Vibrations (Oscillations in Time) Properties of SHM ïƒ F =-k s x ïƒ Period (time...

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Transcript of Vibrations (Oscillations in Time) Properties of SHM ïƒ F =-k s x ïƒ Period (time...

  • Vibrations (Oscillations in Time)Properties of SHMF =-ksxPeriod (time for a full cycle): T=2(m/ks)Frequency (number of cycles per second): f=1/TAmplitude: A = Maximum displacement from equilibriumVelocity: v = speed of massAngle swept: (t) = t and: = 2f = 2/Ty-motion: y(t) = A sin(t) = A sin(2ft)= A sin(2t/T)Review Circular Analog from SHM ppt slides. (next slide should look familiar)

  • Circle Analog to SHM

    Suppose we have an object of mass (m) in uniform circular motion with angular velocity () and radius (A).Angle swept: (t) = t and: = 2f = 2/Ty-motion: y(t) = A sin(t) = A sin(2ft)= A sin(2t/T)Projection or shadow of vertical axis traces out an SHM with period (T).Speed of circular motion: v= A = A(2/T) _____ Max speed of SHM: vmax=(ks/m) A _____Equate to give: T=2(m/ks)

  • Wave MotionHarmonic Motion in Time and Space.Freezing time: See oscillation over spaceLook at one point: See oscillations over time.y(x,t) = A sin(2x/ -2t/T)Waves transport Energy

  • Characteristics of WavesAmplitude (A)Maximum displacement from equilibriumSound (density or pressure)E&M (Electric field) (E)Intensity (I ~ A2 ~ Power)Sound Wave: LoudnessE&M Wave: BrightnessWavelength () (m)Distance between successive crests (time frozen)Period (T) (s)Time between successive crests (distance frozen)Frequency (f) (Hz or 1/s)Number of cycles per second (pitch or color)Speed (v = f ) (m/s)Speed at which energy is transported.

  • Types of WavesTransverse WavesAmplitude is perpendicular to propagation direction Ocean Waves, Waves in strings, (E&M) WavesLongitudinal WavesAmplitude is parallel to propagation directionSound Waves, parallel compression of springs, p-waves of earthquakes.Follow any one particle in the animation; it oscillates in time.

  • Period vs. Wavelength(watch whats plotted on horizontal axis)

  • Behavior of a Wave at an InterfaceInterface: boundary between two media (materials)A wave incident on an interface can beReflected: Surface appears mirror-likeAbsorbed: Surface can appear black (heats up; if hot enough can emit visible radiation)Transmitted: Surface is invisibleCombination of the above: Real world.

  • Superposition and InterferenceWhen two (or more) waves pass through the same region at the same time:they interfere with each other to create a new waveformCombined waveform is the sum of the displacements at each point.Constructive Interference: combined waveform larger than originals.Destructive Interference: combined waveform smaller than originals.

  • Superposition and Interference (contd)Total Constructive InterferenceWaves add in phase Phase shift: 0, 360 or 0, 2, Maximum combined waveformTotal Destructive InterferenceWaves add out of phasePhase shift: 180 , 540 or /2, 3/2, Minimum combined waveformCan result in extinction.Intermediate waveforms dependent on wave properties.

    Phase Shift: Relative Difference in angle between two waves. (See Circular Analog)

  • Mechanical Wave at a Reflective BoundaryWhen a wave reflects off a fixed boundary, it sustains a /2 wave shift or /2 phase shift.Pulse gets inverted by reaction force.

    When a wave reflects off a loose boundary, there is no wave shift.

  • Self-Interference and Standing WavesA wave incident on a reflective boundary Can interfere with its reflectionAmplitudes vibrate up and down (antinodes)Stationary points of zero amplitude (nodes)Net energy flow is zero (flow to right = flow to left)ResonanceAll vibrations and waves have natural or resonant frequencies (ex: pendulum of length L)Leads to highest possible amplitudes whenForces are applied at natural frequencies. Pushing a kid on a swing (or pushing a pendulum)Frequencies match geometry (next slide)

  • Standing Waves (contd)For waves traveling along length (L) between two fixed ends, to form standing waves:Displacement = 0 at endsIntegral number of half wave lengths must fit on L.Leads to: (n = 1, 2, 3, .)Allowed wavelengths: L = n (n/2) or n = 2L/nNatural Frequencies: fn = v/n = nv/2LFundamental Frequency or first harmonic: n=1: f1Second Harmonic or First Overtone: n=2: f2 = 2f1And so on: nth Harmonic or (n-1)th Overtone: fn = n f1Application to stringed instrumentsApplet and Illustrations for Standing Waves on Strings

  • Sound WavesLongitudinal WavesCaused by vibration of materials (e.g. tuning fork)Cannot travel through a vacuum.Travel faster and louder through solids and liquids than in gases (340 m/s in dry air at 20C).Pitch: describes how we hear frequency.Hearing range: 20Hz to 20,000HzLower (infrasonic), Higher (ultrasonic)Loudness: describes how we hear intensity (square of amplitude)

  • Decibel Scale (dB)Measures LoudnessAn increase of 10dB = 10 times increase in intensity.

  • The Doppler EffectHow does our perception of a wave change when its source is moving towards us or away from us?Sound Wave AppletIf the relative motion of the source is towards us we perceive higher frequency (higher pitch or bluer).If the relative motion of the source is away from us perceive lower frequency (lower pitch or redder).The observed shift in frequency is called the Doppler Shift.

  • What if a source of sound moves as fast as the sound wave itself?The source will move with the wave itself.Both source and wave arrive at our ear at the same time. Sound Wave AppletWe hear all of the amplitudes compressed together (LOUD) at one instant (SUDDEN)Its a SONIC BOOMWhat if the source exceeds speed of sound?Source arrives before the sound it makes. We dont hear it until its past us.A CONICAL SHOCK WAVE.

  • A Plane in Supersonic Flight

  • The Doppler Shift also tells us.Whether a storm system is coming to us or not.By bouncing radio waves off rain droplets and analyzing the shift between sent and received waves.Whether were exceeding the speed limit.By bouncing radio waves off your car (RADAR)Whether we have blood clots in our legs or the rate of blood flow through our hearts.By bouncing sound waves off red blood cells.Whether or not the Universe is expanding.What does that mean?

  • Resonance and MusicStringed Instruments (Already discussed)Integral number of half wave lengths must fit on L. Use finger pressure to vary L.Leads to: (n = 1, 2, 3, .)Allowed wavelengths: L = n (n/2) or n = 2L/nNatural Frequencies: fn = v/n = nv/2LFundamental Frequency or first harmonic: n=1: f1Second Harmonic or First Overtone: n=2: f2 = 2f1And so on: nth Harmonic or (n-1)th Overtone: fn = n f1

  • Organ Pipes and Wind InstrumentsClosed Pipe: Open mouth (A): antinodeClosed end (D): nodeBoundary Condition: L = n n/4 or n = 4L/n; n=1,3,5, ...Natural frequencies: f1 = v/4L; fn=n f1; n=1,3,5, ...All even harmonics are missing.Open Pipe: Open mouth (A): antinodeOpen end (D): antinodeBoundary Condition : L = nn/2 or n = 2L/n; n=1,2,3,...Natural Frequencies: f1 = v/2L; fn= f1; n=1,2,3,...

    Antinodes exist at open ends; freer movement

  • Harmonics of Open and Closed PipesPressure and Displacements in open and closed pipesClosed pipeOpen and closed pipes