Question

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\). (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) The periodic time of a simple pendulum increases on the surface of moon. Statement \(-2:\) Moon is very small as compared to Earth. (A) a (B) \(\mathrm{b}\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\)

Short answer

The correct option is (B) Statement 1 is true, Statement 2 is true, but Statement 2 is not the correct explanation for Statement 1.

Step-by-step solution

Analyze Statement 1

Statement 1 states that the periodic time of a simple pendulum increases on the surface of the Moon. To verify this statement, we will examine the formula for the period of a simple pendulum: \(T = 2\pi\sqrt{\frac{l}{g}}\), where T is the period, l is the length of the pendulum, and g is the acceleration due to gravity. The periodic time of the pendulum depends on the value of g. Now, let's compare the acceleration due to gravity on Earth and Moon. Earth's gravity, \(g_E\), is approximately 9.81 m/s², and the Moon's gravity, \(g_M\), is approximately 1.63 m/s². Since \(g_M < g_E\), we can see that the periodic time of the pendulum will indeed be greater on the Moon compared to Earth, and thus, Statement 1 is true.

Analyze Statement 2

Statement 2 states that the Moon is very small compared to Earth. The Moon's diameter is about 3,474 km, and Earth's diameter is about 12,742 km. The Moon's mass is also smaller than Earth's mass (about 1/6th). Therefore, Statement 2 is true.

Determine the correctness of the explanation

Although Statement 2 is true, the size of the Moon is not the correct reason for the increase in periodic time of a simple pendulum on the Moon's surface. The correct explanation is related to the Moon's lower gravity, which directly affects the period of the pendulum according to its formula, \(T = 2\pi\sqrt{\frac{l}{g}}\). Lower gravity results in a greater period, and the Moon has a lower gravity than Earth due to its smaller mass and size.

Select the correct option

Since both Statement 1 and Statement 2 are true, but Statement 2 is not the correct explanation for Statement 1, the correct option is (b) Statement 1 is true, Statement 2 is true, but Statement 2 is not the correct explanation for Statement 1.