Question 44: Gravitation

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-$${\rm I}$$  to the numbers in List-$${\rm II}.$$

LIST - I 
LIST - II

P.
v1/v2
1.
1/8

Q.
L1/L2
2.
1

R.
K1/K2
3.
2

S.
T1/T2
4.
8

Accepted Answer

Given : $${{{m_1}} \over {{m_2}}} = 2$$ and $${{{R_1}} \over {{R_2}}} = {1 \over 4}$$
Now,
$${{GMm} \over {{R^2}}} = {{m{v^2}} \over R}$$
$$ \Rightarrow {{GM} \over R} = {v^2} \Rightarrow v = \sqrt {{{GM} \over R}} $$
$$ \Rightarrow {{{v_1}} \over {{v_2}}} = {{\sqrt {{{GM} \over {{R_1}}}} } \over {\sqrt {{{GM} \over {{R_2}}}} }} \Rightarrow {{{v_1}} \over {{v_2}}} = \sqrt {{{{R_2}} \over {{R_1}}}} $$
$$ \Rightarrow {{{v_1}} \over {{v_2}}} = \sqrt {{4 \over 1}}  \Rightarrow {{{v_1}} \over {{v_2}}} = 2$$
Thus, the correct mapping is P $$	o$$ 3.
Now, L = mvR
$$ \Rightarrow {{{L_1}} \over {{L_2}}} = {{{m_1}{v_1}{R_1}} \over {{m_2}{v_2}{R_2}}} = {{{m_1}} \over {{m_2}}} 	imes {{{v_1}} \over {{v_2}}} 	imes {{{R_1}} \over {{R_2}}}$$
$$ \Rightarrow {{{L_1}} \over {{L_2}}} = 2 	imes 2 	imes {1 \over 4} = 1$$
Thus, the correct mapping is Q $$	o$$ 2.
Now,
$$K = {1 \over 2}m{v^2}$$
$$ \Rightarrow {{{K_1}} \over {{K_2}}} = {{{1 \over 2}{m_1}v_1^2} \over {{1 \over 2}{m_2}v_2^2}} = {{{m_1}} \over {{m_2}}} 	imes {\left( {{{{v_1}} \over {{v_2}}}} ight)^2}$$
$$ \Rightarrow {{{K_1}} \over {{K_2}}} = 2 	imes {(2)^2} = 8$$
Thus, the correct mapping is R $$	o$$ R.
Now,
$$T = {{2\pi R} \over v}$$
$$ \Rightarrow {{{T_1}} \over {{T_2}}} = {{{{2\pi {R_1}} \over {{v_1}}}} \over {{{2\pi {R_2}} \over {{v_2}}}}} \Rightarrow {{{T_1}} \over {{T_2}}} = {{{R_1}} \over {{R_2}}} 	imes {{{v_2}} \over {{v_1}}}$$
$$ \Rightarrow {{{T_1}} \over {{T_2}}} = {1 \over 4} 	imes {1 \over 2} \Rightarrow {{{T_1}} \over {{T_2}}} = {1 \over 8}$$
Thus, the correct mapping is S $$	o$$ 1.
Therefore, option (B) is correct.

Answer

$$P 	o 4;Q 	o 2;R 	o 1;S 	o 3$$

Answer

$$P 	o 3;Q 	o 2;R 	o 4;S 	o 1$$

Answer

$$P 	o 2;Q 	o 3;R 	o 1;S 	o 4$$

Answer

$$P 	o 2;Q 	o 3;R 	o 4;S 	o 1$$

Gravitation

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