The length of a simple pendulum is decreased by `21%`. Find the percentage change in its time period.

Explanation

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

### 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 of the pendulum is decreased by 21%, the new length \( L' \) can be calculated as:
\[ L' = L - 0.21L = 0.79L \]

### Step 3: Calculate the new time period
Using the new length \( L' \), the new time period \( T' \) can be expressed as:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{0.79L}{g}} \]
This can be simplified to:
\[ T' = 2\pi \sqrt{0.79} \sqrt{\frac{L}{g}} = \sqrt{0.79} T \]
where \( T \) is the original time period.

### Step 4: Find the value of \( \sqrt{0.79} \)
Calculating \( \sqrt{0.79} \):
\[ \sqrt{0.79} \approx 0.889 \]
Thus, we have:
\[ T' \approx 0.889 T \]

### Step 5: Calculate the percentage change in time period
The percentage change in the time period can be calculated using the formula:
\[ \text{Percentage Change} = \frac{T' - T}{T} \times 100\% \]
Substituting \( T' \):
\[ \text{Percentage Change} = \frac{0.889T - T}{T} \times 100\% = \frac{-0.111T}{T} \times 100\% \]
This simplifies to:
\[ \text{Percentage Change} = -11.1\% \]

### Conclusion
The time period of the pendulum decreases by approximately 11.1%.