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.

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Wheels

Professor Bloomfield illustrates the physics concept of frictional forces through experiments with wheels.

- Louis A. BloomfieldProfessor of Physics

How do powered wheels propel a bicycle or car forward?

The answer to that question is that the ground exerts a forward static frictional

force on each powered wheel and thereby pushes the entire bicycle or car forward.

In the previous video, however, I pointed out, that the ground exerts static

frictional forces on free wheels as well. So, what could be different between free

wheels and powered wheels? To explain that, let me start by

re-examining free wheels. When you pull a wagon forward so that it

accelerates forward. The bottoms of its wheels are threatening

to slide forward relative to the ground. And so static friction acts to prevent

that sliding. It pushes the bottoms of the wheels

backward, and that backwards force of static friction, the bottoms of the

wheels, has two effects First, it produces torques on the wheels that cause

those wheels to begin their spin, to roll.

And so when I pull the wagon forward and its wheels rolls, that's because of

static friction with the ground. But second, those backward frictional

forces on the bottoms of the wheels affect the entire wagon.

And, you have to pull a little extra hard, and do a little extra work to get

the wagon up to speed. Because its spinning wheels are carrying

extra kinetic energy. Something we'll explain in the next

video. The key point here is, that the ground

pushes free wheels backward as their vehicle accelerates forward.

In contrast the ground pushes power wheels forward as their vehicle

accelerates forward. To explain that observation I need a

powered wheel, wagons don't normally have powered wheels so I have to add one and

here it is. It's a cordless electric drill with a big

rubber stopper. And when I turn on the drill it twists

the rubber stopper and the stopper begins to spin.

Well, in the language of physics, the drill exerts a torque on the rubber

stopper and causes the rubber stopper to undergo angular exceleration.

If I press this stopper against the sidewalk, the stopper becomes a powered

wheel and it propels the vehicle attached to it forward.

At present, the vehicle it's going to power and propel is me.

And when I turn on the drill, it pulls me forward.

Now where does this come from? It comes from the fact that if I were to

spin the wheel above, above the sidewalk here.

The bottom of the wheel will move in the backward direction relative to the

sidewalk. And once I touch the two surfaces

together, and static friction appears, it's going to fight that relative motion.

It does not want the bottom of the wheel moving backward.

And therefore it will push the bottom of that wheel forward to oppose sliding and

try to keep those two surfaces at the same velocity.

That forward force on the bottom of the powered wheel has two effects.

Does that sound familiar? One effect of that forward static

frictional force on the bottom of the wheel is to produce a second torque on

the wheel, that opposes the, the drill torque.

It acts to slow the angular acceleration of that wheel.

It fights the, it's fighting the drill. That's one effect of that forward

frictional force on the bottom of the wheel.

The second effect is it's, it's a forward force on the entire vehicle.

It causes the entire vehicle to accelerate forward, namely me.

Whoah off we go. So.

A powered wheel obtains a forward static frictional force from the sidewalk that

propels the vehicle forward. Well, I'm, I don't make a very good

vehicle. So, what could we do better?

Well, I have a big wagon, I have a powered wheel.

Let's go on a road trip. Here we go, powered wheel on a wagon.

[NOISE] . Full speed ahead, yeah.

[NOISE] That worked out pretty well, and I'm still in one piece.

It's time for a question related to powered wheels.

At a track and field competition the runners in a 100 meter sprint.

Start that race with their feet pressed against the nearly vertical surfaces of

starting blocks, with feet against those blocks.

And they're wearing spikes that project into the track as they run.

Why are the starting blocks and spikes important to the runners in a 100-meter

sprint? Speaker 1: To win a sprint like this, the

runner wants to accelerate forward as rapidly as possible, and may well be

accelerating even as the runner crosses the finish line.

At the start of the race, the runner has to be motionless before the gun goes off,

so the runner has... Feet pressed against these nearly

vertical surfaces of the starting blocks. When the gun goes off, the runner pushes

back hard on the starting blocks with support forces.

Your surfaces, these two surfaces, are attempting to occupy the same space at

the same time. So as the runner's foot pushes back on

the starting blocks Hard. The starting blocks respond by pushing

forwad on the runner's feet hard. And that helps the runner get started

from the moment the gun goes off. Once the runner loses contact with the

starting blocks, however, all they have to push against is the horizontal track

surface. So to get a forward force out of a

horizontal surface, you need friction. And ideally, static friction.

So, having spikes on your shoes as a runner, allows you to get better

traction. That is a higher maximum static

frictional force out of the track. So with those spikes digging into the

track, You can get forward forces that are enormous on a human scale.

And accelerate forward, at dramatics, a dramatic rate.

And so the best sprinters obtain very large support forces out of the starting

blocks. And then maintain their acceleration by

getting very large forward frictional forces of static friction toward the

finish line. And they may well still be picking up

speed in the forward direction as they cross the finish line.

I've talked a lot about powered wheels, but I've never actually defined the term

"power." People often use the words power and energy interchangeably.

But they're actually different, though related, physical quantities.

Energy is the conserved quantit of doing. And power is the rate at which energy,

that conserved quantity, is transferred from one object to another.

In more official language, energy is the capacity to do work and power is the rate

at which that work is being done. To shed more light on this relationship

between energy and power, let's revisit the energy/money analogy.

Just as energy is the conserved quantity of doing, money is the conserved quantity

of spending. If you have energy, you can do work.

If you have money, you can spend. Spending transfers money.

Just as doing work transfers energy. When you buy something, you spend a

specific amount of money. Similarly, when you complete some

mechanical task, you do a specific amount of work.

Money and spending are measured in your local currency, which might be Dollars,

it might be Euros, or something else. Energy and work are measured in the s i

unit of energy. The joule.

But there are other units. What about when you rent something?

You're not buying it outright all at once.

You're paying for it steadily over time. Rather than a one time transfer of money,

there is a substained flow of money. A dollar, a dollar, a dollar, and like

rent salaries wages subscriptions are also these sustained flows of money.

And you usually measure them in your local currency divided by a unit of time.

For example, dollars per hour. Or euros per year.

Or bots per day. Or rupees per month.

Analgously. When you're peddling your bicycle, you're

providing a sustained flow of energy to the bicycle's powered wheel.

The rate at which energy is flowing, that is, the work you do per unit of time is

the power you're providing. A typical bicyclist transfers an energy

of 100 Joules. To the powered wheel in a time of one

second corresponding to a power of 100 joules per second.

The SI unit of power is the joule per second, which is so important that it has

it's own name. It's called the Watt.

A typical bicyclist provides a power of 100 watts to the bicycles powered wheel

or 100 jewels per second. There are other units of power but

they're all related to watts. For example one horsepower is equal to

about 750 watts. A serious bicyclist can provide about 750

watts or 1 horsepower to the power wheel of a bicycle.

At, at least for a short period of time. To help you distinguish between energy

and power and jewels and watts, let me ask you a question.

If your toaster is labeled as using 1,000 watts, how much energy does that toaster

consume in 1 hour? A toaster that consumes power steadily at

a rate of 1,000 watts, that is 1,000 joules per second.

We'll consume 1,000 watts in one second, 2,000 watts in two seconds, 3,000 watts

in three seconds and in an hour which corresponds to 3,600 seconds that toaster

will consume 3.6 million joules of energy coming in the form of electricity.

So, 3,000, 3.6 million joules is kind of an awkward number to talk about, so it

has another name, it's called a kilowatt hour.

So when you see written on a power bill, electric power bill for example, that you

have consumed energy in total of one kilowatt hour.

That's 3.6 million joules of energy. The energy that would be consumed by a

1,000 watt device like a toaster, in the course of a single hour.

Powered wheels are rotating objects. And the work that's done on them is done

in the context of rotation. That's an interesting complication.

Rotary work. We've seen that you can do work on a

translating object by pushing it and having it move in the direction of your

push. You exert a force on it, and it moves a

distance in the direction of that force. But what about a rotating object?

It turns out, that you do work on a rotating object by twisting it, and

having it turn in the direction of your twist.

You exert torque on it, that's twist, and it rotates through an angle in the

direction of that torque. For example, when I spin this bicycle

wheel, I do work on it. I exerted torque on it, I'm going to

twist it, and it rotates in an angle in the direction of my torque.

My torque was toward you and it rotated toward you.

I'm transfering energy to it and you can see that energy because it's moving, it's

got kinetic energy. Getting the units right when doing work

in a rotational context is a bit tricky. You have to be careful to chose the

correct units when measuring angles. Those angles should be measured in

radians, the natural unit of angle. As a reminder, one full rotation like

this. Is two pi radians, where pi is the

mathematical constant. It's approximately equal to 3.14159.

So two pi radians is about six and a quarter.

That means if I rotate this wheel all the way around, that was about six and a

quarter radians. Well, if you choose that unit to measure

your angles in and you use the SI unit for the torques you exert, you end up

with the SI units of energy when you figure out how much work you've done.

So, for example, if I exert a torque of one SI unit, which is one Newton meter --

that's the unit of torque -- On this wheel, as it rotates through an angle of

one radian in the direction of my torque. I'm going to rotate it like that towards

you. And it's going to turn toward you.

So I'm going to follow all the rules. So here you go, about.

I'm going to exert a torque of about one Newton meter.

And I'm going to do it as the wheel rotates through one radian.

So I've gotta go from about here to about there, that's about a radian.

Here we go. Ready, get set, okay.

I did about one Newton meter times one radian, one joule of work on the wheel.

So I transferred about one joule of energy to the wheel.

And that was the goal here. For a powered wheel, like my drill

powered rubber stopper here. The rotary work is done at a steady rate.

So the drill is providing rotary power to the wheel.

[SOUND]. Before I show you how to calculate power

in a rotating system, let me show you how to calculate power in a translating

system, this wagon. If I push the wagon, and it moves

steadily in the direction of my push, I'm providing it with power.

And the amount of power I'm providing it with.

Is the work I do on it per second. In this case that is the force I exert on

the wagon times the distance it travels in the direction of that force per

second. Well, the distance it travels in the

direction of my force per second is its velocity.

At least the part of the velocity that is in the direction of my force.

So, the amount of power I'm providing this translating object, the wagon, is

simply the force I exert on it times its velocity, where we're only considering

the part of velocity that's in the direction of my force.

[SOUND] That's the power in a translating object.

Now, we can return to the power in a rotating[SOUND] object.

That rotary power is the rotary work,[SOUND] the drill does on the wheel,

per second. Which is the torque that the rotary drill

exerts on the wheel. Times the angle through which the wheel

rotates per second. That angle through which it rotates per

second - that part, is just the wheel's angular velocity.

So in short, the rotary power the drill provides to its wheel is the torque it

exerts on the wheel. Times the wheel's angular velocity, the

angle through which it rotates each second with all of the directions done

properly. If some of these details about rotary

work are too complicated to follow, don't worry about them.

The take-away message is that if you twist something and it turns in the

direction of your twist You're doing work on it!

And transferring energy to it. And if you twist it and it keeps moving.

You twist you know, twist and it's steadies, rrrrrrrr, then you're providing

it with rotary power. When you do rotary work on a wheel, you

give it energy. We'll look at what becomes that energy in

the next video.