Vibrations and WavesVibrations and Waves

Chapter 11Chapter 11

Simple Harmonic MotionSimple Harmonic Motion

Chapter 11 Section 1Chapter 11 Section 1

Periodic MotionPeriodic Motion

Any repetitive, or cyclical, types of Any repetitive, or cyclical, types of motionmotion– Examples?Examples?

Simple Harmonic MotionSimple Harmonic Motion (SHM) (SHM) is a is a specialized form of periodic motionspecialized form of periodic motion

Simple Harmonic MotionSimple Harmonic Motion

Periodic vibration about an Periodic vibration about an equilibriumequilibrium position position

Restoring forceRestoring force must be must be

–proportional to displacement proportional to displacement from equilibriumfrom equilibrium

–in the direction of equilibriumin the direction of equilibrium

Simple Harmonic MotionSimple Harmonic Motion

Common examples include:Common examples include:– mass-spring systemmass-spring system

– pendulum for small anglespendulum for small angles

Mass on a SpringMass on a SpringWhen a spring is When a spring is

stretched, the restoringstretched, the restoring

force from the tension inforce from the tension in

The spring is describedThe spring is described

by Hooke’s Law…by Hooke’s Law…

F = kxF = kx

The force acting on the mass is proportional The force acting on the mass is proportional to its displacement from equilibrium and in a to its displacement from equilibrium and in a direction towards equilibrium, thus SHMdirection towards equilibrium, thus SHM

The PendulumThe Pendulum

A A simple pendulumsimple pendulum consists of a mass consists of a mass called a bob, which is attached to a fixed called a bob, which is attached to a fixed string. Effectively, all the mass is in the string. Effectively, all the mass is in the bob.bob.

The The x component of theof the

weight (weight (Fg sin Fg sin )) is the is the

restoring force.restoring force.

The PendulumThe PendulumThe magnitude of the restoring force (The magnitude of the restoring force (FFg g sin sin

)) is proportional to sin is proportional to sin ..

When the angle of displacement When the angle of displacement is relatively is relatively small, sin small, sin is approximately equal to is approximately equal to in in radians… sin 0 = 0radians… sin 0 = 0

So, for small angles, So, for small angles, the restoring force is the restoring force is very nearly proportional to the very nearly proportional to the displacementdisplacement, and the pendulum’s motion is , and the pendulum’s motion is an excellent approximation ofan excellent approximation of simple simple harmonic motionharmonic motion..

Virtual Simple Harmonic MotionVirtual Simple Harmonic Motion

http://phet.colorado.edu/simulations/sims.php?sim=Pendulum_Lab

http://phet.colorado.edu/simulations/sims.php?sim=Masses_and_Springs

Measuring Simple Harmonic Measuring Simple Harmonic MotionMotion

Chapter 11 Section 2Chapter 11 Section 2

AmplitudeAmplitude

The maximum displacement The maximum displacement from equilibrium.from equilibrium.

PeriodPeriod

The time it takes for one complete cycle of The time it takes for one complete cycle of motion.motion.

Represented by the symbol TRepresented by the symbol T

Unit of secondsUnit of seconds

FrequencyFrequency

The number of cycles completed in a unit The number of cycles completed in a unit of time (usually seconds)of time (usually seconds)

Represented by the symbol fRepresented by the symbol f

Unit of sUnit of s-1 -1 (also known as Hertz)(also known as Hertz)

Period and FrequencyPeriod and Frequency

Period and frequency are inversely Period and frequency are inversely related.related.

f = 1/T and T = 1/ff = 1/T and T = 1/f

A mass-spring system vibrates A mass-spring system vibrates exactly 10 times each second. exactly 10 times each second.

What is its period and frequency?What is its period and frequency?

f = 10 cycles per secondf = 10 cycles per second

= = 10 Hz10 Hz

T = 1/f = 1/10 sT = 1/f = 1/10 s

= = 0.1 s0.1 s

Factors Affecting PendulumsFactors Affecting PendulumsFor small amplitudes, the period of a pendulum For small amplitudes, the period of a pendulum does not depend on the mass or amplitude.does not depend on the mass or amplitude.

Length and acceleration due to gravity do affect Length and acceleration due to gravity do affect the period of a pendulum. the period of a pendulum.

Factors Affecting Mass-Spring Factors Affecting Mass-Spring SystemsSystems

The heavier the mass, the longer the period The heavier the mass, the longer the period (more inertia)(more inertia)

The stiffer the spring, the less time it will take to The stiffer the spring, the less time it will take to complete one cycle.complete one cycle.

Properties of WavesProperties of Waves

Chapter 11 Section 3Chapter 11 Section 3

What is a wave?What is a wave?

A wave is an means by which energy is A wave is an means by which energy is transferred from one place to another via transferred from one place to another via periodic disturbancesperiodic disturbances

Some general terminology…Some general terminology…

PulsePulse – a single disturbance, single cycle – a single disturbance, single cycle

Periodic wavePeriodic wave – continuous, repeated – continuous, repeated disturbancesdisturbances

Sine waveSine wave – a wave whose source vibrates with – a wave whose source vibrates with simple harmonic motionsimple harmonic motion

Medium Medium – whatever the– whatever the

wave is traveling throughwave is traveling through

Mechanical WavesMechanical WavesWaves that require a physical medium to travel Waves that require a physical medium to travel through.through.– Examples: sound, disturbance in a slinkyExamples: sound, disturbance in a slinky

Examples of physical Examples of physical mediamedia are water, air, are water, air, string, slinky.string, slinky.

Electromagnetic wavesElectromagnetic waves

Waves that do not require a physical medium.Waves that do not require a physical medium.

Comprised of oscillating electric and magnetic Comprised of oscillating electric and magnetic fieldsfields

Examples include x-rays, visible light, radio Examples include x-rays, visible light, radio waves, etc.waves, etc.

Transverse WavesTransverse WavesParticles of the medium move perpendicular to Particles of the medium move perpendicular to the direction of energy transferthe direction of energy transfer

You should be able to identify You should be able to identify crestscrests, , troughstroughs, , wavelengthwavelength (distance traveled during one full (distance traveled during one full cycle), and cycle), and amplitudeamplitude

Crest

Trough

Longitudinal WavesLongitudinal Waves

Particles of the medium move parallel to the Particles of the medium move parallel to the direction of energy transfer (slinky demo)direction of energy transfer (slinky demo)

Be able to Identify Be able to Identify compressionscompressions, , rarefactionsrarefactions, , wavelengthswavelengths

Compressions Rarefactions

Waves transfer energyWaves transfer energy

Note that, while energy is transferred from point A Note that, while energy is transferred from point A to point B, the particles in the medium do not to point B, the particles in the medium do not move from A to B. move from A to B. – Individual particles of the medium merely Individual particles of the medium merely

vibrate back and forth in simple harmonic vibrate back and forth in simple harmonic motionmotion

The rate of energy transfer is proportional to The rate of energy transfer is proportional to the square of the amplitudethe square of the amplitude– When amplitude is doubled, the energy carried When amplitude is doubled, the energy carried

increases by a factor of 4. increases by a factor of 4.

Wave speedWave speedWave speed is determined completely by the characteristics of the Wave speed is determined completely by the characteristics of the mediummedium– For an unchanging medium, wave speed is constantFor an unchanging medium, wave speed is constant

The speed of a wave can be calculated by multiplying wavelength by frequency.The speed of a wave can be calculated by multiplying wavelength by frequency.

v = f x λ

Practice #1Practice #1

Q: Microwaves travel at the speed of light, Q: Microwaves travel at the speed of light, 3.003.00101088 m/s. When the frequency of m/s. When the frequency of microwaves is 9.00 microwaves is 9.00 101099 Hz, what is their Hz, what is their wavelength?wavelength?

A: 0.0300 mA: 0.0300 m

Practice #2Practice #2

Q: The piano string tuned to middle C Q: The piano string tuned to middle C vibrates with a frequency of 264 Hz. vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound m/s, find the wavelength of the sound waves produced by the string.waves produced by the string.

A: 1.30 mA: 1.30 m

11.3 Problems11.3 Problems

Page 387 1-4Page 387 1-4

Wave InteractionsWave Interactions

Chapter 11 Section 4Chapter 11 Section 4

5 behaviors common to all waves:5 behaviors common to all waves:

1.1. ReflectionReflection

2.2. InterferenceInterference

3.3. Rectilinear PropagationRectilinear Propagation

4.4. RefractionRefraction

5.5. DiffractionDiffraction

1. Reflection1. Reflection

The bouncing of a wave when it The bouncing of a wave when it encounters the boundary between two encounters the boundary between two different mediadifferent media

Fixed End ReflectionFixed End ReflectionAt a fixed boundary, waves are inverted as they At a fixed boundary, waves are inverted as they are reflected.are reflected.

Free End ReflectionFree End Reflection

At a free boundary, waves are reflected on the At a free boundary, waves are reflected on the same side of equilibriumsame side of equilibrium

2. Interference2. Interference

The combination of two or more waves in The combination of two or more waves in a medium at the same time.a medium at the same time.– Physical matter cannot occupy the same Physical matter cannot occupy the same

space at the same time, but energy can.space at the same time, but energy can.

The The Superposition Principle Superposition Principle describes describes what happens when waves interfere…what happens when waves interfere…– Waves (energy) pass through each other Waves (energy) pass through each other

completely unaffectedcompletely unaffected– The medium will be displaced an amount The medium will be displaced an amount

equal to the vector sum of what the waves equal to the vector sum of what the waves would have done individuallywould have done individually

Constructive InterferenceConstructive InterferencePulses on the same side Pulses on the same side of equilibrium.of equilibrium.

Waves meet, combine Waves meet, combine according to the according to the superposition principle, superposition principle, and pass through and pass through unchanged.unchanged.

Displacement of Displacement of medium greater medium greater than originalsthan originals

Destructive InterferenceDestructive Interferencepulses on opposite sides of pulses on opposite sides of equilibrium.equilibrium.

Waves meet, combine Waves meet, combine according to the according to the superposition principle, and superposition principle, and pass through unchanged.pass through unchanged.

Displacement of Displacement of medium less than medium less than at least one originalat least one original

Complete Destructive InterferenceComplete Destructive Interference

Interference patternsInterference patterns

Interference Interference patterns result patterns result from continuous from continuous interference.interference.

http://phet.colorado.edu/en/simulation/wave-interference

Standing WavesStanding Waves

An interference pattern that results when two An interference pattern that results when two waves of the same frequency, wavelength, and waves of the same frequency, wavelength, and amplitude travel in opposite directions and amplitude travel in opposite directions and interfere.interfere.

Standing wave partsStanding wave parts

NodeNode – point that maintains zero displacement, – point that maintains zero displacement, complete destructive interferencecomplete destructive interference

AntinodeAntinode – point at which largest displacement – point at which largest displacement occurs, constructive interferenceoccurs, constructive interference

Standing wavesStanding waves

Only specific Only specific frequency-wavelength frequency-wavelength combinations will combinations will produce standing produce standing wave patterns in a wave patterns in a given medium. given medium.

If a string is 4.0 m long, what are If a string is 4.0 m long, what are three wavelengths that will produce three wavelengths that will produce

standing waves on this string?standing waves on this string?

3. Rectilinear Propagation3. Rectilinear PropagationWaves travel in straight linesWaves travel in straight lines

The direction of travel is perpendicular to The direction of travel is perpendicular to the the wavefrontwavefront. .

WavefrontWavefront - The set of points in space - The set of points in space reached by a wave at the same instant as reached by a wave at the same instant as the wave travels through a medium. the wave travels through a medium.

Direction of a single wave

Direction of a single wave

Parallel Wavefronts: Circular Wavefronts:

4. Refraction4. Refraction

The bending of the path of a wave as it The bending of the path of a wave as it enters a new medium of different wave enters a new medium of different wave speed.speed.

5. Diffraction5. Diffraction

The spreading of wave energy around the The spreading of wave energy around the edges of barriers and obstaclesedges of barriers and obstacles