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

Options

A. increases by 2%
B. decreases by 2%
C. increases by 1%
D. decreases by 1 %

Explanation

To solve the problem of how the time period of a simple pendulum changes when its length is increased by 2%, we will follow these steps:

### Step 1: Understand the formula for the time period of a 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: Differentiate the time period with respect to length
To find how a change in length affects the time period, we can take the logarithm of both sides:
\[ \log T = \log(2\pi) + \frac{1}{2} \log L - \frac{1}{2} \log g \]
Differentiating both sides gives:
\[ \frac{dT}{T} = \frac{1}{2} \frac{dL}{L} \]

### Step 3: Relate the percentage change in length to the percentage change in time period
From the differentiation, we can express the relationship between the change in time period and the change in length:
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L} \]
where \( \Delta T \) is the change in time period and \( \Delta L \) is the change in length.

### Step 4: Convert to percentage changes
Multiplying both sides by 100 to convert to percentage changes:
\[ \frac{\Delta T}{T} \times 100 = \frac{1}{2} \frac{\Delta L}{L} \times 100 \]
This means:
\[ \Delta T \% = \frac{1}{2} \Delta L \% \]

### Step 5: Substitute the given change in length
We know that the length of the pendulum is increased by 2%, so:
\[ \Delta L \% = 2\% \]
Substituting this into the equation gives:
\[ \Delta T \% = \frac{1}{2} \times 2\% = 1\% \]

### Step 6: Conclusion
Since the length is increased by 2%, the time period increases by 1%. Therefore, the correct answer is that the time period increases by 1%.

### Final Answer
The time period increases by 1%.

---