JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Gravitation

Two planets A and B having masses

m_{1}

and

m_{2}

move around the sun in circular orbits of

r_{1}

and

r_{2}

radii respectively. If angular momentum of A is L and that of B is

3 L

, the ratio of time period

(\frac{T _{A}}{T _{B}})

is:

Answer: (C)

Solution

Areal velocity

\frac{d A}{d t} = \frac{L}{2 m}

; over one period the swept area is

π r^{2}

\frac{π r _{1}^{2}}{T _{A}} = \frac{L}{2 m _{1}} (1), \frac{π r _{2}^{2}}{T _{B}} = \frac{3 L}{2 m _{2}} (2)

Dividing (1) by (2):

(\frac{r _{1}}{r _{2}})^{2} \frac{T _{B}}{T _{A}} = \frac{m _{2}}{3 m _{1}} \Rightarrow \frac{T _{A}}{T _{B}} = 3 \frac{m _{1}}{m _{2}} (\frac{r _{1}}{r _{2}})^{2}

.
By Kepler's law

T^{2} \propto r^{3}

, so

(\frac{r _{1}}{r _{2}})^{2} = (\frac{T _{A}}{T _{B}})^{4/3}

. Substituting,

(\frac{T _{A}}{T _{B}})^{1/3} = \frac{m _{2}}{3 m _{1}}

, therefore

\frac{T _{A}}{T _{B}} = (\frac{m _{2}}{3 m _{1}})^{3} = \frac{1}{27} (\frac{m _{2}}{m _{1}})^{3}

Option (C).

JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Single correct option