Question

The figure shows a spherical hollow inside a lead sphere of radius $R;$ the surface of the hollow passes through the centre of the sphere and touches the right side of the sphere. The mass of the sphere before hollowing was $M$. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass m that lies at a distance $d$ from the centre of the lead sphere, on the straight line connecting the centres of the spheres and of the hollow?

• A. $ \frac{GMm}{d^{2}}\left[ 1-\frac{1}{8\left( 1-\frac{R}{2d} \right)^{2}} \right] $
• B. $ \frac{GMm}{d^{2}}\left[ 1+\frac{1}{8\left( 1-\frac{R}{2d} \right)^{2}} \right] $
• C. $ \frac{GMm}{d^{2}}\left[ 1-\frac{1}{4\left( 1+\frac{R}{2d} \right)^{2}} \right] $
• D. $ \frac{GMm}{d^{2}}\left[ 1-\frac{1}{8\left( 1+\frac{R}{2d} \right)^{2}} \right] $

Answer 1

Answer: (A)

From the given figure For solid $F_{1}=\frac{G M m}{d^{2}}$
For cavity $F_{2}=\frac{\frac{G m}{8}(m)}{\left(d-\frac{R}{2}\right)^{2}}$
$=\frac{G M m}{8 d^{2}\left(1-\frac{R}{2 d}\right)^{2}}$
So, resultant force
$F=F_{1}-F_{2}$
$=\frac{G M m}{d^{2}}-\frac{G M m}{8 d^{2}\left(1-\frac{R}{2 d}\right)^{2}}$
$=\frac{G M m}{d^{2}}\left[1-\frac{1}{8\left(1-\frac{R}{2 d}\right)^{2}}\right]$