A second's pendulum clock has a steel wire. The clock is calibrated at `20^@ C`. How much time does the clock lose or gain in one week when the temperature is increased to `30^@ C` ? `alpha_(steel = 1.2 xx 10^-5(^@ C)^-1`.

Explanation

To solve the problem, we need to determine how much time a second's pendulum clock loses or gains in one week when the temperature of the steel wire is increased from 20°C to 30°C.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial temperature, \( T_1 = 20^\circ C \)
- Final temperature, \( T_2 = 30^\circ C \)
- Coefficient of linear expansion for steel, \( \alpha = 1.2 \times 10^{-5} \, ^\circ C^{-1} \)
- Time period of a second's pendulum, \( T = 2 \, \text{s} \)

2. **Calculate the Change in Temperature:**
\[ \Delta T = T_2 - T_1 = 30^\circ C - 20^\circ C = 10^\circ C \]

3. **Calculate the Change in Time Period (\( \Delta t \)):**
The change in time period due to temperature change is given by the formula:
\[ \Delta t = \frac{1}{2} T \alpha \Delta T \]
Substituting the known values:
\[ \Delta t = \frac{1}{2} \times 2 \, \text{s} \times 1.2 \times 10^{-5} \, ^\circ C^{-1} \times 10^\circ C \]
Simplifying:
\[ \Delta t = 1 \times 1.2 \times 10^{-5} \times 10 = 1.2 \times 10^{-4} \, \text{s} \]

4. **Calculate the New Time Period (\( T' \)):**
The new time period \( T' \) can be calculated as:
\[ T' = T + \Delta t = 2 \, \text{s} + 1.2 \times 10^{-4} \, \text{s} = 2.00012 \, \text{s} \]

5. **Determine the Time Lost in One Week:**
To find the time lost in one week, we need to calculate the total time for one week in seconds:
\[ \text{Time in one week} = 7 \, \text{days} \times 24 \, \text{hours/day} \times 3600 \, \text{seconds/hour} = 604800 \, \text{s} \]
The time lost can be calculated using the ratio of the original time period to the new time period:
\[ \text{Time lost} = \left( \frac{T - T'}{T'} \right) \times \text{Total time in one week} \]
Substituting the values:
\[ \text{Time lost} = \left( \frac{2 - 2.00012}{2.00012} \right) \times 604800 \]
\[ \text{Time lost} = \left( \frac{-0.00012}{2.00012} \right) \times 604800 \]
\[ \text{Time lost} \approx -36.28 \, \text{s} \]

### Final Answer:
The clock loses approximately **36.28 seconds** in one week when the temperature is increased from 20°C to 30°C.