A uniform solid sphere of mass M and radius R has its Centre at the origin O of the coordinate system. Two spherical cavities A and B, each of radius $r = R / 4$ are made such that their coordinates are $(2 r, 0, 0) a n d (- 2 r, 0, 0)$ respectively as shown in fig. The gravitational field at O due to the remaining part of the sphere is

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zero

$\frac{7 G M}{R^{2}}$

$\frac{9 G M}{R^{2}}$

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

answer is A.

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Detailed Solution

Mass of complete sphere is $M = \frac{4 π}{3} R^{3} ρ$ . Mass of each spherical cavity is
$m = \frac{4 π}{3} {(\frac{R}{4})}^{3} ρ = \frac{M}{64}$
Gravitational at O due to cavity A is
$I_{A} = \frac{G M}{(2 r)^{2}}$
$= \frac{G M \times 16}{64 \times 4 \times R^{2}} = \frac{G M}{16 R^{2}}$

directed to the left.
Gravitational field at O due to cavity B is
$I_{B} = \frac{G M}{16 R^{2}}$ directed to the right.
If I is gravitational field at O due to the complete sphere, then the gravitational field at O due to the remaining part is
$I_{r} = I - I_{A} + I_{B}$
Now $I_{A} = I_{B}$ and $I = 0$ (because $x = 0$ ), $I_{r} = 0$ . So the correct choice is (a)