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 :

<p>Assertion (A): A simple pendulum transported to a planet where the mass and radius are 4 times and 2 times that of the Earth, respectively, has the same time period as it does on Earth.</p> <p>Reason (R): The mass of the pendulum remains unchanged whether on Earth or the other planet.</p> <p><strong>Explanation:</strong></p> <p>The acceleration due to gravity $ g $ on a planet is given by the formula:</p> <p>$ g = \frac{G M}{R^2} $</p> <p>where $ G $ is the gravitational constant, $ M $ is the mass of the planet, and $ R $ is the radius of the planet.</p> <p>For the new planet, the gravitational acceleration $ g' $ is:</p> <p>$ g' = \frac{G(4M)}{(2R)^2} = \frac{4G M}{4R^2} = \frac{G M}{R^2} = g $</p> <p>Thus, the gravitational acceleration on this new planet is the same as on Earth, $ g $.</p> <p>The time period $ T $ of a simple pendulum is determined by:</p> <p>$ T = 2\pi \sqrt{\frac{\ell}{g}} $</p> <p>where $ \ell $ is the length of the pendulum, which indicates that the time period $ T $ is independent of the mass of the pendulum.</p> <p>Therefore, while Assertion (A) is true and Reason (R) is true, Reason (R) does not correctly explain Assertion (A) because the time period of the pendulum is also independent of the pendulum's mass, and the key factor is the unchanged gravitational acceleration.</p>