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, we need to find the percentage increase in the time period of a simple pendulum when its length is increased by 69%. Let's break down the solution step by step.

### Step 1: Understand the formula for 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 and \( g \) is the acceleration due to gravity.

### Step 2: Determine the new length of the pendulum
If the length \( L \) is increased by 69%, the new length \( L' \) can be calculated as:
\[ L' = L + 0.69L = 1.69L \]

### Step 3: Calculate the new time period
Using the new length \( L' \), the new time period \( T' \) can be expressed as:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{1.69L}{g}} \]
This can be simplified to:
\[ T' = 2\pi \sqrt{1.69} \cdot \sqrt{\frac{L}{g}} = 2\pi \cdot 1.3 \cdot \sqrt{\frac{L}{g}} = 1.3 \cdot T \]
where \( T \) is the original time period.

### Step 4: Calculate the percentage increase in time period
The increase in time period can be calculated as:
\[ \text{Increase} = T' - T = 1.3T - T = 0.3T \]
To find the percentage increase, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{T' - T}{T}\right) \times 100 = \left(\frac{0.3T}{T}\right) \times 100 = 30\% \]

### Conclusion
Thus, the percentage increase in the time period of the pendulum when its length is increased by 69% is **30%**.

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