3. Gravitational Energy
(15 points total)
Suppose the Earth is a sphere of uniform density, mass M and radius R. constant). Show
the total amount of work done by gravity to assemble the Earth (or, alternatively, the total
stored gravitational potential energy in the Earth) is given by:
U = 5R
HINT: Suppose you are assembling the Earth layer by layer. If the radius of the "partly-
built" Earth is r, calculate the amount of work gravity would do to bring a spherical,
differentially thin layer of width dr and radius r from infinity to the surface of the
Earth. Now, integrate this expression from r = 0 to r = R. The negative sign is
because gravity does the work for you.
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