Transcript Chapter 12 Universal Law of Gravity Chapter 12: Universal Gravitation • The earth exerts a gravitational force mg on a mass m. • By the action-reaction.

Chapter 12
Universal
Law of Gravity
Chapter 12:
Universal Gravitation
• The earth exerts a gravitational force mg on a
mass m.
• By the action-reaction law, the mass m exerts a
force mg on the earth.
• By symmetry, since the force mg is proportional to
the mass m, the value of g must also be
proportional to the mass M of the earth.
• Isaac Newton realized that the motion of
projectiles near the earth, the moon around the
earth, the planets around the sun,… could be
described by a universal law of gravitation
Gravity
If two particles of mass m1 and m2 are separated by a
distance r, then the magnitude of the gravitational force
is:
m1m2
F G
r
2
G is a constant = 6.67  10-11 N·m2/kg2
The force is attractive:
The direction of the force on one
mass is toward the other mass.
The gravitational force varies like 1/r2. It
decreases rapidly as r increases, but it never
goes to zero.
Example: The gravitational force between two
masses is 10-10 N when they are separated by 6 m. If
the distance between the two masses is decreased to 3
m, what is the gravitational force between them?
Gravitational Attraction of
Spherical Bodies
If you have an extended object, it behaves as if all of its
mass is at the center of mass. Therefore, to calculate the
gravitational force between two objects, use the distance
between their centers of mass.
Gravitational force
between the Earth and
the moon.
Gravitation of finite objects
 Newton invented Differential Calculus to
interpret his theory: F = ma
 Newton invented Integral Calculus to prove
that the gravitational force of the earth and
motion of the moon is the same as if the
earth and moon were each concentrated in a
single point.
Example
Calculate the gravitational force between a 70 kg man and
the Earth.
F = m g = (70 kg) (9.8 m/s2) = 686 N, but
F
GMm
RE2
 mg
2
24

N

m
5
.
97

10
kg
m
g
 6.67  10 11

9
.
81

2
2
RE 
kg  6.37  10 6 m 2
s2
GM


Variation of g with height
The gravitational force between the
Earth and the space shuttle in orbit
is almost the same as when the
shuttle is on the ground.
g ( h) 
GM
RE  h2
Kepler’s Laws of Orbital Motion
1. Objects follow elliptical
orbits, with the mass being
orbited at one focus of the
ellipse.
A circle is just a special
case of an ellipse.
Kepler’s Laws (cont.)
2. As an object moves in its orbit, it sweeps out an equal
amount of area in an equal amount of time.
This law is just
conservation of
angular momentum.
Gravity does not exert
a torque on the planet
Why?
perigee
apogee
Kepler’s Laws (cont.)
3. The period of an object’s orbit, T, is proportional to
the 3/2 power of its average distance from the thing it is
orbiting, r:
2 3 / 2
T
r
GM
Note: M is the mass that is being orbited. The
period does not depend on the mass of the orbiting
object.
Example
1. The space shuttle orbits the Earth with a period of
about 90 min. Find the average distance of the shuttle
above the Earth’s surface.
answer: 6.65E6 m
Gravitational Potential Energy
The gravitational potential energy of a pair of objects is:
m1m2
U  G
r
When we deal with astronomical objects, we usually
choose U = 0 when two objects are infinitely far away
from each other. In this case, gravitational potential
energy is negative.
The formula we have used in the past, U = mgy, is valid
only near the surface of the Earth (and has a different
location for U=0).
Escape Speed
We can use conservation of energy to calculate the speed
with which an object must be launched from Earth in
order to entirely escape the Earth’s gravitational field.
Initially, the object has
kinetic (velocity v) and
potential energy. In order
to escape, the object must
have just enough energy to
reach infinity with no speed
left. In this case, M = mass
of Earth and R = radius of
Earth.
Ei  E f
Ki  U i  K f  U f
1 mv 2
2
mM
G
 0  0
R
2GM
2
v 
R
v  11,200 m/s
Example
A satellite is orbiting the Earth as shown below. At
what part of the orbit, if any, are the following
quantities largest?
(a) Kinetic energy
(b) Potential energy
(c) Total energy
A
(d) Velocity
(e) Gravitational force
(f) Centripetal acceleration
(g) Momentum
B