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JIPMER 2017
Exam
JIPMER
Year
2017
Subject
Physics
Chapter
Gravitation

JIPMER 2017 · Physics · Gravitation

The planets with radii R_1 and R_2 have densities _1, _2...

The planets with radii R1R_1 and R2R_2 have densities ρ1,ρ2\rho_1, \quad \rho_2 respectively. Their atmospheric pressures are p1p_1 and p2p_2 respectively. Therefore, the ratio of masses of their atmospheres, neglecting variation of gg within the limits of atmosphere is

Options

  1. Aρ1R2ρ1/ρ2R1ρ2\rho_1 R_2 \rho_1 / \rho_2 R_1 \rho_2
  2. Bρ1R2ρ2/ρP2R1ρ1\rho_1 R_2 \rho_2 / \rho P_2 R_1 \rho_1
  3. Cp1R1ρ1/p2R2ρ2p_1 R_1 \rho_1 / p_2 R_2 \rho_2
  4. Dp1R1ρ2/ρ2R2ρ1p_1 R_1 \rho_2 / \rho_2 R_2 \rho_1

Correct answer

D. p1R1ρ2/ρ2R2ρ1p_1 R_1 \rho_2 / \rho_2 R_2 \rho_1

Step-by-step solution

Acceleration due to gravity, g=GMR2=43πρGR∴g1g2=ρ1R1ρ2R2\begin{aligned} & g=\frac{G M}{R^2}=\frac{4}{3} \pi \rho G R \\ & \therefore \frac{g_1}{g_2}=\frac{\rho_1 R_1}{\rho_2 R_2} \end{aligned} Atmospheric pressure can be given by p=WSp=\frac{W}{S} where, W=W= Weight of atmosphere, S=S= Surface area of the planet ∴m1m2=ρ1S1g2ρ2S2g1=p1⋅(4πR12)p2(4πR22)⋅ρ2R1ρ1R1=p1R1ρ2p2R2ρ1\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}
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