cto. ppt.

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