To determine the temperature at which the pendulum clock keeps accurate time, we can analyze how temperature affects the pendulum's timing. The clock runs fast at lower temperatures and slow at higher temperatures, indicating that the length of the pendulum rod is changing with temperature. This is due to the thermal expansion of the materials involved.
Understanding Pendulum Behavior
The period of a pendulum, which is the time it takes to complete one full swing, is influenced by the length of the pendulum and the acceleration due to gravity. The formula for the period (T) of a simple pendulum is:
T = 2π√(L/g)
Where:
- T = period of the pendulum
- L = length of the pendulum
- g = acceleration due to gravity
As the temperature changes, the metal rod of the pendulum expands or contracts, altering its length (L). A longer pendulum swings more slowly, while a shorter pendulum swings more quickly.
Analyzing the Given Data
We know the following:
- At 15 °C, the clock is 5 seconds fast per day.
- At 30 °C, the clock is 10 seconds slow per day.
This means that at 15 °C, the pendulum is shorter than its ideal length, causing it to swing faster. Conversely, at 30 °C, the pendulum is longer, leading to a slower swing. We need to find the temperature at which the pendulum is at its correct length, thus keeping accurate time.
Finding the Correct Temperature
Let’s denote the temperature at which the clock is accurate as T_c. We can set up a linear relationship based on the two data points we have:
- At 15 °C: +5 seconds (fast)
- At 30 °C: -10 seconds (slow)
We can calculate the rate of change of time error with respect to temperature:
Rate of change = (Time error at 30 °C - Time error at 15 °C) / (30 °C - 15 °C)
Substituting the values:
Rate of change = (-10 - 5) / (30 - 15) = -15 / 15 = -1 second per °C
This indicates that for every degree Celsius increase in temperature, the clock becomes 1 second slower. Now, we can find the temperature at which the clock is accurate by setting the total time error to zero:
Time error = 5 seconds (fast) - (T_c - 15 °C) seconds = 0
Solving for T_c gives:
T_c - 15 = 5
T_c = 20 °C
Conclusion
The temperature at which the pendulum clock is designed to keep accurate time is 20 °C. This analysis shows how temperature affects the physical properties of the pendulum, ultimately influencing its timing. Understanding these relationships can help in adjusting clocks and other timekeeping devices for accuracy in varying conditions.
