Question

A fixed sphere of radius $R$ and uniform density ρ 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 $\left(A\right)$ of the cavity. The gravitational field at A is $\frac{2}{3}\pi \rho GR$. Calculate the velocity with which the particle strikes the centre $O$ of the sphere. (Neglect earth's gravity)

• A. ${\left(\frac{2\pi G\rho R^2}{6}\right)}^{\frac{1}{2}}$
• B. ${\left(\frac{2\pi G\rho R^2}{3}\right)}^{\frac{1}{2}}$
• C. ${\left(\frac{2\pi G\rho R^2}{9}\right)}^{\frac{1}{2}}$
  
• D. None of these

Answer 1

The correct option is B ${\left(\frac{2\pi G\rho R^2}{3}\right)}^{\frac{1}{2}}$

Find the velocity at $O$.

Formula used: $F=mE=ma$
Given,
$(E_A{)}_{net}=\frac{2}{3}\pi \rho GR$
$s=\frac{R}{2}$

As we know,
$F=m(E_A{)}_{\text{net}}=ma$

$a=\left(\frac{2}{3}\right)\pi \rho GR$
Acceleration is uniform so, applying equation of motion,
$v^2=u^2+2as=0+2.\left(\frac{2}{3}\right)\pi \rho GR.\frac{R}{2}$
$v=\sqrt{\frac{2}{3}G\rho R^2}\pi$
Final Answer: (𝒂)