If the length of a simple pendulum is increased by 45% what is the percentage increase in its time period?

Answer

To solve the problem of finding the percentage increase in the time period of a simple pendulum when its length is increased by 45%, 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 the increase.
If the length of the pendulum is increased by 45%, the new length \( L' \) can be calculated as:
\[ L' = L + 0.45L = 1.45L \]

### Step 3: Calculate the new time period \( T' \) using the new length.
Substituting \( L' \) into the time period formula gives:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{1.45L}{g}} = 2\pi \sqrt{1.45} \sqrt{\frac{L}{g}} = \sqrt{1.45} \cdot T \]

### Step 4: Calculate \( \sqrt{1.45} \).
Using a calculator, we find:
\[ \sqrt{1.45} \approx 1.20415 \]
Thus,
\[ T' \approx 1.20415 \cdot T \]

### Step 5: Find the percentage increase in the time period.
The percentage increase in the time period can be calculated using the formula:
\[ \text{Percentage Increase} = \frac{T' - T}{T} \times 100 \]
Substituting \( T' \):
\[ \text{Percentage Increase} = \frac{1.20415T - T}{T} \times 100 = (1.20415 - 1) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = 0.20415 \times 100 \approx 20.415\% \]

### Conclusion
The percentage increase in the time period of the pendulum when its length is increased by 45% is approximately **20.41%**.

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