Chapter 13Chapter 13

Vibrations Vibrations

andand

WavesWaves

Hooke’s Law 13.1Hooke’s Law 13.1

FFss = - k x = - k x FFss is the spring force is the spring force k is the spring constantk is the spring constant

It is a measure of the stiffness of the springIt is a measure of the stiffness of the spring– A large k indicates a stiff spring and a small k A large k indicates a stiff spring and a small k

indicates a soft springindicates a soft spring

x is the displacement of the object from its x is the displacement of the object from its equilibrium positionequilibrium position

The negative sign indicates that the force is The negative sign indicates that the force is always directed opposite to the displacementalways directed opposite to the displacement

Hooke’s Law ForceHooke’s Law Force

The force always acts toward the The force always acts toward the equilibrium positionequilibrium position It is called the It is called the restoring forcerestoring force

The direction of the restoring force The direction of the restoring force is such that the object is being is such that the object is being either pushed or pulled toward the either pushed or pulled toward the equilibrium positionequilibrium position

Hooke’s Law Applied to a Hooke’s Law Applied to a Spring – Mass SystemSpring – Mass System

When x is positive When x is positive (to the right), F is (to the right), F is negative (to the negative (to the left)left)

When x = 0 (at When x = 0 (at equilibrium), F is 0equilibrium), F is 0

When x is negative When x is negative (to the left), F is (to the left), F is positive (to the positive (to the right)right)

Motion of the Spring-Mass Motion of the Spring-Mass SystemSystem

Assume the object is initially pulled to x = A Assume the object is initially pulled to x = A and released from restand released from rest

As the object moves toward the equilibrium As the object moves toward the equilibrium position, F and a decrease, but v increasesposition, F and a decrease, but v increases

At x = 0, F and a are zero, but v is a maximumAt x = 0, F and a are zero, but v is a maximum The object’s momentum causes it to overshoot The object’s momentum causes it to overshoot

the equilibrium positionthe equilibrium position The force and acceleration start to increase in The force and acceleration start to increase in

the opposite direction and velocity decreasesthe opposite direction and velocity decreases The motion continues indefinitelyThe motion continues indefinitely

Simple Harmonic MotionSimple Harmonic Motion

Motion that occurs when the net Motion that occurs when the net force along the direction of motion is force along the direction of motion is a Hooke’s Law type of forcea Hooke’s Law type of force The force is proportional to the The force is proportional to the

displacement and in the opposite displacement and in the opposite directiondirection

The motion of a spring mass system The motion of a spring mass system is an example of Simple Harmonic is an example of Simple Harmonic MotionMotion

Simple Harmonic Motion, Simple Harmonic Motion, cont.cont.

Not all periodic motion over the Not all periodic motion over the same path can be considered same path can be considered Simple Harmonic motionSimple Harmonic motion

To be Simple Harmonic motion, the To be Simple Harmonic motion, the force needs to obey Hooke’s Lawforce needs to obey Hooke’s Law

AmplitudeAmplitude

Amplitude, AAmplitude, A The amplitude is the maximum The amplitude is the maximum

position of the object relative to the position of the object relative to the equilibrium positionequilibrium position

In the absence of friction, an object in In the absence of friction, an object in simple harmonic motion will oscillate simple harmonic motion will oscillate between between ±A on each side of the ±A on each side of the equilibrium positionequilibrium position

Period and FrequencyPeriod and Frequency

The period, T, is the time that it The period, T, is the time that it takes for the object to complete takes for the object to complete one complete cycle of motion one complete cycle of motion From x = A to x = - A and back to x = From x = A to x = - A and back to x =

AA The frequency, The frequency, ƒ, is the number of ƒ, is the number of

complete cycles or vibrations per complete cycles or vibrations per unit timeunit time

Acceleration of an Object Acceleration of an Object in Simple Harmonic Motionin Simple Harmonic Motion

Newton’s second law will relate force Newton’s second law will relate force and accelerationand acceleration

The force is given by Hooke’s LawThe force is given by Hooke’s Law F = - k x = m aF = - k x = m a

a = -kx / ma = -kx / m The acceleration is a function of positionThe acceleration is a function of position

Acceleration is Acceleration is notnot constant and therefore constant and therefore the uniformly accelerated motion equation the uniformly accelerated motion equation cannot be appliedcannot be applied

Sample problem 13.1

A spring with constant k=475 N/m stretch 4.50 cm when an object of mass 25.0 kg is attached to the end of the spring. Find the acceleration of gravity in this location. (0.855 m/s2)

Sample Problem 13.2Sample Problem 13.2

A 0.350 kg object attached to a spring of force A 0.350 kg object attached to a spring of force constant 130 N/m is free to move on a constant 130 N/m is free to move on a frictionless horizontal surface. If the object is frictionless horizontal surface. If the object is released from rest at x = 0.100 m find the released from rest at x = 0.100 m find the force and acceleration of the object. force and acceleration of the object. (Ans: -13 (Ans: -13 N; -37 m/sN; -37 m/s22))

Acceleration Defining Acceleration Defining Simple Harmonic MotionSimple Harmonic Motion

Acceleration can be used to define Acceleration can be used to define simple harmonic motionsimple harmonic motion

An object moves in simple An object moves in simple harmonic motion if its acceleration harmonic motion if its acceleration is directly proportional to the is directly proportional to the displacement and is in the displacement and is in the opposite directionopposite direction

Elastic Potential Energy Elastic Potential Energy 13.213.2

A compressed spring has potential A compressed spring has potential energyenergy The compressed spring, when allowed The compressed spring, when allowed

to expand, can apply a force to an to expand, can apply a force to an objectobject

The potential energy of the spring can The potential energy of the spring can be transformed into kinetic energy of be transformed into kinetic energy of the objectthe object

Elastic Potential Energy, Elastic Potential Energy, contcont

The energy stored in a stretched or The energy stored in a stretched or compressed spring or other elastic compressed spring or other elastic material is called material is called elastic potential elastic potential energyenergy PePess = = ½kx½kx22

The energy is stored only when the The energy is stored only when the spring is stretched or compressedspring is stretched or compressed

Elastic potential energy can be added to Elastic potential energy can be added to the statements of Conservation of the statements of Conservation of Energy and Work-EnergyEnergy and Work-Energy

Energy in a Spring Mass Energy in a Spring Mass SystemSystem

A block sliding on A block sliding on a frictionless a frictionless system collides system collides with a light springwith a light spring

The block The block attaches to the attaches to the springspring

Energy TransformationsEnergy Transformations

The block is moving on a frictionless surfaceThe block is moving on a frictionless surface The total mechanical energy of the system The total mechanical energy of the system

is the kinetic energy of the blockis the kinetic energy of the block

Energy Transformations, 2Energy Transformations, 2

The spring is partially compressedThe spring is partially compressed The energy is shared between kinetic energy The energy is shared between kinetic energy

and elastic potential energyand elastic potential energy The total mechanical energy is the sum of the The total mechanical energy is the sum of the

kinetic energy and the elastic potential energykinetic energy and the elastic potential energy

Energy Transformations, 3Energy Transformations, 3

The spring is now fully compressedThe spring is now fully compressed The block momentarily stopsThe block momentarily stops The total mechanical energy is stored The total mechanical energy is stored

as elastic potential energy of the springas elastic potential energy of the spring

Energy Transformations, 4Energy Transformations, 4

When the block leaves the spring, the total When the block leaves the spring, the total mechanical energy is in the kinetic energy of mechanical energy is in the kinetic energy of the blockthe block

The spring force is conservative and the total The spring force is conservative and the total energy of the system remains constantenergy of the system remains constant

Sample problem 13.3Sample problem 13.3

A 13,000 N car starts at rest and rolls down a hill from a A 13,000 N car starts at rest and rolls down a hill from a height of 10.0 m. I then moves across a level surface height of 10.0 m. I then moves across a level surface and collides with a light spring-loaded guardrail. A) and collides with a light spring-loaded guardrail. A) neglecting friction, what is the maximum distance the neglecting friction, what is the maximum distance the spring is compressed if the spring constant is 1.0 x 106 spring is compressed if the spring constant is 1.0 x 106 N/m. B) Calculate the max. acceleration of the car after N/m. B) Calculate the max. acceleration of the car after contact with the spring, again neglect friction. C) if the contact with the spring, again neglect friction. C) if the spring is compressed by only 0.30 m, find the energy spring is compressed by only 0.30 m, find the energy lost through friction.lost through friction.

Velocity as a Function of Velocity as a Function of PositionPosition

Conservation of Energy allows a Conservation of Energy allows a calculation of the velocity of the object calculation of the velocity of the object at any position in its motionat any position in its motion

Speed is a maximum at x = 0Speed is a maximum at x = 0 Speed is zero at x = Speed is zero at x = ±A±A The ± indicates the object can be traveling The ± indicates the object can be traveling

in either directionin either direction

22 xAm

kv

Sample Problem 13.4Sample Problem 13.4

A 0.500 kg object connected to a light spring with a A 0.500 kg object connected to a light spring with a spring-constant of 20.0 N/m oscillates on a frictionless spring-constant of 20.0 N/m oscillates on a frictionless horizontal surface. A) calculate the total energy of the horizontal surface. A) calculate the total energy of the system and max. Speed of the object if the amplitude of system and max. Speed of the object if the amplitude of the motion is 3.00 cm. B) what is the velocity of the the motion is 3.00 cm. B) what is the velocity of the object when the displacement is 2.00 cm? C) compute object when the displacement is 2.00 cm? C) compute the KE and PE of the system when the displacement is the KE and PE of the system when the displacement is 2.00 cm. 2.00 cm. (0.00900 J; 0.190 m/s; 0.141 m/s; 0.00497 J; (0.00900 J; 0.190 m/s; 0.141 m/s; 0.00497 J; 0.00400 J)0.00400 J)

Simple Harmonic Motion Simple Harmonic Motion and Uniform Circular and Uniform Circular Motion 13.3Motion 13.3

A ball is attached to the A ball is attached to the rim of a turntable of rim of a turntable of radius Aradius A

The focus is on the The focus is on the shadow that the ball casts shadow that the ball casts on the screenon the screen

When the turntable When the turntable rotates with a constant rotates with a constant angular speed, the angular speed, the shadow moves in simple shadow moves in simple harmonic motionharmonic motion

Period and Frequency Period and Frequency from Circular Motionfrom Circular Motion

PeriodPeriod

This gives the time required for an object of This gives the time required for an object of mass m attached to a spring of constant k mass m attached to a spring of constant k to complete one cycle of its motionto complete one cycle of its motion

FrequencyFrequency

Units are cycles/second or Hertz, HzUnits are cycles/second or Hertz, Hz

k

m2T

m

k

2

1

T

1ƒ

Angular FrequencyAngular Frequency

The angular frequency is related to The angular frequency is related to the frequencythe frequency

m

kƒ2

Sample problem 13.5Sample problem 13.5 A 1300 kg car is constructed on a frame supported by A 1300 kg car is constructed on a frame supported by

four springs. Each spring has a spring constant of four springs. Each spring has a spring constant of 20,000 N/m. If two people riding in the car have a 20,000 N/m. If two people riding in the car have a combined mass of 160 kg, find the frequency of combined mass of 160 kg, find the frequency of vibration of the car when it is driven over a pothole in vibration of the car when it is driven over a pothole in the road. Find also the period and the angular the road. Find also the period and the angular frequency. frequency. (1.18 Hz; 0.847 s; 7.41 rad/s)(1.18 Hz; 0.847 s; 7.41 rad/s)

Motion as a Function of Time Motion as a Function of Time 13.413.4

Use of a Use of a reference reference circlecircle allows a allows a description of the description of the motionmotion

x = A cos (2x = A cos (2πƒt)πƒt) x is the position at x is the position at

time ttime t x varies between x varies between

+A and -A+A and -A

Graphical Representation Graphical Representation of Motionof Motion

When x is a maximum When x is a maximum or minimum, velocity or minimum, velocity is zerois zero

When x is zero, the When x is zero, the velocity is a maximumvelocity is a maximum

When x is a maximum When x is a maximum in the positive in the positive direction, a is a direction, a is a maximum in the maximum in the negative directionnegative direction

Verification of Sinusoidal Verification of Sinusoidal NatureNature

This experiment This experiment shows the shows the sinusoidal nature sinusoidal nature of simple harmonic of simple harmonic motionmotion

The spring mass The spring mass system oscillates system oscillates in simple harmonic in simple harmonic motionmotion

The attached pen The attached pen traces out the traces out the sinusoidal motionsinusoidal motion

Sample problem 13.6Sample problem 13.6 If the object-spring system is described by If the object-spring system is described by

xx=(0.330m)=(0.330m)cos cos (1.50(1.50tt), find the amplitude, the angular ), find the amplitude, the angular frequency, the frequency, the period, and the position frequency, the frequency, the period, and the position of when t=0.250 s. of when t=0.250 s. (0.330m; 1.50 rad/s; 0.239 Hz; 4.19 (0.330m; 1.50 rad/s; 0.239 Hz; 4.19 s; 0.307 m)s; 0.307 m)

Simple Pendulum 13.5Simple Pendulum 13.5

The simple The simple pendulum is pendulum is another example of another example of simple harmonic simple harmonic motionmotion

The force is the The force is the component of the component of the weight tangent to weight tangent to the path of motionthe path of motion F = - m g sin F = - m g sin θθ

Simple Pendulum, contSimple Pendulum, cont

In general, the motion of a pendulum In general, the motion of a pendulum is not simple harmonicis not simple harmonic

However, for small angles, it becomes However, for small angles, it becomes simple harmonicsimple harmonic In general, angles < 15° are small In general, angles < 15° are small

enoughenough sin sin θ = θθ = θ F = - m g θF = - m g θ

This force obeys Hooke’s LawThis force obeys Hooke’s Law

Period of Simple PendulumPeriod of Simple Pendulum

This shows that the period is This shows that the period is independent of of the amplitudeindependent of of the amplitude

The period depends on the length of The period depends on the length of the pendulum and the acceleration of the pendulum and the acceleration of gravity at the location of the gravity at the location of the pendulumpendulum

g

L2T

Sample problem 13.7Sample problem 13.7 Using a small pendulum of length 0.171 m, a Using a small pendulum of length 0.171 m, a

geophysicist counts 72.0 complete swings in a time of geophysicist counts 72.0 complete swings in a time of 60.0 s. What is the value of g in this location? 60.0 s. What is the value of g in this location? (9.73 (9.73 m/sm/s22))

Simple Pendulum Simple Pendulum Compared to a Spring-Compared to a Spring-Mass SystemMass System

Damped Oscillations 13.6Damped Oscillations 13.6

Only ideal systems oscillate Only ideal systems oscillate indefinitelyindefinitely

In real systems, friction retards the In real systems, friction retards the motionmotion

Friction reduces the total energy of Friction reduces the total energy of the system and the oscillation is the system and the oscillation is said to be said to be dampeddamped

Damped Oscillations, cont.Damped Oscillations, cont.

Damped motion Damped motion varies depending on varies depending on the fluid usedthe fluid used With a low viscosity With a low viscosity

fluid, the vibrating fluid, the vibrating motion is preserved, motion is preserved, but the amplitude of but the amplitude of vibration decreases in vibration decreases in time and the motion time and the motion ultimately ceasesultimately ceases

This is known as This is known as underdampedunderdamped oscillationoscillation

More Types of DampingMore Types of Damping

With a higher viscosity, the object With a higher viscosity, the object returns rapidly to equilibrium after it is returns rapidly to equilibrium after it is released and does not oscillatereleased and does not oscillate The system is said to be The system is said to be critically dampedcritically damped

With an even higher viscosity, the With an even higher viscosity, the piston returns to equilibrium without piston returns to equilibrium without passing through the equilibrium passing through the equilibrium position, but the time required is longerposition, but the time required is longer This is said to be This is said to be over dampedover damped

Damping GraphsDamping Graphs

Plot a shows a Plot a shows a critically damped critically damped oscillatoroscillator

Plot b shows an Plot b shows an overdamped overdamped oscillatoroscillator

Wave Motion 13.7Wave Motion 13.7

A wave is the motion of a disturbanceA wave is the motion of a disturbance Mechanical waves requireMechanical waves require

Some source of disturbanceSome source of disturbance A medium that can be disturbedA medium that can be disturbed Some physical connection between or Some physical connection between or

mechanism though which adjacent mechanism though which adjacent portions of the medium influence each portions of the medium influence each otherother

All waves carry energy and All waves carry energy and momentummomentum

Types of Waves -- Types of Waves -- TransverseTransverse

In a transverse wave, each element that In a transverse wave, each element that is disturbed moves perpendicularly to is disturbed moves perpendicularly to the wave motionthe wave motion

Types of Waves -- Types of Waves -- LongitudinalLongitudinal

In a longitudinal wave, the elements of In a longitudinal wave, the elements of the medium undergo displacements the medium undergo displacements parallel to the motion of the waveparallel to the motion of the wave

A longitudinal wave is also called a A longitudinal wave is also called a compression wavecompression wave

Waveform – A Picture of a Waveform – A Picture of a WaveWave

The red curve is a The red curve is a “snapshot” of the “snapshot” of the wave at some wave at some instant in timeinstant in time

The blue curve is The blue curve is later in timelater in time

A is a A is a crestcrest of the of the wavewave

B is a B is a troughtrough of of the wavethe wave

Longitudinal Wave Longitudinal Wave Represented as a Sine Represented as a Sine CurveCurve

A longitudinal wave can also be A longitudinal wave can also be represented as a sine curverepresented as a sine curve

Compressions correspond to crests and Compressions correspond to crests and stretches correspond to troughsstretches correspond to troughs

Description of a Wave Description of a Wave 13.813.8

Amplitude is the Amplitude is the maximum maximum displacement of displacement of string above the string above the equilibrium positionequilibrium position

Wavelength, Wavelength, λ, is the λ, is the distance between distance between two successive two successive points that behave points that behave identicallyidentically

Speed of a WaveSpeed of a Wave

v = ƒ v = ƒ λλ Is derived from the basic speed Is derived from the basic speed

equation of distance/timeequation of distance/time This is a general equation that can This is a general equation that can

be applied to many types of wavesbe applied to many types of waves

Sample problem 13.8Sample problem 13.8 A wave traveling in the A wave traveling in the

positive x-direction is positive x-direction is pictured. Find the pictured. Find the amplitude, wavelength, amplitude, wavelength, speed and period of the speed and period of the wave if it has a frequency of wave if it has a frequency of 8.00 Hz. 8.00 Hz. (0.150 m; 0.400 m; (0.150 m; 0.400 m; 3.20 m/s; 0.125 s)3.20 m/s; 0.125 s)

Speed of a Wave on a String Speed of a Wave on a String 13.913.9

The speed on a wave stretched The speed on a wave stretched under some tension, Funder some tension, F

The speed depends only upon the The speed depends only upon the properties of the medium through properties of the medium through which the disturbance travelswhich the disturbance travels

L

mwhere

Fv

Sample problem 13.10Sample problem 13.10 To what tension must a string with mass 0.0100 kg and To what tension must a string with mass 0.0100 kg and

length 2.50 m be tightened so that waves will travel on length 2.50 m be tightened so that waves will travel on it at a speed of 125 m/s?it at a speed of 125 m/s?

Interference of Waves Interference of Waves 13.1013.10

Two traveling waves can meet and pass Two traveling waves can meet and pass through each other without being through each other without being destroyed or even altereddestroyed or even altered

Waves obey the Waves obey the Superposition PrincipleSuperposition Principle If two or more traveling waves are moving If two or more traveling waves are moving

through a medium, the resulting wave is through a medium, the resulting wave is found by adding together the displacements found by adding together the displacements of the individual waves point by pointof the individual waves point by point

Actually only true for waves with small Actually only true for waves with small amplitudesamplitudes

Constructive InterferenceConstructive Interference

Two waves, a and Two waves, a and b, have the same b, have the same frequency and frequency and amplitudeamplitude Are Are in phasein phase

The combined The combined wave, c, has the wave, c, has the same frequency same frequency and a greater and a greater amplitudeamplitude

Constructive Interference Constructive Interference in a Stringin a String

Two pulses are traveling Two pulses are traveling in opposite directionsin opposite directions

The net displacement The net displacement when they overlap is the when they overlap is the sum of the sum of the displacements of the displacements of the pulsespulses

Note that the pulses are Note that the pulses are unchanged after the unchanged after the interferenceinterference

Destructive InterferenceDestructive Interference

Two waves, a and b, Two waves, a and b, have the same have the same amplitude and amplitude and frequencyfrequency

They are 180° out They are 180° out of phaseof phase

When they When they combine, the combine, the waveforms cancelwaveforms cancel

Destructive Interference in Destructive Interference in a Stringa String

Two pulses are traveling Two pulses are traveling in opposite directionsin opposite directions

The net displacement The net displacement when they overlap the when they overlap the displacements of the displacements of the pulses subtractpulses subtract

Note that the pulses are Note that the pulses are unchanged after the unchanged after the interferenceinterference

Reflection of Waves – Reflection of Waves – Fixed End 13.11Fixed End 13.11

Whenever a traveling Whenever a traveling wave reaches a wave reaches a boundary, some or all boundary, some or all of the wave is of the wave is reflectedreflected

When it is reflected When it is reflected from a fixed end, the from a fixed end, the wave is invertedwave is inverted

Reflected Wave – Free EndReflected Wave – Free End

When a traveling When a traveling wave reaches a wave reaches a boundary, all or boundary, all or part of it is part of it is reflectedreflected

When reflected When reflected from a free end, from a free end, the pulse is not the pulse is not invertedinverted