Two identical metal spheres are kept in contact with each other, each having radius ' $R$ ' cm and ' $\rho$ ' is the density of material of metal spheres. The gravitational force ' $F$ ' of attraction between them is proportional to

• A

  $R^3 \rho$
• Correct answer
  B

  $R^4 p^2$
• C

  $R^4 \rho$
• D

  $R^3 \rho^2$

Solution

Force due to gravity, $\begin{aligned} & \mathrm{F}=\frac{\mathrm{GMM}}{(2 \mathrm{R})^2} \\ & \mathrm{~F}=\frac{\mathrm{GM}^2}{4 \mathrm{R}^2}...(i) \end{aligned}$
Density, $\rho=\frac{M}{V}$ $M=\rho \times \frac{4}{3} \pi R^3$
Substituting in (i), $\begin{aligned} & F=\frac{G\left(\frac{4}{3} \pi R^3 \rho\right)^2}{4 R^2} \\ \therefore \quad & F \propto R^4 \rho^2 \end{aligned}$

Hence, the correct answer is option: 'B'.