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Gravitational Field Due to a Shell

A point mass ...

Question

A point mass m is situated inside a thick spherical shell of mass M, inner radius R and outer radius 2R. Work required to take this mass m from inside the shell to infinity, slowly is

A

\frac{G M m}{2 R}

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B

\frac{7 G M m}{15 R}

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C

\frac{9 G M m}{14 R}

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D

\frac{3 G M m}{7 R}

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Solution

The correct option is C $\frac{9 G M m}{14 R}$

$d V = \frac{- G d m}{r} = \frac{- G}{r} ρ 4 π r^{2} d r V = \int d V = - G ρ 4 π \int_{R}^{2 R} r d r = \frac{- G ρ 4 π}{2} 3 R^{2} w h e r e ρ = \frac{M}{\frac{4}{3} π (8 R^{3} - R^{3})} = \frac{3 M}{4 π (7 R^{3})} \Rightarrow V = \frac{- 9 GM}{14 R} W o r k d o n e = - V \times m = \frac{9 GMm}{14 R}$

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