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Gravitational Energy
Gravitational Work

The work done by the force of gravity only depends
on the vertical distance.
• The path taken doesn’t matter
• It’s a conservative force
• There is a potential energy
F = -mg
y2
d
q
h
 
W  F  d  -mg (-d sin q )  mgh
h
y1
 
W   F  dr  -mg( y1 - y2 )  mgh
Energy of Position

Raise a block
• The state of the block
changes
• Work is stored in a new
position

Energy due to position is
called potential energy (U).
• An object has the potential
to do work
• The potential energy of
gravity: U = mgy
W  -mg( y2 - y1 )  -mgy
U  mg( y2 - y1 )  mgy
y2
y
y1
Using Friction and Energy
q

The hill is 2.5 km long with a
drop of 800 m.

The skier is 75 kg.

The speed at the finish is
120 km/h.

How much energy was
dissipated by friction?
Friction and Height



Find the total change in
kinetic energy.

Find the total change in
potential energy.

The difference is due to
friction and drag.

K = ½ mv2 - 0
= ½(75 kg)(130 m/s)2
= 5.4 x 105 J
U = mgh
= (75 kg)(9.8 m/s2)(-800 m)
= -5.9 x 105 J
Wnon = K + U
= -0.5 x 105 J
No Absolute

Potential energy reflects the
work that may be done.
• The point U = 0 is arbitrary

At the top of a table of height
h:
• U = mg(y+h)

The same experiment is
shifted by a constant
potential mgh:
• U = mgy + mgh = mgy + C
y2
y
h
y1
Universal 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  r
RE
r
GmM
F
r2
W 
r
RE
GmM
r
r2
1 1
W  GmM   r RE



Universal 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
U r
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
K  -U / 2

The total energy in a circular
orbit can be described in
terms of either the kinetic or
the potential energy.
E  K U
E  (-U / 2)  U
E  U / 2  -K
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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