If the length of a clock pendulum increases by 0.2% due to atmospheric temperature rise, then the loss in time of clock per day is

Options

A. 86.4s
B. 43.2s
C. 72.5s
D. 32.5s

Answer

To solve the problem, we need to determine the loss in time of a clock per day when the length of the pendulum increases by 0.2% due to a rise in atmospheric temperature.

### Step-by-Step Solution:

1. **Understand the relationship between length and time period**:
The time period \( T \) of a 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.

2. **Calculate the change in time period due to change in length**:
When the length \( L \) changes, the change in time period \( \Delta T \) can be expressed as:
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L} \]
where \( \Delta L \) is the change in length.

3. **Substitute the given percentage change in length**:
We are given that the length increases by 0.2%, which can be expressed as:
\[ \frac{\Delta L}{L} = 0.002 \]
Now substituting this into the equation:
\[ \frac{\Delta T}{T} = \frac{1}{2} \times 0.002 = 0.001 \]

4. **Calculate the percentage change in time**:
To find the change in time \( \Delta T \) in terms of the total time \( T \):
\[ \Delta T = T \times 0.001 \]

5. **Calculate the total time in a day**:
The total time in one day is:
\[ T = 24 \times 3600 \text{ seconds} = 86400 \text{ seconds} \]

6. **Calculate the loss in time per day**:
Now substituting \( T \) into the equation for \( \Delta T \):
\[ \Delta T = 86400 \times 0.001 = 86.4 \text{ seconds} \]

### Final Answer:
The loss in time of the clock per day is **86.4 seconds**.

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