Transcript Chapter 12

Chapter 12 - Gravity
Newton’s Law of Gravitation
Kepler’s Laws of Planetary Motion
Newton’s --------
Force of Attraction - Weight
G = 6.67 x 10-11Nm2kg-2
 ME = 5.98 x 1024kg
 RE = 6.37 x 106 m
 m = 70 kg
Calculate your
Solution to Problem
Cavendish Experiment
Henry Cavendish –
English Physicist
Cavendish Experiment
The Gravitational Field
Radial Nature
and Inverse
Square law
g = GME/R2
Calculate the
field strength at
any location
The Acceleration Due to Gravity
at a Height h Above the Earth’s Surface
Orbital Motion
Newton’s Cannon
Newton visualized that
if a projectile were
fired with enough
velocity the Earth
would forever “curve”
away and it would
never hit the ground..
… It would be in
Newton’s Cannon
While falling you still have weight
but your apparent weight is zero!
Kepler’s Laws of Motion
1st Law
2nd Law
Equal Area
3rd Law
Kepler's second law
How fast does the
planet move in its
As the planet moves in
its orbit, a line from the
sun to the planet
sweeps out equal
areas in equal times.
Is Pluto a Planet?
Kepler's second law
Kepler’s 3rd Law
T2 / R3= constant
 Ta2 / Tb2 = Ra3 / Rb3
The ratio of the
squares of the orbital
periods for two
planets is equal to
the ratio of the cubes
of orbital radius.
Useful Links
Kepler’s Laws
Centripetal Forces on Satellites
is supplied by the
gravitational force.
If the equations are set
equal what do you get?
Satellites and Orbital Velocity
An expression
for orbital
velocity !
Quick Calculator
Set the centripetal force
equal to the
gravitational force
Vo2 = (GM/r)
Orbital Velocity & Period
Remember that for
any object moving in a
circular path, we can
relate the speed and
Vo = 2πR/T
Satellites in circular orbits.
Vo2 = (GM/r)
Vo = 2πR/T
If these equations are set
equal what do you get?
Kepler’s 3rd Law - Derived
Square both sides
and set equal.
Vo 2= (2πR/T)2
Vo 2 = (GM/R)
4π2R2/T2 = GM/R
T2 / R3 = 4π2 / GM,
K = 4π2 / GM
Kepler’s 3rd Law
T2 / R3= constant = (4π2/GM)
T2 = (4π2/GM) R3
Where (4π2/GM) is the Kepler constant!
Depends only on the
Mass of the planet
being orbited.
Geosynchronous Orbit
AKA: Geostationary Orbit or Parking orbit
At what altitude must a satellite be placed
in order to be in a geosynchronous orbit?
Binary Star Systems
m1a1  m2 a2
 v 2 
v 2 
1 
 2 
m1 
 r 
 r 
 1 
 2 
   
2r1 2
m1 T
2 r 2 2
m2 T
m1r1  m2 r2
m2 r1
m1 r2
Both stars have the
same orbital period.
Gravitational Potential Energy
Gravitational potential
energy is energy an object
possesses because of its
position in a gravitational
 Gravitational Potential
Energy between two
masses increases as r
increases. (becoming less neg)
At ∞ it becomes 0 Joules
m1m 2
U  G
Gravitational Potential Energy
Work done against the
gravity force in
bringing a mass in
from infinity where the
potential energy is
assigned the value
zero .
m1m 2
U  G
Gravitational Potential Energy
Change in KE = Change in PE
ΔKE + ΔPE = 0
If these equations are set
equal what do you get?
Escape Velocity !
Set Gravitational Potential Energy equal
to Kinetic Energy.
GMm/R = ½ mVe 2
Solve for V.
V = √ (2GM/R)
Gravitational Potential
The potential energy that a unit mass
(usually 1 kg) would have at any point.
M is the mass of the
gravitating object. This is
sometimes useful because it
assigns each point in space a
definite gravitational potential
value, irrespective of mass.
 Units are in Joules/kg
Tides are due to differential
gravitational forces on a body.
Consider the Earth and Moon: the gravitational force on the
Moon due to Earth is stronger on the near side than on the far
This net difference in force will cause the body to stretch
along the line between the bodies.
Tidal flexing
and heating
of Io causes
the most
volcanism in
Tidal Friction
Tides result in a net force which slows Earth’s rotation and
speeds the Moon’s orbital velocity.
As a result the day is getting
longer by ~1 second/century
and the distance between the
Earth and Moon is increasing.
There is evidence for this in
the fossil record on Earth
Tidal Effects
Roche Limit: Tidal deformation
increases as an astronomical object
moves closer to the body it orbits. At the
Roche Limit, the object can break into
small pieces.
 Tidal Locking: occurs when one
astronomical object always points its
tidal bulge at the object it orbits.
Roche limit: tidal forces destroy bodies which come too
close to a source of gravitation
through differential acceleration
Consider an orbiting mass of fluid held together
by gravity. Far from the Roche limit the mass is
practically spherical.
The body is deformed by tidal forces
Within the Roche limit the mass' own gravity
(self-gravity) can no longer withstand the tidal
forces, and the body disintegrates.
Particles closer to the primary orbit move more
quickly than particles farther away, as
represented by the red arrows.
The varying orbital speed of the material
eventually causes it to form a ring.
Tidal Locking
What path does the moon trace during its orbit about the sun?