Gravitational Potential Energy
In Chapter 4, we learned that the potential energy of
a system is equal to the amount of work that must be done to arrange
the system in that particular configuration. We also saw that gravitational
potential energy depends on how high an object is off the
ground: the higher an object is, the more work needs to be done
to get it there.
Gravitational potential energy is not an absolute measure.
It tells us the amount of work needed to move an object from some
arbitrarily chosen reference point to the position it is presently
in. For instance, when dealing with bodies near the surface of the
Earth, we choose the ground as our reference point, because it makes
our calculations easier. If the ground is h = 0,
then for a height h above the ground
an object has a potential energy of mgh.
Gravitational Potential in Outer Space
Off the surface of the Earth, there’s no obvious
reference point from which to measure gravitational potential energy.
Conventionally, we say that an object that is an infinite distance
away from the Earth has zero gravitational potential energy with
respect to the Earth. Because a negative amount of work is done
to bring an object closer to the Earth, gravitational potential energy
is always a negative number when using this reference point.
The gravitational potential energy of two masses,
and
, separated by a distance
r is:
Example
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|
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A
satellite of mass  is launched from the
surface of the Earth into an orbit of radius  , where  is the radius of the Earth. How much work
is done to get it into orbit? |
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The work done getting the satellite from one place to
another is equal to the change in the satellite’s potential energy.
If its potential energy on the surface of the Earth is
and its potential energy when it is in
orbit is
, then the amount of work done is:
Energy of an Orbiting Satellite
Suppose a satellite of mass
is in orbit around the Earth at a radius
R.
We know the kinetic energy of the satellite is
KE =
1/2 mv2.
We also know that we can express centripetal force,
, as
=
mv2/
R.
Accordingly, we can substitute this equation into the equation for
kinetic energy and get:
Because
is equal to the gravitational
force, we can substitute Newton’s Law of Universal Gravitation in
for
:
We know that the potential energy of the satellite is
, so the total energy of the satellite is
the sum,
E = KE + U:
