Vibration and Waves

AP PhysicsChapter 11

11.1 Simple Harmonic Motion

Vibration and Waves

11.1 Simple Harmonic Motion

Periodic motion – when an object vibrates over the same pathway, with each vibration taking the same amount of time

Equilibrium position – the position of the mass when no force is exerted on it

11.1

11.1 Simple Harmonic Motion

If the spring is stretched from equilibrium, a force acts so the object is pushed back toward equilibriumRestoring ForceProportional to the displacement (x)

Called Hooke’s Law

11.1

F kx

11.1 Simple Harmonic Motion

Any vibrating system for which the restoring force is directly proportional to the negative of the displacement (F = -kx) exhibits Simple Harmonic Motion (SHM)

11.1 Simple Harmonic Motion

Amplitude (A) – maximum distance from equilibrium

Period (T) – time for one complete cycleFrequency (f) – number of vibrations per second

11.1

Amplitude

1fT

11.1 Simple Harmonic Motion

A vertical spring follows the same patternThe equilibrium positions is just shifted by gravity

11.1

11.2 Energy in a Simple Harmonic Oscillator

Vibration and Waves

11.2 Energy in a Simple Harmonic Oscillator

Review – energy of a spring

So the total mechanical energy of a spring (assuming no energy loss) is

At maximum amplitude then

So E is proportional to the square of the amplitude

11.2

221 kxU s

2212

21 kxmvE

2212

21 )0( kAmE 2

21 kAE

11.2 Energy in a Simple Harmonic Oscillator

Velocity as a function of position

Since the maximum velocity is when A=0

Factor the top equation for A2, then combine with the bottom equationTake square root

11.2

2212

21 kxmvE

2212

212

21 kxmvkA 222 kxmvkA 222 xvA k

m )( 222 xAv mk

2212

21 )0(kmvE 2

max21 mvE 2

max212

21 mvkA 2

max2 mvkA 2

max2 vAm

k

)1( 2

222Ax

mk Av

)1( 2

22max

2Axvv 2

21max Axvv

11.3 The Period and Sinusoidal Nature of SHM

Vibration and Waves

11.3 The Period and Sinusoidal Nature of SHM

The Period of an object undergoing SHM is independent of the amplitudeImagine an object traveling in a circular pathway If we look at the motion in just the x axis, the motion is analogous to SHM

11.3

11.3 The Period and Sinusoidal Nature of SHM

As the ball moves the displacement in the x changesThe radius is the amplitudeThe velocity is tangent to the circleNow looking at ComponentsIf we put the angles into the triangle, we can see similar triangles

11.3

A

Vmax

V22 xA

q

q

11.3 The Period and Sinusoidal Nature of SHM

So the opposite/hypotenuse is a constant

This can be rewritten

This is the same as the equation for velocity of an object in SHM

11.3

A

Vmax

V22 xA

q

q

AxA

vv 22

max

2

2

max 1 Axvv

11.3 The Period and Sinusoidal Nature of SHM

The period would be the time for one complete revolution

The radius is the same as the amplitude, and the time for one revolution is the period

11.3

txv trv 2

TAv 2

max max

2vAT

11.3 The Period and Sinusoidal Nature of SHM

Using the previous relationship between maximum velocity and amplitude

Substitute in the top equation

11.3

txv trv 2

max

2vAT

2max2

1221 mvkA 2

max2 mvkA k

mvA 2max

2

km

vA max

kmT 2

11.3 The Period and Sinusoidal Nature of SHM

Position as a function of timex displacement is

You don’t know this, but

Where is the frequencySo

11.3

A

q

qcosAx

ftq 2

)2cos( ftAx

11.4 The Simple Pendulum

Vibration and Waves

11.4 The Simple Pendulum

Simple Pendulum – mass suspended from a cordCord is massless (or very small)Mass is concentrated in small volume

11.4

11.4 The Simple Pendulum

Looking at a diagram of a pendulumTwo forces act on the it1. Weight2. TensionThe motion of the bob is at a tangent to the arc

11.4

W

T

11.4 The Simple Pendulum

The displacement of the bob is given by xUsing the triangleFor a complete circle (360o)

Then for our arc it would be

11.4

W

T

x

q L

Ly2x r

2360

x r q2360rx q 360

2x

Lq

11.4 The Simple Pendulum

The component of force in the direction of motion isBut at small angleAnd

So

11.4

W

T

x

q L

Ly

mgsinq

sinq qsinmg mgq qF mgq

3602

xL

q

3602

F mg xL

F kx

11.4 The Simple Pendulum

Usual standard is below 30o

We can then take the equation and reason that 360o=2 rad (we didn’t study angular motion, so take my word for it)

11.4

3602

F mg xL

mgF xL

mg kL

11.4 The Simple Pendulum

We can now combine the equation for period

Mass does not appear in this equationThe period is independent of mass

11.4

kmT 22 mgL

mT 2 LTg

11.5 Damped Harmonic Motion

Vibration and Waves

11.5 Damped Harmonic Motion

The amplitude of a real oscillating object will decrease with time – called damping

Underdamped – takes several swing before coming to rest (above)

11.5

11.5 Damped Harmonic Motion

Overdamped – takes a long time to reach equilibrium

Critical damping – equalibrium reached in the shortest time

11.5

11.6 Forced Vibrations; Resonance

Vibration and Waves

11.6 Forced Vibrations; Resonance

Natural Frequency – depends on the variables (m,k or L,g) of the object

Forced Vibrations – caused by an external force

11.6

11.6 Forced Vibrations; Resonance

Resonant Frequency – the natural vibrating frequency of a system

Resonance – when the external frequency is near the natural frequency and damping is small

11.6

Tacoma Narrow Bridge

11.7 Wave Motion

Vibration and Waves

11.7 Wave Motion

Mechanical Waves – travels through a medium

The wave travels through the medium, but the medium undergoes simple harmonic motion

Wave motionParticle motion

11.7

11.7 Wave Motion

Waves transfer energy, not particlesA single bump of a wave is called a pulse

A wave is formed when a force is applied to one end

Each successive particle is moved by the one next to it

11.7

Tsunami

11.7 Wave Motion

Parts of a waveTransverse wave – particle motion perpenduclar to wave motionWavelength (l) measured in metersFrequency (f) measured in Hertz (Hz)Wave Velocity (v) meters/second

11.7

v f l

11.7 Wave Motion

Longitudinal (Compressional) Wave Particles move parallel to the direction of wave motionRarefaction – where particles are spread outCompression – particles are close

11.7

11.7 Wave Motion

Earthquakes S wave – Transverse P wave – LongitudinalSurface Waves – can travel along the

boundary

Notice the circular motion of the particles11.7

11.9 Energy Transported by Waves

Vibration and Waves

11.9 Energy Transported by Waves

Energy for a particle undergoing simple harmonic motion is

Intensity (I) power across a unit area perpendicular to the

direction of energy flowSo

11.9

221 kAE

22 1

21 2

I rI r

24PIr

11.11 Reflection and Transmission of Waves

Vibration and Waves

11.11 Reflection and Transmission of Waves

When a wave comes to a boundary (change in medium) at least some of the wave is reflected

The type of reflection depends on if the boundary is fixed (hard) - inverted

11.11

11.11 Reflection and Transmission of Waves

When a wave comes to a boundary (change in medium) at least some of the wave is reflected

Or movable (soft) – in phase

11.11

11.11 Reflection and Transmission of Waves

For two or three dimensional we think in terms of wave fronts

A line drawn perpendicular to the wave front is called a ray

When the waves get far from their source and are nearly straight, they are called plane waves

11.11

11.11 Reflection and Transmission of Waves

Law of Reflection – the angle of reflection equals the angle of incidence

Angles are always measured from the normal

11.11

i rq q

11.12 Interference; Principle of Superposition

Vibration and Waves

11.12 Interference; Principle of Superposition

Interference – two waves pass through the same region of space at the same time

The waves pass through each otherPrinciple of Superposition – at the point where

the waves meet the displacement of the medium is the algebraic sum of their separate displacements

11.12

11.12 Interference; Principle of Superposition

Phase – relative position of the wave crestsIf the two waves are “in phase” Constructive interference

If the two waves are “out of phase”Destructive Interference

11.12

11.12 Interference; Principle of Superposition

For a wave (instead of a single phase)Interference is calculated by adding amplitudeIn real time this looks like

11.12

11.13 Standing Waves; Resonance

Vibration and Waves

11.13 Standing Waves; Resonance

In a specific case of interference a standing wave is produced

The areas with complete constructive interference are called loops or antinodes (AN)

The areas with complete destructive interference are called nodes (N)

11.13

11.13 Standing Waves; Resonance

Standing waves occur at the natural or resonant frequency of the medium

In this case, called the first harmonic, the wavelength is twice the length of the medium

The frequency of is called the fundamental frequency

11.13

L21 l

11.13 Standing Waves; Resonance

The second harmonic is the next standing wave formed

Then the third harmonic would be

11.13

L2l

L323 l

11.13 Standing Waves; Resonance

The basic form for the wavelength of harmonics isEach resonant frequency, is an integer

multiple of the fundamental frequencyOvertone – all the frequencies above the

fundamentalThe first overtone is the second harmonic

11.13

Lnn2l

Harmonic Applet