A clock has an aluminum pendulum with a period of 1.000 s at 20.2 °C. Suppose the clock is moved to a location where the average temperature is 28.1 °C. (The linear expansion coefficient for aluminum is 2.20 10-5 °C−1.)
(a) Determine the new period of the clock's pendulum. (Enter your answer to six siginficant figures.)
b)Determine how much time the clock will lose in 2 weeks.
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To determine the new period of the clock's pendulum, we need to consider the change in temperature and the linear expansion coefficient of aluminum. We can use the following formula:
ΔT = α * T * ΔL
Where:
ΔT = change in temperature
α = linear expansion coefficient
T = original temperature
ΔL = change in length
(a) To find the new period of the clock's pendulum, we need to determine the change in length. The change in length is given by:
ΔL = α * L * ΔT
Where:
L = original length
Since the period is directly proportional to the square root of the length, the new period (T') can be calculated as:
T' = T * sqrt(L' / L)
Where:
T = original period
L' = original length + ΔL
Let's calculate the new period:
Given:
T = 1.000 s
T' = ?
α = 2.20 * 10^(-5) °C^(-1)
T = 20.2 °C
T' = 28.1 °C
First, calculate ΔL:
ΔL = α * L * ΔT
= (2.20 * 10^(-5) °C^(-1)) * L * (T' - T)
Next, calculate L':
L' = L + ΔL
Substitute the values into the formula for T':
T' = T * sqrt(L' / L)
Finally, substitute the known values and solve for T'. Round your answer to six significant figures.
(b) To determine how much time the clock will lose in 2 weeks, we need to find the change in period. The change in period can be calculated using the following formula:
ΔT' = T - T'
Where:
ΔT' = change in period
Once we have the change in period, we can determine the time lost by multiplying the change in period by the number of periods in 2 weeks.
Let's calculate the time lost:
Given:
T = 1.000 s
T' = ? (calculated in part a)
ΔT' = ?
Number of periods in 2 weeks = 2 weeks * (7 days/week) * (24 hours/day) * (3600 seconds/hour) / T
Substitute the known values into the formula for ΔT' and then calculate the time lost by multiplying ΔT' by the number of periods in 2 weeks.