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{GMm_{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 \times2 \times \frac{1}{4} = 1 $
(R) $K = \frac{1}{2} mv^{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} $