Kepler's Laws and Newton's Law of Gravitation | JEE Main previous year questions with Solutions

Q 1 :
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 [2024]

$\frac{1}{27} {(\frac{m_{2}}{m_{1}})}^{3}$
${(\frac{r_{1}}{r_{2}})}^{3}$
${(\frac{r_{2}}{r_{1}})}^{\frac{3}{2}}$
$27 {(\frac{m_{1}}{m_{2}})}^{3}$

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(A) We know, $T^{2} \propto r^{3} \Rightarrow {(\frac{T_{A}}{T_{B}})}^{2} = {(\frac{r_{1}}{r_{2}})}^{3}$

$\Rightarrow \frac{T_{A}}{T_{B}} = {(\frac{r_{1}}{r_{2}})}^{3 / 2}$ ...(1)

Given $L_{B} = 3 L_{A} \Rightarrow m_{2} v_{2} r_{2} = 3 (m_{1} v_{1} r_{1})$

$\Rightarrow \frac{v_{2}}{v_{1}} = 3 \frac{m_{1} r_{1}}{m_{2} r_{2}}$ ...(2)

Orbital velocity, $v = \sqrt{\frac{G M_{s}}{r}} \Rightarrow \frac{v_{2}}{v_{1}} = \sqrt{\frac{r_{1}}{r_{2}}}$ ...(3)

From eq. (2) and eq. (3)

$\sqrt{\frac{r_{1}}{r_{2}}} = \frac{3 m_{1}}{m_{2}} \frac{r_{1}}{r_{2}} \Rightarrow \frac{r_{1}}{r_{2}} = (\frac{m_{2}}{3 m_{1}})$ ...(4)

Using eq. (4) in eq. (1)

$\frac{T_{A}}{T_{B}} = {(\frac{m_{2}}{3 m_{1}})}^{3} = \frac{1}{27} {(\frac{m_{2}}{m_{1}})}^{3}$

Q 2 :
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): The angular speed of the moon in its orbit about the earth is more than the angular speed of the earth in its orbit about the sun.

Reason (R): The moon takes less time to move around the earth than the time taken by the earth to move around the sun.

In the light of the above statements, choose the most appropriate answer from the given below:   [2024]

(A) is correct but (R) is not correct
Both (A) and (R) are correct and (R) is the correct explanation of (A)
Both (A) and (R) are correct but (R) is not the correct explanation of (A)
(A) is not correct but (R) is correct

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(B) $ω = \frac{2 π}{T} \Rightarrow ω \propto \frac{1}{T}$

$T_{m o o n} =$ 27 days

$T_{e a r t h} =$ 365 days 4 hour

$\Rightarrow ω_{m o o n} > ω_{e a r t h}$

Q 3 :
A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution [2024]

20
25
50
100

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(B) $T^{2} \propto r^{3} \Rightarrow \frac{T_{1}^{2}}{r_{1}^{3}} = \frac{T_{2}^{2}}{r_{2}^{3}}$

$\frac{{(200)}^{2}}{r^{3}} = \frac{T_{2}^{2}}{{(\frac{r}{4})}^{3}}$

$\frac{200 \times 200}{4 \times 4 \times 4} = T_{2}^{2} \Rightarrow T_{2} = \frac{200}{4 \times 2}$

$T_{2} = 25 d a y s$

Q 4 :
A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{- 3 / 2}$ then choose the correct option [2024]

$T^{2} \propto R^{5 / 2}$
$T^{2} \propto R^{7 / 2}$
$T^{2} \propto R^{3 / 2}$
$T^{2} \propto R^{3}$

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(A) $∵$ $F \propto \frac{1}{R^{3 / 2}}$ or $F = \frac{K}{R^{3 / 2}}$

Gravitational force provides centripetal force.

$\frac{K}{R^{3 / 2}} = m ω^{2} R$

$ω^{2} = \frac{K}{m R^{5 / 2}}$

${(\frac{2 π}{T})}^{2} = \frac{K}{m R^{5 / 2}} \Rightarrow T^{2}$

$= \frac{4 π^{2} m R^{5 / 2}}{K} \Rightarrow T^{2} \propto R^{5 / 2}$

Topic Question Set

Q 1 :
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 [2024]

Q 3 :
A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution [2024]

Q 4 :
A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{- 3 / 2}$ then choose the correct option [2024]

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Topic Question Set

Q 1 : Two planets A and B having masses m1 and m2 move around the sun in circular orbits of r1 and r2 radii respectively. If angular momentum of A is L and that of B is 3 L, the ratio of time period (TATB) is [2024]

Q 3 : A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution [2024]

Q 4 : A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to R-3/2 then choose the correct option [2024]

Want Us to Email you About Special Offers & Updates?

Q 1 :
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 [2024]

Q 3 :
A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution [2024]

Q 4 :
A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{- 3 / 2}$ then choose the correct option [2024]