The pendulum on a cuckoo clock is 5.00 cm long. (a) Determine the period of this pendulum. (b) What is its frequency?
L = 5 cm = .05 m
similar to k-m problem but here deflection angle = z sin (2 pi f t)
angular speed = (2 pi f)z cos (2 pi f t)
angle z from vertical
h above bottom = .05 (1-cos z)
cos z = 1 - z^2/2 + z^3/3! etc
for small angle z, cos z = 1 - z^2/2
so
h = .05(z^2/2)
potential energy at top of swing = m g h
= (m g/2) (.05 z^2)
kinetic energy at bottom = (1/2)m v^2 = (1/2)m (2 pi f .05 z)^2
so
g = (2 p f)^2 (.05)
2 pi f = sqrt (9.81/.05)
f = 2.23 Hz part b ans
T = 1/f = .449 second part a answer
To determine the period of a pendulum, we can use the formula:
T = 2π√(L/g)
where:
T = Period of the pendulum
L = Length of the pendulum
g = Acceleration due to gravity (approximately 9.8 m/s²)
(a) Let's substitute the known values into the formula:
L = 5.00 cm = 0.05 m
g = 9.8 m/s²
T = 2π√(0.05/9.8)
T ≈ 0.628 s
So, the period of this pendulum is approximately 0.628 seconds.
(b) The frequency (f) of a pendulum is the reciprocal of its period (T):
f = 1/T
Substituting the value of T we found earlier:
f = 1/0.628
f ≈ 1.59 Hz
Therefore, the frequency of this pendulum is approximately 1.59 Hz.
To determine the period and frequency of a pendulum, we need to remember that the period is the time taken for one complete swing (back and forth) of the pendulum, while the frequency is the number of complete swings per second.
(a) To find the period of the pendulum, we can use the formula T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity (approximately 9.81 m/s^2).
Given that the length of the pendulum is 5.00 cm (or 0.05 m), we can plug these values into the formula:
T = 2π√(0.05/9.81)
T = 2π√(0.0051)
T ≈ 2π(0.071)
T ≈ 0.448 seconds
Therefore, the period of the pendulum is approximately 0.448 seconds.
(b) The frequency is the reciprocal of the period, which means it is the number of swings per second. In other words, it represents how many complete oscillations the pendulum makes in one second.
To calculate the frequency, we can use the formula f = 1/T, where f is the frequency and T is the period.
Given that the period is 0.448 seconds, we can calculate:
f = 1/0.448
f ≈ 2.232 Hz
Therefore, the frequency of the pendulum is approximately 2.232 Hz.