The length of a simple pendulum executing simple harmonic motion is increased by `21%`. The percentage increase in the time period of the pendulum of increased length is

Options

A. `11%`
B. `12%`
C. `42%`
D. `10%`

Explanation

To solve the problem of finding the percentage increase in the time period of a simple pendulum when its length is increased by 21%, we can follow these steps:

### Step 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.

### Step 2: Calculate the New Length
If the original length \( L \) is increased by 21%, the new length \( L' \) can be calculated as:
\[ L' = L + 0.21L = 1.21L \]

### Step 3: Calculate the New Time Period
Using the new length \( L' \), the new time period \( T' \) can be expressed as:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{1.21L}{g}} \]

### Step 4: Express the New Time Period in Terms of the Original
We can rewrite the new time period \( T' \) as:
\[ T' = 2\pi \sqrt{1.21} \sqrt{\frac{L}{g}} = \sqrt{1.21} \cdot T \]
where \( T \) is the original time period.

### Step 5: Calculate the Percentage Increase in Time Period
To find the percentage increase in the time period, we need to calculate:
\[ \text{Percentage Increase} = \frac{T' - T}{T} \times 100 \]

Substituting \( T' \):
\[ \text{Percentage Increase} = \frac{\sqrt{1.21} \cdot T - T}{T} \times 100 \]
\[ = (\sqrt{1.21} - 1) \times 100 \]

### Step 6: Calculate \( \sqrt{1.21} \)
Calculating \( \sqrt{1.21} \):
\[ \sqrt{1.21} = 1.1 \]

### Step 7: Substitute and Calculate the Final Percentage Increase
Now substituting back:
\[ \text{Percentage Increase} = (1.1 - 1) \times 100 = 0.1 \times 100 = 10\% \]

### Conclusion
The percentage increase in the time period of the pendulum when the length is increased by 21% is **10%**.

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