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Chapter 17. circular motion (part 2)

# Chapter 17. circular motion (part 2)

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artificial gravity

Houston ... we have a problem

Astronauts at the Head First space station are

threatening to go on strike. They’re fed up with floating

around all the time. The astronauts want to be able to

walk around the station just like they can on Earth.

You’ve been called in to create artificial gravity for an

add-on to the station... and keep those astronauts happy.

Space station

Fi

rst

e

The astronauts ngar.

ti

oa

tired of fl feel like

They want to on the

they’re walking e!

Earth - in spac

Astronaut

664   Chapter 17

circular motion (part 2)

How can the space station be in orbit

around the Earth when gravity attracts it

towards the Earth?

If an object

is in freefall,

the only force

acting on the

object is its

own weight.

1

The space station is in FREEFALL.

The only force acting on the space station is the force of

its weight - the gravitational attraction it experiences

from the Earth. It’s not touching anything else, so there

are no contact forces on it. And the lack of atmosphere

means that there’s no friction either as it orbits the Earth.

If an object is in freefall, then the only force acting

on the object is its own weight. So the space station is in

freefall. The same is true for the astronauts - the only force

acting on each astronaut is that astronaut’s weight.

But when the force of an object’s weight acts down

towards the center of the Earth, how can the object

possibly orbit the Earth by going around the Earth?!

Suppose you fire a cannon. The

cannonball is in freefall, because the

only force acting on the cannonball is its

own weight. So the cannonball follows

a curved path as it falls and hits the

Earth.

In freefall - only force acting

on cannonball is its weight.

Lands here if

Earth is flat.

2

Locally, the Earth appears flat, but it’s

really round. If the cannonball is very

fast or starts off very high, it goes further

before it lands because the surface of the

Earth curves away from the cannonball

as it falls.

Where surface of Earth

would be if it was flat.

Earth is round, so the

cannonball actually goes

further and lands here.

3

If the cannon is high

enough and the cannonball

is fast enough, the surface

of the Earth keeps on

curving away as the

cannonball’s flightpath

curves down. The

cannonball keeps on

freefalling round the Earth in orbit - just like the space

station and astronauts.

For a tall cannon and

high speed, cannonball

freefalls forever - and

orbits the Earth.

Cannonball is

orbiting, and is

in freefall too.

The astronaut is in freefall - the only force

acting on him is his weight. So why does

he feel weightless in the space station?

you are here 4   665

freefall

When you’re in freefall, objects

appear to float beside you

Suppose you’re midway through a parachute jump, but you

haven’t opened your chute yet. If you let go of an apple as

you fall, the apple will fall with the same velocity as

you. It looks like the apple’s just floating there.

You and apple are falling at

the same rate.- so the apple

stays beside you.

Of course, if you’re doing a parachute jump, there are

some clues that indicate that you and the apple are both

falling. The Earth gradually looks bigger, and you can feel

the wind rushing past you!

The box is also

falling with the

same velocity

- so you float

around the box!

Apple’s

velocity

But if you and the apple were in a soundproof

windowless box, you wouldn’t have any visual or

audible clues that you’re falling. The apple would

appear to float next to you! And as the box is also

falling at the same rate as you, you would appear

to float around inside the box. If you pushed

yourself up off the floor, you wouldn’t fall back

down again like you would on Earth.

Your

velocity

rst

It’s the same for an astronaut in the space

station. He’s continually falling around the

Earth. So if he lets go of an apple, it floats

around the space station rather than falling

towards the ground.

Fi

Box’s

velocity

Your

velocity

This time you

can’t see that

you’re falling

because of the

box. So the apple

appears to float

next to you.

Apple’s

velocity

This makes it look as though the apple is

weightless - and as though the astronaut is

weightless. The astronaut is used to the force

of his weight attracting him towards the

ground. But here, the “ground” (the wall of

the spacestation) is falling at the same rate as

the astronaut

The space station is like a

box that’s in freefall. So

everything appears to float

around in the space station

because everything’s falling

at the same rate as it.

This means that the astronaut can’t walk

around the space station like he can on Earth.

Astronaut’s

velocity

666   Chapter 17

Space station’s

velocity

circular motion (part 2)

What’s the astronaut missing,

compared to when he’s on Earth?

If an astronaut is in freefall, then he and the objects in the space station

will appear to float around because everything’s falling at the same rate.

But why does a dropped apple in an orbiting space station appear to act

so differently from a dropped apple on Earth - when they’re both falling

because of the force of their weight, as usual? And why isn’t it possible

to walk around normally in the space station like you can on Earth?

When you’re dealing with forces, always start with a free body diagram...

a. Draw a free body diagram of all the forces the

astronaut would experience while standing on Earth.

For problems

involving forces,

free body diagram.

b. Draw a free body diagram of all the forces an

astronaut would experience while in freefall.

c. When you compare the free body diagrams in parts a and b, which force is missing?

d. How does a person on Earth experience their weight differently from a person in freefall?

e. How might you introduce a new force to compensate for the missing one you spotted in parts c and d,

so the astronaut can walk around as he would on Earth?

you are here 4   667

contact force

a. Draw a free body diagram of all the forces the

astronaut would experience while standing on Earth.

Normal / contact

force from ground.

Weight

mg

b. Draw a free body diagram of all the forces an

astronaut would experience while in freefall.

These two forces are

equal and in opposite

directions, so there’s

no net force and you

don’t accelerate.

Weight

mg

In freefall, the only

force acting on you

c. When you compare the free body diagrams in parts a and b, which force is missing?

The astronaut in the space station doesn’t experience a contact force from the ground.

d. How does a person on Earth experience their weight differently from a person in freefall?

They don’t feel a contact force from the ground - they’re in freefall, and the only force

they experience is their weight. This means that they can’t walk around like they can on

Earth, as you need a normal force to have enough friction to walk like they usually do.

e. How might you introduce a new force to compensate for the missing one you spotted in parts c and d,

so the astronaut can walk around as he would on Earth?

Newton’s 2nd law is F = ma. You could make the astronaut experience a force by

accelerating the space station.

Q:

So why do we need artificial gravity

when the space station still feels the

effect of the Earth’s gravity?

A:

If you’re in freefall, you feel weightless

because you - and other objects - appear to

float around, as they’re not going anywhere

with respect to each other.

Q:

A:

But you still have a weight, right?

Yes - just like someone doing a

parachute jump still has a weight. It’s the

force of their weight that makes them fall!

Go back and look at the first

WeightBotchers machine in

chapter 11 if you’re not sure

why this is.

668   Chapter 17

Q:

So why is someone in freefall

called “weightless” when they have a

weight? Isn’t that confusing?

A:

Yes, it is confusing! “Weightless” is

an everyday way of saying that they aren’t

experiencing any kind of contact force

from a surface as a result of their weight. If

the person had scales under their feet, the

circular motion (part 2)

Can you mimic the contact

force you feel on Earth?

The difference between standing on the

ground and freefalling is the contact force

that the ground exerts on you. This is the

force that the astronauts want to experience.

If you can make each astronaut experience a

contact force equal to the size of his weight,

as he does on Earth, he should be able to

walk around the space station just like he can

on Earth.

But how can you make someone experience

a contact force like this?

On Earth, you experience

this contact force.

Contact

force

Weight = mg

In freefall, the only

force acting on you

Weight = mg

On Earth, you experience

contact force from the ground.

The key thing is to close your

eyes and ask “WHAT DO I

FEEL PUSHING ON ME?”

Imagine yourself in these scenarios. Draw the contact force you experience in each situation as a result of

the acceleration, and write down what you FEEL. For instance, “Something’s pushing me in the back.”

All passengers are

wearing seatbelts.

Train is sitting

still, then

accelerates to the

right as it pulls

out of a station.

Train is moving

to the right, then

decelerates to a

stop as it pulls

into a station.

Does this give you an idea about how you might make the astronaut experience a contact force from the

inside wall of the space station that would feel similar to the one he experiences on Earth?

you are here 4   669

imagine yourself

Imagine yourself in these scenarios. Draw the contact force you experience in each situation as a result of

the acceleration, and write down what you FEEL. For instance, “Something’s pushing me in the back.”

Train is sitting

still, then

accelerates to the

right as it pulls

out of a station.

The back of

the seat is

pushing on me.

Train is moving

to the right, then

decelerates to a

stop as it pulls

into a station.

F

The seatbelt

is digging

into me.

The seatbelt

is digging

into me.

F

All passengers are

wearing seatbelts.

F

The back of

the seat is

pushing on me.

F

Does this give you an idea about how you might make the astronaut experience a contact force from the

inside wall of the space station that would feel similar to the one he experiences on Earth?

You could accelerate the space station. That would make the astronaut experience a contact

force - like I do when a train accelerates.

2

If you accele

rate it at 9.8 m/s , then this

contact force will be exactly the same

size as the one he experiences on Earth.

Q:

I can imagine the seat pushing into

my back when a train accelerates. But

why does that happen?

A:

Newton’s 1st law says that an object

will continue to move at a constant velocity

unless it’s acted on by a net force. If you

were sitting on the platform, the fact that the

train is accelerating wouldn’t affect you, as

there’s no contact between you and it.

But because you’re sitting on the train, the

back of your seat is able to mediate a net

contact force that causes you to accelerate.

You feel the seat pushing into you.

670   Chapter 17

Q:

A:

How large is the contact force?

Newton’s 2nd Law says that F = ma.

You can work out the size of the contact

Q:

How can I make the astronaut feel a

contact force?

A:

If you accelerate the space station

upward, the astronaut will experience a

contact force from its wall because he is

inside it - just like you do when you’re on

the train.

In the context of forces,

‘mediate’ means ‘transmit’.

Q:

Why will accelerating the space

station mean that a dropped apple will fall

like it does on Earth?

A:

The apple will continue at the velocity it

no contact force on it while it is falling.

Meanwhile, the space station will accelerate

up to meet it at a rate of 9.8 m/s2. So if

you’re in the space station, it feels like you’re

on Earth (because of the contact force you

experience), and it looks like you’re on Earth

because objects accelerate towards the

ground at the same rate.

circular motion (part 2)

Accelerating the space station allows

you to experience a contact force

If you lie on the ground with your

eyes shut, you can feel a contact

force from the ground pushing

Contact

force

If the train pulls away from the

station with an acceleration of

exactly 9.8 m/s2, then you would

feel exactly the same size of

contact force as you do when you

lie on the ground.

Train

accelerating

from left

to right.

a. Einstein’s Theory of Relativity says that nothing can move

faster than the speed of light, 3.0 × 108 m/s. If you accelerate a

space station from rest at a rate of 9.8 m/s2, what time would it

take it to reach a speed of 3.0 × 108 m/s (assume for a moment

that this is possible and there are no relativistic effects)?

b. What distance would the space station cover in that time?

Contact

force

So if you accelerate the space

station at a rate of 9.8 m/s2, the

astronauts will experience the

same size of contact force as they

usually experience when they’re

standing on Earth. This creates

the artificial gravity that the

astronauts want!

Space station

accelerating

upwards.

Don’t be intimidated

by the first four

words of this problem!

Contact

force

c. The distance from the Earth to the Moon is 4 × 108 m (1 sd),

and the distance to the edge of the Solar System is 5.7 × 1012 m.

How does the distance you worked out in part b compare?

d. How practical do you think this idea is for creating artificial gravity

in a space station?

But how practical is this?

you are here 4   671

impractical linear acceleration

a. Einstein’s Theory of Relativity says that nothing can move

faster than the speed of light, 3.0 × 108 m/s. If you accelerate a

space station from rest at a rate of 9.8 m/s2, what time would it

take it to reach a speed of 3.0 × 108 m/s (assume for a moment

that this is possible and there are no relativistic effects)?

Work out t: v = v0 + at

v0 = 0 m/s

v = 3.0 × 108 m/s

a = 9.8 m/s2

t = ? x - x0 = ?

But v0 = 0

3.0 × 108

t = va = 9.8

t = 3.1 × 107 s (2 sd)

b. What distance would the space station cover in that time?

Work out x :

x = x0 + v0t + ½at2

x = 0 + 0 + 0.5 × 9.8 × (3.1 × 107)2

x = 4.7 × 1015 m (2 sd)

1000 times further than the

edge of the Solar System?

That’s waaay too far!

c. The distance from the Earth to the Moon is 4 × 108 m (1 sd),

and the distance to the edge of the Solar System is 5.7 × 1012 m.

How does the distance you worked out in part b compare?

This distance is around 10 million times greater than the

Earth-Moon distance and a thousand times greater than

the edge of the Solar System.

d. How practical do you think this idea is for creating artificial gravity

in a space station?

It’s not practical because it’s not possible to sustain it for a

good length of time, and you end up very far away from the

Earth. It must take a lot of fuel too.

Linear means “along

a straight line.”

672   Chapter 17

It’s (theoretically) possible to

make the astronaut experience

a contact force similar to the

one he experiences on Earth by

accelerating the space station

along a straight line at 9.8 m/s2.

But it’s not practical. It’s

impossible to do this indefinitely,

since the space station can’t go

faster than the speed of light,

you’d run out of fuel, and you’d

wind up a ridiculously long way

away from the Earth.

So if linear acceleration isn’t

practical, what might another

option be?

circular motion (part 2)

So if we can’t accelerate the space

station linearly, what can we do?!

Jim: I wonder if there’s another way of experiencing a contact

force, apart from accelerating or decelerating along a straight line?

Joe: Hmmm ... what about those carnival rides where you go

around in a circle? You kinda feel the side of the car pushing on

you when they spin really fast ...

Frank: You feel the side of the car pushing you, so there must be a

contact force. But where does it come from?

Jim: Yeah, it’s not like the ride gets faster and faster. It spins at the

same rate, so you keep going at a constant speed, yet you still feel

this contact force from the side of the car. How can you feel a force

if your speed is constant - doesn’t that break Newton’s 1st Law?

Joe: But the direction you’re traveling in is changing all the time.

constant. Velocity is a vector. Newton’s 1st Law says that you move

with a constant velocity unless there’s a force acting on you.

Frank: So I guess the contact force changes your direction of

travel - which changes your velocity - so causes you to accelerate.

Jim: But where does the force come from?! It’s not like there’s a

train engine sitting behind you making you accelerate!

Frank: Well, you’re thrown to the outside of the ride, aren’t

you? So there must be some kind of mysterious force pushing you

outwards that’s only there when you’re spinning.

Frank: Hang on! When you’re thinking about contact forces,

you’re meant to shut your eyes and ask, “What do I feel pushing

on me?” And when I do that, I feel the side of the car pushing me

inwards, not a ‘ghost force’ pushing me outwards.

If a contact force is

acting on you, you can

feel the direction it’s

pushing you in.

Jim: But you slide kinda outwards across your seat before you make

contact with the side of the car. If there isn’t a force pushing you

outwards, then why does that happen?!

Joe: If the side of the car wasn’t there, you’d go straight on and fly

out of the car. You only go in a circle because of the contact force

from the side of the car pushing you inwards.

Frank: Ah ... you mean that sliding outwards feeling is just you

continuing on at your current velocity (Newton’s 1st Law) before

you make contact with the side of the car - which exerts a force on

you that lets you move in a circle?

Can you imagine yourself

in the spinning carnival

ride and the contact force

acting on you?

you are here 4   673

centripetal force

You can only go in a circle

because of a centripetal force

If there’s no net force acting on you,

you just keep on going with this velocity.

v

Rotating

carnival ride

Newton’s 1st Law says that you continue with a

constant velocity unless there’s a net force acting

on you. In other words, you keep on going at the

same speed in the same direction.

v

If you’re going around in

stay the same, but the

vector changes.

v

You actually wind up here with a

different velocity, so there must

be a net force acting on you.

If you’re going around in a circle, your speed may be

constant, but the direction of your velocity is certainly

changing! This means that a force must be acting on

you in order to make you go around in a circle - and

stop you from going off along a straight line with the

A force that allows you to go in a circle like this is called

a centripetal force.

Err ... I can walk in a

circle without needing a

centripetal force to do it!

When you walk in a circle, the centripetal

force is provided by friction.

Centripetal force is the name given to a net force that

allows you to change the direction of your velocity

so that you follow a circular path. Depending on the

context, centripetal force can be provided by a number

of things.

You’re able to walk because of the friction between

your feet and the ground. Without friction, you wouldn’t

be able to change the horizontal component of your

velocity at all. You couldn’t speed up. You couldn’t slow

down. And you couldn’t change direction to follow a

circular path. So in this case, friction provides the net

force that enables you to follow a circular path - the

centripetal force.

674   Chapter 17

If the net force

acting on you

changes the

direction of your

velocity so that

you travel in a

circle, it’s called a

centripetal force.

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Chapter 17. circular motion (part 2)

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