Question

A planet of mass M has two natural satellites with masses $m_1$ and $m_2$ the radii of their circular orbits are $R_1$ and $R_2$ respectively. Ignore the gravitational force between the satellites. Define $V_1, L_1, U_1$ and $T_1$ to be respectively, the escape velocity, angular momentum, potential energy and time period of revolution of satellite -1 and $V_2, L_2, U_2$ and $T_2$ to be the corresponding quantities of satellite-2.

Given $\frac{m_1}{m_2}=3$ and $\frac{R_1}{R_2}=\frac{1}{9}$. Match the ratios in List-I to the numbers in List-II.

	List-I		List-II
P	$\frac{V_1}{V_2}$	I	3
Q	$\frac{L_1}{L_2}$	II	$\frac{1}{27}$
R	$\frac{U_1}{U_2}$	III	1
S	$\frac{T_1}{T_2}$	IV	27

• A. P $\to$ I, Q $\to$ III, R $\to$ IV, S $\to$ II
• B. P $\to$ III, Q $\to$ I, R $\to$ II, S $\to$ IV
• C. P $\to$ I, Q $\to$ II, R $\to$ III, S $\to$ IV
• D. P $\to$ IV, Q $\to$ III, R $\to$ I, S $\to$ II

Answer 1

Answer: (A)

P $\to$ I, Q $\to$ III, R $\to$ IV, S $\to$ II

Answer 2

For a satellite in a circular orbit around a planet of mass $M$: Orbital velocity $v = \sqrt{\frac{GM}{R}}$, so $v \propto R^{-1/2}$. Escape velocity $V_e = \sqrt{\frac{2GM}{R}}$, so $V_e \propto R^{-1/2}$. Angular momentum $L = mvr = m \sqrt{\frac{GM}{R}} R = m \sqrt{GM} R^{1/2}$, so $L \propto m R^{1/2}$. Potential energy $U = -\frac{GMm}{R}$, so $U \propto m R^{-1}$. Time period $T = \frac{2\pi R}{v} = \frac{2\pi R}{\sqrt{GM/R}} = \frac{2\pi}{\sqrt{GM}} R^{3/2}$, so $T \propto R^{3/2}$.

Given $\frac{m_1}{m_2} = 3$ and $\frac{R_1}{R_2} = \frac{1}{9}$.

P. $\frac{V_1}{V_2}$: $\frac{V_1}{V_2} \propto \frac{R_2^{1/2}}{R_1^{1/2}} = \sqrt{\frac{R_2}{R_1}} = \sqrt{9} = 3$. Match: P $\to$ I (3)

Q. $\frac{L_1}{L_2}$: $\frac{L_1}{L_2} = \frac{m_1 R_1^{1/2}}{m_2 R_2^{1/2}} = \frac{m_1}{m_2} \sqrt{\frac{R_1}{R_2}} = 3 \sqrt{\frac{1}{9}} = 3 \times \frac{1}{3} = 1$. Match: Q $\to$ III (1)

R. $\frac{U_1}{U_2}$: $\frac{U_1}{U_2} = \frac{m_1 R_1^{-1}}{m_2 R_2^{-1}} = \frac{m_1}{m_2} \frac{R_2}{R_1} = 3 \times 9 = 27$. Match: R $\to$ IV (27)

S. $\frac{T_1}{T_2}$: $\frac{T_1}{T_2} \propto \frac{R_1^{3/2}}{R_2^{3/2}} = \left(\frac{R_1}{R_2}\right)^{3/2} = \left(\frac{1}{9}\right)^{3/2} = \left(\left(\frac{1}{3}\right)^2\right)^{3/2} = \left(\frac{1}{3}\right)^3 = \frac{1}{27}$. Match: S $\to$ II ($\frac{1}{27}$)