If the length of simple pendulum is increased by 21%, then its time period is

Options

A. Increases by 10%
B. Decreases by 10%
C. Increases by 21%
D. Decreases by 21%

Answer

To solve the problem, we will 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:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

### Step 2: Determine the new length after the increase
If the length of the pendulum is increased by 21%, we can express the new length \( L_2 \) as:
\[ L_2 = L_1 + 0.21L_1 = 1.21L_1 \]
where \( L_1 \) is the original length.

### Step 3: Find the ratio of the new time period to the original time period
Using the formula for the time period, we can write the new time period \( T_2 \) as:
\[ T_2 = 2\pi \sqrt{\frac{L_2}{g}} = 2\pi \sqrt{\frac{1.21L_1}{g}} \]
We can express the ratio of the new time period to the original time period:
\[ \frac{T_2}{T_1} = \frac{2\pi \sqrt{\frac{1.21L_1}{g}}}{2\pi \sqrt{\frac{L_1}{g}}} = \sqrt{\frac{1.21L_1}{L_1}} = \sqrt{1.21} \]

### Step 4: Calculate the square root of 1.21
Calculating \( \sqrt{1.21} \):
\[ \sqrt{1.21} \approx 1.1 \]

### Step 5: Relate the new time period to the original time period
Now we can express \( T_2 \) in terms of \( T_1 \):
\[ T_2 = 1.1 T_1 \]
This means that the new time period \( T_2 \) is 1.1 times the original time period \( T_1 \).

### Step 6: Determine the percentage increase in the time period
The increase in time period can be calculated as:
\[ T_2 - T_1 = 1.1T_1 - T_1 = 0.1T_1 \]
This indicates that the time period has increased by:
\[ \frac{0.1T_1}{T_1} \times 100\% = 10\% \]

### Final Answer
Thus, the time period of the pendulum increases by 10%.

---