Unit10HarmonicMotionandWaves - WordPress

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Name:Unit 10: 1Harmonic MotionPeriod:Harmonic Motion ismotion that repeats itself,oscillating back and forth.Eventually it will lose energy (called dampening)and come to rest in themiddle, known as itsequilibrium position.A bird flyingis notharmonicmotion: oneforce pullsup and adifferent force pulls down.Also, each force pulls fromthe ends not the middle.To be harmonic motion there must be arestoring force that tries to return anobject to its equilibrium position.EquilibriumWhen a pendulum is disturbedposition(moved), gravity pulls down torestore the pendulum back to the center.Because of momentum, it goes past thecenter to the other side and back again.A pendulumHarmonic Motion BasicsCycle: the repeated part of the motion; mustinclude all of the steps of the motion.endstartAPeriod (T in sec): length of time for one cycle;how long it takes for one repetition. A slowerobject has a bigger (longer) period .B0:01.0Frequency (f in Hz): number of cycles persecond. Motion that repeats more often is morefrequent and has a higher frequency.Period and Frequency are inversely related.Period(in secs)T 1fORf 1TPeriod(in secs)Frequency(in hertz)From C to A isthe second halfof the cycle.From A to Cis only halfa cycle.ACThe period (T) is thetime from A backto A. T 2 sec.BEx: A wheel has a period of2 seconds. Find its frequency.T 1/fAmplitude ½(distance side-to-side)Harmonic Motion Graphs40º20ºAmplitude 20 oMore energyAmplitude 10 oLess energyAmplitude never affects periodor frequency! A pendulum withmore amplitude moves fast, buttravels a long distance. Apendulum with less amplitudemoves slow, but only travels asmall distance. Either way, theperiod is the same.Imagine a pen attached to the bottom of a pendulum. If a piece of paper is movedbeneath the pendulum as it swings, a harmonic motion graph is drawn.Cycle—from any point on the line to that samepoint going the same way. This graph shows2 complete cycles.Period 1 cyclePosition vs.Time– 0.75) (1.75 1 sec4Position (cm)Period—measure the time for one cycle betweenany two identical points on the graph (top-totop, bottom-to-bottom, etc.).Amplitude ½(side-to-side) ½(3 3) ½(6)Frequency—count the number of cycles in 3 cm1 second OR find the period and use f 1/T.32Start10-1AAEnd of2nd cycleEnd of1st cycle-2-321.751.51.251Legal copying of this worksheet requires written permission.0.750.50.250Amplitude—measure the total distance fromside-to-side (or top-to-bottom) and divide by two-4OR measure the distance from the equilibriumposition (halfway between the peaks) to oneof the peaks.End of 1st cycle period (T) 1 sec Time (sec)1 cycle in 1 sec frequency (f) 1 Hzcstephenmurray.comf 1/Tf 1/2f 0.5 HzT 2 secf T 1/4T 0.25 secAs period increases, the frequency decreases.As period decreases, the frequency increases.Amplitude (A in m, cm, or degrees): maximum distance or angle from the equilibrium (center) position.Wider swing more energy more amplitude.Only half of the cycle occurs in the first second, sothe frequency is ½ cycle persecond. f 0.5 Hz.0:02.0Ex: A pendulum has a frequencyof 4 Hz. Find its period.f 4 HzT CEquilibriumPosition(halfwaybetweenpeaks)Copyright 2014, C. Stephen Murray

Name:Unit 10: 1Period:Harmonic Motion: Yes or No?1. PeriodA. The number of cycles per second.Pendulum:A bouncing ball:2. EquilibriumpositionB. A unit of one cycle per second.Ocean waves:A ruler pulled from one sideand released:3. AmplitudeD. Time it takes to complete one cycle.4. DampingE. A part of motion that repeats over andover with a set series of events.A child on a swing:Jumping Jacks:Bouncing spring:A person jumping up anddown:A spinning ball:Period, Frequency, or Amplitude?Doesn’t change period.More of this means more energy.Increases as a pendulum swings back and forth faster.Measured in cycles per second.Measured in meters or centimeters.This decreases with a smaller swing.If the frequency increases, this decreases.Measured in Hertz.Measured in seconds.If it swings back and forth slower, this decreases.As it dampens, this decreases.A moving springC. The size or strength of a cycle.5. Frequency6. CycleF. Halfway between the two sides andwhere the motion comes to rest.7. HertzG. The motion dying out over time.Where is the equilibrium position forthis pendulum?If the pendulum starts at C going tothe right, where does 1 cycle end?From letter to letter wouldbe the amplitude.If the pendulum starts at A, how many times does it pass pointC in 1 cycle?An spring has a period of 4 seconds. What is its frequency?Where is its equilibrium position?A pendulum has a frequency of 3 Hz. What is its period?If the spring starts at position A, howmuch of a cycle does it completefrom A to C?A.B.If the spring moves 10 cm from Cto A (side to side), how big is it’samplitude?C.A pendulum takes 10 seconds to complete 2 cycles.A) What is its period?B) What is its frequency?10 cmPosition vs. Time8AEIM6)m4c( 2nBFDHJL0ioits -2oP -4-6CGK-8 0 0 0 0 1 1 1 1 2 2 2 2 3.2 .5 .7.2 .5 .7.2 .5 .7555555Position vs. Time54)3mc(2no1tiis0oP-1-2-3 00.511.522.5Tim e (sec)1 cycle after A is ;2 cycles after D is .1/2 cycle after G is ;1/4 cycle before M is .# of complete cycles shown is .Period (T) Frequency (f) Equilibrium position Amplitude (A) cstephenmurray.com33 4 4.5.5Tim e (sec)55.566.577.5Mark 1 cycle of the harmonic motion.Starting at 1.5 secs, when does the 2nd cycle end:Number of cycles shown is .Period (T) Frequency (f) Equilibrium position Amplitude (A) Legal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Name:Unit 10:Period:cstephenmurray.comLegal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Name:Unit 10: 2Period:WavesBall 1Individual water molecules(like the ball) move upand down, butdo not moveforward.Waves are harmonic motion that movesthru a medium (matter). Water is the medium for water waves; a slinky is the medium when you shake a slinky. The particles in the medium vibrate, but do notmove. Only the energy of the wavemoves, transferring the energy. This iswhy waves seem to go thru things, likesound moving thru air.Types of WavesEnergyEnergyWhen ball 1 is moved, waves transfer theenergy thru the water to ball 2. Ball 2 willvibrate with the same frequency as ball 1.Cell phones and radios work the same way:microwaves moving thru the air.Only the wave’s energymoves forward.There are two simple forms of waves: transverse (across) and longitudinal (the long way).Longitudinal (compression) wavesvibrate parallel to (same direction as)the direction of motion. Sound is alongitudinal wave: a speaker vibratesin and out pushing the sound forward.Only longitudinal waves can movethru fluids (liquids and gases).vibrationBall 2FappliedTransverse waves vibrate perpendicular(90º) to the direction of motion. Becausethe energy moves forward while the vibration is up and down, water waves look liketransverse waves, but are actually surfacewaves, which occur between materials (airand water).vibrationmotionEarthquakes (seismic waves)Earthquakes are made up of both typesof waves. Longitudinal waves are thefastest and hit first, so they are calledprimary waves (P waves). Transverse(T) waves are slower, but do moredamage because the up and downbreak thing by shearing (cutting), soare called S-waves. Only the P wavestravel thru the earth’s center, whichproves the earth has a liquid center.motionWavelength (λ)crest1 wavelengthAmplitude does not affectwavelength, just like amplitudedoesn’t affect period andfrequency. Likewise, in theocean bigger waves (greateramplitude) don’t overtake(catch) smaller waves.λThe wavelength—λ [lambda] (in m)is the length of one wave between anytwo identical points on the wave (crestto-crest or trough-to-trough, etc).Amplitude2Aλtrough1 wavelengthWave Speed (v)Because waves move, it is obvious that they must have a speed. However, you may be surprised toknow that amplitude, frequency, and wavelength don’t change speed: only the medium it travels thru.Since f 1/T,the wave speedequation couldalso be written as:The Speed (velocity) of a Wavefrequency (Hz)velocity(m/sec)v fλwavelength (m)Wave speed equals frequency times wavelength.λv TThe speed of a wave changes only if the medium changes. Sound moves fasterin more elastic substances. Sound is faster in colder water and in solids (ratherthan liquids) because the molecules are closer. The wave on a slinky moves fasterif the slinky is pulled tighter. Yet, if the medium stays the same, the speed staysthe same. Different waves will have the same speed in the same medium.Closer Molecules—Faster WaveA push:giving energy(starts the wave)cstephenmurray.comf 20 Hzλ 5mv v fλv (20 Hz) x (5 m)v 100 m/sChanging frequency or wavelengthdoes not change speed. Changing thewave changes what moves thru themedium, not the medium itself!More Distant Molecules—Slower WaveFasterTighterSolidsWarm airClose dominos will fall quickly becausethey hit each other quickly.Ex. What is the speed of a 20 Hz wavethat has a 5 meter wavelength.SlowerLiquidsLooserCool airEnergyDominos that are farther apart will fall slowerbecause they take more time to hit each other.Legal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Name:Unit 10: 2Period:1. Transverse wave A. A wave where the oscillation is perpendicular to the direction of motion.2. LongitudinalB. The bottom of a wave.waveC. The top of a wave.3. CrestD. A wave where the oscillation is in the4. Troughsame direction (parallel) as the motion.E. The length of one wave cycle.5. Wavelengthf is the variable for and is measured in .λ is the variable for and is measured in .T is the variable for and is measured in .v is the variable for and is measured in .A wave is 8 meters long and has a frequency of 3 Hz. Find speed.Wave Motion, Yes or No?FM radio:Music:A car going 70 m/s:A bulldozer:Clock pendulum:Earthquakes:Ocean waves:Cellphones:Transverse or Longitudinal Waves?A. You move the slinky left and right.B. You push the slinky forward.C. Sound, if a radio’s speaker moves in and out.D. Earthquakes.E. Vibrates up and down and moves to the right.1Which number shows:A. Double the amplitudeB. AmplitudeWave A has a wavelength of 2 meters and a frequency of 1.5 Hz.Calculate the wave’s speed.2C. WavelengthD. Half λ34Faster or slower wave speed?A. The medium gets colder.B. The amplitude gets bigger.C. A slinky gets looser.D. The medium turns from solid to liquid.E. The wavelength gets shorter.Wave B has a frequency of 18 Hz in the same medium.What is Wave B’s speed?Calculate Wave B’s wavelength.Wave 1: f 25 Hz; Wave 2: f 40 Hz. Which one will be fasterin water?So, as f increases in the same medium, λ .Displacement vs. Position5) 43(mt 2ne 1me 0cal -1p -2si -3D-4-5 00.250 0.5 .7511.251.51.752 2.252.52.75Displacement vs. Pos ition33.253 3.5 .7545) 43(mt 2ne 1me 0cal -1p -2siD -3-4-5 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4.2 .5 .7.2 .5 .7.2 .5 .7.2 .5 .755555555Position (m)Mark 1 cycle of the wave.Starting at 0.75 m, where does the 2nd cycle end:Number of complete cycles:Mark the third crest.Wavelength:Amplitude:If f 4 Hz, find speed:cstephenmurray.comPositio n (m)Mark 1 cycle of the harmonic motion.Starting at 1.5 secs, when does half a cycle end:Number of complete cycles:Number of troughs:Wavelength:Amplitude:If f 50 Hz, find speed:Legal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Name:Unit 10:Period:cstephenmurray.comLegal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Name:Unit 10:Period:Teacher explanation:A teacher asked me some questions that he and I thought would help everyone.Teacher:I noticed that on the Waves worksheet you state in your answer key that a clock pendulum is not an example ofwave motion. Can you explain why not?Me:A pendulum is an example of oscillating, or repeating, motion. Wave motion requires the energy to move, like awater wave or sound. The pendulum doesn't move anywhere. Also, circular motion isn't harmonic either. It repeats, but doesn't follow a path through the "equilibrium position". For a pendulum, the equilibrium position iswhere it comes to rest.Teacher:That makes sense when you talk about energy moving. Initially I was thinking differently since when you graph apendulum it takes a wave-like form but I understand what you are saying. Thanks.Me:Yes, the graphs for SHM (pendulums, springs) and waves look the same.For SHM it is possible to have a position vs time graph, where position is for the back and forth motion, centeredat the equilibrium point. Think of a pendulum with a pen attached. To make this graph you would have to pullthe paper one direction to signify time.For wave motion you could make the same graph, but this time the back and forth of the pendulum would be theup and down of the amplitude of the wave motion (like the top of a wave going up and down) over time.But for wave motion you could also make a graph of position vs location (or displacement vs position). Hereposition would be the up and down and the location would be how far away from your starting position. In thiscase the distance between two crests gives the wavelength.All of the above graphs are sinusoidal (sin or cos-like).cstephenmurray.comLegal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Name:Unit 10: 3Standing WavesPeriod:Sometimes waves are trapped in boundaries.If the length of a wave matches the space it isin, resonance occurs, causes maximum amplitude. The wave seems to stand still. Standingwaves occur only at certain frequencies.Looks like multiple strings.Resonance—When an object vibratessympathetically and amplifies the energyof a wave.Actually, one alternating string.Guitar strings would be quiet without theresonance (amplification) of the guitar’s body.The places of no amplitude are called nodes. Theplaces of greatest amplitude are called anti-nodes.Anti-nodecrestAnti-nodeA jump rope looks like a standing wave, butis not because it moves in a circle and canexist at any frequency (you can speed up alittle at a time). A standing wave can’t existat any frequency.troughNodeAnti-nodeNodeNode1 wavelength (λ) 2 AN (antinodes)When a string isplucked it willvibrate with only one anti-node. This isknown as the natural frequency and alwaysequals one half of a wavelength. The naturalfrequency is also called the fundamentalfrequency (ff) or harmonic one (H1).Natural FrequencyNatural frequency λ/2 1 ANBefore it isplucked.A string of length LAfterλ 2LThe wavelength of the fundamental always equals 2L!Harmonics are standing waves that fit in the same boundaries as the fundamental (natural frequency). Aswith any wave, changing the frequency does not change the wave speed. So if f changes, λ changes, not v.HarmonicsFrequency of a HarmonicFirst 5 Harmonics of a Vibrating StringH1Some standingwaves have openboundaries, likea tuning fork.Open boundariesmove and mustbe anti-nodes.H2H3H4H5NodeAnti-nodeFrequency of theharmonic N(in Hz)fHN N(H)Frequency of thefundamental (in Hz)Node1 wavelength# of Anti-nodeEx. Find the frequency of thethird harmonic (H3) of a4 Hz fundamental.H 4 HzN 3H3 ?Ex. If the fifth harmonic hasa frequency of 55 Hz, find thefundamental frequency.fH5 55 HzN 5fH1 H ?fHn N(H)fH3 3(4)fH3 12 HzfHn N(H)55 5HH fH1 11 HzNodeFundamental1stharmonicf ff H2ndharmonicf2 2H3rdharmonicf3 3H4th5thharharmonic monicf4 f5 4H 5HExamples of Fundamentals and their HarmonicsH1 (ff )H2H3H4H5H2H3H4H5H2 Hz4 Hz6 Hz8 Hz10 Hz5 Hz10 Hz15 Hz20 Hz25 Hz10 Hz20 Hz30 Hz40 Hz50 Hzcstephenmurray.comSpeed of a Standing WaveTo find the speed of a fixed string you would need to know thefrequency of any harmonic and that harmonic’s wavelength.6mf 21 Hzλ 3mRemember thatλ (wavelength) 2 antinodes!Legal copying of this worksheet requires written permission.λ 3mf 21 Hzv v fλv 21(3)v 63 m/sCopyright 2014, C. Stephen Murray

Name:Unit 10: 3Period:A. Where wave’s amplitude is greatest.2. HarmonicB. Where the wave has no motion.3. FundamentalC. A wave that is a multiple of anotherwave.4. NaturalFrequencyD. A wave that is trapped within boundaries.E. The first harmonic of a standing wave,equal to 1/2 its wavelength.Amplitude:# of Anti-nodes:1.51.2510.750.5The frequency at which any space willvibrate when disturbed.Wavelength:0.25F.6. Anti-node# of cycles:43210-1-2-3-405. NodePosition vs. DisplacementDisplacem ent (m )1. Standing waveHarmonic #:Position (m)Why does a violin have a wood body instead of just strings?A string has a fundamental (first harmonic) of 15 Hz, find thefrequency of harmonic 3 (H3).Sometimes when talking or singing in a room, certain notes getvery loud. Why?If 20 Hz is the fundamental, find H6.If 35 Hz is H7, what is the fundamental frequency?String A has a fundamental with a period of 0.25 seconds.A) What is the fundamental’s frequency?LB) How many antinodes does it have?C) If the fundamental is on a 6 m long string, what is its wavelength?D) Find the speed of the wave on that string.ABCDEIs the second harmonic.Has a wavelength of L.Has 4 anti-nodes.Is the highest frequency.Has 3 nodes.Longest wavelength.Has a length of 1.5λ.Fastest wave speed.Is the fundamental.Is the natural frequency.The following table shows the frequencies of the first5 harmonics of different strings. Fill in the blank spaces.E) What would be the frequency of the third harmonic?F) What is the wave speed of the fourth harmonic?Find its period:40 HzMark the nodes and anti-nodes.What harmonic is this?14 Hz6 Hz2345Fundamental frequency 3m3rd harmonic frequency 4 Hz36 Hz44 HzA fellow student shows you the frequencies of four harmonics of astring. Which one would you question and why?Frequencies: 12 Hz; 24 Hz; 29 Hz; 48 HzWavelength Speed of the wave Speed of 5th harmonic cstephenmurray.comLegal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Name:Unit 10:Period:cstephenmurray.comLegal copying of this worksheet requires written permission.Copyright 2014, C. Stephen Murray

Change the frequency ofthe oscillator until you finda harmonic. You willknow because the amplitude (antinode) will be bigand the oscillator will bequieter.1 wavelength (λ) 2 antinodes.You will need to find thefirst 6 harmonics for yourstring.cstephenmurray.com907060H(# of AN)30201090807060λ (m)V ( f λ)(in m/s)λ (m)V ( f λ)(in m/s)3f4 f3 456Answer the questions on the back.Fill in the following table for each harmonic.NodeAnti-nodeNodeAnti-nodeMeasuring thewavelengthDifference betweenfrequencies:H(# of AN)f (Hz)1f2 f1 23f4 f3 4f5 f4 4050V ( f λ)(in m/s)1 wavelength5f6 f5 3020λ (m)2f6 f5 61 wavelengthAnswer the questions on the back.1090f (Hz)1f2 f1 f5 f4 4050Measuring thewavelength5th harmonic80Standing Wave Lab70You will need to find thefirst 6 harmonics for yourstring.Anti-nodeDifference betweenfrequencies:f3 f2 601 wavelength (λ) 2 antinodes.Node5th harmonicFill in the following table for each harmonic.NodeAnti-nodeNodeAnti-nodeDifference betweenfrequencies:H(# of AN)f2 f1 f3 f2 50Change the frequency ofthe oscillator until you finda harmonic. You willknow because the amplitude (antinode) will be bigand the oscillator will bequieter.Anti-nodeMeasuring thewavelengthf4 f3 f5 f4 40Standing Wave LabNodef6 f5 30You will need to find thefirst 6 harmonics for yourstring.201 wavelength (λ) 2 antinodes.Unit 10:Fill in the following table for each harmonic.f3 f2 10Change the frequency ofthe oscillator until you finda harmonic. You willknow because the amplitude (antinode) will be b

Harmonic Motion Graphs Cycle —from any point on the line to that same point going the same way. This graph shows 2 complete cycles. Period —measure the time for one cycle between any two identical points on the graph (top-to-top, bottom-to-bottom, etc.). Frequency —count the number of cycles in 1 second OR find the period and use f 1/T.

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