### Universal Gravitation

```UNIVERSAL GRAVITATION
History of Gravity

Ptolemy c. AD 90 – c. 168
 All objects go to their natural place
 Geocentric universe
 Main text for 1600 years

Copernicus 1473 – 1543
 Heliocentric universe

Brahe & Keppler (Keppler - (1571 –1630)
 Keppler used Brahe’s observations to formulate 3 laws of planetary
motion

Galileo 1564 –1642
 Tested with gravitational affects on Earth, acceleration of gravity
 First to come up w/ idea for inertia
 Contemporary of Keppler

Newton 1643-1727
 Law of Universal Gravitation

Einstein 1879–1955)
 Theory of General Relativity
 Punked out Isaac Newton
Isaac Newton
 First credited with the idea that Gravity
extends through the universe
 Newton knew the concept of inertia, and that
if something changed its speed or direction a
force must be responsible
 Apple falling from tree… apple changed its
motion, so a force must be responsible-Gravity
 He also noticed the moon, and that it was
traveling in circular motion
 Circular motion means changing direction and this
also required a force
 He reasoned the same force was responsible for
the Moon as was the apple
The Falling Moon- Satellite
Motion
 Newton realized the moon must be falling or
else it would travel away in a straight line
 So if it is falling, why is it never getting
closer??
 hypothesized that moon is just a projectile
falling under the influence of gravity.
 And it never falls to Earth because of its
speed….
 Launched projectiles will follow a curved path
(see below) until they hit the ground
 But the ground also curves (Earth is not flat)
 Most projectiles have a curve that are much steeper
than the curve of the Earth
 BUT, if you throw a projectile fast enough, then curved
path of the projectile will match the curve of the Earth
 Then, the projectile will never reach the surface and
forever be in an orbit
 Projectile must also be above the Earth’s
atmosphere, so that air resistance does not slow
it down
Projectile being thrown a different speeds, from
the top of a mountain above the atmosphere
 If you could throw something as fast as you
wanted and there was no air resistance could
you make something orbit?/
Newton’s Law of Universal
Gravitation
 Force of gravity acting between 2 objects is directly
proportional to their mass and inversely proportional to
the square of the distance between them
 m1
m2 represents the masses of the two objects
involved
 d - distance between the center of the two objects
 G- Universal Gravitation Constant, value of
G = 6.67 x 10-11 Nm2/kg2
 F - represents the Force of Gravity between any 2 objects
and
 Remember that Force of Gravity is the same thing as Weight
What this law means
 Whatever factor the mass is increased by, the force
of gravity is increased by the same amount
 Ex. Triple the mass, force of gravity also is tripled
 Whatever factor the distance between 2 objects is
increased by, the force of gravity is multiplied by the
inverse of the square of that number
 Ex. Triple the distance, force of gravity is 1/9th as much
 Inverse square proportion
Examples
 The force of gravity acting on you while on the
surface of the Earth is 100 N.
 What would be the force of gravity acting on you if the
Earth’s mass was doubled?
 Force Directly proportional to mass,
So
200 N
 What would it be if your mass was doubled and the
Earths stayed the same?
 Force Directly proportional to mass,
So
200 N
 What would the force of gravity acting on you be if you
were twice as far from the center of the Earth as you
are now?
 Force is Inversely proportional to the square of the distance, so
distance is increased by 2, inverse square of 2 is 1/22 or 1/4th , so force
of gravity would be 1/4th as much or
25 N.
Why this happens?? Inverse
Square Law… Surface Area of
a sphere
More examples….
 A spaceship with a weight of 10,000 N on the
surface of the Earth moves to a distance that is 3
times further away from the center of the Earth,
what is the force of gravity on it now?
 Distance increased by 3 x , so Fg decreases by the inverse
square of this number so Fg goes down by 1/9th.
10,000/9 = 1111.1 N
R
Space ship
weighs
1111.1N
here
Space ship
weighs
10,000N
here
M – mass of central object (ie
Earth or Sun)
m-mass of small object (person
or satellite)
objects
 Fg = GMm/r2 and Fg =mg
 So …. Fg =GMm/d2 = mg
 After cancelling out ‘m’
we get
 g = GM/r2
 This gives us the acceleration of gravity, g,
(also known as the Gravitational Field
Strength) at a distance, r, away from the
center of the Earth.
 so acc. of gravity also follows inverse square rule
Object in orbit around Earth requires a Fc to keep in circular path. This Fc is the force
of gravity from the Earth. So Fc = Fg. So if we want to figure out the speed, vt, that an
object must go in order to travel in orbit at a distance, d, away from the center of the
Earth… we can solve as follows…
“d” and “r” are representative of the same thing, so just turn ‘d’ into ‘r’ to simplify math
After we cross multiplied,
divided and cancelled out like
terms we get…
And this tells us the speed, vt,
that we would need to travel in
an orbit with radius, r, around an
object with mass, mE.
Everyone thought…
 All satellites travel in perfectly circular orbits
 NOT TRUE
 Planets, moons, etc. Were not where they were
supposed to be!
 Planets did not follow these predicted paths
 So something must be wrong
 Then…… along came Johannes Kepler
Kepler’s 3 Laws of planetary
motion
 1) The paths of the planets are ellipses, with the
sun at one focus (the other focus is just a point in
space)
 http://www.astro.utoronto.ca/~zhu/ast210/kepler.h
tml
 Keppler’s 2nd Law
An Imaginary line from the sun to a planet sweeps
out equal areas in equal time intervals.
This means planets move faster when they are closer
to the sun and slower when they are further away
http://surendranath.tripod.com/Applets/Dyna
mics/Kepler/Kepler1Applet.html
 Keppler’s Third Law
 The square of the ratio of the periods of any
two planets revolving about the sun is equal
to the cube of the ratio of their average
distances from the sun. Thus, if Ta and Tb are
the planets periods, and ra and rb are their
average distances from the sun we get
 (Ta/Tb)2 =(ra/rb)3
Elliptical paths & Kepler
 Kepler came up with Three Laws of Satellite Motion
 These laws explained that planets & other satellites
travel in ellipses (ovals), not circles
 When tested Kepler’s laws almost perfectly
predicted paths of planets and moons
 This means planets are closer to the Sun at some
points in their orbit, and farther from the Sun at
other points
Orbits
 Earth’s orbitVery
nearly circular, but
elliptical like all other
planets
 Elliptical path of Earth
does not affect
weather at all
 Other planets, like
Mercury, have more
exaggerated elliptical
orbits
Conservation of Energy and
Satellite motion
 Based on the fact that KE + PE always stays constant…..
 Planets travel faster when they are closer to the Sun, and
slower when they are further away
Slow
Fast
,
Escape Speed
 Speed that something needs to travel in order to
permanently escape an objects gravitational pull
without getting pulled back
 Escape speed for the Earth is roughly 7 miles per sec.
Escape velocity equation is derived from conservation of Energy Equation
-KEi + PEI = KEF + PEF
-Basically the initial kinetic energy has to be greater than or larger than the initial
potential energy relative to the center of the object that is being escaped from
-So
KEi + PEi = 0
to find minimum escape speed
-Plugging in equations for KE and PE we get…
½ mve2 +-mgri = 0
- and substituting in
-we get
-And solving for ve, we get
g= GM/r2
and cancelling out ‘r’ ‘s
½ mve2 + - GMm/r = 0
M – mass of central object
m- mass of projectile
Big Bang vs. Big Crunch
• If KE from big bang is larger than PE before big
bang then universe will expand forever
• However if KE from big bang is smaller than PE
before, then universe will eventually collapse
back onto itself, thus causing a BIG CRUNCH
• In other words…. If matter had the escape
speed of the universe when the big bang
began… then it will expand forever, if not big
crunch
How do we know universe is still expanding?
• Red Shift of galaxies in all directions
– All galaxies are moving away from us
• Color of galaxies is shifted to red end (long wavelengths) of visible light
• This is caused by Doppler Effect
– Squishing together of wavelengths as object approaches (blue shift)
– Spreading apart of wavelengths as object leaves (red shift)
– Can see this with sound waves as an ambulance approaches
– If we saw a blue shift in galaxies…. That would mean big crunch is happening
and everything is moving towards us
GRAVITATIONAL INTERACTIONS
Gravitational Fields
 Areas of a certain distance away from a gravitational source
experience the same ‘field strength’
 Like ripples in a pond after stone dropped in
 The field strength on the surface of the Earth is about 9.8
 ‘g’ also changes by the inverse square rule
g= Gm
d2
If you are twice as
far away from the
center of the Earth
‘g’ = (9.8)/4 m/s2
=2.45 m/s2
Hypothetically Going to the center of
the Earth
 According to Newton, the closer you get to an object
the stronger gravity is, but if you were travelling to
the center of the Earth, all of the mass above you
would start to cancel the gravity below you, so as you
travel towards the center of earth gravity will
become less and less strong
Weight & Weightlessness
 How you feel weight, is different than your actual weight
 As long as you are near the surface of the Earth you will always
have the same weight but you may “feel” like you have a
different weight
 This can happen if you are accelerating up or down
 Imagine an elevator……..
Ocean Tides From the Moon
 Side of the Earth closest to the moon, has a high tide
 As well as the side of earth furthest from the moon
 This happens because…

Water flows

There is a large difference in gravitational strength from the moon on either
side of the Earth… because of difference in distance from the moon

Gravity from the moon tugs on Earth just a bit
 Gravity from moon is of course stronger when moon is close
 And weaker when far away
Earth
Earth
The Sun also causes Tides
 The Sun, like the Moon pulls on Earth, so the side of
Earth closest to Sun, receives a high tide from Sun
 Tides from the Sun are weaker than from the moon
 Why is this?
 Sun actually has a lot stronger pull on Earth than the
moon does, so why aren’t solar tides larger??
 Not that big of a difference in how far the sun is from
sunny side compared to dark side
 So difference in gravity strength is small
Tidal Effects… spring tides, Neap Tides
Earth-Moon Orbit
 Earth and moon orbit a common point in space
 Point is center of gravity of Earth Moon system
Perturbations
 Small irregularities in the orbit of a planet
caused by a close proximity to another planet
 Gravity of one planet slightly tugging on
another
 This is how Neptune was discovered
 http://csep10.phys.utk.edu/astr161/lect/histor
y/perturbations.html
Earth Tides and Atmosphere Tides
 Tides don’t just happen with water
 Occur w/ the ground
 Earth is really just a thin solid crust on top of molten
(liquid) rock
 Ground rises and lowers small amount depending on
moon’s location
 and with the atmosphere
 Atmosphere has much larger tides than oceans do
Gravity differences on Earth??
-Universal Law of Gravitation tells us if we are closer or further from
Center of the Earth gravity will change
-gravity slightly weaker at the top of a tall mountain… ie you weigh less
-other inconsistencies…..
http://www.sciencefriday.com/news/051407/news0514071.html
-Big Earthquakes change mass distribution of Earth…
-2004 Indonesia Earthquake was so big the rate at which the Earth
orbits slightly changed
http://www.uwgb.edu/dutchs/platetec/rotationqk2004.htm
Determine the escape speed in km/s
of Pluto, Earth, and Jupiter using
below data.
1) Determine the escape velocity from
a position of 150 km above the Earth.
• 2) Use Kepler’s 3rd Law to determine
Mercury’s orbital period in days?
• mass of the Moon = 7.36 × 1022 kilograms
• radius of the moon = 1 737.4 kilometers
• Question… how fast would an astronaut have to
throw a moon rock from the surface of the moon
for it to orbit around the moon without falling to
the surface?
What is the acceleration of the moon
as it “falls” toward Earth?
• Distance from Earth to moon = 384,403 kilometers
Determine the speed at which an
object would have to be travelling at in
order to orbit Earth at an altitude of
300 km above Earth’s surface.
• Mass of the Earth --- 5.97 x 1024 kg
• Radius of the Earth – 6400 km
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
0.382
0.949
1
0.532
11.209
9.44
4.007
3.883
mass (Earth=1
=5.97 x 1024 kg)
2,438
6,052
6,378
3,397
71,400
60,000
25,559
24,764
0.055
0.815
1
0.107
318
95
15
17
mean distance from
Sun (AU)
0.39
0.72
1
1.52
5.20
9.54
19.18
30.06
orbital period (Earth
years)
0.24
0.62
1
1.88
11.86
29.46
84.01
164.8
orbital eccentricity
0.2056
0.0068
0.0167
0.0934
0.0483
0.0560
0.0461
0.0097
mean orbital
velocity (km/sec)
47.89
35.03
29.79
24.13
13.06
9.64
6.81
5.43
rotation period (in
Earth days)
58.65
-243*
1
1.03
0.41
0.44
-0.72*
0.72
mean temperature
at surface (C)
-180 to 430
465
-89 to 58
-82 to 0
-150
-170
-200
-210
gravity at equator
(Earth=1)
0.38
0.9
1
0.38
2.64
0.93
0.89
1.12
escape velocity
(km/sec)
4.25
10.36
11.18
5.02
59.54
35.49
21.29
23.71
mean density
(water=1)
5.43
5.25
5.52
3.93
1.33
0.71
1.24
1.67
number of moons
0
0
1
2
63
62
27
13
rings?
no
no
no
no
yes
yes
yes
yes
Mass of sun = 1.98 x 1030 kg
Radius of the Sun = 695,500 km
```