How is energy present in a wheel? The answer to that question is that the

wheel has kinetic energy in both its translational and rotational motions.

It can have other forms of energy. Such as gravitational potential energy if

it's high in the air. Or nuclear energy if its made out of

uranium, but the energy I have in mind here is kinetic energy, energy of motion.

And, as I'm about to explain, a spinning wheel on a vehicle carries an especially

large amount of kinetic energy. Let's start by looking at the kinetic

energy the wheel carries in its translational motion.

That is, the kinetic energy that's associated with the wheel's velocity.

We can determine that kinetic energy by calculating the amount of work we would

have to do on the wheel. To bring it from rest to its current

velocity. So, we'll start with the wheel at rest,

we'll push on it as it begins to pick up speed and move in our direction, and

we'll keep pushing until it's moving at the final velocity, and if we know how

much work we did We know how much energy we've invested in that wheel.

And therefore, we know, its kinetic energy.

That's the program here. So, rather than doing that calculation

completely, which is, more complicated than is appropriate, for, for this class,

this course. I'm going to explain the most important

issues. To do this, make these observations, I

need a wheel that can translate without rotating, and without friction.

So I don't want to have, if I do this, that wheel, yeah it's translating, but

it's also rotating, that just complicates the story.

So I can't make this wheel translate without making it rotate.

I have friction otherwise. So what I'm going to use is this wheel

which is actually a toy puck that rides on a little cushion of air.

It's a miniature hovercraft that. The cushion of air is created by a

battery powered fan so when I turn it on, it rides essentially without any

important friction. And if I start it from rest and do work

on it like this, it carries that work with it as kinetic energy.

Nonetheless[SOUND] It's a wheel. Off it goes, right?

I don't have to roll it though to get the friction less effect.

So, first observation to make using this interesting little wheel is that the

direction of the final velocity I end up with doesn't matter.

It takes just as much work. To go from rest to moving to your right

at a certain speed as it does to go from rest to moving to your left at a certain

speed or toward you at a certain speed. The kinetic energy of this wheel, then,

doesn't depend on the direction of its velocity.

It depends only on the amount of its velocity.

Which is speed. So it's important that kinetic energy

doesn't depend on the direction of motion.

I've been called energy the conserve quantity of doing because it's about

doing work, not about moving. In the episode on bumper cars, we will

encounter the conserved quantity of moving.

A physical quantity officially known as momentum.

Since momentum is all about moving. Going somewhere in a particular

direction. It does depend on a direction.

But energy has no direction, and no dependence on the direction of motion.

So so far we know that the kinetic energy of a wheel depends on its speed.

Second important observation is the kinetic energy of a wheel is proportional

to its mass. How can I show you that?

Well, I have one me, wheel, and a second wheel.

Isn't this great? These are two identical wheels.

And if it takes one unit of work to get that wheel moving the right at certain

speed how much work do you think to get two wheels move from to, to go from rasp

to moving at that speed. It takes twice as much work.

Each wheel takes one unit or work. Boof, off they go.

So, without further ado it's pretty clear that kinetic energy, the kinetic energy

of a wheel or any other object is proportional to to its mass.

[NOISE] My third important observation Is that the work required to bring a

motionless wheel up to its final speed is proportional to that speed squared.

That's a rather startling observation. It means that if I double the final speed

of the, of the wheel. So, say from this to this, twice the

speed, one quadruple the kinetic energy. A huge increase in kinetic energy.

Now, rather than going into all the details behind that relationship, I'll

show you the basic issues behind it. First, let's look at how the, the wheel's

velocity evolves over time. If I exert a steady force on the wheel.

So, I'm going to push it with a constant force.

It accelerates at a constant rate. The acceleration is constant.

And so its velocity increases steadily. And basically, then the velocity is

proportional to, to the time over which I've been pushing it.

If I double the time over which I push it, I double its velocity.

That doubling is not going to work for work.

Watch what happens if I, to the energy I invest in that wheel, as time goes on.

Once again I'm going to push it with a steady force and its gone to a steady

acceleration. But as it accelerates it travels faster

and faster. So i have to chase it.

At first its easy to push on it, but then it goes faster and faster.

I have to go i have to move faster and faster its moving faster and faster.

The distance it travels with each new portion of time each new second, is

bigger than during the previous second. While the work I'm doing, is equal to the

force sizer times the distance it travels.

During teh first second it doesn't travel very far.

So I don't do very much work on it. But the second second it travels farther.

The third second it travels father still. With each passing second, I'm doing more

work on it than the previous second. That means that although its speed and

velocity are increasing in proportion to the time over which I push it.

The work I'm doing is skyrocketing. It's going faster and faster, I could

chase after this poor wheel and push and push and push longer and longer

distances. So, that's the origin of this explosion

in energy that occurs as the puck's speed and velocity increase.

So[SOUND] When all the dust settles and the calculation is done in its full

glory, it turns out that the kinetic energy that the wheel has by virtue of

its velocity, that is kinetic energy associated with translational motion

alone. That kinetic energy is equal to one half.

The wheel's mass times the square of it's speed.

That's the final story. And that's true of any translating

object. When I begin to move to your right or to

your left, remember direction doesn't matter, my kinetic energy at any moment

is 1/2 my mass times my speed squared. I happen to be a complicated, moveable,

you know, shape changeable object. So in my translational motion you really

would concentrate that on my center of mass.

And how is my center of mass moving? And take one half of my total mass and

multiply it by the square of the speed of my center of mass.

Well this observation, that, that kinetic energy increases in proportion to the

square of speed, goes a long way to explaining why high speeds are

potentially so hazardous. If you compare two cars, two identical

cars, one traveling 60 miles an hour, which is essentially the same as 100

kilometers an hour. To a second car that's traveling only

half that fast, 30 miles an hour, or 50 kilometers per hour.

They're different by a factor of two in speed and velocity, but they're different

by a factor of four in kinetic energy. That faster moving car carries four times

as much kinetic energy in its translational motion as the slower moving

car. And that explains why collisions that

occur at those high speeds are a problem and modern cars are designed to dissipate

that enormous excess in energy, in this case in the form of kinetic energy.

They dissipate both by, well by grinding it up in part into thermal energy and in

part into the deformation and destruction of safety zones in the car.

Portions of the car that are intended to have work done on them during collisions

and to absorb that energy in a way that's safe.

It's time for a question. A ten year old child, who is a good

baseball pitcher can throw the baseball at 50 miles an hour as a fastball, that's

about 80 kilometers an hour, but it takes a world class professional pitcher to

throw a fastball that travels 100 miles an hour or 160 kilometers an hour.

Compare the kinetic energies in those two pitches.

The faster moving baseball is carrying four times as much kinetic energy with

it. Even though it's only traveling twice as

fast. That enourmous increase in kinetic energy

explains why it's so hard to throw 100mph fast ball and why only a few people have

been able to do it. Moreover, they throw that ball in a

shorter period of time than a person throws a 50mph fastball.

They're pouring their energy into, they're pouring four times the energy

into it, In roughly half the time, which involves a much more rapid transfer of,

of energy, that is more power. Throwing a 100 mile an hour fastball

requires something like eight times as much mechanical power as throwing a 50

mile an hour fastball. Now let's look at the kinetic energy

associated with the wheel's rotational motion.

That kinetic energy is the work it takes to bring the motionless wheel up to its

final angular velocity. [SOUND] .

The rotational story will be almost exactly the same as the translational

story. But this time, I'm going to do rotational

work by exerting a torque on the wall as it rotates through an angle.

Well, as before, as with a translational story.

I'm not going to do everything. I'm going to look instead at some

important observations. And the first of those, there's three of

them again, the first of those is that the direction of angular velocity doesn't

matter, whether i spin it from my perspective route you know clockwise

or[SOUND] Counterclockwise, that's, this is rotation down.

This is rotation up. It doesn't matter.

I do the same work spinning it this way as spinning it this way.

So, what matters is not angular velocity, the whole thing, but rather angular

speed. The ammount part of angular velocity.

The direction part of directional velocity, not important.

Because once again, energy is about doing, not about moving, direction

doesn't matter. Second observation, this one is about

rotational mass. It turns out that the kinetic energy

associated with rotation in a wheel is proportional to the wheel's rotational

mass. To show you that, I'm going to get my

second wheel. So here we have a second wheel, same as

the first, but instead of. Moving around separately.

I'm going to sit this wheel on top of the first wheel.

Well if it took me one unit of work to get this guy spinning like that, it's

going to take two units of work to get this stack, if I can do it, spinning like

that. Basically, the amount of work I have to

do rotationally, to get one of these wheels spinning is proportional to it's

rotational mass. If I double the rotational mass, takes

twice as much work. Ten times the rotational mass?

Ten times the work. We know now that the kinetic energy in an

objects rotational motion is proportional to its rotational mass.

The third part, the tough one, put this guy away.

Is how the kinetic energy in the rotational motion of the wheel depends on

rote on angular speed and once again that kinetic energy is proportional to angular

speed squared for the same reason, if I exert a steady torque on this wheel it

undergoes angular acceleration steadily, constant angular acceleration.

And it spins faster and faster and faster, so its angular velocity is then

proportional to how long I twist it, the amount of time over which I twist it.

If I twist it twice as long it ends up spinning at twice the angular velocity.

[NOISE] But the work I do doesn't go up proportionately with time for exactly the

same reason that we saw with translational motion.

As I spin this little guy, I exert a torque on it, a constant torque, and it

undergoes a major acceleration. It goes faster and faster.

The work that I'm doing depends on the angle through which it rotates.

It's the torque I exert times the angle through which it rotates.

And as it picks up angular speed it rotates farther and farther with each

additional second. So I'm chasing it around in a circle.

The first second it doesn't turn very far, and I do relatively little work on

it. The second second it turns farther and I

do more work on it. Third second, farther still, and so on,

I'm chasing it around the circle. So the energy, the work I have to do on

it skyrockets and so does its kinetic energy.

The kinetic energy in a wheel's rotational motion turns out to be equal

to one half the wheel's rotational mass times the square of its angular speed.

Fast-spinning wheels can therefore carry enormous kinetic energies, even when

they're rotating in place. The wheels on a moving vehicle however

are translating and rotating at the same time.

What about their kinetic energies? It turns out that those kinetic energies

add up. When your bicycle is moving forward, each

wheel has translational kinetic energy associated with the velocity of its

center of mass. And rotational kinetic energy associated

with its angular velocity about its center of mass.

That double dose of kinetic energy distinguishes the wheels from the rest of

the bicycle. Bicycle wheels carry more kinetic energy

per kilogram than any other part of the bicycle.

Since you have to provide that energy as work when you peddle the bicycle forward

from rest. The wheels make your job more difficult.

Similarly, the bicycle's brakes have to get rid of that extra energy when they

slow the bicycle to a stop. Ao, because of these effects, extra

energy in, extra energy out, in the wheels.

Those wheels are generally designed very carefully to reduce both their masses,

and their rotational masses. Replacing a solid rubber tire with an air

filled pneumatic one, not only reduces the wheel's weight, so it's easier to

lift and carry uphill. It also reduces the wheel's mass and

rotational mass and thus reduces the kinetic energy that the wheel carries as

it moves.

How is Energy Present in a Wheel?

生活中的物理

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