A clock has a copper pendulum with a period of 1.000 s at 17.3°C. Suppose the clock is moved to a location where the average temperature is 33.9°C.How much time will the clock lose (positive result) or gain (negative result) in 4 days?
To determine the change in time that the clock will experience in 4 days due to the change in temperature, we need to consider the effect of temperature on the period of the pendulum. The formula to calculate the change in time is as follows:
Δt = (t2 - t1) * (L / g) * (αΔT)
Where:
Δt is the change in time,
t2 is the final temperature (33.9°C),
t1 is the initial temperature (17.3°C),
L is the length of the pendulum,
g is the acceleration due to gravity, and
α is the coefficient of linear expansion of the material (copper in this case),
ΔT is the change in temperature (t2 - t1).
First, we need to determine the coefficient of linear expansion (α) for copper. Copper has a coefficient of linear expansion of approximately 0.000016 per degree Celsius.
Next, we need to determine the length of the pendulum (L). We'll assume a standard length of 1 meter.
Finally, we need the acceleration due to gravity (g), which is approximately 9.8 m/s².
Now we can plug in the values into the formula and calculate the change in time (Δt):
Δt = (33.9 - 17.3) * (1 / 9.8) * (0.000016 * ΔT) * 4
Where ΔT is the change in temperature in degrees Celsius over the 4 days. Since the average temperature is given, we can assume that ΔT = t2 - t1 = 33.9 - 17.3.
Δt = (33.9 - 17.3) * (1 / 9.8) * (0.000016 * (33.9 - 17.3)) * 4
Simplifying the equation:
Δt = 16.6 * (1 / 9.8) * (0.000016 * 16.6) * 4
Δt ≈ 0.000113 seconds
Therefore, the clock will lose approximately 0.000113 seconds in 4 days when moved to a location with an average temperature of 33.9°C.