If length of a simple pendulum is increased by 69%, then the percentage increase in its time period is

Options

A. 0.69
B. 0.3
C. 0.5
D. 0.1

Answer

To solve the problem of finding the percentage increase in the time period of a simple pendulum when its length is increased by 69%, 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. **Calculate the New Length**:
If the length of the pendulum is increased by 69%, the new length \( L' \) can be calculated as:
\[ L' = L + 0.69L = 1.69L \]

3. **Calculate the New Time Period**:
Substitute the new length \( L' \) into the time period formula:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{1.69L}{g}} \]
This can be simplified to:
\[ T' = 2\pi \sqrt{1.69} \sqrt{\frac{L}{g}} = 2\pi (1.3) \sqrt{\frac{L}{g}} = 1.3 T \]
where \( T \) is the original time period.

4. **Calculate the Increase in Time Period**:
The increase in time period \( \Delta T \) is:
\[ \Delta T = T' - T = 1.3T - T = 0.3T \]

5. **Calculate the Percentage Increase**:
The percentage increase in the time period is given by:
\[ \text{Percentage Increase} = \left(\frac{\Delta T}{T}\right) \times 100 = \left(\frac{0.3T}{T}\right) \times 100 = 30\% \]

### Final Answer:
The percentage increase in the time period of the pendulum is **30%**.

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