First slide

Gravitational field intensity

Question

A cavity of radius R/2 is made inside a solid sphere of radius R. The centre of the cavity is located at a distance R/2 from the centre of the sphere. The gravitational force on a particle of mass ‘m’ at a distance R/2 from the centre of the sphere on the line joining both the centres of sphere and cavity is (opposite to the centre of cavity) [Here $g = GM / R^{2}$ , where M is the mass of the sphere]

Difficult

A

$\frac{mg}{2}$
B

$\frac{3 mg}{8}$
C

$\frac{mg}{16}$
D

None of these

Solution

$g_{net} = g_{1} - g_{2}$
$\begin{array}{l} = \frac{Gm \frac{R}{2}}{R^{3}} - \frac{G (\frac{m}{8})}{R^{2}} \\ = \frac{Gm}{2 R^{2}} - \frac{Gm}{8 R^{2}} \\ = \frac{Gm}{2 R^{2}} [1 - \frac{1}{4}] \\ = \frac{Gm}{2 R^{2}} [\frac{3}{4}] \\ = \frac{3 Gm}{8 R^{2}} \end{array}$
$\begin{array}{l} F = mg \\ = \frac{3 GMm}{8 R^{2}} = \frac{3 mg}{8} \end{array}$

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