Circular Motion Gravitation Force Moment of Inertia
Question
A sphere shell has inner radius R, outer radius R, and mass M, distributed uniformly throughout the shell. Find the magnitude of the gravitational force exerted on the shell by a point particle of mass m located at a distance d from the center, outside the inner radius and inside the outer radius.
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Hint
The mass of the spherical shell that will cause force will be mass located between the spheres of radius d and radius R1. As mass is distributed uniformly in the shell we can determine the density. Then we can find the force exerted.Answer
Mass of the sphere =
M
Point P is at a distance of d from the center and there is a particle of mass m located there.
The mass of the spherical shell that will cause force will be mass located between the spheres of radius d and radius R1.
That is mass of volume
4/3Πd3 - 4/3ΠR13 = 4/3Π(d3 - R13)
Mass is distributed uniformly in the shell. Thus density of the shell is
M/[4/3p(R23 - R13)]
Thus
M' = M/[4/3Π(R23 - R13)] x 4/3Π(d3 - R13) = [(d3 - R13)M]/(R23 - R13)
Thus force exerted on m is =
G.m.M' / d2
or