Question

The planets with radii $R_1$ and $R_2$ have densities ${\rho }_1$, ${\rho }_2$ respectively. Their atmospheric pressures are $p_1$ and $p_2$ respectively. Therefore, the ratio of masses of their atmospheres, neglecting variation of g within the limits of atmosphere is?

• A. ${\rho }_1{R}_2{p}_1/{\rho }_2{R}_1{p}_2$
• B. $p_1{R}_2{\rho }_2/p{P}_2{R}_2{\rho }_1$
• C. $p_1{R}_1{\rho }_1/p_2{R}_2{\rho }_2$
• D. $p_1{R}_1{\rho }_2/p_2{R}_2{\rho }_1$

Answer 1

The correct option is D $p_1{R}_1{\rho }_2/p_2{R}_2{\rho }_1$

Acceleration due to gravity,
$g=\frac{GM}{R^2}=\frac{4}{3}\pi \rho GR$
$\therefore \frac{g_1}{g_2}=\frac{{\rho }_1{R}_1}{{\rho }_2{R}_2}$
Atmospheric pressure can be given by $p=\frac{W}{S}$
where, $W=$ weight of atmosphere,
$S=$ Surface area of the planet
$\therefore \frac{m_1}{m_2}=\frac{{\rho }_1{S}_1{g}_2}{{\rho }_2{S}_2{g}_1}=\frac{p_1\cdot \left(4\pi R_1^2\right)}{p_2\left(4\pi R_2^2\right)}\cdot \frac{{\rho }_2{R}_1}{{\rho }_1{R}_1}=\frac{p_1{R}_1{\rho }_2}{p_2{R}_2{\rho }_1}$.