Imagine a isolated uniform solid spherical rigid planet (ignore its rotation) of mass M and radius R. A smaller sphere of radius 2 R whose periphery coincides with centre of original planet can be s

Question

Imagine a isolated uniform solid spherical rigid planet (ignore its rotation) of mass M and radius R. A smaller sphere of radius $\frac{R}{2}$ whose periphery coincides with centre of original planet can be scooped out as shown. If the scooped out portion somehow is slowly kept back in its place, then what is contact force between this scooped out portion and rest of the planet?

TestHub · Accepted Answer

Gravitational field inside the cavity is constant: $E = \frac{GM}{R ^{3}} (O O^{'}) = \frac{GM}{R ^{3}} \cdot \frac{R}{2} = \frac{GM}{2 R ^{2}}$ directed toward the planet's centre.

Mass of scooped sphere: $m = \frac{M}{\frac{4}{3} π R ^{3}} \times \frac{4}{3} π (\frac{R}{2})^{3} = \frac{M}{8}$

Contact force when kept back: $F = m E = \frac{M}{8} \times \frac{GM}{2 R ^{2}} = \frac{G M ^{2}}{16 R ^{2}}$ Hence (A).

Physics - Gravitation Question with Solution | TestHub

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