Download Gravitation - Cedarville University

Gravitation
Flat Earth
• Up to now we have parameterized the
weight of an object as W  mg
• This is true for a flat earth assumption.
• Is the earth flat?
• What evidence is there that it is not?
Gravitational Force
• Central Force - Directed between the
centers of mass for the two objects.
• Inverse Square Law Force - Strength drops
off as 1/r².

m1m2
F21  G 2 rˆ
r
G  6.672 1011 Nm2 / kg 2
r
m1
m2
Earth & the Moon
• What is the gravitational force between the
rm
earth and the moon?
M e  5.97 10 kg
24
mm  7.35 10 kg
22
rem  3.85  10 m
8
re  6.37 10 m
6
rm  1.74  10 m
6
re
me
rem
mm
Potential Energy
• Calculate work needed to change height
U  W   
rf
ri
U   
rf
ri


F ( r )dr
 1 1
 GMm ˆ 
  2 r   drrˆ   GMm  
r


 rf ri 
If Ui=0 at ri=,
 GMm
U
r
Kepler’s Laws
• All planets move in elliptical orbits with the sun at
one focal point.
• Planets sweep out equal areas in equal time intervals.
– Conservation of angular momentum
• Square of the orbital period is proportional to the
cube of the semi-major axis of the orbit.
A1
A2
Orbital Motion
• Assume only gravitational force and
circular motion at constant speed.
Newton’s 2nd Law T = orbital time period


 F  ma
GMm
v2
m
2
r
r
GM
2
v 
r
d 2 r
v 
t
T
GM
 2 r 

 
r
 T 
2
v
m
r
M
2 3
4

r
2
T 
GM
Earth Satellites
• Landsat-7 has an orbit altitude of 705 km.
What is its acceleration, speed, and orbital
period?
re  6.37 10 m
6
M e  5.97 10 kg
24
Geostationary
Satellites
• One orbit = One Day
• Types of Satellites
– Weather
– Communications
Digital Globe - Quickbird
• 2 foot resolution
• Altitude of 280 mi
q = 1.3 mrad
Moon’s Orbit
• What is the period of the moon’s orbit?
29.5 days Synodic (w.r.t. sun)
27.3 days
Sidereal (w.r.t. stars)
0 days
M e  5.97 10 24 kg
rem  3.85  108 m
4 2 r 3
6
T
 2.38 10 s
GM e
T = 27.5 days
Escape Velocity
• Escape velocity - Velocity a projectile
needs to escape the gravitational force.
Ki  U i  U f
At rmax = , v 
M e  5.97 10 24 kg
Re  6.37  10 6 m
GMm
GMm
mv 

R
rmax
2GM
R
1
2
2
ve = 11.2 km/s
Tides
• Tides--rise and fall of ocean levels
– 12-hour cycle
– Interaction between the moon’s gravity and the earth’s
centripetal acceleration.
Gravitational Acceleration
GM
g 2
r
Centripetal Acceleration
a  r
2
a>g
a=g
a<g