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Gravitational Potential Energy
As discussed in a previous chapter, gravity is a conservative force so
that we can assign gravitational potential energy to any object moving
under the influence of gravity. However, in the previous discussion
the gravitational acceleration of the object was assumed constant, and
we now know that this is only approximately true for objects that move
through vertical distances that are small compared to the radius of
the Earth. More generally, (for example for satellite orbits) we must take into account the
fact that the gravitational acceleration changes as the
object moves. Instead of graviational potential energy being simply
proportional to the height above the Earth's surface, one finds that
it is inversely proportional to the distance from the object to the
center of the Earth. It is reasonable that gravitational potential
energy depends on Newton's constant and the Earth's mass in the same
way that the gravitational force does. The correct expression for
gravitational potential energy for an object a distance r from the
center of the Earth is:
U = 
The minus sign ensures that the potential energy decreases (gets more
and more negative) as the
object falls towards the center of the Earth. The dependence of
the gravitational potential energy U on the distance r from
the center of the Earth is illustrated in the Figure below.
Figure 6.1:
Gravitational Potential Energy

Notice that the
gravitational potential energy has been defined so that an object very far from
Earth's center (i.e. infinite r) has zero potential energy. This
observation enables us to calculate in a simple way the escape
velocity of any object sitting on the Earth.
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19990929