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
Radians
Normal force
Be part of it
Vector
Speed
Gravitational potential energy
Mechanical energy
Velocity
Radius
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
help you think through and solve problems. You’ve learned to ask
“How can I use what I know to work out what I don’t know (yet)?”
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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
already in the center.
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
in order for your answer to part a to make sense?
Newton’s
2nd Law
Newton’s
1st Law
Velocity
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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
in order for your answer to part a to make sense?
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
proportional to your displacement from
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
whether you’ve ast by asking
similar in the p ’s it like?”
yourself: “What
x
Equilibrium
position
Tunnel entrance
Center of Earth
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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
arrive with your pizza?
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.
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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,
think about the
CONTEXT. Is
it OK to use the
equation here?
You can only
use your simple
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?
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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
So beneath you, there’s a sphere with radius r. You already
know how to calculate the gravitational force exerted on you
Gm1m2
by a sphere if you know its radius and mass: FG = –
r2
You already know an equation
for the gravitational force
from a sphere!
Sphere with
radius 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?
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
Radius of the
Earth is RE.
If it’s SHM, can use SHM equations
to calculate time that trip takes.
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what about the shell?
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
add to zero
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
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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
radius and with surface
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:
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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
that add to zero.
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.
Horizontal components 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.
Vertical components add
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.
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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!
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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
sphere with radius r.
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
sphere with radius r.
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Q:
Why choose that particular radius,
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!