From a solid sphere of mass M and radius R, a spherical portion of radius is removed, as shown in the figure

Question

From a solid sphere of mass M and radius R, a spherical portion of radius $\frac{R}{2}$ is removed, as shown in the figure. Taking gravitational potential $V = 0$ at $r = \infty$ , the potential at the centre of the cavity thus formed is:

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(4)

$Due to complete solid sphere, potential at point P$

$V_{sphere} = \frac{- G M}{2 R^{3}} [3 R^{2} - {(\frac{R}{2})}^{2}]$

$= \frac{- G M}{2 R^{3}} (\frac{11 R^{2}}{4}) = - \frac{11 G M}{8 R}$

$Due to cavity part, potential at point P$

$V_{cavity} = - \frac{3}{2} \frac{\frac{M}{8}}{\frac{R}{2}} = - \frac{3 G M}{8 R}$

$So potential at the centre of cavity$

$= V_{sphere} - V_{cavity} = - \frac{11 G M}{8 R} - (- \frac{3 G M}{8 R}) = \frac{- G M}{R}$

Solution

Q.
From a solid sphere of mass M and radius R, a spherical portion of radius $\frac{R}{2}$ is removed, as shown in the figure. Taking gravitational potential $V = 0$ at $r = \infty$ , the potential at the centre of the cavity thus formed is:

1 $\frac{- 2 G M}{3 R}$

2 $\frac{- 2 G M}{R}$

3 $\frac{- G M}{2 R}$

4 $\frac{- G M}{R}$

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Solution

Q. From a solid sphere of mass M and radius R, a spherical portion of radius R2 is removed, as shown in the figure. Taking gravitational potential V=0 at r=∞, the potential at the centre of the cavity thus formed is:

1 -2GM3R

2 -2GMR

3 -GM2R

4 -GMR