
Harmonic motion repeats in identical patterns called cycles. An oscillator, such as a pendulum, is a system that features repeating cycles. The period of an oscillator is the time to complete one full cycle. The frequency is the number of cycles per second. The amplitude of an oscillation is the maximum displacement from equilibrium.

oscillator, cycle, period, frequency, hertz (Hz), amplitude, damping, phase



Review problems and questions 

 The Earth exhibits several different types of harmonic motion. Describe at least three examples of harmonic motions associated with our planet.

Answers will vary. Some possible answers are the following:  The Earth revolves on its axis with a period of 24 hours.
 The Earth rotates around the Sun with a period of 365 days.
 The Moon rotates around the Earth with a period of 30 days.
 The Moon causes tides on Earth with a period of about 12 hours.

 A swing is observed to move back and forth 12 times in 30 s.
 What is the period of the motion?
 What is the frequency of the motion?

Answer:  2.5 s
 0.40 Hz
Solution: Asked: period T of one cycle of the harmonic motion of the swinging boy
Given: 12 cycles in 30 s Relationships: T = seconds per cycle Solve: Divide the number of seconds by the number of cycles to get the period (that is, the number of seconds per cycle):$$T=\frac{\text{seconds}}{\text{cycle}}=\frac{30\text{s}}{12\text{cycle}}=2.5\text{s}$$Answer: The period of the boy’s motion is 2.5 s.  Asked: frequency f of the boy’s motion
Given: period T = 2.5 s Relationships: f = 1/T Solve: Calculate the frequency using the equation:$$f=\frac{1}{T}=\frac{1}{2.5\text{s}}=0.40\text{Hz}$$Answer: The frequency of the boy’s motion is 0.40 Hz.

 Determine the following information from the graph above.
 How many cycles occur in 10 s?
 What is the amplitude of the motion?
 What are the period and frequency of the motion?

 There are 2.5 cycles in 10 s. A cycle is completed when the oscillator returns to its original position.
 The amplitude is 7.5 cm. This is the maximum distance the oscillator moves from its equilibrium position.
 The period T is the time to complete one cycle. T = 4 s.
The frequency, f, is the number of cycles per second. It is the inverse of the period, T. f = 1/T = 1/(4 s) = 0.25 s^{−1} = 0.25 Hz.

 Systems undergoing harmonic motion contain energy that oscillates between different forms. What forms of energy are changing in the systems described below?
 a simple pendulum
 a mass on a spring that is sliding on a frictionless, horizontal surface

 kinetic energy and gravitational potential energy
 kinetic energy and elastic potential energy

 A simple pendulum and a mass on a spring are both moving in simple harmonic motion. In which of these systems does the period of the harmonic motion depend on mass? On amplitude?

Only the period of the mass on a spring depends on the mass. Neither system has a period that depends on amplitude.


If an oscillator has a frequency of 60 Hz, how many complete oscillations take place in 2 s?

A frequency of 60 Hz means that the oscillator makes 60 cycles per second. Therefore, in 2 s the oscillator makes 120 complete cycles.


A quartz crystal in a watch has a period of 0.00350 s. What is its frequency in hertz?

Answer: The frequency of the crystal is 286 Hz.
Asked: frequency f in hertz of the quartz crystal Given: period T = 0.0035 s Relationships: f = 1/T Solve:$$f=\frac{1}{T}=\frac{1}{0.00350\text{s}}=285.7\text{Hz}$$

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