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S10 PHY321: Midterm 3
April 16, 2010
NOTE: Show all your work.
No credit
for unsupported answers
.
Moreover,
the work can earn you credit for problems that are not fully completed!
Turn the front page only when advised by the instructor!
Total points for this exam:
25
Useful formulas can be found at the end of this exam.
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1. A spherical planet of radius
R
has a density that is largest in its center
and decreases with distance
r
from the center as
ρ
(
r
) =
B
(
R
2

r
2
),
where
B
is a constant with the appropriate units.
(a) [3 pts] Determine the mass of the planet in terms of
B
and
R
.
(b) [2 pts] Use the shell theorem to find the gravitational field at
distance
r
≥
R
.
(c) [3 pts] Use the shell theorem to find the gravitational field at
distance
r
≤
R
.
Check whether the answers for (1b) and (1c)
agree at
r
=
R
.
(d) [4 pts] Use your answer to part (1c) and the differential form
of Gauss’ law (
~
∇·
~g
(
~
r
) =

4
πGρ
(
~
r
)) to derive the original density.
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 Spring '08
 B.Pope
 mechanics, Mass, Work, pts, Gravitational field, Celestial mechanics

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