Physics for Scientists and Engineers Lecture 14

1Physics for Scientists &

Engineers, with Modern

Physics, 4th edition

Giancoli

Piri Reis University 2011-2012/ Physics -I

2Chapter 14-15-16

Oscillations, Wave Motion and Sound

3Units of Chapter 14

Simple Harmonic Motion

Energy in the Simple Harmonic Oscillator

The Period and Sinusoidal Nature of SHM

The Simple Pendulum

Damped Harmonic Motion

Forced Vibrations; Resonance

414-2 Simple Harmonic Motion

If an object vibrates or oscillates back

and forth over the same path, each

cycle taking the same amount of

time, the motion is called periodic.

The mass and spring system is a

useful model for a periodic system.

514-2 Simple Harmonic MotionWe assume that the surface is frictionless. There is a point where the

spring is neither stretched nor compressed; this is the equilibrium

position. We measure displacement from that point (x = 0 on the

previous figure).

The force exerted by the spring depends on the displacement:


The minus sign on the force indicates that it is a restoring force it is directed to restore the mass to its equilibrium position.

k is the spring constant

The force is not constant, so the acceleration is not constant either


Displacement is measured from the equilibrium point

Amplitude is the maximum displacement

A cycle is a full to-and-fro motion; this figure shows half a cycle

Period is the time required to complete one cycle

Frequency is the number of cycles completed per second


If the spring is hung vertically, the only

change is in the equilibrium position,

which is at the point where the spring

force equals the gravitational force.


Any vibrating system where the restoring force is proportional

to the negative of the displacement is in simple harmonic

motion (SHM), and is often called a simple harmonic oscillator.

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14-3 Energy in the Simple Harmonic Oscillator

We already know that the potential energy of a spring is given by:

The total mechanical energy is then:

The total mechanical energy will be conserved, as we are

assuming the system is frictionless.

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14-3 Energy in the Simple

Harmonic Oscillator

If the mass is at the limits of its

motion, the energy is all potential.

If the mass is at the equilibrium

point, the energy is all kinetic.

We know what the potential energy is at

the turning points:

12

14-3 Energy in the Simple Harmonic Oscillator

The total energy is, therefore

And we can write:

This can be solved for the velocity as a function of position:

where

13

The Period and Sinusoidal

Nature of SHM

If we look at the projection onto the x axis of

an object moving in a circle of radius A at a

constant speed vmax, we find that the x

component of its velocity varies as:

This is identical to SHM.

14


Therefore, we can use the period and frequency of a particle moving in a

circle to find the period and frequency:

15

The Period and Sinusoidal Nature of SHMWe can similarly find the position as a function of time:

16


The top curve is a graph of the

previous equation.

The bottom curve is the same,

but shifted period so that it is a sine function rather than a

cosine.

17

The Period and Sinusoidal

Nature of SHM

The velocity and acceleration can

be calculated as functions of time;

the results are below, and are

plotted at left.

18

14-5 The Simple Pendulum

A simple pendulum consists of a

mass at the end of a lightweight

cord. We assume that the cord

does not stretch, and that its

mass is negligible.

19


In order to be in SHM, the restoring force must

be proportional to the negative of the

displacement. Here we have:

which is proportional to sin and not to itself.

However, if the angle is small,

sin .

20


Therefore, for small angles, we have:

where

The period and frequency are:

21


So, as long as the cord can be

considered massless and the amplitude

is small, the period does not depend on

the mass.

22

14-7 Damped Harmonic Motion

Damped harmonic motion is harmonic motion with a frictional or

drag force. If the damping is small, we can treat it as an

envelope that modifies the undamped oscillation.

23


However, if the damping is large, it no

longer resembles SHM at all.

A: underdamping: there are a few small

oscillations before the oscillator comes to

rest.

B: critical damping: this is the fastest way to get to equilibrium.

C: overdamping: the system is slowed so much that it takes a long time to

get to equilibrium.

24


There are systems where damping is unwanted, such as clocks and

watches.

Then there are systems in which it is wanted, and often needs to be as

close to critical damping as possible, such as automobile shock absorbers

and earthquake protection for buildings.

25

14-8 Forced Vibrations; Resonance

Forced vibrations occur when there is a periodic driving force. This

force may or may not have the same period as the natural frequency of

the system.

If the frequency is the same as the natural frequency, the amplitude

becomes quite large. This is called resonance.

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14-8 Forced Vibrations; Resonance

The sharpness of the resonant

peak depends on the damping. If

the damping is small (A), it can be

quite sharp; if the damping is larger

(B), it is less sharp.

Like damping, resonance can be wanted or unwanted. Musical instruments

and TV/radio receivers depend on it.

27

HOMEWORK

Giancoli, Chapter 14

4, 6, 11, 13, 17, 22, 24, 25, 28, 32

References

o Physics For Scientists &Engineers with Modern Physics Giancoli 4th edition,

Pearson International Edition

28

Units of Chapter 15

Wave Motion

Types of Waves: Transverse and Longitudinal

Energy Transported by Waves

Intensity Related to Amplitude and Frequency

Reflection and Transmission of Waves

Interference; Principle of Superposition

Standing Waves; Resonance

Refraction

Diffraction

Mathematical Representation of a Traveling Wave

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15-1 Wave Motion

A wave travels along its

medium, but the individual

particles just move up and

down.

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15-1 Wave Motion

All types of traveling waves transport energy.

Study of a single wave pulse shows

that it is begun with a vibration and

transmitted through internal forces in

the medium.

Continuous waves start with vibrations

too. If the vibration is SHM, then the

wave will be sinusoidal.

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15-1 Wave Motion

Wave characteristics:

Amplitude, A

Wavelength,

Frequency f and period T

Wave velocity

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15-2 Types of Waves: Transverse and

Longitudinal

The motion of particles in a wave can either be perpendicular to the wave

direction (transverse) or parallel to it (longitudinal).

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Longitudinal

Sound waves are longitudinal waves:

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Longitudinal

Earthquakes produce both longitudinal and transverse waves. Both types

can travel through solid material, but only longitudinal waves can

propagate through a fluid in the transverse direction, a fluid has no restoring force.

Surface waves are waves that travel along the boundary between two

media.

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15-3 Energy Transported by Waves

Just as with the oscillation that starts it, the energy transported by a

wave is proportional to the square of the amplitude.

Definition of intensity:

The intensity is also proportional to the square of the amplitude:

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15-3 Energy Transported by Waves

If a wave is able to spread out three-dimensionally from its source, and the

medium is uniform, the wave is spherical.

Just from geometrical considerations, as

long as the power output is constant, we

see:

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Intensity Related to Amplitude and Frequency

By looking at the energy of a

particle of matter in the medium of

the wave, we find:

Then, assuming the entire medium has the same density, we find:

Therefore, the intensity is proportional to the square of the frequency and

to the square of the amplitude.

(11-17)

38

15-4 Mathematical Representation of a

Traveling Wave

To the left, we have a snapshot of

a traveling wave at a single point

in time. Below left, the same wave

is shown traveling.

39

15-4 Mathematical Representation of a

Traveling Wave

A full mathematical description of the wave describes the displacement

of any point as a function of both distance and time:

40

15-7 Reflection and Transmission of Waves

A wave reaching the end

of its medium, but where

the medium is still free to

move, will be reflected (b),

and its reflection will be

upright.

A wave hitting an obstacle will be reflected

(a), and its reflection will be inverted.

41

15- 7 Reflection and Transmission of Waves

A wave encountering a denser medium will be partly reflected and partly

transmitted; if the wave speed is less in the denser medium, the wavelength will

be shorter.

42

15-7 Reflection and Transmission of WavesTwo- or three-dimensional waves can be represented by wave fronts,

which are curves of surfaces where all the waves have the same

phase.

Lines perpendicular to the wave

fronts are called rays; they point in

the direction of propagation of the

wave.

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15-7 Reflection and Transmission of WavesThe law of reflection: the angle of incidence equals the angle of reflection.

44

15-8 Interference; Principle of Superposition

The superposition principle says that when two waves pass

through the same point, the displacement is the arithmetic

sum of the individual displacements.

In the figure below, (a) exhibits destructive interference and

(b) exhibits constructive interference.

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15-8 Interference; Principle of Superposition

These figures show the sum of two waves. In (a) they add constructively;

in (b) they add destructively; and in (c) they add partially destructively.

46

15-9 Standing Waves; Resonance

Standing waves occur when both

ends of a string are fixed. In that case,

only waves which are motionless at

the ends of the string can persist.

There are nodes, where the amplitude

is always zero, and antinodes, where

the amplitude varies from zero to the

maximum value.

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The frequencies of the standing waves

on a particular string are called resonant

frequencies.

They are also referred to as the

fundamental and harmonics.

48


The wavelengths and frequencies of standing waves are:

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15-10 RefractionIf the wave enters a medium where the wave speed is different, it will be

refracted its wave fronts and rays will change direction.

We can calculate the angle of refraction, which

depends on both wave speeds:

50

15-10 RefractionThe law of refraction works both ways a wave going from a slowermedium to a faster one would follow the red line in the other direction.

51

15-11 Diffraction

When waves encounter an obstacle,

they bend around it, leaving a shadow region. This is called diffraction.

52

15-11 DiffractionThe amount of diffraction depends on the size of the obstacle compared

to the wavelength. If the obstacle is much smaller than the wavelength,

the wave is barely affected (a). If the object is comparable to, or larger

than, the wavelength, diffraction is much more significant (b, c, d).

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Chapter 16

Sound

54

Units of Chapter 16

Characteristics of Sound

Intensity of Sound: Decibels

The Ear and Its Response; Loudness

Sources of Sound: Vibrating Strings and Air Columns

Quality of Sound, and Noise; Superposition

Interference of Sound Waves; Beats

Doppler Effect

Shock Waves and the Sonic Boom

Applications: Sonar, Ultrasound, and Medical Imaging

55

16-1 Characteristics of Sound

Sound can travel through any kind of matter, but not through a vacuum.

The speed of sound is different in different

materials; in general, it is slowest in gases,

faster in liquids, and fastest in solids.

The speed depends somewhat on

temperature, especially for gases.

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16-1 Characteristics of Sound

Loudness: related to intensity of the sound wave

Pitch: related to frequency.

Audible range: about 20 Hz to 20,000 Hz; upper limit decreases with age

Ultrasound: above 20,000 Hz; see ultrasonic camera focusing below

Infrasound: below 20 Hz

57

16-2 Intensity of Sound: Decibels

The intensity of a wave is the energy

transported per unit time across a unit

area.

The human ear can detect sounds with an

intensity as low as 10-12 W/m2 and as high

as 1 W/m2.

Perceived loudness, however, is not

proportional to the intensity.

58


The loudness of a sound is much more closely related to the logarithm of

the intensity.

Sound level is measured in decibels (dB) and is defined:

I0 is taken to be the threshold of hearing:

59


An increase in sound level of 3 dB,

which is a doubling in intensity, is a

very small change in loudness.

In open areas, the intensity of

sound diminishes with distance:

However, in enclosed spaces this is complicated by reflections, and if sound

travels through air the higher frequencies get preferentially absorbed.

60

12-3 The Ear and Its Response; Loudness

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12-3 The Ear and Its Response; Loudness

Outer ear: sound waves travel down the ear canal to the

eardrum, which vibrates in response

Middle ear: hammer, anvil, and stirrup transfer vibrations to

inner ear

Inner ear: cochlea transforms vibrational energy to electrical

energy and sends signals to the brain

62

16-3 The Ear and its Response; Loudness

The ears sensitivity varies with frequency. These curves translate the intensity into sound level at different frequencies.

63

16-4 Sources of Sound: Vibrating Strings

and Air Columns

Musical instruments produce sounds in various ways vibrating strings, vibrating membranes, vibrating metal or wood shapes, vibrating air

columns.

The vibration may be started by plucking, striking, bowing, or blowing.

The vibrations are transmitted to the air and then to our ears.

64


and Air Columns

The strings on a guitar can be

effectively shortened by fingering,

raising the fundamental pitch.

The pitch of a string of a given length

can also be altered by using a string

of different density.

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and Air ColumnsA piano uses both methods to cover its more than seven-octave range the lower strings (at bottom) are both much longer and much thicker than

the higher ones.

66


and Air Columns

Wind instruments create sound through standing waves in a column of

air.

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and Air Columns

A tube open at both ends (most wind instruments) has pressure nodes, and

therefore displacement antinodes, at the ends.

68


and Air Columns

A tube closed at one end (some organ pipes) has a displacement node

(and pressure antinode) at the closed end.

69

16-5 Quality of Sound, and Noise;

Superposition

So why does a trumpet sound different from a flute? The answer lies in

overtones which ones are present, and how strong they are, makes a big difference.

The plot below shows frequency spectra for a clarinet, a piano, and a

violin. The differences in overtone strength are apparent.

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16-6 Interference of Sound Waves; Beats

Sound waves interfere in the same way that other

waves do in space.

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16-6 Interference of Sound Waves; Beats

Waves can also interfere in time, causing a phenomenon called beats.

Beats are the slow envelope around two waves that are relatively close in frequency.

72

16-7 Doppler Effect

The Doppler effect occurs when a source of sound is moving with respect

to an observer.

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16-7 Doppler Effect

As can be seen in the previous image, a source

moving toward an observer has a higher frequency

and shorter wavelength; the opposite is true when a

source is moving away from an observer.

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16-7 Doppler Effect

If we can figure out what

the change in the

wavelength is, we also

know the change in the

frequency.

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16-7 Doppler Effect

The change in the wavelength is given by:

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16-7 Doppler Effect

And the change in the frequency:

If the source is moving away from the observer:

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16-7 Doppler Effect

If the observer is moving with respect to the source, things are a bit

different. The wavelength remains the same, but the wave speed is

different for the observer.

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16-7 Doppler Effect

We find, for an observer moving towards a stationary source:

And if it is moving away:

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16-8 Shock Waves and the Sonic Boom

If a source is moving faster than the wave speed in a medium, waves

cannot keep up and a shock wave is formed.

The angle of the cone is:

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Shock waves are analogous to the bow waves produced by a boat going

faster than the wave speed in water.

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Aircraft exceeding the speed of sound in air will produce two sonic booms,

one from the front and one from the tail.

82

16-9 Applications: Sonar, Ultrasound, and

Medical Imaging

Sonar is used to locate objects underwater by measuring the time it

takes a sound pulse to reflect back to the receiver.

Similar techniques can be used to learn about the internal structure of

the Earth.

Sonar usually uses ultrasound waves, as the shorter wavelengths are

less likely to be diffracted by obstacles.

83


Medical Imaging

Ultrasound is also used for

medical imaging. Repeated

traces are made as the

transducer is moved, and a

complete picture is built.

84


Medical Imaging

Ordinary ultrasound gives a good picture; high-resolution ultrasound is

excellent.

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Summary of Chapter 16

Sound is a longitudinal wave in a medium.

The pitch of the sound depends on the frequency.

The loudness of the sound depends on the intensity and also on the sensitivity of the ear.

The strings on stringed instruments produce a fundamental tone whose wavelength is twice the length of the string; there are also various

harmonics present.

Wind instruments have a vibrating column of air when played. If the tube is open, the fundamental is twice its length; if it is closed the fundamental

is four times the tube length.

Sound waves exhibit interference; if two sounds are at slightly different frequencies they produce beats.

The Doppler effect is the shift in frequency of a sound due to motion of the source or the observer.

86

HOMEWORK

Giancoli, Chapter 16

3, 6, 10, 16, 18, 69

References

o Physics For Scientists &Engineers with Modern Physics Giancoli 4th edition,

Pearson International Edition

Documents

Physics for Scientists and Engineers Lecture 14