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 $v_1,L_1,K_1$ and $T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and $v_2,L_2,K_2$ and $T_2$ to be the corresponding quantities of satellite 2. Given $m_1/m_2=2$ and $R_1/R_2=1/4$, match the ratios in List-I to the numbers in List-II.

List-I		List-II
P.	$\frac{v_1}{v_2}$	1.	$\frac{1}{8}$
Q.	$\frac{L_1}{L_2}$	2	1
R.	$\frac{K_1}{K_2}$	3	2
S.	$\frac{T_1}{T_2}$	4	8

• A. P $\to$ 4; Q $\to$ 2; R $\to$ 1; S $\to$ 3
• B. P $\to$ 3; Q $\to$ 2; R $\to$ 4; S $\to$ 1
• C. P $\to$ 2; Q $\to$ 3; R $\to$ 1; S $\to$ 4
• D. P $\to$ 2; Q $\to$ 3; R $\to$ 4; S $\to$ 1

Answer 1

Answer: (B)

$\frac{GM{m}_1}{R_1^2}=\frac{m_1{v}_1^2}{R_1}$
$v_1^2=\frac{GM}{R_1},v_2^2=\frac{GM}{R_2}$
$\frac{v_1^2}{v_2^2}=\frac{R_2}{R_1}=4$
(P) $\frac{v_1}{v_2}=2$
(Q) $L=mvR$
$\frac{L_1}{L_2}=\frac{m_1{v}_1{R}_1}{m_2{v}_2{R}_2}=2\times 2\times \frac{1}{4}=1$
(R) $K=\frac{1}{2}m{v}^2$
$\frac{K_1}{K_2}=\frac{m_1{v}_1^2}{m_2{v}_2^2}=2\times {\left(2\right)}^2=8$
(S) $T=2\pi R/V$
$\frac{T_1}{T_2}=\frac{R_1}{V_1}\times \frac{v_2}{R_2}=\frac{R_1}{R_2}\times \frac{v_2}{v_1}=\frac{1}{4}\times \frac{1}{2}=\frac{1}{8}$