Question

The spherical planets have the same mass but densities in the ratio $1:8$. for these planets, the

• A. acceleration due to gravity will be in the ratio $4:1$
• B. acceleration due to gravity will be in the ratio $1:4$
• C. escape velocities from their surfaces will be in the ratio $\sqrt{2}:1$
• D. escape velocities from their surfaces will be in the ratio $1:\sqrt{2}$

Answer 1

Answer: (B), (C), (D)

escape velocities from their surfaces will be in the ratio $1:\sqrt{2}$

Given:
${\rho }_1:{\rho }_2=1:8$ and $m_1=m_2$

We know that, $\rho =\frac{m}{V}$
$\Rightarrow {\rho }_1{V}_1={\rho }_2{V}_2$

$\Rightarrow {\rho }_1\frac{4}{3}\pi R_1^3={\rho }_2\frac{4}{3}\pi R_2^3$

$\Rightarrow {\left(\frac{R_1}{R_2}\right)}^3=\frac{{\rho }_2}{{\rho }_1}=\frac{8}{1}$

$\Rightarrow \left(\frac{R_1}{R_2}\right)=\frac{2}{1}$

As we know that the acceleration due to gravity, $g=\frac{GM}{R^2}$

$g=\frac{G}{R^2}\times \rho \times \frac{4}{3}\pi R^3=\frac{4}{3}G\pi \rho R$

$\Rightarrow \frac{g_1}{g_2}=\left(\frac{{\rho }_1}{{\rho }_2}\right)\left(\frac{R_1}{R_2}\right)$

$\therefore \frac{g_1}{g_2}=\frac{1}{8}\times 2=\frac{1}{4}$

Now escape velocity,
$V_e=\sqrt{\frac{2GM}{R}}=\sqrt{\frac{2G}{R}\times \frac{4}{3}\pi R^3\rho }$

$V_e\propto R\sqrt{\rho }$

$\Rightarrow \frac{{\left(V_e\right)}_1}{{\left(V_e\right)}_2}=\frac{R_1}{R_2}\left(\sqrt{\frac{{\rho }_1}{{\rho }_2}}\right)$

$\therefore \frac{{\left(V_e\right)}_1}{{\left(V_e\right)}_2}=2\left(\sqrt{\frac{1}{8}}\right)=\frac{1}{\sqrt{2}}$