a large cuckoo clock has a pendulum with a period of 1.0 sec. The clock loses 10 min a day. how would you fix it? (Hint: change l)
a. Find the length l of the pendulum.
b. How much adjustment in l, up or down, is needed?
To fix the clock, we need to determine the length of the pendulum (l) and the required adjustment in l.
a. To find the length (l) of the pendulum, we can use the relationship between the period (T) and the length of the pendulum (l) given by the formula:
T = 2π * √(l / g)
Where T is the period of the pendulum, π is a mathematical constant (approximately equal to 3.14159), and g is the acceleration due to gravity (approximately equal to 9.8 m/s²).
In this case, we are given that the period of the pendulum is 1.0 second. Plugging this value into the formula, we have:
1.0 = 2π * √(l / 9.8)
We can now solve for l:
√(l / 9.8) = 1.0 / (2π)
Squaring both sides of the equation, we get:
(l / 9.8) = (1.0 / (2π))²
l / 9.8 = 1.0 / (2π)²
l = (1.0 / (2π)²) * 9.8
Evaluating this expression results in the length of the pendulum, l.
b. To determine the adjustment needed in l, we compare the actual period of the pendulum to the desired period. In this case, the clock loses 10 minutes (or 600 seconds) per day, which means the actual period is longer than the desired period (1.0 second).
To calculate the adjustment in l, we can use the formula:
Adjustment = (Actual Period - Desired Period) / (Desired Period)
In this case, the actual period is the time lost per day (600 seconds) divided by the number of swings per day (assuming one swing per second):
Actual Period = 600 seconds / (24 hours * 60 minutes * 60 seconds)
Substituting the values and calculating the adjustment will give us the required adjustment in l.
Note: Changing the length of the pendulum can be done by either adjusting the pendulum rod or adding/removing weights to the pendulum bob.