going to stop it from spinning.

The question is, during which of those 3 steps was the bicycle wheel undergoing non

zero angular acceleration? All three steps involved angular

acceleration of the wheel. When I started it spinning it went from

having an angullar velocity of zero to having an angular velocity toward you.

Remember the right hand rule here. When I pivoted it.

Around like this. I reverse the direction of the wheels

angular velocity from toward you to toward me.

That's angular acceleration. I had to, to make the wheel undergo

angular acceleration to reverse its direction of, of rotation.

And finally, when I stop the wheel from spinning, I take its angular velocity from

>> Toward me to zero. So, all three steps require the wheel to

undergo angular acceleration. So, we see that a seesaw responds to a net

torque by undergoing angular acceleration. Why then does a low rider sitting at the

end of the seesaw board. Exert a torque on that board.

After all, the rider has a weight, which is a force.

And If I hold the seesaw in place now, the rider and seesaw are pushing on each other

with forces. The seesaw to support the rider's weight

and the rider pushing back on the seesaw in response.

It's all forces out here. Where does the torque come from?

Well it turns out that forces and torques are related and that a force can produce a

torque and a torque can produce a force. To see how that all works, let's go

experiment with a door. ...because doors are a wonderful example

of rotational motion and the use of a force to produce a torque.

So here I am outside the physics building, opening and closing doors in a light rain.

The things we do for science. Doors are a nice example of rotational

motion. After all, they don't go anywhere.

They simply rotate open and closed about their hinges.

(End of transcription.) >> They have all of the characteristics we've come to

expect of rotating objects. They have angular postions, they have

angular velocities, and they even have angular accelerations.

But that brings us to the issue at hand, which is when you open a door you do it by

exerting a force on the door handle. And yet the door undergoes angular

acceleration. Well, angular accelerations are produces

by torques, not by forces. So how is it that a force exerted on the

door handle produces a torque On the door. To show you how that works, I first have

to define a center of rotation. Now, the obvious place to put the center

of rotation is somewhere along the hinge line.

'Because that's the line about which all the ro-, door's rotation occurs.

But I have to be more specific than that. Because center of rotation is act-.

Actually a point. Not a line.

So I'm going to put the center of rotation in line horizontally with the door handle

for reasons that we'll come to eventually. And that's going to be our center rotation

right there on the hinge line aligned nicely with the door handle.

Having done that then, let's look at ways in which not to produce a torque about

that center rotation. Starting with a force, so these are all the unsuccessful ways to

try to open a door, some of which you may have encountered by accident.

So, first unsuccessful way to produce a torque, starting the force, is to push the

door handle toward. The center of rotation.

So I'm pushing right at that center of rotation.

No effect. I'm producing no torque.

How about reversing my force? Instead of pushing toward the center of

rotation, let me pull away from the center of rotation.

Also, no luck. Doesn't do anything.

So we see that. Pushing toward or away from the center of

rotation is, is unsuccessful. How about pushing on the center of

rotation. Let me come over here to the center of

rotation and push right on it. I'll try to pull right on it, all the kids

of forces, none of it works. So you can't move the door by exerting

your force toward, away from or on the center of rotation.

Okay. Now it's time to be successful.

We can only take so much frustration. So now, I'm going to exert a force out

here on the door handle, not toward or away from the center of rotation, but at

right angles to a special line. It's actually a vector.

It's called the lever arm. And this is the, this is what the lever

arm is. The lever arm is going to be the vector

that, that extends from the center of rotation to the point at which I'm going

to exert my force. Namely on the door handle.

So there is a vector that points along this lineto this point here.

It has a length of about 1 meter like that and it's direction is exactly to your

left. And I'm going to exert my force Not along

that vector or, you know, with it or against it, but at right angles to it,

perpendicular to that lever arm. I'm going to exert my force toward you,

and watch what happens. The door undergoes angular acceleration

and begins to rotate open. That is how to produce a torque starting

with a force. If you'll exert your force At a lever arm.

From the center rotation, that is the vector that extends from a center of

rotation to where you exert your force. And you exert your force at perpendicular

to that lever arm. Then you produce a torque.

And the torque has a specific direction. Its direction follows yet another right

hand rule. If you take your right hand and extend

your index finger. In the direction of the lever arm towards

your left right now. And then you swepp the index finger of

your right hand in the direction of the force which is towards you.

So that's the sweep. Look what my thumb is doing.

My thumb is pointing up. That is the direction of the torque I

produced in pulling toward you. With, on the door handle.

The lever arm is that, that direction. The force is toward you.

The, the torque I exert is up, and so it causes upward angular acceleration in the

door which swings open. Now the amount of that torque that I

produce depends on two things. One is how much force I exert.

The torque is proportional to the force I exert.

A gentle force produces a gentle torque. A big force produces a big torque.

So, that's the first observation. Second observation is, the length of the

lever arm matters. The torque I produce is proportional to

the length of that lever arm. Here I have a lever arm about that long,

but if I go inside and I push near the hinges, I can make the lever arm very

short; and watch what happens. That was hard.

So, I'm exerting my force here, very close to the pivot, Pivot.

Therefore at a very short lever arm, and I'm obtaining a very small torque until I

really crank up my force. We can combine these observations to

relate the force to the torque it produces, quantitatively.

That torque is equal to the lever arm times the force.

Where only the component of force that is perpendicular to the lever arm is

included. And where the torque is in the direction

determined by the right hand rule. So in this case if the lever arm is

pointing to your left and the force is pointing toward you The torque is up.

Now this door is complicated because it has a closing mechanism, like many doors.

It has a system to try and keep that door closed when you leave it alone.

So it's not free to exhibit rotational inertia and has all kinds of, of its own

trouble and I had to overcome that resistance, That, that The mechanism

trying to keep the door closed. That's a lot easier to overcome if I'm out

here at a, with a big lever arm. I can exert relatively gentle force on the

door handle and get the door to open despite the closing mechanism.

If I try to push very close to the hinges, that closing mechanism is hard to beat.

And you may have had this experience that if you go to a door that isn't very well

labeled and you have to push it open, you can't tell which side of the door has the

hinges. If you push near the hinge side of the

door, the door doesn't open very easily. It's very resistant to openeing because

you're producing so little torque with your force.

You need to go out to the other side of the door where you have a big lever arm to

work with. And therefore can really create a lot of

torque with a small force. To produce a torque with a force then, all

we need is a lever arm. For an unconstrained seesaw like this.

One that can rotate in any possible direction, the options are limitless.

I'm going to choose as our center rotation The seesaw's center of mass just for

convenience here, right about there. And now, let me show you a couple of

torques. Things you've seen before, maybe some you

haven't. If I come out here to a lever arm Towards

your left and then I push down with my force, which is at right angles to that

lever arm or, in fact I don't have to be perfectly right angles I have to just,

just have to have some component that's at right angles to the lever arm.

And I push down, I cause an angular acceleration toward you.

Right hand rule again. On the other hand, if I come out to a

lever arm, same lever arm. But I push my force towards you.

Watch what happens. I cause angular acceleration up.

And if I come out to a level arm toward you, very short one but it's there, and

push down, I caused angular acceleration to your right.

I flipped the board like that. Well.

This is exciting, but very complicated. There are too many options with our

unconstrained seesaw. So fortunately, we're going to focus on a

constrained seesaw, one that has a pivot shot through the center that forces it to,

to rotate in a very simple manner. >> This seesaw board down here cannot do

this kind of rotation, or this kind of rotation.

And so, it operates in a more simple fashion, like this.

And it still exhibits the same sorts of behaviors.

To produce a torque on this seesaw, I come out to a lever arm and push at right

angles, or partly at right angles to that lever arm down and I cause angular

acceleration toward you. Because my torque was toward you.

We've seen how to produce torques with forces in the context of doors, in the

context of seesaws. But what about another important household

use of torques, putting in or taking out screws or bolts?

You rotate a bolt into place and you rotate it the other direction to take it

out of place. Well suppose you have a big bolt like

this. >> That has rusted in place, and you're

trying to get it out but it won't turn when you grab it with your hand and try to

twist. You need more torque.

So, in that case you get a wrench. This is a device, and you will have to

figure out how it works. This is a device that when you put it on

the head of the, the bolt. >> It allows you to produce more torque.

And by now, you should be thinking about how this works.

But what if this is really, really stuck? And you need a bigger wrench?

Well, that's already a pretty big wrench, you think.

And you're probably thinking that I'm going to go over and get this wrench.

To show you the bigger wrench. But no.

I have in mind this wrench. And so we take this wrench, put it on our

stuck bolt. And lo and behold, it's a lot easier.

To produce a large torque on that bolt and remove it from wherever it's stuck.

The question then, is this, why is using this larger wrench more effective it, why

does it enable you to remove that bolt when this wrench didn't to the job?

This wrench has a longer handle, and it provides a longer lever arm with which to

produce a torque using a force. So when you come out here to the end of

the handle and push perpendicular to that handle and therefore perpendicular to the

lever arm. Your force produces a larger torque as

compared to this wrench. There's just not as much length here to

work with. It's got a shorter lever arm and so when

you push on the handle of this wrench with that shorter lever arm your force produces

less torque. So we see, whenever a lone rider goes out

to a lever arm on the seesaw and sits down, the rider's weight gives rise to a

torque on the seesaw that causes it to undergo angular acceleration, such that

the rider ends up pretty much sitting on the ground.

The rider's weight. Is a force and that weight causes the

rider to push on the seesaw with a force, but the force acting at a lever arm from

the center rotation produces the torque that causes all this to happen.

Pretty much the only place a, a single low rider can, can sit or stand and not

produce a torque. On the seesaw is exactly on top of the

pivot, which is kind of an interesting place to stand.

And I must admit to having done that myself, from time to time.

But it's much more fun to have 2 riders on a seesaw, and that is the subject for the

next video.