Vibrations and Waves Chapter 12. 12.1 – Simple Harmonic Motion.

Vibrations and WavesVibrations and Waves

Chapter 12Chapter 12

12.1 – Simple Harmonic 12.1 – Simple Harmonic MotionMotion

Remember…Remember…

Elastic Potential Energy (PEElastic Potential Energy (PEee)) is the is the

energy stored in a stretched or energy stored in a stretched or compressed elastic objectcompressed elastic object

Gravitational Potential Energy (PEGravitational Potential Energy (PEgg)) is is

the energy associated with an object due the energy associated with an object due to it’s position relative to Earthto it’s position relative to Earth

Useful DefinitionsUseful Definitions

Periodic MotionPeriodic Motion – A repeated motion. If it is – A repeated motion. If it is back and forth over the same path, it is called back and forth over the same path, it is called simple harmonic motion. simple harmonic motion. – Examples: Wrecking ball, pendulum of clockExamples: Wrecking ball, pendulum of clock

Simple Harmonic MotionSimple Harmonic Motion – Vibration about an – Vibration about an equilibrium position in which a restoring force is equilibrium position in which a restoring force is proportional to the displacement from equilibriumproportional to the displacement from equilibriumhttp://http://www.ngsir.netfirms.com/englishhtm/SpringSHM.www.ngsir.netfirms.com/englishhtm/SpringSHM.htmhtm

Useful DefinitionsUseful Definitions

A A spring constant (k)spring constant (k) is a measure of is a measure of how resistant a spring is to being how resistant a spring is to being compressed or stretched.compressed or stretched.

(k) is always a positive number(k) is always a positive number

The The displacement (x)displacement (x) is the distance from is the distance from equilibrium.equilibrium.

(x) can be positive or negative. In a spring-mass (x) can be positive or negative. In a spring-mass system, positive force means a negative system, positive force means a negative displacement, and negative force means a positive displacement, and negative force means a positive displacement. displacement.

Hooke’s LawHooke’s Law

Hooke’s LawHooke’s Law – for small displacements – for small displacements from equilibrium:from equilibrium:

FFelasticelastic = -(kx) = -(kx)

Spring force = -(spring constant x displacement)Spring force = -(spring constant x displacement)

This means a stretched or compressed This means a stretched or compressed spring has elastic potential energy.spring has elastic potential energy.

Example: Bow and ArrowExample: Bow and Arrow

Example ProblemExample Problem

If a mass of 0.55 kg attached to a vertical If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its spring stretches the spring 2.0 cm from its original equilibrium position, what is the original equilibrium position, what is the spring constant?spring constant?

Example AnswerExample Answer

Given: m = 0.55 kgGiven: m = 0.55 kg x = -0.020mx = -0.020mg = 9.81g = 9.81 k = ?k = ?

Fg = mg = 0.55 kg x 9.81 = 5.40 NFg = mg = 0.55 kg x 9.81 = 5.40 N

Hooke’s Law: F = -kxHooke’s Law: F = -kx

5.40 N = -k(-0.020m)5.40 N = -k(-0.020m) k = 270 N/mk = 270 N/m

12.2 – Measuring simple 12.2 – Measuring simple harmonic motionharmonic motion

Useful DefinitionsUseful Definitions

AmplitudeAmplitude – the maximum angular – the maximum angular displacement from equilibrium.displacement from equilibrium.

PeriodPeriod – the time it takes to execute a – the time it takes to execute a complete cycle of motioncomplete cycle of motion– Symbol = TSymbol = T SI Unit = second (s)SI Unit = second (s)

FrequencyFrequency – the number of cycles or – the number of cycles or vibrations per unit of timevibrations per unit of time– Symbol = fSymbol = f SI Unit = hertz (Hz)SI Unit = hertz (Hz)

Formulas - PendulumsFormulas - Pendulums

T = 1/fT = 1/f or or f = 1/Tf = 1/T

The period of a pendulum depends on the The period of a pendulum depends on the string length and free-fall acceleration (g)string length and free-fall acceleration (g)

T = 2T = 2π√π√(L/g)(L/g)Period = 2Period = 2ππ x square root of (length divided by free-fall acceleration) x square root of (length divided by free-fall acceleration)

Formulas – Mass-spring systemsFormulas – Mass-spring systems

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

A heavier mass has a greater period, thus A heavier mass has a greater period, thus as mass increases, the period of vibration as mass increases, the period of vibration increases.increases.

T = 2T = 2π√π√(m/k)(m/k)Period = Period = 22ππ x the square root of (mass divided by spring constant) x the square root of (mass divided by spring constant)

Example Problem- PendulumExample Problem- Pendulum

You need to know the height of a tower, You need to know the height of a tower, but darkness obscures the ceiling. You but darkness obscures the ceiling. You note that a pendulum extending from the note that a pendulum extending from the ceiling almost touches the floor and that its ceiling almost touches the floor and that its period is 12s. How tall is the tower?period is 12s. How tall is the tower?

Example AnswerExample Answer

Given: Given: T = 12 sT = 12 s g = 9.81g = 9.81 L = ?L = ?

T = 2T = 2π√π√(L/g)(L/g)

12 = 2 12 = 2 π√π√(L/9.81)(L/9.81)

144 = 4144 = 4ππ22L/9.81L/9.81

1412.64 = 41412.64 = 4ππ22LL

35.8 m = L35.8 m = L

Example Problem- Mass-SpringExample Problem- Mass-Spring

The body of a 1275 kg car is supported in The body of a 1275 kg car is supported in a frame by four springs. Two people riding a frame by four springs. Two people riding in the car have a combined mass of 153 in the car have a combined mass of 153 kg. When driven over a pothole in the kg. When driven over a pothole in the road, the frame vibrates with a period of road, the frame vibrates with a period of 0.840 s. For the first few seconds, the 0.840 s. For the first few seconds, the vibration approximates simple harmonic vibration approximates simple harmonic motion. Find the spring constant of a motion. Find the spring constant of a single spring. single spring.

Example answerExample answer

Total mass of car + people = 1428 kgTotal mass of car + people = 1428 kg

Mass on 1 tire: 1428 kg/4= 357 kgMass on 1 tire: 1428 kg/4= 357 kg

T= 0.840 sT= 0.840 s

T = 2T = 2π√π√(m/k) (m/k)

K=(4K=(4ππ22m)/Tm)/T22

K= (4K= (4ππ22(357 kg))/(0.840 s)(357 kg))/(0.840 s)22

k= 2.00*10k= 2.00*104 4 N/mN/m

12.3 – Properties of Waves12.3 – Properties of Waves

Useful DefinitionsUseful Definitions

Crest:Crest: the highest the highest point above the point above the equilibrium positionequilibrium position

Trough:Trough: the lowest the lowest point below the point below the equilibrium positionequilibrium position

Wavelength Wavelength λλ : the : the distance between two distance between two adjacent similar adjacent similar points of the wavepoints of the wave

Wave MotionWave Motion

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

Medium:Medium: the material through which a the material through which a disturbance travelsdisturbance travels

Mechanical waves:Mechanical waves: a wave that requires a wave that requires a medium to travel througha medium to travel through

Electromagnetic waves:Electromagnetic waves: do not require a do not require a medium to travel through medium to travel through

Wave TypesWave TypesPulse wave:Pulse wave: a single, non-periodic a single, non-periodic

disturbancedisturbance

Periodic wave:Periodic wave: a wave whose source is some a wave whose source is some form of periodic motionform of periodic motion– When the periodic motion is simple harmonic motion, When the periodic motion is simple harmonic motion,

then the wave is a SINE WAVE (a type of periodic then the wave is a SINE WAVE (a type of periodic wave)wave)

Transverse wave:Transverse wave: a wave whose particles a wave whose particles vibrate perpendicularly to the direction of wave vibrate perpendicularly to the direction of wave motionmotionLongitudinal wave:Longitudinal wave: a wave whose particles a wave whose particles vibrate parallel to the direction of wave motion vibrate parallel to the direction of wave motion

Transverse WaveTransverse Wave

Longitudinal WaveLongitudinal Wave

Speed of a WaveSpeed of a Wave

Speed of a wave= frequency x wavelengthSpeed of a wave= frequency x wavelength

v = v = ffλλ

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

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

v = v = ffλλ

343 m/s = (264 Hz)(343 m/s = (264 Hz)(λλ))

1.30 m = 1.30 m = λλ

The Nature of Waves VideoThe Nature of Waves Video 2:20 2:20

12.4 – Wave Interactions12.4 – Wave Interactions

Constructive vs Destructive Constructive vs Destructive InterferenceInterference

Constructive Interference:Constructive Interference: individual displacements on individual displacements on the same side of the the same side of the equilibrium position are equilibrium position are added together to form the added together to form the resultant waveresultant wave

Destructive Interference:Destructive Interference: individual displacements on individual displacements on the opposite sides of the the opposite sides of the equilibrium position are equilibrium position are added together to form the added together to form the resultant waveresultant wave

Wave Interference DemoWave Interference Demo

When Waves Reach a Boundary…When Waves Reach a Boundary…

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

Standing WavesStanding Waves

Standing wave:Standing wave: a wave pattern that results a wave pattern that results when two waves of the same frequency, when two waves of the same frequency, wavelength, and amplitude travel in opposite wavelength, and amplitude travel in opposite directions and interferedirections and interfereNode:Node: a point in a standing wave that always a point in a standing wave that always undergoes complete destructive interference undergoes complete destructive interference and therefore is stationaryand therefore is stationaryAntinode:Antinode: a point in a standing wave, halfway a point in a standing wave, halfway between two nodes, at which the largest between two nodes, at which the largest amplitude occursamplitude occurs

Ruben's Tube VideoRuben's Tube Video 1:57 1:57