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

The correct option is B $P\to 3;Q\to 2;R\to 4;S\to 1$

Force due to gravity provides the necessary centripital force. So,
$\frac{GM{m}_1}{R_1^2}$ = $\frac{m_1{v}_1^2}{R_1}$

$v_1^2=\frac{GM}{R_1},v_2^2=\frac{GM}{R_2}$

$\frac{v_1^2}{v_2^2}=\frac{R_2}{R_1}=4$

(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 = $\frac{1}{2}m{v}^2$
$\frac{K_1}{K_2}=\frac{m_1{v}_1^2}{m_2{v}_2^2}=2\times (2{)}^2=8$

(S) T = $2\pi R/v$

$\frac{T_1}{T_2}=\frac{R_1}{v_1}\times \frac{v_2}{R_2}=\frac{R_1}{R_2}\times \frac{v_2}{v_1}=\frac{1}{4}\times \frac{1}{2}=\frac{1}{8}$