Simple pendulum is executing simple harmonic motion with time period `T`. If the length of the pendulum is increased by `21%`, then the increase in the time period of the pendulum of the increased length is:

Options

A. `22%`
B. `13%`
C. `50%`
D. `10%`

Explanation

To solve the problem, we need to understand the relationship between the length of a simple pendulum and its time period. The formula for the time period \( T \) of a simple pendulum is given by:

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

where:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

### Step 1: Determine the new length of the pendulum
If the length of the pendulum is increased by 21%, we can express the new length \( L' \) as:

\[ L' = L + 0.21L = 1.21L \]

### Step 2: Calculate the new time period
Now, we can find the new time period \( T' \) using the new length \( L' \):

\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{1.21L}{g}} \]

### Step 3: Simplify the expression for the new time period
We can factor out the \( 1.21 \) from the square root:

\[ T' = 2\pi \sqrt{1.21} \sqrt{\frac{L}{g}} = 2\pi \cdot 1.1 \sqrt{\frac{L}{g}} = 1.1 \cdot T \]

### Step 4: Calculate the increase in the time period
The increase in the time period \( \Delta T \) is given by:

\[ \Delta T = T' - T = 1.1T - T = 0.1T \]

### Step 5: Calculate the percentage increase in the time period
To find the percentage increase in the time period, we use the formula:

\[ \text{Percentage Increase} = \left(\frac{\Delta T}{T}\right) \times 100 = \left(\frac{0.1T}{T}\right) \times 100 = 10\% \]

### Final Answer
The increase in the time period of the pendulum of the increased length is **10%**.

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