Chapter 14
Study Guide for Gravitation
14.1
Universal gravitation
Skill 14.1 Understand the role of the gravitational force in maintaining the orbital
motion of the planets.
Don’t get
rotational
velocity and
orbital
velocity confused.
Rotational velocity
is the velocity of rotation of an
object about an axis through its center.
Orbital velocity
is the tangential velocity of an object orbiting another
object.
In the absence of external forces, all planets would follow straightline trajectories. The fact that they move
in approximately circular paths means that there must be a centripetal force acting on them.
As Newton Frst postulated, the centripetal force that holds the planets in their orbits is the gravitational force.
Skill 14.2 Understand the fundamental properties of the gravitational force.
A solid sphere exerts a gravitational force as if all the matter in the sphere were concentrated at its center.
±urthermore, this gravitational force drops o² as
1
r
2
(the larger the distance between the interacting objects, the
weaker the gravitational attraction).
By equating gravitational acceleration to centripetal acceleration, we Fnd that the square of the period of a
planetary orbit is proportional to the cube of the orbit’s radius (Kepler’s Third Law):
1
R
2
∝
a
g
=
a
c
∝
v
2
R
=
p
2
πR
T
P
2
1
R
∝
R
2
T
2
.
(14.1)
=
⇒
T
2
∝
R
3
(14.2)
The gravitational force exerted by the earth on an object is proportional to the object’s
gravitational mass
,
which is equivalent to the object’s inertia.
Similarly, the gravitational force exerted by an object on the earth is proportional to the earth’s gravitational
mass.
By putting together all of this information, we are able to arrive at
Newton’s law of universal gravitation
:
F
g
12
∝
m
1
m
2
r
2
.
(14.3)
Note: Newton’s law of universal gravitation applies to
all
the gravitational mass in the universe.
1
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Gravitation and angular momentum
Skill 14.3 Understand how information about the orbital motion of planets is obtained
from the fact that gravitation is a central force.
Gravitation is a type of
central force
 a force whose line of action always lies along the line connecting the two
interacting objects.
The central force is directed to a point called the
force center
.
The torque caused by any central force about the force center is always zero
. This is because the central force
always lies along the radius vector from the force center.
As a result,
any object subject to a central force has a constant angular momentum about the force center
(remember, change in angular momentum is due to a net external torque).
The angular momentum of any particle about the origin is proportional to the rate at which area is swept
out by the particle’s position vector (Look at Figure 14.9).
Since angular momentum is constant for planetary orbits, the above rate must also be constant. This is stated
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 Spring '08
 Stewart
 Physics, Force, Mass, Potential Energy, General Relativity, Gravitational Force

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