If the length of a simple pendulum is increased by 2%, then the time period

Options

A. increases by 1%
B. Decreases by 1%
C. Increases by 2%
D. Decreases by 2%

Answer

To solve the problem of how the time period of a simple pendulum changes when its length is increased by 2%, 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 after a 2% increase
If the length \( L \) is increased by 2%, the new length \( L' \) can be calculated as:
\[ L' = L + 0.02L = 1.02L \]

### Step 3: Substitute the new length into the time period formula
Now, we substitute \( L' \) into the time period formula to find the new time period \( T' \):
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{1.02L}{g}} = 2\pi \sqrt{1.02} \sqrt{\frac{L}{g}} \]

### Step 4: Relate the new time period to the original time period
We can express \( T' \) in terms of the original time period \( T \):
\[ T' = T \sqrt{1.02} \]

### Step 5: Calculate the percentage increase in the time period
To find the percentage increase in the time period, we can use the formula:
\[ \text{Percentage Increase} = \left(\frac{T' - T}{T}\right) \times 100 \]
Substituting \( T' \):
\[ \text{Percentage Increase} = \left(\frac{T \sqrt{1.02} - T}{T}\right) \times 100 = (\sqrt{1.02} - 1) \times 100 \]

### Step 6: Calculate \( \sqrt{1.02} \)
Using a calculator, we find:
\[ \sqrt{1.02} \approx 1.00995 \]
Thus,
\[ \text{Percentage Increase} \approx (1.00995 - 1) \times 100 \approx 0.995\% \]

### Conclusion
The time period of the pendulum increases by approximately 1%.

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