Chapter 11Chapter 11

Vibrations & WavesVibrations & Waves

11.1 Simple Harmonic Motion11.1 Simple Harmonic Motion

Hooke’s LawHooke’s LawRepeated motion is “Repeated motion is “periodicperiodic motion”. motion”.

Like a pendulum, back and forth over Like a pendulum, back and forth over

the same path.the same path.

At equilibrium position, speed reaches maximumAt equilibrium position, speed reaches maximumAs the mass is pulled away, it As the mass is pulled away, it displacesdisplaces the spring to a the spring to a certain distance, x = ? certain distance, x = ? This exerts a force towards the equilibrium position.This exerts a force towards the equilibrium position.

At (b) force = 0 and so does displacement. At (b) force = 0 and so does displacement.

AccelerationAcceleration also equals “0”. also equals “0”.

However, speed is at However, speed is at maxmax due to momentum. due to momentum.

At (c) due to momentum, mass overshoots At (c) due to momentum, mass overshoots equilibriumequilibrium and compresses spring the other way.and compresses spring the other way.

At maximum displacement, spring force and acceleration At maximum displacement, spring force and acceleration reach a maximumreach a maximum

Beyond equilibrium the force and acceleration Beyond equilibrium the force and acceleration increaseincrease, , however speed slows down.however speed slows down.

At maximum displacement, speed is 0 but acceleration At maximum displacement, speed is 0 but acceleration and force are at max. Then, back and forth (oscillates).and force are at max. Then, back and forth (oscillates).

Friction brings the vibrating mass to rest (Friction brings the vibrating mass to rest (dampingdamping).).

In simple harmonic motion, restoring force In simple harmonic motion, restoring force

is proportional to displacementis proportional to displacementThis pushing and pulling is sometimes called This pushing and pulling is sometimes called

“ “restoringrestoring force”, it is directly proportional to the force”, it is directly proportional to the

displacement of a mass.displacement of a mass.

Determined by Robert Hooke in 1678… Determined by Robert Hooke in 1678…

The The negativenegative sign means that the spring force is opposite sign means that the spring force is opposite the direction the mass is displaced.the direction the mass is displaced.

Harmonic Motion Animation

kxFelastic

kk is the spring constant and is based on the is the spring constant and is based on the stiffnessstiffness of of the spring in N/m.the spring in N/m.

The motion of the vibrating mass is an example of The motion of the vibrating mass is an example of “simple harmonic motion” and any periodic motion that is “simple harmonic motion” and any periodic motion that is a result of a a result of a restoringrestoring force. force.

Spring Constant Animation

Practice APractice AHooke’s LawHooke’s Law

A mass of 0.55 kg, attached to a vertical A mass of 0.55 kg, attached to a vertical spring, stretches the spring -0.020 m from its spring, stretches the spring -0.020 m from its original equilibrium position.original equilibrium position.

What is the spring constant?What is the spring constant?

AnswerAnswer

Given: m = 0.55 kg g = 9.81 m/s2

x = 0.020 mUnknown: k = ?

270 N/m

A stretched or compressed spring has A stretched or compressed spring has

elastic potential energyelastic potential energy

A bent bow is the same as a stretched A bent bow is the same as a stretched

springspring..

Stretched or compressed springs Stretched or compressed springs storestore elastic potential elastic potential energy (PE).energy (PE).

Once released the PE becomes KE, or moving arrow!Once released the PE becomes KE, or moving arrow!

InsideInside

The Simple PendulumThe Simple PendulumThe swinging motion of a pendulum is The swinging motion of a pendulum is periodicperiodic

vibration.vibration.

A simple pendulum consists of a mass called A simple pendulum consists of a mass called

a a bobbob which is attached to a fixed string. which is attached to a fixed string.

The restoring force of a pendulum is a component The restoring force of a pendulum is a component of the bob’s weightof the bob’s weight

If the restoring force is proportional to the If the restoring force is proportional to the displacementdisplacement, , then the pendulum’s motion is simple harmonic.then the pendulum’s motion is simple harmonic.

Any displacement from equilibrium can be Any displacement from equilibrium can be

resolved with both the x and y components. resolved with both the x and y components.

Simple Pendulum Animation

For small angles, the pendulum’s motion is simple harmonicFor small angles, the pendulum’s motion is simple harmonic

When the maximum angle for displacement When the maximum angle for displacement θθ is relatively is relatively

small (<15small (<1500), sin ), sin θθ is approximately equal in radians. is approximately equal in radians.

Pendulum's motion is an excellent Pendulum's motion is an excellent approximationapproximation of simple of simple

harmonic motion.harmonic motion.

Because a Because a simplesimple pendulum pendulum

vibrates with simple harmonic vibrates with simple harmonic

motion, many of our earlier motion, many of our earlier

conclusions for a mass-springconclusions for a mass-spring

system apply here.system apply here.

Gravitational potential increases as a pendulum’s Gravitational potential increases as a pendulum’s displacement increasesdisplacement increases

This diagram shows how a pendulum’s mechanical This diagram shows how a pendulum’s mechanical energy changes as the pendulum energy changes as the pendulum oscillatesoscillates..

As the pendulum swings toward equilibrium, it gains As the pendulum swings toward equilibrium, it gains kinetic energy and losses kinetic energy and losses potentialpotential energy. energy.

QuestionsQuestions1. Repeated motion is _______ motion.1. Repeated motion is _______ motion.

2. When the mass/spring is at maximum displacement, 2. When the mass/spring is at maximum displacement, speed is 0 but acceleration and force are at _____.speed is 0 but acceleration and force are at _____.

3. The motion of a vibrating mass is an example of “simple 3. The motion of a vibrating mass is an example of “simple harmonic motion” and any periodic motion that is a result harmonic motion” and any periodic motion that is a result of a _________ force.of a _________ force.

4. The restoring force of a pendulum’s motion is _______4. The restoring force of a pendulum’s motion is _______

5. As the pendulum swings toward equilibrium, it gains 5. As the pendulum swings toward equilibrium, it gains kinetic energy and losses ________ energy.kinetic energy and losses ________ energy.

periodic

max

restoring

gravity

potential

11.2 11.2 Measuring Simple Harmonic MotionMeasuring Simple Harmonic Motion

Amplitude, Period, and Amplitude, Period, and FrequencyFrequencyA moving trapeze always returns to the A moving trapeze always returns to the same displacement from equilibrium, this is same displacement from equilibrium, this is the the amplitudeamplitude..A pendulum’s amplitude can be A pendulum’s amplitude can be measured by the angle between the measured by the angle between the pendulum’s equilibrium position and its pendulum’s equilibrium position and its maximum maximum displacementdisplacement..For a mass-spring system, this is its For a mass-spring system, this is its stretched or compressed position. stretched or compressed position.

Period and frequency measure timePeriod and frequency measure timeFrom one side of max displacement to the other, From one side of max displacement to the other,

one complete cycle, is one complete cycle, is periodperiod, , TT

If one complete cycle takes 20 seconds, then If one complete cycle takes 20 seconds, then the period of motion is 20 s.the period of motion is 20 s.

The number of complete cycles in a unit of time The number of complete cycles in a unit of time is is frequencyfrequency..

If it takes 20 s to complete one cycle, the If it takes 20 s to complete one cycle, the frequency is 1/20 cycles or 0.05 cycles.frequency is 1/20 cycles or 0.05 cycles.

Frequency is sFrequency is s-1-1 or or hertzhertz (Hz) (Hz)

So…So…

-period is time per cycle-period is time per cycle

-frequency is number of cycles -frequency is number of cycles

per unit of time.per unit of time.

Tf

fT

1

1

Animation

The period of a simple pendulum depends on The period of a simple pendulum depends on pendulum length and free-fall accelerationpendulum length and free-fall acceleration

Simple pendulums and mass-spring systems vibrate with Simple pendulums and mass-spring systems vibrate with harmonicharmonic motion. motion.

To calculate period (To calculate period (TT) and frequency ) and frequency ƒ ƒ inin (Hz), requires a (Hz), requires a separate formula.separate formula.

A pendulum with the same A pendulum with the same lengthlength ( (LL) but with bobs of two ) but with bobs of two different masses or amplitude, the different masses or amplitude, the periodperiod will be the same. will be the same.

If free-fall acceleration (gravity) changes, If free-fall acceleration (gravity) changes,

so will the period.so will the period.

g

LT 2

When two pendulums have When two pendulums have differentdifferent lengths but the lengths but the same amplitude, the shorter pendulum will have a same amplitude, the shorter pendulum will have a smaller arc to travel through.smaller arc to travel through.

Mass and amplitude do Mass and amplitude do notnot affect the period for the affect the period for the same reason all objects fall at the same rate.same reason all objects fall at the same rate.

The reason for this is, the more you increase the The reason for this is, the more you increase the amplitude, the more amplitude, the more restoringrestoring force there is (even though force there is (even though it has a greater distance to cover)it has a greater distance to cover)

Practice B Simple Harmonic Motion of Practice B Simple Harmonic Motion of a Simple Penduluma Simple Pendulum

You need to know the height of a tower, but darkness You need to know the height of a tower, but darkness obscures the ceiling. obscures the ceiling.

You note that a pendulum extending from the ceiling You note that a pendulum extending from the ceiling almost touches the floor and that it has a period of 12 s.almost touches the floor and that it has a period of 12 s.

How tall is the tower?How tall is the tower?

AnswerAnswer 36 m

Period of a mass-spring system depends on mass Period of a mass-spring system depends on mass and spring constantand spring constant

The period of a mass-spring system uses Hook’s Law.The period of a mass-spring system uses Hook’s Law.

Because heavier objects have more Because heavier objects have more inertiainertia, they take , they take longer to speed up.longer to speed up.

This causes them to have This causes them to have

longer longer periodsperiods..

kxFelastic

The greater the spring constant (k), the greater the force The greater the spring constant (k), the greater the force needed to stretch or compress the spring.needed to stretch or compress the spring.

When force is When force is greatgreat, so is the acceleration., so is the acceleration.

This makes the time required to make one cycle less.This makes the time required to make one cycle less.

So, So, stiffstiff spring = short period. spring = short period.

Also, changing amplitude does Also, changing amplitude does notnot effect the period. effect the period.

k

mT 2

Practice C Simple Harmonic Motion of Practice C Simple Harmonic Motion of a Mass-Spring Systema Mass-Spring SystemThe body of a 1275 kg car is supported The body of a 1275 kg car is supported

on a frame by four springs. Two people on a frame by four springs. Two people riding in the car have a combined mass of 153 kg. riding in the car have a combined mass of 153 kg.

When driven over a pothole in the road, the frame When driven over a pothole in the road, the frame vibrates with a period of 0.840 s.vibrates with a period of 0.840 s.For the first few seconds, the vibration For the first few seconds, the vibration

approximates simple harmonic motion.approximates simple harmonic motion.Find the spring constant of a Find the spring constant of a singlesingle spring. spring.

AnswerAnswer 20,000 N/m

QuestionsQuestions1. A pendulum’s _________ can be measured by the angle 1. A pendulum’s _________ can be measured by the angle between the pendulum’s equilibrium position and its between the pendulum’s equilibrium position and its maximum displacement.maximum displacement.

2. The number of complete cycles in a unit of time is known 2. The number of complete cycles in a unit of time is known as _________.as _________.

3. A pendulum with the same length but with bobs of two 3. A pendulum with the same length but with bobs of two different masses or amplitude, the ______ will be the same.different masses or amplitude, the ______ will be the same.

4. Mass and amplitude (do / do not) affect the period of a 4. Mass and amplitude (do / do not) affect the period of a pendulum.pendulum.

5. The stiffer the spring, the (longer / shorter) the period.5. The stiffer the spring, the (longer / shorter) the period.

do not

shorter

amplitude

frequency

period

11.3 Properties of Waves11.3 Properties of WavesWave MotionWave MotionIf there is a disturbance in a pond, you see a If there is a disturbance in a pond, you see a circularcircular pattern pattern move outward in all directions.move outward in all directions.If there is a leaf floating near by, it moves a little but does If there is a leaf floating near by, it moves a little but does notnot travel with the wave.travel with the wave.

Breaking waves are different Breaking waves are different

A wave is the motion of a disturbanceA wave is the motion of a disturbance

This disturbance causes the pattern to This disturbance causes the pattern to move out in a circular pattern.move out in a circular pattern.

Water in the pond is the Water in the pond is the mediummedium through which the disturbance travels.through which the disturbance travels.

The medium (water) does not actually The medium (water) does not actually travel with the wave.travel with the wave.

Sound waves require Sound waves require airair as their as their medium. In space there is no sound.medium. In space there is no sound.

Waves that require a material medium Waves that require a material medium are called are called mechanicalmechanical waves. waves.

Animation

Wave TypesWave TypesA wave that consists of a single traveling pulse is called a A wave that consists of a single traveling pulse is called a

pulsepulse wave. wave.

If you continue to generate pulses, this will create a If you continue to generate pulses, this will create a periodicperiodic wave. wave.

Sine waves describe particles vibrating with simple Sine waves describe particles vibrating with simple harmonic motionharmonic motion

Periodic waves can show simple harmonic motion on a Periodic waves can show simple harmonic motion on a string.string.

A wave whose source vibrates with simple harmonic A wave whose source vibrates with simple harmonic motion is called a motion is called a sinesine wave. wave.

These are called sine waves because a graph of These are called sine waves because a graph of trigonometric function y = sin x produces this curve when trigonometric function y = sin x produces this curve when plotted.plotted.

Vibrations of a transverse wave are perpendicular Vibrations of a transverse wave are perpendicular to the wave motionto the wave motion

When vibrations are perpendicular to the direction of the When vibrations are perpendicular to the direction of the wave’s motion they are called wave’s motion they are called transversetransverse waves. waves.

Displacement of a single particle as time passes creates a Displacement of a single particle as time passes creates a wavewave form.form.

Wave measures include crest, trough, amplitude, Wave measures include crest, trough, amplitude, and wavelengthand wavelength

A wave can be measured in terms of its displacement A wave can be measured in terms of its displacement from equilibrium.from equilibrium.

The highest point is called the The highest point is called the crestcrest..

The lowest point the The lowest point the troughtrough..

Remember, amplitude is a measure of maximum Remember, amplitude is a measure of maximum displacement from equilibrium.displacement from equilibrium.

The distance the wave travels in one cycle along its path The distance the wave travels in one cycle along its path is called is called wavelengthwavelength, (, (λλ).).

Vibrations of a longitudinal wave are parallel to the Vibrations of a longitudinal wave are parallel to the wave motionwave motion

When the displacement of the medium vibrates parallel to When the displacement of the medium vibrates parallel to the direction of wave motion is called a the direction of wave motion is called a longitudinallongitudinal wave. wave.

Longitudinal waves can also be described by a sine Longitudinal waves can also be described by a sine curve.curve.

The type of wave represented above is often called a The type of wave represented above is often called a density or density or pressurepressure wave. wave.

The crests are where the spring coils are The crests are where the spring coils are compressedcompressed..

Animation

Period, Frequency, and Wave SpeedPeriod, Frequency, and Wave Speed

Sound waves may begin with the Sound waves may begin with the vibrations of your vibrations of your vocalvocal cords. The cords. The source of wave motion is a source of wave motion is a vibrating object.vibrating object.

When the vibrating particles of the When the vibrating particles of the medium complete one full cycle, medium complete one full cycle, one wavelength passes any given one wavelength passes any given point.point.

Thus, wave frequency describes Thus, wave frequency describes the number of waves that pass a the number of waves that pass a given point in a unit of given point in a unit of timetime..

The The periodperiod of a wave is the time of a wave is the time required for one complete cycle of required for one complete cycle of vibration of the mediums particles. vibration of the mediums particles.

Wave speed equals frequency times wavelength Wave speed equals frequency times wavelength We can now derive an expression for the speed of a wave We can now derive an expression for the speed of a wave

in terms of its period or frequency.in terms of its period or frequency.

Animation

The The speedspeed of a mechanical wave is of a mechanical wave is

constant for any given medium.constant for any given medium.

Even though all sounds are different, Even though all sounds are different,

they reach your ears at the same speed.they reach your ears at the same speed.

As a result, when the frequency increasesAs a result, when the frequency increases

its wavelength must its wavelength must decreasedecrease..

Practice D Practice D Wave SpeedWave Speed

The piano string tuned to middle C vibrates with a The piano string tuned to middle C vibrates with a frequency of 264 Hz. frequency of 264 Hz.

Assuming the speed of sound in air is 343 m/s, find the Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string.wavelength of the sound waves produced by the string.

AnswerAnswer 1.30 m

Waves transfer energyWaves transfer energyWaves transfer energy by the Waves transfer energy by the vibrationvibration of matter rather than by of matter rather than by

the transfer of matter itself.the transfer of matter itself.

For this reason, waves are often able to transport energy For this reason, waves are often able to transport energy efficiently. efficiently.

The rate at which a wave transfers energy depends on the The rate at which a wave transfers energy depends on the amplitudeamplitude, the greater the amplitude, the more energy carried in , the greater the amplitude, the more energy carried in a given time interval.a given time interval.

When the amplitude is When the amplitude is doubleddoubled, the , the

energy carried increases by a factor of 4.energy carried increases by a factor of 4.

QuestionsQuestions1. Waves that require a material medium are called 1. Waves that require a material medium are called __________ waves.__________ waves.

2. A wave whose source vibrates with simple harmonic 2. A wave whose source vibrates with simple harmonic motion is called a _____ wave.motion is called a _____ wave.

3. When the displacement of the medium vibrates parallel 3. When the displacement of the medium vibrates parallel to the direction of wave motion is called a __________ to the direction of wave motion is called a __________ wave. wave.

4. When the frequency increases, its wavelength must? 4. When the frequency increases, its wavelength must?

5. When the amplitude is doubled, the 5. When the amplitude is doubled, the energy carried increases by a factor of ______. energy carried increases by a factor of ______.

mechanical

sine

longitudinal

decrease.

four

11.4 11.4 Wave InteractionsWave Interactions

Wave InterferenceWave InterferenceWhen two waves come together, they do not bounce back When two waves come together, they do not bounce back

like bumper boats.like bumper boats.With sound waves, you can distinguish the sounds of With sound waves, you can distinguish the sounds of

different instruments.different instruments.This is because sound waves (mechanical waves) are not This is because sound waves (mechanical waves) are not

matter but matter but displacementsdisplacements of matter. of matter.Two waves Two waves cancan occupy the same space occupy the same space

at the same time.at the same time.As they pass through one another, they As they pass through one another, they

interact to form an interact to form an interferenceinterference pattern pattern..

Displacements in the same direction produce constructive Displacements in the same direction produce constructive interferenceinterference

When two pulses meet, a When two pulses meet, a resultantresultant wave forms. wave forms.

The amplitude of the resultant wave is equal to the The amplitude of the resultant wave is equal to the sumsum of the of the amplitudes of each pulse.amplitudes of each pulse.

Summing the displacements of waves is known as the Summing the displacements of waves is known as the superpositionsuperposition principleprinciple..

After the two pulses pass, they have their original shape.After the two pulses pass, they have their original shape.

If the displacements are on the same side of equilibrium, when added If the displacements are on the same side of equilibrium, when added together, we get together, we get constructiveconstructive interference interference..

Displacements in opposite directions produce Displacements in opposite directions produce destructive interferencedestructive interference

The following shows what happens when pulses are on The following shows what happens when pulses are on opposite sides of equilibrium.opposite sides of equilibrium.

When the positive and negative displacements are added When the positive and negative displacements are added we get we get destructivedestructive interference. interference.

When two pulses coincide, their resultant wave can have When two pulses coincide, their resultant wave can have a displacement of zero.a displacement of zero.

This is known as complete This is known as complete destructivedestructive interference. interference.Animation

The superposition principle is valid for longitudinal The superposition principle is valid for longitudinal ((compressioncompression) waves.) waves.

In a compression, particles move closer together, while In a compression, particles move closer together, while rarefactionrarefaction, particles spread apart., particles spread apart.

When a compression and rarefaction interfere, there is When a compression and rarefaction interfere, there is destructive interference.destructive interference.

In the case of sound, these wave can In the case of sound, these wave can

cancelcancel causing a lack of sound. causing a lack of sound.

ReflectionReflectionAt a free boundary, waves are reflectedAt a free boundary, waves are reflected At a free boundary, the rope is At a free boundary, the rope is

free to move up and down sending free to move up and down sending

the wave the wave backback in the same way. in the same way. This is called This is called reflectionreflection..

At a fixed boundary, waves are reflected and invertedAt a fixed boundary, waves are reflected and inverted

When the pulse reaches the When the pulse reaches the wall, the pulse exerts an wall, the pulse exerts an upward force on the wall.upward force on the wall.

The wall in turn exerts an The wall in turn exerts an equal and equal and oppositeopposite reaction reaction force on the rope.force on the rope.

As a result, the pulse is As a result, the pulse is invertedinverted. .

Standing WavesStanding WavesStandingStanding waves occur when a string is waves occur when a string is

attached to one ridged end and shaken attached to one ridged end and shaken

in a regular motion.in a regular motion.

This will produce a wave of a certain This will produce a wave of a certain frequency, wavelength, and amplitude traveling frequency, wavelength, and amplitude traveling down the string.down the string.

As the waves reach the other end they As the waves reach the other end they

are reflected back toward the oncoming are reflected back toward the oncoming waves.waves.

If the string is vibrated at a certain frequency, a If the string is vibrated at a certain frequency, a standing wave or standing wave or resultantresultant wave pattern wave pattern appears on the string.appears on the string.

The standing wave consists of alternating The standing wave consists of alternating regions of constructive and destructive regions of constructive and destructive interferenceinterference..

Standing waves have nodes and antinodesStanding waves have nodes and antinodesFour possible standing waves are shown below.Four possible standing waves are shown below.

Points where complete destructive interference happen Points where complete destructive interference happen are called are called nodesnodes..

Midway between two adjacent nodes, where the string Midway between two adjacent nodes, where the string vibrates with the largest amplitude are the vibrates with the largest amplitude are the antinodesantinodes..

In the second example In the second example

below, on the right, shows below, on the right, shows

where there are 3 nodes(N) where there are 3 nodes(N)

and 2 antinodes(A).and 2 antinodes(A).

Animation

Only Only certaincertain frequencies, and frequencies, and wavelengths, produce standing waves.wavelengths, produce standing waves.

A standing wave can only be produced A standing wave can only be produced for any wavelength that allows both for any wavelength that allows both ends of the string to be ends of the string to be nodesnodes..

Example (b)Example (b) is half a wave length, so to is half a wave length, so to find wave length you multiply the string find wave length you multiply the string length by two (2length by two (2LL).).

Example (c)Example (c) is one wave length so we is one wave length so we just use (just use (LL).).

Example (d)Example (d) has 4 nodes and 3 has 4 nodes and 3 antinodes. We would have to go another antinodes. We would have to go another half wavelength to have two full wave half wavelength to have two full wave lengths. In this case, to find the length of lengths. In this case, to find the length of one wave, we just multiply by (2/3one wave, we just multiply by (2/3LL).).

QuestionsQuestions1. Sound waves (mechanical waves) are not matter but 1. Sound waves (mechanical waves) are not matter but _____________ of matter._____________ of matter.

2. In a compression, particles move closer together, while 2. In a compression, particles move closer together, while __________, particles spread apart.__________, particles spread apart.

3. At a free boundary, the rope is free to move up and 3. At a free boundary, the rope is free to move up and down sending the wave back in the same way, this is called down sending the wave back in the same way, this is called _________._________.

4. Midway between two adjacent nodes, where the string 4. Midway between two adjacent nodes, where the string vibrates with the largest amplitude are the _________.vibrates with the largest amplitude are the _________.

5.5. If a standing wave is half a wave length, to find its wave If a standing wave is half a wave length, to find its wave length you multiply the string length by ____.length you multiply the string length by ____.two

displacements

rarefaction

reflection

antinodes

EndEnd