To determine the temperature at which the pendulum clock will keep accurate time in a different gravitational field, we need to consider how the length of the pendulum changes with temperature and how that affects the period of the pendulum. The period of a simple pendulum is given by the formula:
Understanding the Pendulum's Period
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
In this case, we have two different values for g: 9.8 m/s² at the original location and 9.788 m/s² at the new location. The clock is calibrated to keep accurate time at 20°C, which means the length of the pendulum at this temperature is optimal for the gravitational pull of 9.8 m/s².
Effect of Temperature on Length
The length of the pendulum will change with temperature due to thermal expansion. The change in length (ΔL) can be calculated using the formula:
ΔL = L₀ * α * ΔT
Where:
- L₀ = original length of the pendulum
- α = coefficient of linear expansion (12 x 10⁻⁶ /°C for steel)
- ΔT = change in temperature (T - 20°C)
Finding the New Length for Accurate Timekeeping
To keep accurate time at the new location with g = 9.788 m/s², we need to find the new effective length (L') that corresponds to the period that matches the new gravitational pull. Since the period must remain the same for the clock to keep accurate time, we can set the periods equal:
2π√(L/g₁) = 2π√(L'/g₂)
Squaring both sides and simplifying gives:
L/g₁ = L'/g₂
From this, we can express L' in terms of L:
L' = L * (g₂/g₁)
Substituting the values:
L' = L * (9.788 / 9.8)
Calculating the Change in Length
Now, we can find the change in length required to maintain accurate time:
ΔL = L' - L = L * (9.788 / 9.8 - 1)
Calculating the ratio:
ΔL = L * (-0.0012248)
This negative value indicates that the pendulum needs to shorten slightly to maintain the correct period.
Relating Change in Length to Temperature
Now we can relate this change in length back to temperature:
L * α * ΔT = -0.0012248 * L
Canceling L (assuming it is not zero) gives:
α * ΔT = -0.0012248
Substituting the coefficient of linear expansion:
(12 x 10⁻⁶) * ΔT = -0.0012248
Solving for ΔT:
ΔT = -0.0012248 / (12 x 10⁻⁶) ≈ -102.07°C
Final Temperature Calculation
Since ΔT is the change from 20°C, we find the new temperature:
T = 20°C + ΔT = 20°C - 102.07°C ≈ -82.07°C
This means that at approximately -82.07°C, the pendulum clock will give the correct time at the new location where g = 9.788 m/s². This example illustrates how temperature and gravitational changes can affect timekeeping mechanisms, particularly in pendulum clocks.
