# 15.1 Simple harmonic motion

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By the end of this section, you will be able to:
• Define the terms period and frequency
• List the characteristics of simple harmonic motion
• Explain the concept of phase shift
• Write the equations of motion for the system of a mass and spring undergoing simple harmonic motion
• Describe the motion of a mass oscillating on a vertical spring

When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time ( [link] ). The string vibrates around an equilibrium position, and one oscillation is completed when the string starts from the initial position, travels to one of the extreme positions, then to the other extreme position, and returns to its initial position. We define periodic motion    to be any motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by a child swinging on a swing. In this section, we study the basic characteristics of oscillations and their mathematical description.

## Period and frequency in oscillations

In the absence of friction, the time to complete one oscillation remains constant and is called the period ( T )    . Its units are usually seconds, but may be any convenient unit of time. The word ‘period’ refers to the time for some event whether repetitive or not, but in this chapter, we shall deal primarily in periodic motion, which is by definition repetitive.

A concept closely related to period is the frequency of an event. Frequency ( f ) is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is

$f=\frac{1}{T}.$

The SI unit for frequency is the hertz (Hz) and is defined as one cycle per second :

$1\phantom{\rule{0.2em}{0ex}}\text{Hz}=1\frac{\text{cycle}}{\text{sec}}\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}1\phantom{\rule{0.2em}{0ex}}\text{Hz}=\frac{1}{\text{s}}=1\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}.$

A cycle is one complete oscillation    .

## Determining the frequency of medical ultrasound

Ultrasound machines are used by medical professionals to make images for examining internal organs of the body. An ultrasound machine emits high-frequency sound waves, which reflect off the organs, and a computer receives the waves, using them to create a picture. We can use the formulas presented in this module to determine the frequency, based on what we know about oscillations. Consider a medical imaging device that produces ultrasound by oscillating with a period of $0.400\phantom{\rule{0.2em}{0ex}}\mu \text{s}$ . What is the frequency of this oscillation?

## Strategy

The period ( T ) is given and we are asked to find frequency ( f )    .

## Solution

Substitute $0.400\phantom{\rule{0.2em}{0ex}}\mu \text{s}$ for T in $f=\frac{1}{T}$ :

$f=\frac{1}{T}=\frac{1}{0.400\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{s}}.$

Solve to find

$f=2.50\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{Hz}\text{.}$

## Significance

This frequency of sound is much higher than the highest frequency that humans can hear (the range of human hearing is 20 Hz to 20,000 Hz); therefore, it is called ultrasound. Appropriate oscillations at this frequency generate ultrasound used for noninvasive medical diagnoses, such as observations of a fetus in the womb.

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