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 :

Step 1: Understanding the Time Period of a Simple Pendulum

The time period ($T$) of a simple pendulum is given by the formula:

$$T = 2\pi \sqrt{\frac{l}{g}}$$

where:

• $l$ is the length of the pendulum,
• $g$ is the acceleration due to gravity.

Since $g$ depends on the planet’s mass $M$ and radius $R$:

$$g = \frac{GM}{R^2}$$

where:

• $G$ is the universal gravitational constant,
• $M$ is the mass of the planet,
• $R$ is the radius of the planet.

Step 2: Finding Gravity on the New Planet

The new planet has:

• Mass $M' = 4M_{\text{Earth}}$
• Radius $R' = 2R_{\text{Earth}}$

The acceleration due to gravity on the new planet is:

$$g' = \frac{G M'}{R'^2}$$

Substituting $M' = 4M$ and $R' = 2R$:

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

Thus, the acceleration due to gravity on the new planet is the same as on Earth.

Step 3: Comparing Time Periods

Since $T = 2\pi \sqrt{\frac{l}{g}}$, and we found that $g' = g$, the time period remains:

$$T' = 2\pi \sqrt{\frac{l}{g'}} = 2\pi \sqrt{\frac{l}{g}} = T$$

Thus, the time period remains the same on both Earth and the new planet. This confirms the Assertion (A) is correct.

Step 4: Evaluating the Reason (R)

The Reason (R) states that "The mass of the pendulum remains unchanged at Earth and the other planet."

• While this statement is true, it is not the correct reason for why the time period remains the same.
• The time period does not depend on the mass of the pendulum, but rather on gravity.

Thus, Reason (R) is true, but it does not correctly explain Assertion (A).