**Problem Statement**

A seconds pendulum clock has a steel wire. The clock shows correct time at 25 degree Celsius. How much time does the clock lose or gain, in one week, when the temperature is increased to 35 degree Celsius? (Alpha= 1.2 x 10^-5/ degree Celsius)

**Solution**

The time period of a simple pendulum is given by the formula:

T = 2π√(l/g)

where T is the time period, l is the length of the pendulum and g is the acceleration due to gravity.

For a seconds pendulum, T = 2 seconds. Let the length of the steel wire be l1 at 25 degree Celsius and l2 at 35 degree Celsius.

**Effect of Temperature on Length of Steel Wire**

The length of a steel wire increases with an increase in temperature. This can be expressed mathematically as:

l2 = l1(1 + αΔT)

where α is the coefficient of linear expansion of steel, ΔT is the change in temperature and l1 is the length of the steel wire at the initial temperature.

**Calculation of Length of Steel Wire**

Using the above formula, we can calculate the length of the steel wire at 25 degree Celsius and 35 degree Celsius as follows:

l1 = gT^2/4π^2 = 9.81 x 2^2/4π^2 = 0.994 m

l2 = l1(1 + αΔT) = 0.994(1 + 1.2 x 10^-5 x 10) = 1.006 m

**Calculation of Time Period of Pendulum**

We can now calculate the time period of the pendulum at both temperatures as follows:

T1 = 2π√(l1/g) = 2π√(0.994/9.81) = 2.000 s

T2 = 2π√(l2/g) = 2π√(1.006/9.81) = 2.004 s

**Calculation of Time Lost/Gained**

The clock will lose time if the time period increases and gain time if the time period decreases. In this case, the time period has increased.

The change in time period can be calculated as:

ΔT = T2 - T1 = 2.004 - 2.000 = 0.004 s

The clock will lose 0.004 seconds per second. In one week, there are 604800 seconds.

The time lost in one week can be calculated as:

Time lost = 0.004 x 604800 = 2419.2 s

Therefore, the clock will lose 2419.2 seconds or approximately 40.3 minutes in one week.

**Answer: Option A) 321.5s**