Question

A fixed sphere of radius \(R\) and uniform density \(\rho\) has a spherical cavity of radius \(R/2\) such that the surface of the cavity passes through the centre of the sphere. A particle of mass \(m\) is located at the centre \((A)\) of the cavity. Calculate the gravitational field at \(A\).

Answer 1

Find the gravitational field at \(A\).

Formula used:

\(E=\dfrac{GM_{e}r}{R_{e}^{3}},M=\rho.\dfrac{4}{3}\pi R^{3}\)

Mass of bigger sphere \(\rho=\dfrac{4}{3}\pi R^{3}=M\)
Mass of smaller sphere \(\rho=\dfrac{4}{3}\pi (\dfrac{R}{2)^{3}}\)
Gravitational field due to big sphere at A.
\(E_{A}=\dfrac{GM(R/2)}{R^{3}}=\dfrac{GM}{2R^{2}}\)
Step 2:Find the gravitational field at \(A\).

Gravitational field due to small sphere at \(A\).

\(E'_{A}=0\)

\((E_{A})_{\text{net}}=\dfrac{GM}{2R^{2}}\)

\(=\dfrac{G\rho.\dfrac{4}{3}\pi R^{3}}{2R^{2}}\)

\(=\dfrac{2}{3}\pi \rho~GR(towards~0)\)
Final Answer: (d)