Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. Reason (R) : The mass of the pendulum remains unchanged at Earth and the other planet. In the light of the above statements, choose the correct answer from the options given below :

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

Both (A) and (R) are true and (R) is the correct explanation of (A)

Simple Harmonic Motion PYQ — JEE Main 2025 · 22 Jan · Shift 2 · Q51 with Solution

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Correct answer: D

Time period of a simple pendulum

For small oscillations, $T = 2\pi \sqrt{\frac{l}{g}}$ where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity.
Compare gravity on Earth and on the planet

Given for the new planet:
- Mass $M_p = 4M_E$
- Radius $R_p = 2R_E$
Using $g = \frac{GM}{R^2}$

For the planet, $g_p = \frac{G(4M_E)}{(2R_E)^2} = \frac{4GM_E}{4R_E^2} = \frac{GM_E}{R_E^2} = g_E$

So, the acceleration due to gravity is the same on Earth and on the planet.
Check Assertion (A)

Since $g_p = g_E$ and the pendulum length is unchanged, $T_p = 2\pi\sqrt{\frac{l}{g_p}} = 2\pi\sqrt{\frac{l}{g_E}} = T_E$

Therefore, Assertion (A) is true.
Check Reason (R)

The mass of the pendulum bob does remain unchanged when taken from Earth to another planet.

So, Reason (R) is true.
Does (R) explain (A)?

The time period of a simple pendulum does not depend on the mass of the bob. It depends on $l$ and $g$ only.

The assertion is true because $g$ remains the same on the planet, not because the bob's mass remains unchanged.

Hence, (R) is not the correct explanation of (A).
Correct option

$\boxed{\text{D}}$

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

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Correct answer: D

Time period of a simple pendulum

For small oscillations, $T = 2\pi \sqrt{\frac{l}{g}}$ where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity.
Compare gravity on Earth and on the planet

Given for the new planet:
- Mass $M_p = 4M_E$
- Radius $R_p = 2R_E$
Using $g = \frac{GM}{R^2}$

For the planet, $g_p = \frac{G(4M_E)}{(2R_E)^2} = \frac{4GM_E}{4R_E^2} = \frac{GM_E}{R_E^2} = g_E$

So, the acceleration due to gravity is the same on Earth and on the planet.
Check Assertion (A)

Since $g_p = g_E$ and the pendulum length is unchanged, $T_p = 2\pi\sqrt{\frac{l}{g_p}} = 2\pi\sqrt{\frac{l}{g_E}} = T_E$

Therefore, Assertion (A) is true.
Check Reason (R)

The mass of the pendulum bob does remain unchanged when taken from Earth to another planet.

So, Reason (R) is true.
Does (R) explain (A)?

The time period of a simple pendulum does not depend on the mass of the bob. It depends on $l$ and $g$ only.

The assertion is true because $g$ remains the same on the planet, not because the bob's mass remains unchanged.

Hence, (R) is not the correct explanation of (A).
Correct option

$\boxed{\text{D}}$

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