Question

A fixed sphere of radius R and uniform density $\rho$ has a spherical cavity of radius $\frac{R}{2}$ such that the surface of the cavity passes through the centre of the sphere. A particle of mass m is released at the centre (A) of the cavity.
Choose the incorrect statement (x is distance of a point inside cavity from centre of the cavity)

• A. The particle strikes the centre O of the sphere
• B. Gravitational field inside the cavity is proportional to x
• C. Acceleration of particle inside the cavity is independent of x
• D. Gravitational field inside the cavity is same at different points

Answer 1

The correct option is B Gravitational field inside the cavity is proportional to x

The gravitational field strength at P due to large sphere is $-\frac{4}{3}\pi G\rho (\frac{R}{2}-x)$


The gravitational field strength at P due to small sphere is $+\frac{4}{3}\pi G\rho x$

Net gravitational field at $P=\frac{2}{3}\pi G\rho R$(towards O) which is independent of x

$\Rightarrow$ Acceleration of mass m is $a=\frac{2}{3}\pi G\rho R$