Bonus question for exam 3 (+10%)
G. Berkolaiko
March 26, 2008
Find the gravitational force exerted upon a body b of mass m (kg) by a planet P of
radius R (m) and uniform density ρ (kg/m3 ), if the body b is inside the planet at distance
d meters (d < R) from its center. The solution must involve setting up and evaluating a
3d integral. Do not use any facts that you cannot prove using calculus (e.g. “gravitational
field inside a hollow sphere is zero”).
Some notes:
1. We can assume the body is a point and neglect the tunnel that got the body inside
the planet.
2. According to the Newton’s Law of Gravity, the force on the body due to an infinitesimal
piece of the planet of volume dV (m3 ) is
~
Fgrav =
γmρdV
,
r2
where γ (N-m2 /kg2 ) is the gravitational constant, r (m) is the distance from b to the
piece dV and ρdV is the mass of the piece dV .
3. F~grav is a vector and it acts on the body in the direction towards the piece dV . From
symmetry considerations, the total force must pull the body towards the center of the
planet. Thus we need to sum only the component of the vector F~grav that acts towards
the center. Be careful to actually calculate this component.
Rules:
1. You must not copy the solution from any source (such as Internet or a book). Doing
so is plagiarism and therefore cheating.
2. If you choose to collaborate on this project, you must acknowledge your collaboration and the resulting bonus (if any) will be split evenly among the collaborators.
3. Not acknowledging your collaboration is cheating and will not be tolerated.
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