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`.

Explanation

To solve the problem of how much time a clock with an iron pendulum will lose or gain in a day when the temperature changes from \(20^\circ C\) to \(40^\circ C\), we can follow these steps:

### Step 1: Understand the relationship between temperature and pendulum length
The time period of a pendulum is influenced by its length. As the temperature increases, the length of the pendulum also increases due to thermal expansion. This change in length affects the time period of the pendulum.

### Step 2: Use the formula for change in time
The change in time (\(\Delta t\)) can be calculated using the formula:
\[ \Delta t = \frac{1}{2} \alpha \Delta \theta \cdot T \]
where:
- \(\alpha\) is the coefficient of linear expansion,
- \(\Delta \theta\) is the change in temperature,
- \(T\) is the total time period in seconds for one day.

### Step 3: Calculate the change in temperature
Given:
- Initial temperature = \(20^\circ C\)
- Final temperature = \(40^\circ C\)
- Therefore, the change in temperature (\(\Delta \theta\)) is:
\[ \Delta \theta = 40^\circ C - 20^\circ C = 20^\circ C \]

### Step 4: Calculate the total time in seconds for one day
One day has:
\[ T = 24 \text{ hours} \times 3600 \text{ seconds/hour} = 86400 \text{ seconds} \]

### Step 5: Substitute the values into the formula
Now, substituting the values into the formula:
- \(\alpha = 0.000012 \, \text{per}^\circ C\)
- \(\Delta \theta = 20^\circ C\)
- \(T = 86400 \text{ seconds}\)

The change in time (\(\Delta t\)) is:
\[ \Delta t = \frac{1}{2} \times 0.000012 \times 20 \times 86400 \]

### Step 6: Perform the calculations
Calculating step-by-step:
1. Calculate \(0.000012 \times 20 = 0.00024\)
2. Then, calculate \(0.00024 \times 86400 = 20.7936\)
3. Finally, multiply by \(\frac{1}{2}\):
\[ \Delta t = \frac{1}{2} \times 20.7936 = 10.3968 \text{ seconds} \]

### Step 7: Interpret the result
Since the pendulum length increases with temperature, the time period of the pendulum increases, meaning the clock will lose time. Therefore, the clock will lose approximately \(10.40\) seconds in a day.

### Final Answer
The clock will lose approximately \(10.40\) seconds in a day when the temperature changes from \(20^\circ C\) to \(40^\circ C\).

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