A solid sphere of uniform density and mass M has radius 4 m. Its centre is at the origin of the coordinate system. Two spheres of radii 1 m are taken out so that their centres are P (0, -2,0) and Q (0,2,0) respectively as shown in fig This leaves two spherical cavities.
What is the gravitational field at the origin of the coordinate axis ?

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31 GM

$\frac{31 GM}{1024}$

$\frac{GM}{1024}$

Zero

answer is D.

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Detailed Solution

Let p be the density of sphere, R radius of the sphere and r, radius of spherical cavities
Mass of complete sphere $= \frac{4}{3} {πR}^{3} ρ = M$
Mass of sphere removed $= \frac{4}{3} {πr}^{3} ρ = m$
Here $m = \frac{{Mr}^{3}}{R^{3}} = \frac{M (1)^{3}}{(4)^{3}} = \frac{M}{64}$
Now , $I_{R} = I + I_{P} + I_{Q}$
$\begin{array}{l} 0 = I + I_{P} - I_{P} (∵ I_{Q} = - I_{P} and I_{P} = 0) \\ ∴ I = 0 \end{array}$