Chapter 13Chapter 13

Vibrations Vibrations

andand

WavesWaves

Hooke’s LawHooke’s Law

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 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)

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

Amplitude Period and Amplitude Period and FrequencyFrequency Amplitude, AAmplitude, A

The amplitude is the maximum position of the The amplitude is the maximum position of the object relative to the equilibrium positionobject relative to the equilibrium position

Oscillation between ±A on each side of the Oscillation between ±A on each side of the equilibrium positionequilibrium position

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

The frequency, ƒ, is the number of The frequency, ƒ, is the number of complete cycles or vibrations per unit complete cycles or vibrations per unit timetime

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

Elastic Potential EnergyElastic Potential Energy

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

Energy in a Spring Mass Energy in a Spring Mass SystemSystem

elastic potential elastic potential energyenergy PePess = ½kx = ½kx22

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

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 = ±ASpeed is zero at x = ±A The ± indicates the object can be traveling The ± indicates the object can be traveling

in either directionin either direction

22 xAm

kv

Simple Harmonic Motion Simple Harmonic Motion and Uniform Circular and Uniform Circular MotionMotion

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 to mass m attached to a spring of constant k to complete one cycle of its motioncomplete one cycle of its motion

FrequencyFrequency

Units are cycles/second or Hertz, HzUnits are cycles/second or Hertz, Hz The angular frequency is related to the The angular frequency is related to the

frequencyfrequency

k

m2T

m

k

2

1

T

1ƒ

m

kƒ2

Motion as a Function of Motion as a Function of TimeTime

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

x = A cos (2πƒt)x = A cos (2πƒt) 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 of sinusoidal nature of simple harmonic simple harmonic motionmotion

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

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

Simple PendulumSimple Pendulum

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 θF = - m g sin θ

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 θ = θsin θ = 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

Damped OscillationsDamped Oscillations Only ideal systems Only ideal systems

oscillate oscillate indefinitelyindefinitely

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

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

Wave MotionWave Motion

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

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 WaveDescription of a Wave Amplitude is the Amplitude is the

maximum maximum displacement of string displacement of string above the equilibrium above the equilibrium positionposition

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

v = ƒ λ v = ƒ λ (for all types of waves)(for all types of waves)

Speed of a Wave on a Speed of a Wave on a StringString

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

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 EndFixed End

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