JEE Main 2025 Jan 22 Shift 2, Physics Q11: SHM in Pendulums

JEE Main2025Jan 22, Shift 2Physics

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 :

A
Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
B
(A) is true but (R) is false
C
(A) is false but (R) is true
D
Both (A) and (R) are true and (R) is the correct explanation of (A)

Solution

Gravitational acceleration at the surface of a planet: $g = \frac{GM}{R ^{2}}$ .

For the new planet with $M^{'} = 4 M$ and $R^{'} = 2 R$ :

$g^{'} = \frac{G ( 4 M )}{( 2 R ) ^{2}} = \frac{4 GM}{4 R ^{2}} = \frac{GM}{R ^{2}} = g$ .

Since the time period of a simple pendulum is $T = 2 π \frac{ℓ}{g}$ , and $g$ is unchanged, $T$ remains the same. So (A) is true.

The mass of the pendulum bob does not appear in the formula for $T$ , so the reason given (that the bob's mass is unchanged) is a true statement, but it is not the cause of the time period being unchanged. The actual cause is that $g$ remains equal. Therefore (R) is true but is not the correct explanation of (A).

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JEE Main 2025 Jan 22 Shift 2, Physics Q11: SHM in Pendulums

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