GPS Module 3 - Part 2

Chapter 3 - Newton’s Laws
Weight
W = m g
The pull of gravity at the earth’s surface was given before by:
. . . but this is not very complicated. What happens if you
go into earth orbit or to the moon? We need something
more complex to handle other possibilities.

Newton’s Law of Universal Gravitation:
The classic cartoon image of this subject is Newton
sitting under a tree; an apple falls; hits him on the head
and ‘aha gravity’. But the pull of gravity is obvious to
all. The Greek theory was that objects naturally seek
the earth as their source (specifically ‘earth’ element
objects are attracted to the earth – flames are attracted
to the sun, water to the ocean, etc.).
In Newton’s time, they were just beginning to see the moon as a big ‘rock’ in
the sky and they were trying to figure out solar system structure and motion.
An apple falls off the tree, but why doesn’t the moon fall to the ground.
What would happen if you went to the moon? Would you still be hanging off
dangling toward the earth? These were the big questions in Newton’s time.
In the mid-1600’s a radical new idea was being bounced around (by Newton
and perhaps others).

The two members of every pair of
objects in the universe exert a
gravitational force of attraction on
each other. Gravity is not just an
‘earth’ thing, but is a universal
property of all matter.
This figure is supposed to
illustrate this idea: there is a
pull of gravity between you and
your book, you and a tree, your
book and the moon, etc., etc.

The force of gravity between two
masses is given by:
This is a strong statement and not
immediately obvious. A fair question is
‘why haven’t I noticed this before?’. It is
one thing to make a qualitative statement,
but another to be able to back it up with
mathematical predictions. That is the real
core contribution of Newton; the invention
of the physics formula and the birth of the
modern approach to the physical sciences.
Looks messy, but we’ll go
through this piece by piece.
F =
r2
m1G m2

F =
r2
m1G m2
‘m1’ and ‘m2’ are the two masses involved.
The force of gravity between two
masses is given by:

F =
r2
m1G m2The force of gravity between two
masses is given by:
‘r’ is the separation distance between the masses.
If the masses are not just tiny points, then which
distance; there are many distances between the
objects. Newton had to invent Calculus to fully
handle this issue – but for our purposes the ‘center
to center’ distance is usually an excellent
approximation. In fact for spheres, the center is
the perfect reference point.

F =
r2
m1G m2The force of gravity between two
masses is given by:
G is just a number. G = ( 6.67 x 10-11
2
2kg
n m
)
G is a very tiny number, with very messy units! That’s why we
just give it a one letter abbreviation. It is what is known as a
universal physical constant. A basic measurement of the
universe, such as speed of light or mass of an electron. It is a
measure of the strength of the gravitational force.

What is the force of gravitational
attraction between two 100 kg
masses, 1m apart?
Example 3.3:

masses, 1m apart?
Example 3.3:
F =
r2
m1G m2

masses, 1m apart?
Example 3.3:
F =
2( 1 m )
( 6.67 x 10 -11
( 100 kg )
2
( 100 kg )2kg
n m
)
F =
r2
m1G m2
Everything on the right is
known, so just plug it in.

masses, 1m apart?
Example 3.3:
F =
2( 1 m )
( 6.67 x 10 -11
( 100 kg )
2
( 100 kg )2kg
n m
)
F =
2m
( 6.67 x 10 -7
kg
2
kg2kg
n m
)
The number part and the units part can be
handled separately. Gathering all the numbers
together yields this.

masses, 1m apart?
Example 3.3:
F =
2m
( 6.67 x 10 -7
kg
2
kg2kg
n m
)
The units are a huge mess, but turns out that most
everything cancels. And you don’t have to do it
all in one step either. Note that all the kg cancel.

masses, 1m apart?
Example 3.3:
F =
2m
( 6.67 x 10 -7
2
n m
)
With kg out of the way, it becomes more obvious
that m2 cancels too.
So only ‘newton’ is left – which is good because
the ‘newton’ is a force unit and this is supposed to
be a force we are finding.

masses, 1m apart?
Example 3.3:
F = ( 6.67 x 10 -7
n )
Microscopically small! Less than one
millionth of a pound. No wonder you
haven’t noticed being drawn in by gravity
towards your car or a tree.
or 0.00000015 lb

Some Features of Gravity:
The calculations in the previous example are messy. It is less
important that you be able to follow every single step, then it is to
understand some basic consequences of Newton’s Universal Law of
Gravity. Even without doing lots of messy examples, some basic
behavior of the Law can be understood. These important features are
discussed in the following list.

1.) Since G is so small, the gravitational attraction between
ordinary objects at typical separation distances is microscopic.
If you ask why the force in Example 3.3 is so small, it all goes back to the fact
that G is so tiny. For any mass or separation distance for everyday objects
(people, books, trees, cars,…) the force of gravity is always millionths to
billionths of a pound. These were not even measureable in Newton’s time, but
the one billionth of a pound force between two lead weights is easily measured
with modern techniques.

2.) If one or both masses is large enough, that can overcome the
tiny ‘G’ value and then the force of gravity may be substantial.
F =
r2
m1G m2
Tiny G
Huge mass (whole
moon or planet)
Gravity is actually very weak compared to say the magnetic or
electric force on a particle for particle basis. It is only when a
whole moon or planet sized mass is involved that gravity is felt.

150 lb =
r2
m1G m2
Mass of you Mass of earth
Distance from you
to center of earth
Your weight can actually be calculated this way from
astronomical values. The weight of a 150 lb person
comes from all the pieces of the earth’s mass
contributing a small pull.

The larger the masses involved, the larger the force and vice versa.
The same person (and hence same mass) will weigh less on the
moon, because the moon has much less mass to pull on them.

Same person weighs about 1/6 as much on the moon as on earth. This was
handy for the astronauts needing to carry heavy gear. Would also allow people
to jump 6 times as high or hit a golf ball 6 times farther, etc.
Force between the same person and …

On Jupiter, this same person weighs about 2.5 times more than on earth. Most
people would have trouble even carrying around their own weight.

On smaller solar system objects, such as small moons and asteroids, a person
may weigh only a few ounces. So little that they could jump off and not come
back down.

The sun would be a hostile destination due to the heat, but perhaps just as bad
would be the gravity, which would probably be lethal. This same person
would weigh a couple of tons.

3.) The force gets weaker (gradually) as the separation distance
between the objects increases.
A person weighing 150 lbs at sea level…

...weighs slightly less on a mountain-top.

The same person would weigh even less up in orbit (although note that the
weight is still a long way from being zero).

The force gets very small at large distances (although technically it never goes
exactly to zero). Even on the moon, this same person is still pulled to earth by
a few hundredths of a pound. Don’t confuse this with the pull of the moon on
that person, which is a lot bigger (25 lb).

This was all a pretty amazing leap forward. It is the 1680’s, hundreds of years before
space travel could truly test many of these ideas, but the theoretical framework was
already there to figure out the pull of gravity on different planets and how quickly gravity
would decrease as one left the earth.
Recall that in Newton’s time, in particular, they were interested in the motion of the
moon and the planets. This lead to the idea of satellite motion…
1.) Since G is so small, the gravitational attraction between
ordinary objects at typical separation distances is microscopic.

Earth
Satellite Motion
An object taken several
hundred miles above the
earth (well beyond the
earth’s atmosphere) will
weigh less than it did on
the ground, but not that
much less.

v = 0
hr
miAn object released from
rest at typical orbital
height would plummet
to the ground.
Earth
Satellite Motion

v = 0
hr
mi
500
hr
mi
Suppose instead an initial
‘sideways’ push is given.
The object still falls to the
ground, but in an arc that
will cover some horizontal
distance first.
Earth
Satellite Motion
An object released from
to the ground.

v = 0
hr
mi
5000
hr
mi
500
hr
mi
Earth
The larger the sideways
velocity, the longer of an
arc the falling object will
make.
Satellite Motion
An object released from
to the ground.
Suppose instead an initial
‘sideways’ push is given.
The object still falls to the
ground, but in an arc that
will cover some horizontal
distance first.

v = 0
hr
mi
16000
hr
mi
500
hr
mi
5000
hr
mi
But the earth’s surface is
curved too. So if the
object is pushed sideways
with enough speed, the
arc can be so gentle that
the object doesn’t get any
closer to the ground. The
object is constantly
‘falling’, but is moving
so fast horizontally that
instead of hitting the
earth, it just returns to its
starting point and
continues to orbit around.
Earth
Satellite Motion

If the pull of gravity is still quite large in low earth orbit, then
why do objects/astronauts on the space station or space
shuttle appear completely weightless?

On earth, the ground must push up
on the bottom of your feet to keep
you from falling. Your sense of
weight comes from these support
forces and not from ‘feeling’ gravity
itself.
Drop something and that object, no
longer supported by anything, will
fall away from you.

On earth, the ground must push up
on the bottom of your feet to keep
you from falling. Your sense of
weight comes from these support
forces and not from ‘feeling’ gravity
itself.
Drop something and that object, no
longer supported by anything, will
fall away from you.
Suppose instead you are in a freely
falling box. If the box (and hence all the
contents) are already falling with the full
acceleration of gravity, there is no need
for the floor to push on your feet. The
support forces and your sense of having
any weight will vanish.
Drop something and it will just
hover in front of you. It is falling,
but it was already falling before you
let go (and so are you).

Objects inside a falling box or an orbiting spaceship are all ‘falling’
at the same rate. They appear weightless because nothing needs to
rest on top of anything else.
People paying good money to experience
brief freefall and the associated sense of
weightlessness.

An airplane in a dive* can also
mimic weightlessness.
Astronauts take these training
flights and it is one of the few
space tourism things that a
private individual can purchase.
Photo is from a training vehicle
affectionately known as the
‘vomit comet’.
If you don’t like roller coasters
or airplane turbulence, then
astronaut training may not be
for you.
* A powered dive. Because of air resistance, just cutting
the engines isn’t enough. You have to mimic the path of a
rock falling out of the sky.

Space walking astronaut is still pulled
fairly strongly by the earth, but is
‘falling’ at the same rate as the nearby
ship, the tools, etc.

Figure explaining satellite motion from
Newton’s main work, Principia
Mathematica, circa 1680’s
Satellite Motion
So Newton figured out how to put an artificial satellite around the
earth about 300 yrs before technology made it possible. But he
also realized that the moon orbiting the earth and the planets
orbiting the sun are natural examples of satellite motion.
We’ll finish chapter 3 with some miscellaneous related items.

The lower a satellite orbits, the faster it orbits.
v = 17000
hr
mi
Period = 1.5 hour
Low Earth Orbit
Low satellites feel a stronger gravity, so they must
move faster sideways to stay ‘up’. The period of
an orbit is the time required to complete one orbit.

v = 17000
hr
mi
Period = 1.5 hour
Low Earth Orbit
v = 15000
hr
mi
Period = 3 hour

v = 17000
hr
mi
Period = 1.5 hour
Low Earth Orbit
v = 15000
hr
mi
Period = 3 hour
v = 7000
hr
mi
Period = 24 hour
Geosynchronous Orbit

v = 17000
hr
mi
Period = 1.5 hour
Low Earth Orbit
v = 15000
hr
mi
Period = 3 hour
v = 7000
hr
mi
Period = 24 hour
v = 2000
hr
mi
Period = 28.5 days
Moon’s Orbit

occurs at the height where
satellites orbit in a period of
24 hrs. At this height an
interesting effect occurs.

occurs at the height where
satellites orbit in a period of
24 hrs. At this height an
interesting effect occurs.
The satellite has to keep
moving, but at this rate it
keeps up exactly with the
daily turning of the earth.
So it appears to hang
motionless in the same place
in the sky, as seen by a
ground observer.

Having a 24 hr orbital period is not enough to be geosynchronous.
The geometry must be correct also (orbit is over the equator and
headed ‘east’).

Falling Satellites:
Satellite in a low orbit
(400 mile high),
drawn to scale.

Virtually all of the earth’s
atmosphere is within 100
miles of the ground. But
it doesn’t just abruptly
end and the microscopic
amount left in low earth
orbit does provide a tiny
air resistance. A satellite
can very slowly loose
speed and drop lower
each orbit. This
accelerates toward the
end and the satellite
comes crashing down.
Falling Satellites:
Satellite in a low orbit
(400 mile high),
drawn to scale.

Fell 1979
Launched 1973
Satellites in low earth orbit routinely fall out of the sky. The Russians put up
the first two artificial satellites, but they both fell fairly quickly. The first US
satellite, Explorer 1, was placed in a much higher orbit and remains the oldest
artificial satellite around the planet.
Probably the most famous satellite
to fall out of the sky was our first
space station - Skylab

Watching Satellites:
Satellites can be
observed with the
un-aided eye. Under
dark conditions,
usually a few can be
seen every night
during a couple
different time
windows.

The first hour or two after sunset (and an hour or two
before sunrise) it is dark at ground level, but still
‘sunny’ 400 miles above.

An hour after sunset (and an hour before sunrise) it is
dark at ground level, but still ‘sunny’ 400 miles above.
Satellites will appear as a tiny dot of white light (like a
star) that slowly moves uniformly across the sky.
Ground observer shortly after nightfall,
can still see sunlight glinting off of a
satellite passing overhead. Satellites have
no lights of their own, as they are un-
manned. Blinking lights, colored lights, or
multiple lights means an airplane.

Artificial Gravity:
Objects don’t change velocity without being pushed. In a box
moving in a curve, ‘loose’ objects tend to keep going in a straight
line, but this will appear to observer’s in the box as a tendency for
them to be thrown to the outside of the curve.

This is known as the centrifugal effect.
Artificial Gravity:
Objects don’t change velocity without being pushed. In a box
moving in a curve, ‘loose’ objects tend to keep going in a straight
line, but this will appear to observer’s in the box as a tendency for
them to be thrown to the outside of the curve.

This could be the basis for an
artificial gravity in orbit.
Spinning a 100 m wide drum
at 3.3 rev/min yields the effect
of objects being ‘pressed’ to
the outside with equivalent of
earth’s gravity.

GPS Module 3 - Part 2

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