First slide

Gravitational field intensity

Question

A solid sphere of mass M and radius R has a spherical cavity of radius R/2 such that the centre of cavity is at a distance R/2 from the centre of the sphere. A point mass m is placed inside the cavity at a distance R/4 from the centre of sphere. The gravitational force on mass m is

Moderate

A

$\frac{11 G M m}{R^{2}}$
B

$\frac{14 G M m}{R^{2}}$
C

$\frac{G M m}{2 R^{2}}$
D

$\frac{G M m}{R^{2}}$

Solution

Field strength is uniform inside the cavity. Let us find its value at centre.
$E_{T} = E_{O} + E_{C} (T = Total, O = Remaining, C = Cavity)$
$∴ E_{O} = E_{T} - E_{C} = \frac{G M}{R^{3}} \frac{R}{2} - 0 = \frac{G M}{2 R^{2}}$
$∴ F = m F_{R} = \frac{G M m}{2 R^{2}}$

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