Chapter 25

Vibration and Waves

Simple Harmonic Motion When a vibration or an oscillation

repeats itself back and forth over the same path, the motion is said to be periodic.

The most common oscillation come from springs and you will recall from earlier chapters that the description of a spring’s oscillation requires some vocabulary.

Simplest Form of OscillationA Pendulum

If you suspend any object from a string and swing it back and forth, you have created a pendulum. Each back and forth motion of this pendulum is one cycles, and the time it takes to make that motion is called its period. Which is calculated according to the equation:

g

LT π2=

Sample Problem

A pendulum is 2 meters long. What is its period on earth where gravity is 9.8 m/s2?

What would the period of the same pendulum be on the moon where gravity is 1.63 m/s2?

Solution

On Earth

83.28.9

222 === ππ

gL

T

On the moon

95.663.1

222 === ππ

gL

T

Oscillation of a Mass on a Spring

Top picture is “rest position”; x = 0

Bottom picture is “stretched position”

Here x represents the displacement.

Maximum displacement is called the amplitude.

One cycle refers to one complete to and fro motion.

The period, T represents the time for one cycle.

The frequency, f is the number of cycles in a given time period, usually one second.

Relationship between Frequency and Period

Frequency – the number of cycles in one second

Period – the time required to complete one cycle.

Hence the relationship between period and frequency is:

F = 1/T or T = 1/F Where period is measured in seconds and

frequency is measured in hertz (hz) which is 1/seconds.

Waves Waves are a form of periodic motion. Two types of Waves

(classified by movement) Transverse

Wave moves perpendicular to amplitude Longitudinal

Wave moves parallel to the amplitudeClassified by medium

Mechanical Require a Medium

Electromagnetic Do not require a medium

Wave Vocabulary

For a Transverse Wave Top – Crest Bottom – trough Wavelength (λ) – distance from crest to

crest or trough to trough Frequency – number of waves or cycles

per second Velocity – speed of wave

What they look like

http://superphysics.netfirms.com/wave_m9.jpg

Wave Vocabulary

For a Longitudinal Wave front – compression Back – rarefaction Wavelength (λ) – distance from

compression to compression or rarefaction to rarefaction

Frequency – number of waves or cycles per second

Velocity – speed of wave

The Wave Equation

By DefinitionV = fλ

Where v = wave velocity (meters/second) f = wave frequency (hertz) λ = wavelength in meters.

Sample Problem

A boy sitting on a beach notices that 10 waves come to shore in 2 minutes. He also notices that the waves seem to be about 20 meters apart as they travel on the ocean. What is the frequency of the waves? What is the velocity of the waves?

Solution

f = waves/second = 10/120 = 0.083 hertz

V =fλ =(0.083 hz)(20 meters) =1.66 m/s

Interference

When two waves pass through each other they are said to form an interference pattern.

There are two types of interference pattern: Constructive interference

Waves reinforce each other Destructive interference

Waves cancel each other

Standing Waves

When a wave and its reflection reinforce each other they form a standing wave. In a standing wave the parts which don’t

move are called nodes and the parts which move are called anti-nodes.

Nodes are a results of destructive interference and anti-nodes come from constructive interference.

The Doppler Effect When a person listening to a sound is

moving and/or the source of the sound is moving you get the Doppler effect.

When they are getting closer together the sound that is heard is of a higher frequency than the original.

When they are moving apart, the sound that is heard is of a lower frequency than the original.

Bow and Shock Waves

When a source moves as fast or faster than a wave in a media it creates a bow wave. If this is in air then the shock wave is three dimensional and is called a sonic boom.

Chapter 26

Sound

The Origin of Sound Sound is a longitudinal, mechanical

wave. You can hear sound with a frequency

of 20 – 20,000 Hz. Under 20 hz is infrasonic, and above 20,000 hz is ultrasonic.

We talk about the frequency of sound when it is produced, and the pitch of sound when we hear it.

The Speed of Sound

The speed of sound depends upon the media in which it travels.

The speed of sound in air is 330 m/s at 0° Centigrade.

The speed of sound increases by 0.6 m/s for every 1°C increase in temperature in air.

Loudness When a sound is produced it has a certain

intensity. This is defined as:I = Power/Area

Or intensity is measured as the ratio of power divided by the area when the sound is produced.

1. Loudness is a sensation when we hear a sound. Different people react differently to the same intensity. In other words the same level of sound has a different “loudness” to different people.

Forced Vibration and Natural Frequency

When a vibrating object is placed in contact with another object, the second object will also begin to vibrate. This is known as a force vibration.

An object’s natural frequency is one at which it takes a minimum energy to cause it to vibrate.

All object have a natural frequency at which they vibrate easily and if that frequency is within the range of human hearing – the object makes a sound.

http://www.youtube.com/watch?v=j-zczJXSxnw

Resonance When a force vibration matches an

objects natural frequency – an increase in amplitude occurs which is known as resonance.

Resonance in an instrument occurs when reflected waves are multiples of the natural frequency and these harmonics make a stronger, richer sound.

Law of Pipes

For an Open Pipe (open at both ends) λ ≈ 2l

For a Closed Pipe (open at one end) λ ≈ 4l

In an open pipe all harmonics are present and in a closed pipe only the odd harmonics are present.

Sample Problem

If a pipe is 2 meters long at 0° C: What is its fundamental frequency and

first two harmonics if it is: Open closed

Solution Open pipe:

λ≈2l = 2(2 m) = 4 metersf = V/λ = 330/4 = 82.5 Hz

2nd Harmonic = 2(82.5) = 165 Hz3rd Harmonic = 3(82.5) = 247.5 Hz

Closed Pipeλ≈4l = 4(2 m) = 8 metersf = V/λ = 330/8 = 41.25 Hz

3rd Harmonic = 3(41.25) = 123.75 Hz5th Harmonic = 5(41.25) = 206.25 Hz

Law of Strings There are four

laws which govern the frequency of a string: Length:

Diameter:

Tension:

Density:

l

l

f

f '

'

d

d

f

f '

'

'' F

F

f

f

D

D

f

f '

'

Sample Problem

A violin string has a frequency of 340 Hz when it is 1 meter long. What is its frequency when it is shortened to ½ meter?

When a guitar string is under a tension of 200 newtons it plays a frequency of 330 hz, what will it play if it is tightened to 450 newtons?

Solution

hzffl

l

f

f680

1

5.340'

'

hzffF

F

f

f495

450

200

'

330

''