A clock with a pendulum made of steel has a period of 1.000 s at 20.0°C. The temperature is decreased to 4°C.
By how much does the period change? How much time does the clock gain or lose in one week?
The pendulum length gets shorter and the frequency of oscillation increases.
Compute the new length using the coefficent of thermal expansion of steel, 13*10^-6 C^-1.
Call it the new length L2 and the original length L1.
L2/L1 = 1 -13*10^-6*16 = 0.99979
The new period P2 gets multiplied by sqrt(L2/L1)
P2 = 1.0000*0.999896 = 0.999896 s
One week is 10,080 minutes or 6.04800*10^5 s
The new number of oscillations in one week will be 6.04800*10^5/0.999896 = 6.04863*10^5
63 seconds or about one minute will be gained by the clock.
To find the change in period for the clock with a pendulum made of steel as the temperature is decreased from 20.0°C to 4°C, we can use the formula for the change in length of a material with temperature:
ΔL = αL₀ΔT
Where:
- ΔL is the change in length
- α is the coefficient of linear expansion for steel
- L₀ is the original length
- ΔT is the change in temperature
Since the pendulum's length determines its period, we can use the formula for the period of a simple pendulum:
T = 2π√(L/g)
where:
- T is the period
- L is the length of the pendulum
- g is the acceleration due to gravity
From these two formulas, we can derive the relationship between the change in length and the change in period:
ΔT = (T₀/T) * ΔL
Where:
- ΔT is the change in period
- T₀ is the original period
- T is the new period
Let's calculate the change in period:
1. Determine the original length of the pendulum.
2. Determine the change in length using the coefficient of linear expansion for steel and the change in temperature.
3. Use the change in length to calculate the change in period using the relationship derived from the formulas mentioned above.
To calculate how much time the clock gains or loses in one week, we need to calculate the number of periods that can fit within one week and multiply it by the change in period.
1. Calculate the duration of one period.
2. Calculate the number of periods in one week.
3. Multiply the change in period by the number of periods in one week.
Let's proceed with the calculations.