A pendulum clock with a pendulum made of brass is designed to keep accurate time at 20ºC. How much will the error be, in seconds per hour, if the clock operates at 0ºC?

To determine how much time a pendulum clock will lose or gain when operating at a temperature different from its designed temperature, we need to consider how temperature affects the length of the pendulum. The pendulum's length changes with temperature due to thermal expansion, which in turn affects the period of the pendulum's swing and thus its timekeeping accuracy.

Understanding Pendulum Mechanics

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

T = 2π√(L/g)

Where:

• T = period of the pendulum (time for one complete swing)
• L = length of the pendulum
• g = acceleration due to gravity (approximately 9.81 m/s²)

Effect of Temperature on Pendulum Length

As temperature decreases, the brass pendulum will contract, leading to a decrease in its length. The change in length can be calculated using the formula:

ΔL = L₀ * α * ΔT

Where:

• ΔL = change in length
• L₀ = original length of the pendulum
• α = coefficient of linear expansion for brass (approximately 19 x 10⁻⁶ /°C)
• ΔT = change in temperature (final temperature - initial temperature)

Calculating the Change in Length

In this case, the clock is designed to operate at 20ºC but is functioning at 0ºC. Thus:

• ΔT = 0ºC - 20ºC = -20ºC

Assuming the original length of the pendulum (L₀) is known, we can calculate ΔL:

ΔL = L₀ * (19 x 10⁻⁶) * (-20)

Impact on the Period of the Pendulum

As the length of the pendulum decreases, the period T will also change. A shorter pendulum swings faster, meaning it will complete more cycles in the same amount of time. The new period T' can be calculated using the new length (L - ΔL).

Calculating the Time Error

To find out how much time the clock will lose, we need to compare the original period with the new period. The difference in periods will give us the time error per swing, and we can then scale this up to an hour.

For example, if the original period T is 1 second and the new period T' is 0.99 seconds, then the clock will gain time:

• Time gained per swing = T - T' = 1 - 0.99 = 0.01 seconds

In one hour, there are 3600 seconds, so the total time gained would be:

Time gained in one hour = (0.01 seconds/swing) * (3600 swings) = 36 seconds

Final Thoughts

In summary, if you know the original length of the pendulum, you can calculate the exact time error. The clock will run faster at lower temperatures due to the contraction of the pendulum, leading to a gain in time. This example illustrates the intricate relationship between temperature, material properties, and timekeeping accuracy in pendulum clocks.