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 :

Assertion (A): A simple pendulum is taken to a planet with mass $$4M_E$$ and radius $$2R_E$$. The time period remains the same.

Reason (R): The mass of the pendulum remains unchanged.

Since gravitational acceleration is given by $$g = \frac{GM}{R^2}$$, on Earth we have $$g_E = \frac{GM_E}{R_E^2}$$ and on the planet

$$g_P = \frac{G \cdot 4M_E}{(2R_E)^2} = \frac{4GM_E}{4R_E^2} = \frac{GM_E}{R_E^2} = g_E\,. $$

From this it follows that $$g_P = g_E$$. Since the time period of a simple pendulum is $$T = 2\pi\sqrt{\frac{l}{g}}$$, the equality of gravitational acceleration implies that the time period remains the same on both Earth and the planet. Thus Assertion (A) is true.

The mass of the pendulum does remain unchanged, as mass is invariant under such a transfer, so Reason (R) is true.

However, the pendulum’s time period does not depend on its mass but only on $$g$$ and $$l$$, so the unchanged mass does not explain why the time period remains the same. Therefore, although Reason (R) is true, it does not correctly explain Assertion (A).

The correct answer is Option 4: Both (A) and (R) are true but (R) is NOT the correct explanation of (A).