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 due to an increase in the length of the pendulum. Here's a step-by-step solution:

### Step 1: Understand the relationship between pendulum length and time period
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: Calculate the change in length
Given that the length of the pendulum increases by 0.2%, we can express this mathematically. If the original length is \( L \), then the new length \( L' \) is:
\[ L' = L + 0.002L = 1.002L \]

### Step 3: Calculate the new time period
Now, we can find the new time period \( T' \) using the new length:
\[ T' = 2\pi \sqrt{\frac{L'}{g}} = 2\pi \sqrt{\frac{1.002L}{g}} = 2\pi \sqrt{\frac{L}{g}} \sqrt{1.002} \]
Using the approximation \( \sqrt{1+x} \approx 1 + \frac{x}{2} \) for small \( x \), we get:
\[ T' \approx T \left(1 + \frac{0.002}{2}\right) = T(1 + 0.001) = 1.001T \]

### Step 4: Calculate the change in time period
The change in time period \( \Delta T \) is:
\[ \Delta T = T' - T \approx 1.001T - T = 0.001T \]

### Step 5: Determine the loss in time over one day
The clock is supposed to keep time for one day (86400 seconds). The loss in time per day can be calculated as:
\[ \text{Loss in time} = \Delta T \cdot \frac{86400 \text{ seconds}}{T} \]
Since \( T \) for a clock pendulum is approximately 2 seconds (for a simple pendulum of reasonable length), we can substitute:
\[ \text{Loss in time} = 0.001 \cdot 86400 \text{ seconds} = 86.4 \text{ seconds} \]

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

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