Problem of the Week # 7  Solution
In the rough approximation that the density of the Earth is uniform throughout its interior,
the gravitational force on a mass
m
inside the Earth at a distance
r
from the center is
mgr
R
, where
R
is the radius
R
of the Earth. (Note that at the surface
r = R
, and the
gravitational force is
mgR
R
=
mg
.) Using this uniformdensity approximation, calculate
the amount of energy required to move a mass
m
from the center of the Earth to the
surface, through a tiny hole, without changing its speed. Also calculate the additional
amount of energy required to move the mass from the surface of the Earth to a great
distance away, without changing its speed.
In what follows, we will take the mass,
m
, to be the
system
, and the Earth and an
external agent
to be in the
surroundings
.
We first need to calculate the gravitational force on mass
m
as a function of its (radial)
distance
r
, from the Earth’s center. This force depends only on the mass within the sphere
of radius
r
.
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 Fall '08
 ?
 Force, Mass, Potential Energy, Fundamental physics concepts, Earth’s center

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