Download Gravitational Energy

Gravitational Energy
Gravitational Work

Gravity on the surface of the Earth is a local
consequence of universal gravitation.

How much work can an object falling from very far
from the Earth do when it hits the surface?
 
W   F  dr
r
F
GmM
r2
GmM
dr
2
r
r
1 1 

W  GmM  
 r RE 
W 
RE
RE
Gravitational Potential

The work doesn’t depend on the path.
• Universal gravity is a conservative force

The potential is set with U = 0 at an infinite distance.
• Gravity acts at all ranges
• Gravity is weakest far from the source
 GmM
dr
2

r
GmM
U 
r
U  
r
Kinetic Energy in Circles

For circular motion there
speed is related to the
centripetal acceleration.
• a = v2/r

An object moving in a circle
has kinetic energy.
• K = ½mv2
• K = ½ mar
• K=½Fr

The kinetic energy is equal
to half the work that could be
done.
F
r
v
Kinetic Energy in Orbit

The kinetic energy for a
circular orbit is related to the
potential energy.
mv2 GmM

r
r2
1 2 GmM
mv 
2
2r
 2K  U

The total energy in a circular
orbit can be described in
terms of either the kinetic or
the potential energy.
E  K U
E  K  (2 K )
E  K
E U /2
Escape Velocity



Negative total energy can be
viewed as being captured by
the force of gravity.
To get away from the
influence of gravity requires
zero or positive energy.
The minimum velocity is
called the escape velocity.
E  K U  0
1 2 GmM
mv 
0
2
r
vesc 
2GM
r
On earth, vesc = 11.2 km/s
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