Two planets, A and B are orbiting a common star in circular orbits of radii $R_{A}$ and $R_{B}$, respectively, with $R_{B}=2 R_{A}$. The planet $B$ is $4 \sqrt{2}$ times more massive than planet $A$. The ratio $\left(\frac{L_{B}}{L_{A}}\right)$ of angular momentum $\left(\mathrm{L}_{\mathrm{B}}\right)$ of planet B to that of planet $\mathrm{A}\left(\mathrm{L}_{\mathrm{A}}\right)$ is closest to integer $\_\_\_\_$ .

JEE Main · 2025 · 29 Jan 2025 · Shift 2 · Gravitation

Two planets, A and B are orbiting a common star in circular orbits of radii

R_{A}

and

R_{B}

, respectively, with

R_{B} = 2 R_{A}

. The planet

B

42

times more massive than planet

A

. The ratio

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

of angular momentum

(L_{B})

of planet B to that of planet

A (L_{A})

is closest to integer

____

Answer: 8

Solution

A planet of mass

m

in a circular orbit of radius

R

around a star of mass

M

has gravity supplying the centripetal force:

\frac{GM m}{R ^{2}} = \frac{m v ^{2}}{R} ⟹ v = \frac{GM}{R}

Its angular momentum about the star is

L = m v R = m GM R

For the two planets,

L_{A} = m_{A} GM R_{A}, L_{B} = m_{B} GM R_{B}

\frac{L _{B}}{L _{A}} = \frac{m _{B}}{m _{A}} \frac{R _{B}}{R _{A}}

With

m_{B} = 42 m_{A}

and

R_{B} = 2 R_{A}

\frac{L _{B}}{L _{A}} = 42 \times 2 = 4 \times 2 = 8

JEE Main · 2025 · 29 Jan 2025 · Shift 2 · Integer answer

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