The difference in temperature between summer and winter causes the length of a pendulum in a clock to change by one part in 27000. If the period of the pendulum is 1 s, what time-difference error will this make in one hour?
t=2pisqrt( L/g)
t'=2PI/2* l'/sqrt(l/g)
t'= PI l'/(t/2PI)
t'=2pi^2 l'/t
so l'=1/27000, t=1
t'=0.000731081807 sec, per second.
In an hour, t'=3600*t'
= 2.63 sec check that.
To find the time-difference error in one hour, we need to consider the change in the length of the pendulum due to the difference in temperature.
The given problem states that the difference in temperature causes the length of the pendulum to change by one part in 27000.
Let's calculate the change in length of the pendulum:
Change in length = 1 / 27000
Now, we need to calculate the change in time period due to the change in length of the pendulum:
Change in time period = (Change in length) / (Original length)
Here, the original length of the pendulum does not change and remains constant.
As per the problem statement, the period of the pendulum is 1 second, which means the original length of the pendulum gives a period of 1 second.
So, we can calculate the change in time period as follows:
Change in time period = (1 / 27000) / 1 = 1 / 27000
Finally, we need to calculate the time-difference error in one hour, which is the change in time period multiplied by the number of oscillations in one hour:
Time-difference error = (Change in time period) * (Number of oscillations in one hour)
One hour consists of 60 minutes and each minute consists of 60 seconds. As the period of the pendulum is 1 second, the number of oscillations in one hour is equal to the number of seconds in one hour:
Number of oscillations in one hour = 60 minutes * 60 seconds = 3600
So, the time-difference error in one hour can be calculated as follows:
Time-difference error = (1 / 27000) * 3600
Now, we can calculate the time-difference error in one hour:
Time-difference error = (1 / 27000) * 3600 = 0.1333 seconds
Therefore, the time-difference error caused by the difference in temperature in one hour is approximately 0.1333 seconds.
To determine the time difference error caused by the change in pendulum length, we need to 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, and g is the acceleration due to gravity.
Given that the period is 1 second, we can rearrange the formula to solve for the length of the pendulum:
L = (T^2 * g) / (4π^2).
We also know that the length of the pendulum is affected by the temperature difference between summer and winter. The change in length per unit length is given as 1 part in 27000.
Therefore, the change in length (∆L) of the pendulum can be calculated as:
∆L = (1/27000) * L.
Now, we want to find the time-difference error caused by this change in length over one hour.
To find that, we need to determine the change in period (∆T) caused by the change in length. The change in period can be calculated using:
∆T = 2π √((L + ∆L)/g) - 2π √(L/g).
Since our initial period is 1 second, we can calculate the time-difference error (∆T_total) over one hour as:
∆T_total = ∆T * (3600 seconds).
Now, let's plug in the values and calculate the time-difference error:
1. Calculate the length of the pendulum (L):
L = (T^2 * g) / (4π^2)
= (1 * 9.8) / (4 * π^2) (assuming g = 9.8 m/s^2)
≈ 0.249 m (rounded to three decimal places)
2. Calculate the change in length (∆L):
∆L = (1/27000) * L
≈ (1/27000) * 0.249
≈ 9.22 × 10^(-6) m (rounded to three decimal places)
3. Calculate the change in period (∆T):
∆T = 2π √((L + ∆L)/g) - 2π √(L/g)
≈ 2π √((0.249 + 9.22 × 10^(-6))/9.8) - 2π √(0.249/9.8)
≈ 7.05 × 10^(-7) s (rounded to three decimal places)
4. Calculate the time-difference error (∆T_total) over one hour:
∆T_total = ∆T * (3600 seconds)
≈ (7.05 × 10^(-7)) * 3600
≈ 2.54 × 10^(-3) seconds (rounded to three decimal places)
Therefore, the time-difference error caused by the change in pendulum length over one hour is approximately 0.00254 seconds.