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

Q 11 :

Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length $a$ . When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length $2 a$ and also the masses of the spheres are made 2m, then they will collide after ________ seconds. [2025]

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(8)

$T \propto m^{x} G^{y} a^{z}$

$T \propto M^{x} {[M^{- 1} L^{3} T^{- 2}]}^{y} {[L]}^{z}$

$T \propto M^{x - y} L^{3 y + z} T^{- 2 y}$

$x - y = 0 \Rightarrow x = y$

$- 2 y = 1 \Rightarrow y = - \frac{1}{2}, x = - \frac{1}{2}$

$3 y + z = 0 \Rightarrow z = - 3 y = \frac{3}{2}$

Hence, $T \propto m^{- 1 / 2} G^{- 1 / 2} a^{3 / 2}$

$T \propto {(\frac{a^{3}}{m})}^{1 / 2} \Rightarrow T = 4 \times {(\frac{2^{3}}{2})}^{1 / 2} = 8 s$

Q 12 :

If the distance of the Earth from the Sun is $1.5 \times 10^{6}$ km, then the distance of an imaginary planet from the Sun, if its period of revolution is 2.83 years is: [2023]

$6 \times 10^{6} km$
$3 \times 10^{6} km$
$6 \times 10^{7} km$
$3 \times 10^{7} km$

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(2)

$T^{2} \propto R^{3} \Rightarrow {(\frac{T_{1}}{T_{2}})}^{2} = {(\frac{R_{1}}{R_{2}})}^{3}$

$\Rightarrow {(\frac{1}{2.83})}^{2} = {(\frac{1.5 \times 10^{6}}{R_{2}})}^{3}$

$\Rightarrow R_{2} = {[{(2.83)}^{2} \times {(1.5 \times 10^{6})}^{3}]}^{1 / 3}$

$= 8^{1 / 3} \times 1.5 \times 10^{6} = 3 \times 10^{6} km$

Q 13 :

Every planet revolves around the Sun in an elliptical orbit: [2023]

A. The force acting on a planet is inversely proportional to the square of the distance from the Sun.

B. Force acting on a planet is inversely proportional to the product of the masses of the planet and the Sun.

C. The centripetal force acting on the planet is directed away from the Sun.

D. The square of the time period of revolution of a planet around the Sun is directly proportional to the cube of the semi-major axis of the elliptical orbit.

Choose the correct answer from the options given below:

A and D only
B and C only
A and C only
C and C only

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(1)

$F = \frac{G m_{1} m_{2}}{r^{2}}$

$\Rightarrow F \propto \frac{1}{r^{2}}$

$\Rightarrow F \propto m_{1} m_{2}$

$\Rightarrow This force provides centripetal force and acts towards the Sun.$

$T^{2} \propto a^{3} (Kepler’s third law)$

Q 14 :

A planet has double the mass of the Earth. Its average density is equal to that of the Earth. An object weighing W on Earth will weigh on that planet [2023]

$2^{1 / 3} W$
$W$
$2^{2 / 3} W$
$2 W$

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(1)

$m = ρ \times \frac{4}{3} π R^{3}$

$R \propto m^{1 / 3} (ρ = constant)$

$Weight = W \propto g \propto \frac{G m}{R^{2}}$

$W \propto \frac{m}{m^{2 / 3}} \propto m^{1 / 3}$

$So, W' = {(2)}^{1 / 3} W$

Q 15 :

Choose the incorrect statement from the following: [2023]

The speed of a satellite in a given circular orbit remains constant.
For a planet revolving around the Sun in an elliptical orbit, the total energy of the planet remains constant.
When a body falls towards the Earth, the displacement of the Earth towards the body is negligible.
The linear speed of a planet revolving around the Sun remains constant.

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(4)

Planets revolve in elliptical paths around sun. Thus their linear speed is not constant.

Q 16 :

The radii of two planets 'A' and 'B' are 'R' and '4R' and their densities are $ρ$ and $ρ / 3$ respectively. The ratio of acceleration due to gravity at their surfaces $(g_{A} : g_{B})$ will be [2023]

1 : 16
3 : 16
3 : 4
4 : 3

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(3)

$g = \frac{G M}{R^{2}} = \frac{G}{R^{2}} \times ρ \times \frac{4 π}{3} R^{3} = (\frac{4 π}{3} G) ρ R$

$\frac{g_{A}}{g_{B}} = \frac{R \times ρ}{4 R \times \frac{ρ}{3}} = \frac{3}{4}$

Q 17 :

Two identical particles each of mass $' m'$ go round a circle of radius $a$ under the action of their mutual gravitational attraction. The angular speed of each particle will be [2023]

$\sqrt{\frac{G m}{2 a^{3}}}$
$\sqrt{\frac{G m}{8 a^{3}}}$
$\sqrt{\frac{G m}{4 a^{3}}}$
$\sqrt{\frac{G m}{a^{3}}}$

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(3)

[IMAGE 67]

$F = m ω^{2} r$

$\Rightarrow \frac{G m m}{{(2 a)}^{2}} = m ω^{2} a$

$\Rightarrow ω = \sqrt{\frac{G m}{4 a^{3}}}$

Q 18 :

If the Earth suddenly shrinks to $\frac{1}{64}$ th of its original volume with its mass remaining the same, the period of rotation of the Earth becomes $\frac{24}{x}$ h. The value of $x$ is ________ . [2023]

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(16)

From conservation of angular momentum

[IMAGE 68]

$\frac{2}{5} M R^{2} ω_{1} = \frac{2}{5} M {(\frac{R}{4})}^{2} ω_{2}$

$\Rightarrow M R^{2} ω_{1} = \frac{M R^{2}}{16} ω_{2}$

$\Rightarrow \frac{ω_{1}}{ω_{2}} = \frac{1}{16}$

$\Rightarrow \frac{T_{2}}{T_{1}} = \frac{1}{16} \Rightarrow \frac{T_{2}}{T_{1}} = \frac{16}{1} = \frac{t_{1}}{x}$

$∴$ $t_{1} = 24 \Rightarrow \frac{16}{1} = \frac{24}{t_{2}} \Rightarrow x = 16$

Topic Question Set

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Q 12 :

If the distance of the Earth from the Sun is $1.5 \times 10^{6}$ km, then the distance of an imaginary planet from the Sun, if its period of revolution is 2.83 years is: [2023]

Q 14 :

A planet has double the mass of the Earth. Its average density is equal to that of the Earth. An object weighing W on Earth will weigh on that planet [2023]

Q 15 :

Choose the incorrect statement from the following: [2023]

Q 16 :

The radii of two planets 'A' and 'B' are 'R' and '4R' and their densities are $ρ$ and $ρ / 3$ respectively. The ratio of acceleration due to gravity at their surfaces $(g_{A} : g_{B})$ will be [2023]

Q 17 :

Two identical particles each of mass $' m'$ go round a circle of radius $a$ under the action of their mutual gravitational attraction. The angular speed of each particle will be [2023]

Q 18 :

If the Earth suddenly shrinks to $\frac{1}{64}$ th of its original volume with its mass remaining the same, the period of rotation of the Earth becomes $\frac{24}{x}$ h. The value of $x$ is ________ . [2023]

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Topic Question Set

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Q 15 : Choose the incorrect statement from the following: [2023] .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

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Q 15 :

Choose the incorrect statement from the following: [2023]