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Chapter 21. think like a physicist

Chapter 21. think like a physicist

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a wonderful physics world

You’ve come a long way!

Back in chapter 1, the words on the globe

looked like random jargon that didn’t make

sense. But as you’ve gone through the book

learning to relate physics to everyday life,

your confidence has grown and the words

have become less like an alien language.

Units

Falling

Energy Conservation

Scalar

Inelastic collision

Special points

Frequency

Centripetal force

Angular frequency

Component

Weight

Spring

Torque

Elastic potential energy

Impulse

Equation

Constant acceleration

Distance

Scientific notation

Displacement

Friction

Trigonometry

Angular velocity

Freefall

Kinetic energy

“How can I use

what I know

to work out

what I don’t

know (yet)?”

Symmetry

Inverse square law

Acceleration

Graph

Experiment

Period

Force

Elastic collision

Pendulum

Simple Harmonic Motion

Gravitational field

Momentum conservation

Circumference

Slope

Internal energy

Area

Pythagoras

Substitution

Pulley

Time

Tension

Energy

Equations of motion

Normal force

Be part of it

Vector

Speed

Gravitational potential energy

Mechanical energy

Velocity

Work

Amplitude

Volume

Power

Free body diagram

Newton’s Laws

Does it SUCK?

Mass

Now you’re in chapter 21 - and you’re able to use these same words to

“How can I use what I know to work out what I don’t know (yet)?”

840   Chapter 21

think like a physicist

Now you can finish off the globe

What better way to use your physics superpowers than to

revisit the tunnel through the center of the Earth and really

get to grips with what happens there.

Back in chapter 1, you learned to be part of it by putting

yourself at the heart of the problem and asking “What

happens next?” and “What’s it like?” Now you can also

describe what happens using physics terms and concepts.

You spotted that there’s a special point in

the center where there’s no net force on you

because everything is symmetrical - you’re

equally attracted in all directions, and as

gravity is a non-contact force, you don’t feel

crushed. You also realized that you’re always

attracted towards the center unless you’re

Be part of it

You can use physics

terms and concepts

to describe what

happens next.

a. Use your increased physics knowledge to revisit

the question “What’s it like?” What does the trip

through the Earth now remind you of? Be sure to

mention all the parallels you can see.

Special points

Symmetry

Acceleration

Force

This means that you initially accelerate as

you fall due to the net force on you at the top

of the tunnel, briefly move with a constant

velocity through the center, then decelerate as

the gravitational force continues to attract you

towards the center. After briefly emerging at the

other end, you do the same thing again in reverse.

Newton’s Laws

b. Are there any requirements that need to be met

Newton’s

2nd Law

Newton’s

1st Law

Velocity

you are here 4   841

shm revisited

a. Use your increased physics knowledge to revisit

the question “What’s it like?” What does the trip

through the Earth now remind you of? Be sure to

mention all the parallels you can see.

b. Are there any requirements that need to be met

It looks like simple harmonic motion.

There’s an equilibrium point in the center where

there’s no net force on you.

The force on you always acts towards the

equilibrium position in the center, so is in the

opposite direction from the displacement.

You move slowly at the edges and quickly

through the center.

The restoring force would have to be

the equilibrium position, and in the opposite

direction from the displacement.

That’s the requirement for simple harmonic

motion.

Displacement

Force

Simple Harmonic Motion

The round-trip looks like simple harmonic motion

Way back in chapter 1, you figured out that

you’d fall into the tunnel, travel through the

Earth, and appear at the other end of the

tunnel. Then you’d fall back in again, go

through the tunnel in the opposite direction,

and end up where you started... and so on.

That sounds a lot like simple harmonic motion,

something you learned about in chapter 20. Now

that you know how to tackle simple harmonic motion

problems, we can add a lot of detail to what we

figured out before.

?

What’s it ssLaIKsitEuation you

ro

If you come ac fore, think about

e

b

haven’t seen seen something

similar in the p ’s it like?”

yourself: “What

x

Equilibrium

position

Tunnel entrance

Center of Earth

842   Chapter 21

Other side

of Earth

t

think like a physicist

But what time does the round-trip take?

A trip through the center of the Earth

can be pretty tiring. Suppose you want to

place a pizza order so that once you’ve

gone through the Earth and come back

again, you can have a nice snack.

45

Break Neck Pizza promises delivery in

45 minutes... but will you be able to get

through the Earth and then back again

in time to meet Alex the delivery guy?

What time does your journey through

the Earth take?

Yeah - are you gonna

get back before I

Who arrives first

- you or Alex the

delivery guy?

First of all, you’ll have to work out if your journey through the

Earth IS simple harmonic motion. The journey might only look

like simple harmonic motion without actually being simple

harmonic motion.

Then, if your journey is simple harmonic motion, you can use the

equations you worked out in chapter 20 to calculate the time that

the round-trip takes - and whether you’ll be back in time for pizza.

Back in time for pizza?

Trip through Earth looks like SHM.

Is it SHM? Is the restoring force

proportional to the displacement?

If it’s SHM, can use SHM equations

to calculate time that trip takes.

you are here 4   843

context matters

So that’s that! Going through

the Earth and back again is

simple harmonic motion.

simple harmonic motion

The net force on the

object that always points

in the direction of the

equilibrium position.

Joe: Hang on. It’s only SHM if the restoring force is directly

proportional to the displacement from the equilibrium position

of the object that’s moving to and fro. We don’t know whether the

restoring force follows that pattern or not yet.

Frank: Yeah, if the trip through the Earth isn’t SHM, we can’t use our

SHM equations to calculate the time it takes.

Joe: The gravitational force is an inverse square law, isn’t it?

Gm1m2

. So if you double the displacement, the force is only a

FG = –

r2

quarter of what it was before. The force isn’t directly proportional to the

displacement. The force gets smaller as the displacement gets larger!

Frank: Oh yeah. For SHM, the restoring force needs to get larger as

the displacement gets larger.

Equation

Inverse square

Does it SUCK?

Anytime you want

to use an equation,

CONTEXT. Is

it OK to use the

equation here?

You can only

harmonic motion

equations from

chapter 20 if the

restoring force is

proportional to

the displacement.

Jim: Hey - didn’t we say before that the force is zero in the center of

Gm1m2

works at the center of the

the Earth?! If the equation FG = –

r2

Earth (where r = 0), you’re dividing by zero. If you divide by a very

small number, you get a very large answer. And if you divide by zero,

you get an answer of infinity! Computer says no!

Gm1m2

only works when you’re outside the

Joe: Yes ... maybe FG = –

r2

Earth. When you’re outside, all of the Earth is below you.

Frank: But when you’re inside the tunnel, some of the Earth is below

you and attracts you downwards. The rest of the Earth is above you

and attracts you upwards.

Jim: So the net force on you in the center is zero, as there are equal

masses of Earth above and below you. And the net force on you

somewhere else in the tunnel depends on how much Earth is above you

and how much Earth is below you.

Joe: Looks like we need to work out a different equation for when

Gm1m2

isn’t going to work.

you’re inside the Earth then, if FG = –

r2

Calculating the net force on you when you’re inside the Earth is a complicated

problem. How might you break the problem down into smaller parts?

844   Chapter 21

think like a physicist

You can treat the Earth like a sphere and a shell

Gm1m2

The gravitational force between two spheres is FG = –

. You

r2

can treat each sphere as if its entire mass was concentrated at a single

point in the center of the sphere.

If you treat the human body like a very small sphere, you can use this

equation to calculate the gravitational force that the Earth exerts

on you - as long as you’re outside the Earth.

But when you’re inside the tunnel, there’s Earth above you and

below you. Calculating the gravitational force that the Earth exerts on

you when you’re inside it is a complicated problem!

Try to break

down complicated

problems into

smaller parts.

You are here.

But you can break down this problem into two parts by

thinking of the Earth in two parts. Anytime you’re inside the

Earth, you’re a certain distance (let’s call this distance r) from

the equilibrium position in the center of the Earth.

r

know how to calculate the gravitational force exerted on you

Gm1m2

by a sphere if you know its radius and mass: FG = –

r2

for the gravitational force

from a sphere!

Sphere with

Back in time for pizza?

Trip through Earth looks like SHM.

Is it SHM? Is the restoring force

proportional to the displacement?

Force from shell?

Force from sphere?

We’re defining the radius as the displacement away from

the center of the Earth. The force acts towards the

center of the Earth - hence the minus sign.

You are here.

The rest of the Earth forms

a ‘shell.’ Some of the shell

is below you and some of the

shell is above you.

If you can work out an

equation for the force exerted

on you by the shell, you can

add it to the force exerted

on you by the sphere. This

gives you the net force that

the Earth exerts on you while

you’re inside the tunnel.

‘Shell’

r

RE

Earth is RE.

If it’s SHM, can use SHM equations

to calculate time that trip takes.

you are here 4   845

You can deal with the sphere - but what about the shell?

Use the magnets to work out the force from the shell. Annotate the diagrams as you go.

There’s more shell

is

you than there

you, so there’s more

you than there is

the shell

You are here.

you. But

you is (on average) a much larger

away from you than the shell

you.

above

below

distance

mass

left

below

above

upwards

right

The shell above you is totally                  . So for every small piece of Earth

to the                   of you there’s an equivalent piece to the                  .

The                   components of the gravitational force from these two pieces

of Earth are equal but in                   directions, so they                  .

The                   components add together, so the net force on you from

the shell above you is                  . The same argument applies to the

shell below you, which exerts a net                   force on you once the

components                  . So the part of the shell                  you attracts

you                   and the part                   you attracts you

symmetrical

downwards

opposite

horizontal

vertical

If you take a piece of very thin shell, its                  will be

its                 multiplied by its                 . So the

mass

of the very thin shell will depend on its surface area.

thickness

surface area

846   Chapter 21

volume

think like a physicist

This thin shell is like a tiny layer on the outside.

a is this

If you take a small slice of thin shell from

a

short

distance

and the equivalent from                  , you can

think of them as being tiny slices from spheres with

b

areas                  and                  respectively.

b is this

long

distance

And as the                  of a thin shell depends on its

surface area, the masses of the slices are proportional

to                  and

inverse square

below

a

Gravitation is an                  law. The slice of Earth

above you is distance                  away, so the force from

a2

1 kg of it is proportional to                   The slice of

Earth below you is distance                  away, so the force

b2

You’re

mapping

every

point on

the top to

every point

on the

bottom.

from 1 kg of it is proportional to                  Therefore,

the force from the slice at distance ‘a’ is proportional to

1

a2

×                  and the force from the slice

at distance ‘b’ is proportional to                  ×

b

volume

1

b2

above

So the force from the slice above you is                  to

the force from the slice below you. And the net force from

the whole thin shell is                  If you’re further

Now we’ve filled

in the shell with

many tiny thin

shells like the one

in the box above.

inside the Earth, you can think of the thick shell being

made up of many many thin shells. So the net force from the

thick shell is also

zero

equal

volume

Use this space to sum up

what you’ve discovered:

you are here 4   847

pay attention to symmetry

Magnets Solution

There’s more shell below you than

there is above you, so there’s more

mass below you than there is above

you. But the shell below you is (on

average) a much larger distance away

from you than the shell above you.

Shell

above

you.

You are here.

Shell below you.

In problems

that involve

SYMMETRY,

there are often

force components

The shell above you is totally symmetrical. So for every small piece of

Earth to the left of you, there’s an equivalent piece to the right. The

horizontal components of the gravitational force from these two pieces

of Earth are equal but in opposite directions, so they add to zero.

The vertical components add together, so the net force on you from the

shell above you is upwards. The same argument applies to the shell below

you, which exerts a net downwards force on you once the horizontal

components add to zero. So the part of the shell above you attracts you

upwards, and the part below you attracts you downwards.

to produce a net force.

If you take a piece of very thin shell,

its volume will be its surface area

multiplied by its thickness. So the

mass of the very thin shell will

depend on its surface area.

848   Chapter 21

Surface area

Thickness

Mass depends

on surface area.

think like a physicist

Here, we’re using bold to show you where the magnets

went. The distances a and b are both scalars.

If you take a small slice of thin shell from above and the

equivalent from below, you can think of them as being tiny slices

from spheres with radius a and b with surface areas a2 and b2

respectively. And as the mass of a thin shell depends on its surface

area, the masses of the slices are proportional to a2 and b2.

Gravitation is an inverse square law. The slice of Earth above you

is distance a away, so the force from 1 kg of it is proportional to

1

a2 . The slice of Earth below you is distance b away, so the force

1

from 1 kg of it is proportional to b2 .Therefore, the force from the

1

slice at distance ‘a’ is proportional to a2 × a2 and the force from

1

the slice at distance ‘b’ is proportional to b2 × b2 .

Force from

slice above you.

Force from

slice below you.

Small mass,

very close

to you.

a

b

Larger mass - but

also larger distance.

So the force from the slice above you is equal to

the force from the slice below you. And the net force

from the whole thin shell is zero. If you’re further

inside the Earth, you can think of the thick shell

being made up of many many thin shells. So the net

force from the thick shell is also zero.

Forces are equal and

opposite, so net force

from shell is zero.

Use this space to sum up

what you’ve discovered:

The net force from the shell is zero. This is because the forces from the small, close mass

above you and the large, far away mass below you are the same size, but in opposite directions.

The net force from the shell is zero!

you are here 4   849

zero net force

The net force from the shell is zero

This means that the net force that the shell exerts on you is

zero. So the net force exerted on you when you’re inside the

tunnel must come entirely from the sphere, radius r.

If that force is proportional to r, then you’ll move through

the Earth with simple harmonic motion and can use the

equations you already know to calculate the time you take to

get back home again.

You are here.

Net force

from shell

is zero.

r

Back in time for pizza?

Trip through Earth looks like SHM.

Is it SHM? Is the restoring force

proportional to the displacement?

Force from shell?

This is zero!!

Force from sphere?

If it’s SHM, can use SHM equations

to calculate time that trip takes.

Net force on you at

position r comes from

Q:

Do I need to understand and

reproduce all of that?!

A:

Don’t worry - you won’t be asked to

do something that difficult in an exam. The

big take-away is that the net force from the

shell is zero because the forces from a small

mass close by and a large mass far away

added to zero. If you got that, you’re great!

Q:

But the Earth isn’t a sphere and a

shell ... is it?!

A:

Treating the Earth like a sphere and a

shell is a mathematical tool. In the same

way, moving objects don’t have velocity

vector arrows and components drawn on

them in real life, but vector arrows are very

useful tools in physics.

Q:

A:

What’s so special about a shell?

You’ve worked out that the force the

shell exerts on you is zero. So the net force

on you must come entirely from the sphere.

Q:

A:

What’s so special about a sphere?

You already know how to calculate the

gravitational force an object experiences as

a result of being outside a sphere.

Q:

Doesn’t the equation for the

gravitational force exerted by a sphere

only work if I’m outside the sphere?

A:

When your displacement is r from the

center of the Earth, then you’re outside the

850   Chapter 21

Q:

r, as the place to draw the boundary

between the sphere and shell?

A:

r is your displacement from the

equilibrium position in the center of the Earth.

When we did SHM in chapter 18, we called

this displacement x. Here it’s better to use r

so that you remember that the displacement

from the equilibrium position is also a radius.

If, when your displacement is r, the force

exerted on you by the Earth is proportional

to r, then your trip through the Earth is SHM.

The period of the SHM is the same as the

time it takes you to get back to where you

started - the time you want to calculate!

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