If length of simple pendulum is increased by 6% then percentage change in the time-period will be

Options

A. `3%`
B. `9%`
C. `6%`
D. `1//9 %`

Explanation

To solve the problem of finding the percentage change in the time period of a simple pendulum when its length is increased by 6%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Time Period**:
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 and \( g \) is the acceleration due to gravity.

2. **Identify the Relationship between Length and Time Period**:
The time period depends on the square root of the length \( L \). We can express the change in time period in terms of the change in length:
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L} \]
where \( \Delta T \) is the change in time period and \( \Delta L \) is the change in length.

3. **Convert Changes to Percentage**:
To find the percentage change in the time period, we can multiply both sides of the equation by 100:
\[ \frac{\Delta T}{T} \times 100 = \frac{1}{2} \frac{\Delta L}{L} \times 100 \]
Here, \( \frac{\Delta T}{T} \times 100 \) represents the percentage change in the time period, and \( \frac{\Delta L}{L} \times 100 \) represents the percentage change in length.

4. **Substitute the Given Percentage Change in Length**:
We are given that the length of the pendulum is increased by 6%. Therefore:
\[ \frac{\Delta L}{L} \times 100 = 6\% \]
Substituting this value into the equation gives:
\[ \frac{\Delta T}{T} \times 100 = \frac{1}{2} \times 6\% \]

5. **Calculate the Percentage Change in Time Period**:
Now, we can calculate:
\[ \frac{\Delta T}{T} \times 100 = 3\% \]

6. **Final Answer**:
Thus, the percentage change in the time period of the pendulum when the length is increased by 6% is:
\[ \text{Percentage change in time period} = 3\% \]

### Summary:
The percentage change in the time period of the simple pendulum when its length is increased by 6% is 3%.