= T. Oscillations and Waves. Example of an Oscillating System IB 12 IB 12

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1 Oscillation: the vibration of an object Oscillations and Waves Eaple of an Oscillating Syste A ass oscillates on a horizontal spring without friction as shown below. At each position, analyze its displaceent, velocity and acceleration. Wave: a transfer of energy without a transfer of atter Force fro the Spring: F s = -k restoring force tends to restore syste to equilibriu position opposite in direction of displaceent Eaples of oscillations:. ass on spring (eg. bungee juping). pendulu (eg. swing) 3. object bobbing in water (eg. buoy, boat) 4. vibrating cantilever (eg. diving board) 5. earthquake 6. bouncing ball 7. usical instruents (eg. strings, percussion, brass, woodwinds, vocal chords) 8. heartbeat Mean Position (Equilibriu Position) position of object at rest Displaceent (, eters) distance in a particular direction of a particle fro its ean position Aplitude (A or 0, eters) aiu displaceent fro the ean position Period (T, seconds) tie taken for one coplete oscillation Frequency (f, Hertz) nuber of oscillations that take place per unit tie Phase Difference difference in phase between the particles of two oscillating systes Relationship between period and frequency: f = T f = cycles/sec T = sec/cycle Angular Frequency - product of π ties frequency Forula: ω = πf Sybol: ω ω = π/t Units: rad/sec s -. A pendulu copletes 0 swings in 8.0 seconds. a) Calculate its period. T = 0.8 s b) Calculate its frequency. f =.5 Hz =.3 Hz c) Calculate its angular frequency. ω = 7.8 rad/s = 7.8 s -. When is the velocity of the ass at its aiu value? When the displaceent = 0 at equilibriu position. When is the acceleration of the ass at its aiu value? When the displaceent and force = a at etree positions

2 The Displaceent Function A ass on a spring is allowed to oscillate up and down about its ean position without friction. Two traces of the displaceent () of the ass versus tie (t) are shown. Initial condition: starts at ean position Velocity and Acceleration for Siple Haronic Motion a) Displaceent Function Function: = 0 sin ωt b) Velocity Function Initial condition: starts at aplitude position Function: = 0 cos ωt Analyzing the Displaceent Function. Analyze the displaceent function shown at right. a) What is the aplitude? 0 = a) What is the period? c) Acceleration Function T = 4.0 s. What is the displaceent of the ass when: b) What is the frequency? f = 0.5 Hz c) What is the angular frequency? ω = π/ s - a) t =.0 s? = (.080 )sin (π/)t = (.080 )sin (π/)() =.080 b) t =.0 s? = (.080 )sin (π/)t = (.080 )sin (π/)() = 0 Defining Equation for SHM: ω ( sin ω ) a= 0 t a= ω a α Negative Sign:. acceleration is in opposite direction of displaceent. directed back towards ean position e) Write the displaceent function. = (.080 )sin (π/)t c) t =.5 s? = (.080 )sin (π/)t = (.080 )sin (π/)(.5) =.080 sin (3.96 ) = (RADIAN MODE!!!) Siple Haronic Motion (SHM) otion that takes place when the acceleration of an object is proportional to its displaceent fro its equilibriu position and is always directed toward its equilibriu position 3 4

3 . The graph shown at right shows the displaceent of an object in SHM. Use the graph to find the: a) period of oscillation b) aplitude of oscillation c) displaceent function Alternate Velocity Function F net F s Eaple of SHM Mass on a Horizontal Spring A ass oscillates horizontally on a spring without friction, as shown. Is this SHM? = a = a k= a k a= a α Yes this is SHM since a α -. d) aiu velocity e) velocity at.3 seconds Angular frequency, period, and frequency for a ass on a spring k a= a= ω k ω = ω= k π T = ω T = π k k f = = T π f) aiu acceleration. Use the alternate for of the velocity function to find the velocity of the object at.3 s.. A.00 kg ass oscillates back and forth fro its rest position on a horizontal spring whose constant is 40.0 N/. a) Calculate the angular frequency, period and frequency of this syste. ω= ω= 4.47s.00 T = π 40.0 T =.40s - f =.40 f = 0.7 Hz g) acceleration at.3 seconds b) Write the displaceent, velocity and acceleration functions for this syste. = (0.500) sin(4.47 t) v= (.4) cos(4.47 t) a= ( 9.99) sin(4.47 t) 5 6

4 Alternate Fors of the Equations of Motion for SHM Eaple of SHM Siple Pendulu. Write the equations of otion for the graphs shown below.. Write the equations of otion for the graphs shown below.. A ass is allowed to swing freely fro the end of a light-weight string. Show that the otion of this siple pendulu is approiately siple haronic otion. Fnet = a g = a a= g for sall angles θ a= gθ s a= g L g a= s L a α s for sall angles s g a= L a α -. Deterine the angular frequency, period and frequency for the pendulu. 3. What is the difference between the otions described by the two sets of equations? # - = 0 at t = 0 ω = L ω= g g L π T = ω L T = π g f = T g f = π L # = 0 at t = 0 3. A 0.0 g pendulu on an 80.0 c string is pulled back 5.0 c and then swings. Deterine its: 4. a) Write the equations of otion for the syste whose displaceent is shown on the graph at right. a) angular frequency d) aiu velocity b) displaceent function e) aiu acceleration b) State two ties when the: i) speed is aiu ii) agnitude of the acceleration is aiu. c) velocity function 7 8

5 Energy and Siple Haronic Motion A ass oscillates back and forth on a spring. Analyze the energy in the syste at each location.. A.00 kg ass is oscillating on a spring and its displaceent function is shown. a) At what tie(s) does the ass have the ost kinetic energy? b) Deterine the aiu kinetic energy of the ass. When the ass is at its ean position... c) At what tie(s) does the ass have aiu potential energy? Deterine this value. d) What is the total energy of the syste at.5 seconds? When the ass is at any position... e) Deterine the kinetic and potential energy of the syste at.5 seconds. 9 0

Energy Graphs and SHM Daping in Oscillations Energy-Displaceent Functions Daping: a dissipative force acts on a syste in the opposite direction to the direction of otion of the oscillating particle

6 Energy Graphs and SHM Daping in Oscillations Energy-Displaceent Functions Daping: a dissipative force acts on a syste in the opposite direction to the direction of otion of the oscillating particle EP = ω EK = ω ET = ω 0 ( 0 ) Effect of daping: syste loses energy and aplitude (energy α apl ) Sketch the displaceent function for a syste without and with daping. Energy-Tie Functions EP = = 0 t EP = ω 0 sin ωt E α sin ωt P ω ω ( sinω ) Without Daping Degrees of Daping With Daping EK = v = ( v cos ωt ) EK = v0 cos ωt E α cos ωt K 0 Light daping (under-daping): sall resistive force so only a sall percentage of energy is reoved each cycle period is not affected can take any cycles for oscillations to die out eg. car shock absorbers Note that in siple haronic otion, the energy of a syste is proportional to:. ass Heavy daping (over-daping): large resistive force can copletely prevent any oscillations fro taking place takes a long tie for object to return to ean position eg.- oscillations in viscous fluid. aplitude squared 3. frequency squared Critical daping: interediate resistive force so tie taken for object to return to ean position is iniu inial or no overshoot eg. electric eters with pointers, autoatic door closers

Resonance Natural Frequency of Vibration: when a syste is displaced fro equilibriu and allowed to oscillate freely, it will do so at its natural frequency of vibration Waves Both pulses and traveling

7 Resonance Natural Frequency of Vibration: when a syste is displaced fro equilibriu and allowed to oscillate freely, it will do so at its natural frequency of vibration Waves Both pulses and traveling waves: Forced Oscillations a syste ay be forced to oscillate at any given frequency by an outside driving force that is applied to it Resonance a transfer of energy in which a syste is subject to an oscillating force that atches the natural frequency of the syste resulting in a large aplitude of vibration Pulse single oscillation or disturbance Continuous traveling wave succession of oscillations (series of periodic pulses) transfer energy though there is no net otion of the ediu through which the wave passes. Mechanical Waves: require a ediu to transfer energy eg. sound waves, water waves, waves on strings, earthquake waves Electroagnetic Waves: do not require a ediu to transfer energy eg. light waves, all EM waves Aplitude vs. frequency graph for forced oscillations Factors that affect the frequency response and sharpness of curve: ) frequency of driving force ) natural frequency of syste 3) aplitude of driving force 4) aount of daping A transverse wave is one in which the direction of the oscillation of the particles of the ediu is perpendicular to the direction of travel of the wave (the energy). Eaples: light, violin and guitar strings, ropes, earthquake S waves A longitudinal wave is one in which the direction of the oscillation of the particles of the ediu is parallel to the direction of travel of the wave (the energy). Eaple: sound, earthquake P waves Copression: region where particles of ediu are close together Rarefaction: region where particles of ediu are far apart. Sketch the frequency response for a lightly daped syste whose natural frequency is 0 Hz that eperiences forced oscillations. Note that transverse echanical waves cannot propagate (travel) through a gas only longitudinal waves can. Displaceent (, eters) distance in a particular direction of a particle fro its ean position Aplitude (A or 0, eters) aiu displaceent fro the ean position Period (T, seconds) tie taken for one coplete oscillation - tie for one coplete wave (cycle) to pass a given point Frequency (f, Hertz) nuber of oscillations that take place per unit tie Wavelength (λ, eters) shortest distance along the wave between two points that are in phase -the distance a coplete wave (cycle) travels in one period. Copare the otion of a single particle to the otion of the wave as a whole (the otion of the energy transfer). Particle Speed: not constant speed = SHM Average speed: v = d/t in tie t = period: v = 4A/T Wave Speed: constant speed v = d/t in tie t = period: v = λ/t v = (/T) λ v = f λ 3 4

$. Motion of the Wave λ. Motion of a Particle T Reflection and Refraction Sketch the incident and reflected rays as well as the reflected wavefront.$ To change wave speed, you ust change the ediu or its properties.

To change wave speed, you ust change the ediu or its properties.

$Light: Sound: Mirror Refraction: the change in direction of a wave (due to a change in speed) when it crosses a boundary between two different edia at an angle Wavelength is inversely proportional to$ frequency Waves in Two Diensions Wavefront line (or arc) joining neighboring points that have the sae phase or displaceent Ray line indicating direction of wave otion (direction of energy transfer).

frequency Waves in Two Diensions Wavefront line (or arc) joining neighboring points that have the sae phase or displaceent Ray line indicating direction of wave otion (direction of energy transfer).

Air to glass: Fast to slow = bends toward the noral n < n v > v λ > λ Glass to air: Slow to fast = bends away fro the noral n > n v < v λ < λ Intensity - power received per unit area Forula: I = P/A

8 . Motion of the Wave λ. Motion of a Particle T Reflection and Refraction Sketch the incident and reflected rays as well as the reflected wavefront. Law of Reflection Control variable: in one ediu - wave speed Wave speed depends on the properties of the ediu, not how fast the ediu vibrates. To change wave speed, you ust change the ediu or its properties. Control variable: across a boundary - frequency As a wave crosses a boundary between two different edia, the frequency of a wave reains constant not the speed or wavelength. θ i θ r The angle of incidence is equal to the angle of reflection when both angles are easured with respect to the noral line (and the incident ray, reflected ray and noral all lie in the sae plane). Light: Sound: Mirror Refraction: the change in direction of a wave (due to a change in speed) when it crosses a boundary between two different edia at an angle Wavelength is inversely proportional to frequency Waves in Two Diensions Wavefront line (or arc) joining neighboring points that have the sae phase or displaceent Ray line indicating direction of wave otion (direction of energy transfer). Rays are perpendicular to wavefronts. Wavelength is proportional to speed At great distances, the wavefronts are approiately parallel and are known as plane waves. Air to glass: Fast to slow = bends toward the noral n < n v > v λ > λ Glass to air: Slow to fast = bends away fro the noral n > n v < v λ < λ Intensity - power received per unit area Forula: I = P/A Sybol: I Units: W/ NOTE: for a wave, its intensity is proportional to the square of its aplitude. 0-5 W of sound power pass through each surface as shown. Surface has area 4.0 and surface is twice as far away fro the source. Calculate the sound intensity at each location. 5 Refractive Inde (Inde of refraction)(n): ratio of sine of angle of incidence to sine of angle of refraction, for a wave incident fro air v n= = v c n= v Snell s Law: the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, for a given frequency n v λ = = = n v λ n = n 6

Oscillations and Waves

Oscillations and Waves Oscillation: Wave: Examples of oscillations: 1. mass on spring (eg. bungee jumping) 2. pendulum (eg. swing) 3. object bobbing in water (eg. buoy, boat) 4. vibrating cantilever (eg.