If in a simple pendulum experiment the length of the inextensible string is increased by a factor of four, its period is increased by a factor of?

1. Analyze the question

We are looking at the relationship between the length ($l$) of a simple pendulum and its period ($T$). We are told the length is increased by a factor of 4, and we need to determine the factor by which the period increases.

2. Explain the concept

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

Where:

• $T$ = Period
• $l$ = Length of the string
• $g$ = Acceleration due to gravity

From this formula, we can see that the period is directly proportional to the square root of the length:

3. Show the solution

Let the initial period be $T_1$ and the initial length be $l_1$.

Using the laws of indices/surds, we can simplify $\sqrt{4l_1}$ as $\sqrt{4} \times \sqrt{l_1}$:

Since $T_1 = k\sqrt{l_1}$, we substitute it into the equation:

This shows that the new period is 2 times the original period.

4. Identify the answer

The period increases by a factor of 2. Comparing this to the given options: A. 4 B. $\frac{\pi}{2}$ C. $\frac{1}{4}$ D. 2

The correct option is D.

5. Tip

Whenever you see a "factor of" question involving a square root (like in pendulums or wave equations), remember: if the variable inside the root increases by $x$, the result increases by $\sqrt{x}$. Here, $\sqrt{4} = 2$. This quick mental check will save you time in the exam hall!