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 $F_1=\frac{GMm}{d^2}$
For cavity $F_2=\frac{\frac{Gm}{8}(m)}{{\left(d-\frac{R}{2}\right)}^2}$
$=\frac{GMm}{8d^2{\left(1-\frac{R}{2d}\right)}^2}$
So, resultant force
$F=F_1-F_2$
$=\frac{GMm}{d^2}-\frac{GMm}{8d^2{\left(1-\frac{R}{2d}\right)}^2}$
$=\frac{GMm}{d^2}\left[1-\frac{1}{8{\left(1-\frac{R}{2d}\right)}^2}\right]$