A clock with an iron pendulum keeps correct time at `20^(@)C`. How much time will it lose or gain in a day if the temperature changes to `40^(@)C`. Thermal coefficient of liner expansion `alpha = 0.000012 per^(@)C`.

Options

A. 10.368 sec gain
B. 10.368 sec loss
C. 5.184 sec gain
D. 5.184 sec loss

Explanation

To solve the problem of how much time a clock with an iron pendulum will lose or gain when the temperature changes from 20°C to 40°C, we will follow these steps:

### Step 1: Understand the relationship between the pendulum's length and temperature
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. The length \( L \) changes with temperature due to thermal expansion.

### Step 2: Determine the change in length due to temperature change
The change in length \( \Delta L \) due to a change in temperature \( \Delta \theta \) can be expressed as:
\[ \Delta L = \alpha L \Delta \theta \]
where \( \alpha \) is the coefficient of linear expansion and \( \Delta \theta \) is the change in temperature.

### Step 3: Calculate the change in temperature
Given:
- Initial temperature \( \theta_1 = 20°C \)
- Final temperature \( \theta_2 = 40°C \)

The change in temperature is:
\[ \Delta \theta = \theta_2 - \theta_1 = 40°C - 20°C = 20°C \]

### Step 4: Relate change in time to change in length
From the differentiation of the time period formula, we have:
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L} \]
Substituting for \( \Delta L \):
\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\alpha L \Delta \theta}{L} = \frac{1}{2} \alpha \Delta \theta \]

### Step 5: Calculate the total time period in seconds for one day
The total time in seconds for one day is:
\[ T = 24 \text{ hours} = 24 \times 60 \times 60 = 86400 \text{ seconds} \]

### Step 6: Substitute the known values
Given:
- \( \alpha = 0.000012 \, \text{per} \, °C \)
- \( \Delta \theta = 20°C \)

Now substituting these values into the equation:
\[ \Delta T = \frac{1}{2} \alpha \Delta \theta \cdot T \]
\[ \Delta T = \frac{1}{2} \times 0.000012 \times 20 \times 86400 \]

### Step 7: Calculate \( \Delta T \)
Calculating:
\[ \Delta T = \frac{1}{2} \times 0.000012 \times 20 \times 86400 \]
\[ = 0.000006 \times 20 \times 86400 \]
\[ = 0.000006 \times 1728000 \]
\[ = 10.368 \text{ seconds} \]

### Conclusion
The clock will gain approximately **10.368 seconds** in a day when the temperature changes from 20°C to 40°C.