This document discusses Newton's law of gravitation and Kepler's laws of planetary motion. It provides explanations of gravitational force, including that it is always attractive and inversely proportional to the square of the distance between objects. Kepler's laws are summarized as: 1) Planets orbit in ellipses with the sun at one focus. 2) A line between a planet and sun sweeps equal areas in equal times. 3) The square of a planet's orbital period is proportional to the cube of its semi-major axis. The document also covers gravitational potential energy and energy in circular orbits.

1. Gravitation
Newton’s Law of Gravitation;
Kepler’s Laws of Planetary
Motion.
Lecture 14
Monday: 1 March 2004

2. Physical Principles of Design
Forces and Motion
• Learning to analyze and predict the motion of
objects moving with constant accelerations.-
Topics of kinematics, projectile motion ….
• Coming to understand the causes of motions and
changes in motion-topics of (linear and rotational)
forces, momentum, energy…
• Forces we have encountered:frictional force,
normal force, applied forces of pushes/pulls,
tension in strings, and the GRAVITATIONAL
FORCE

3. How Many Different Forces are
There?
• All of the forces we have worked with or
will work with are specific examples of
following general types of forces:
The Fundamental Forces of Nature:
1. Gravitational Force
2. Electromagnetic Force
3. Strong Force (Nuclear)
4. Weak Force (Nuclear)

4. A General Expression for the
Gravitational Force
2
2
11
2
2
1
kg
/
m
N
10
67
.
6
)
Attractive
always
is
force
(This
generally,
More





G
r
m
m
G
F


5. Revisiting Gravitational Force
g
m
F



More Specifically,
g
m
F object
From



Object
an
on
Earth
Where , is the acceleration due to the
earth’s gravitational attraction.
It is not only the earth that attracts other objects.
Any object with mass will attract any other object with mass.
g


6. At the surface of
the Earth
earth)
on the
gravity
to
due
on
accelerati
(
with
since
But
Earth
the
of
Radius
the
is
R
object
the
of
mass
the
is
m
Earth,
of
mass
the
is
M
where
2
2
2
R
GM
g
R
Mm
G
F
g
m
F
R
Mm
G
F










7. Characteristics of the
Gravitational Force
2
2
1
r
m
m
G
F 

•The force is always attractive.
•There is a Newton’s third law force pair involved.
•It acts along a line connecting the centers of the two objects
(called a Central Force)
•It is inversely proportional to r2 (called a “one over r squared”
force)
•Experimental measurement show us that it is a conservative
force (the gravitational force on earth is conservative-remember?
This is a general expression of that same force)

8. Defining the Potential Energy
Associated with this Force

 







b
a
b
a
ab
ab
a
b
dr
F
d
W
W
U
U
U
s
F
2
2
1
r
m
m
G
F 


9. POTENTIAL ENERGY
• Choose U = 0 at r = 
U F dr
U
GMm
r
r
 
 


10. Gravitational Potential Energy
Near Earth
R
GMm
U 


11. KEPLER'S LAWS
1. The Law of Orbits: All planets move in
elliptical orbits having the Sun at one
focus.
2. The Law of Areas: A line joining any planet
to the Sun sweeps out equal areas in equal
times.

13. The Law of Areas
 
A r r
dA
dt
r
d
dt
r
L
L rmv rm r
L mr

 

 


1
2
1
2
2 1
2
2
2
( )





constant

15. KEPLER'S LAWS
3. The Law of Periods: The square of the
period of any planet about the Sun is
proportional to the cube of the semimajor
axis of its orbit.

16. The Law of Periods
F ma
GMm
r
m r
GM
r T
T
GM
r


  






2
2
3
2
2
2
2
3
2
2




( )

17. ENERGY IN CIRCULAR
ORBITS
K mv m
GM
r
K
GMm
r
U
GMm
r
E U K
GMm
r
 

 
   
1
2
2 1
2
2
2