Final answer:
By calculating the change in the pendulum's period due to thermal contraction at lower temperatures using the coefficient of linear expansion, the time gained or lost by a pendulum clock in a day can be determined. The clock will gain time when the temperature drops, and the exact amount of time can be found by performing the necessary calculations.
Step-by-step explanation:
The change in period of a pendulum clock due to temperature can be determined by considering the coefficient of linear expansion. When the temperature decreases, an aluminum pendulum shaft contracts, shortening the pendulum length and thereby decreasing the period of the pendulum's swing, which causes the clock to run faster. However, we must calculate the exact time gained or lost due to a temperature change from 22°C to 5°C for the pendulum clock with an aluminum shaft that has a coefficient of linear expansion of 24.0 x 10⁻⁶ K⁻¹. The formula to find the new period of the pendulum (T') is:
T' = T \(\sqrt{1 + \alpha \Delta T}\)
Where T is the original period, \(\alpha\) is the coefficient of linear expansion, and \(\Delta T\) is the change in temperature. Given that the original period T = 1.00 s, \(\alpha\) = 24.0 x 10⁻⁶ K⁻¹, and \(\Delta T\) = 22°C - 5°C = 17°C, we can calculate the new period and then determine how much time the clock will gain or lose in a day (86,400 seconds). The slight difference in period over each second will compound over the day to result in a total time gain or loss, which can be found by the following equation:
Time gain/loss = (T' - T) x (Number of seconds in a day / T)
After performing the calculations, we can identify if the clock gains or loses time and by how much, and consequently select the correct answer from the provided options.