Oscillation the vibration of an object. Wave a transfer of energy without a transfer of matter

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1 Oscillation the vibration of an object Wave a transfer of energy without a transfer of matter

2 Equilibrium Position position of object at rest (mean position) Displacement (x) distance in a particular direction of a particle from its mean position Units - meters

3 Amplitude (A or x max ) maximum displacement from the mean position Units - meters

4 Period (T) time taken for one complete oscillation Units seconds (s / cycle) Frequency (f) number of oscillations that take place per unit time Units Hertz (Hz) (cycles/s) T = 2π ω = 1 f

5 Angular Frequency (ω) - a scalar measure of rotation rate ω = 2πf = 2π T Units radians/sec or s -1

6 Spring Force (F s ) restoring force tends to restore system to equilibrium position opposite in direction of displacement! F s = k! x

7 x = v = F = a = x = v = F = a = x = v = F = a = x = v = F = a =

8 x = +A v = 0 m/s F = -F max (Fs = kx) a = - a max (a = F net /m) x = 0 m v = -v max F = 0 N a = 0 m/s 2 x = -A v = 0 m/s F = +F max (Fs = kx) a = + a max (a = F net /m) x = 0 m v = +v max F = 0 N a = 0 m/s 2

9 When is the magnitude of the velocity of the mass at its maximum value? When is the acceleration of the mass at its maximum value?

10 When is the magnitude of the velocity of the mass at its maximum value? At the equilibrium position When is the acceleration of the mass at its maximum value? At the extreme positions

11 Initial condition: starts at equillibrium position Function: x = Asin(2πft) = Asinωt Initial condition: starts at amplitude position Function: x = Acos(2πft) = Acosωt

12

13 cos -sin -cos sin cos -sin

14 Displacement Function Analyze the displacement function shown at right. What is the amplitude? Write the displacement function. What is the period? What is the frequency? What is the displacement of the mass when: t = 1.0 s? What is the angular speed? t = 2.0 s?

15 Displacement Function Analyze the displacement function shown at right. What is the amplitude? A = m What is the period? T = 4.0 s What is the frequency? f = 1 T = s = 0.25 Hz What is the angular speed? ω = 2πf = 2π(0.25Hz) = 1.57 rad s Write the displacement function. x = (0.080m)sin(1.57 rad s )t What is the displacement of the mass when: t = 1.0 s? x = (0.080m)sin(1.57 rad )(1.0s) = 0.080m s t = 2.0 s? x = (0.080m)sin(1.57 rad )(2.0s) = 0m s

16 Displacement Function x = A sin (ωt) Velocity Function v = (Aω) cos ωt v = v cos ωt max Acceleration Function v = (-Aω 2 ) sin ωt v = -a sin ωt max

17 Simple Harmonic Motion (SHM) motion that takes place when the acceleration of an object is proportional to its displacement from its equilibrium position and is always directed toward its equilibrium position

18

19 Period of a Spring T s = 2π m k = 2π ω

20 A 2.00 kg mass oscillates back and forth 0.500m from its rest position on a horizontal spring whose constant is 40.0 N/m. Calculate the angular speed, period and frequency of this system.

21 A 2.00 kg mass oscillates back and forth 0.500m from its rest position on a horizontal spring whose constant is 40.0 N/m. Calculate the angular speed, period and frequency of this system. 2π ω = 2π m k ω = k m = 40.0 N m 2.00kg = 4.47 rad s T = 2π m k = 2π 2.00 kg 40.0 N/m = 1.40 s Or T = 2π/ω f = 1 T = s = Hz

22 T p = 2π l g = 2π ω

23 A 20.0 g pendulum on an 80.0 cm string is pulled back 5.0 cm and then swings. Determine its angular speed, maximum velocity and maximum acceleration.

24 A 20.0 g pendulum on an 80.0 cm string is pulled back 5.0 cm and then swings. Determine its angular speed, maximum velocity and maximum acceleration. 2π ω = 2π l g ω = g l = 9.81 m s m = 3.5 rad s v max = Aω = 0.18 m/s a max = Aω 2 = (0.050m)(3.5 rad s )2 = 0.61 m s 2

25 Energy & Simple Harmonic Motion x = v = F = a = x = v = F = a = x = v = F = a = x = v = F = a =

26 Energy & Simple Harmonic Motion x = max v = 0 m/s Us = max K = 0 J x = 0 m v = max Us = 0 J K = max x = max v = 0 m/s Us = max K = 0 J x = 0 m v = max Us = 0 J K = max

27 Energy & SHM When the mass is at its mean position... E T = K + U s E T = K max 2 E T = 1/2 mv max E T = 1/2 m( ω i A ) 2 E T = 1/2 mω 2 A 2 When the mass is at any position... E T = K + U s E T = 1/2 mv 2 + 1/2 kx 2

28 Energy-Displacement Function

29 Energy-Time Function

30 Waves Both pulses and traveling waves: Pulse single oscillation or disturbance Transfer energy though there is no net motion of the medium through which the wave passes. Continuous traveling wave succession of oscillations (series of periodic pulses)

31 Waves Mechanical Waves: require a medium to transfer energy eg. sound waves, water waves, waves on strings, earthquake waves Electromagnetic Waves: do not require a medium to transfer energy eg. light waves, all EM waves

32 Transverse Waves A transverse wave is one in which the direction of the oscillation of the particles of the medium is perpendicular to the direction of travel of the wave (the energy). Examples: light, violin and guitar strings, ropes, earthquake S waves

33 Longitudinal Waves A longitudinal wave is one in which the direction of the oscillation of the particles of the medium is parallel to the direction of travel of the wave (the energy). Example: sound, earthquake P waves

34 Displacement (x) distance in a particular direction of a particle from its mean position Amplitude (A or x max ) maximum displacement from the mean position Period (T) time taken for one complete oscillation - time for one complete wave (cycle) to pass a given point Frequency (f) number of oscillations that take place per unit time Wavelength (λ) shortest distance along the wave between two points that are in phase -the distance a complete wave (cycle) travels in one period.

35 Compression: region where particles of medium are close together Rarefaction: region where particles of medium are far apart

36 Waves in Motion Compare the motion of a single particle to the motion of the wave as a whole (the motion of the energy transfer).

37 Waves in Motion Compare the motion of a single particle to the motion of the wave as a whole (the motion of the energy transfer). Particle motion is perpendicular to the wave motion

38 Particle Speed Wave Speed Speed is not constant = SHM average speed = v = d t v = 4 A T λ = v f constant speed = v = d t v = λ T = 1 T λ v = fλ

39 Sound v = 3kT m

40 Sound Mechanical Wave Longitudinal Amplitude Volume Energy Frequency Pitch

41 Light Electromagnetic Wave Transverse Amplitude Brightness / Intensity Frequency Color / Type Energy Speed = c = 3.00 x 10 8 m/s

42 Phase Phase the position of any particle in its cycle of oscillation. In phase: (A,E,I) (B,F) (D,H) (C,G) The phase difference between any points in phase is 0 or λ Completely out of phase: (A,C) (B,D) (A,G) (B,H) The phase difference between any points completely out of phase is π or 180 or λ/2

43 Reflection at a Boundary Between Two Media

44 Reflection at a Boundary Between Two Media

45 Superposition & Interference Principle of Linear Superposition When two or more waves (pulses) meet, the resultant displacement is the vector sum of the individual displacements.

46 Constructive Interference

47 Destructive Interference

48 Beats Beats periodic variations in loudness resulting from the superposition of two sound waves of slightly different frequencies

49 What frequency is heard when the two tuning forks pictured above are sounded together? What is the frequency of the beats that they produce?

50 What frequency is heard when the two tuning forks pictured above are sounded together? f sound = f 1 + f 2 2 = 440Hz + 438Hz 2 = 439Hz What is the frequency of the beats that they produce? f beat = f 1 f 2 = 440Hz 438Hz = 2Hz

51 Standing Waves A traveling wave moving in one direction in a medium is reflected off the end of the medium. This sends a reflected wave traveling in the opposite direction in the medium. This second wave is (nearly) identical with the first traveling wave. (same frequency, same wavelength, almost same amplitude)

52 Node: location of constant complete destructive interference Anti-Node: location of maximum constructive interference

53 Transverse Standing Wave 1 st Harmonic (fundamental) 2 nd Harmonic (1 st overtone) 3 rd Harmonic (2 nd overtone) 6 meters v = 1200 m/s L = λ 1 = f 1 = v = 1200 m/s L = λ 2 = f 2 = v = 1200 m/s L = λ 3 = f 3 =

54 Transverse Standing Wave 1 st Harmonic (fundamental) 2 nd Harmonic (1 st overtone) 3 rd Harmonic (2 nd overtone) 6 meters v = 1200 m/s L = ½ λ 1 λ 1 = 2 L = 12 m f 1 = v / λ 1 = 1200 / 12 = 100 Hz v = 1200 m/s L = λ 2 λ 2 = L = 6 m f 2 = v / λ 2 = 1200 / 6 = 200 Hz = 2 f 1 v = 1200 m/s L = (1.5) 3/2 λ 3 λ 3 = 2/3 L = 4 m f 3 = v / λ 3 = 1200 / 4 = 300 Hz = 3 f 1

55 Fundamental wavelength and frequency: L = ½ λ 1 so λ 1 = 2L f 1 = v λ 1 = v 2L Other natural frequencies (Resonant modes): f n = n f 1 = n v 2L where n = 1,2,3,4,

56 Waves on a String Under Tension linear density = µ = m L v = F T µ

57 Boundary conditions for a pipe open at both ends: 2 free ends antinode at each end Pipe open at both ends

58 Fundamental wavelength and frequency: L = ½ λ 1 so λ 1 = 2L f 1 = v λ 1 = v 2L Other natural frequencies (Resonant modes): f n = n f 1 = n v 2L where n = 1,2,3,4,

59 Boundary conditions for a pipe closed at one end: 1 fixed and one free end one node and one antinode Pipe open at both ends

60 Fundamental wavelength and frequency: Other natural frequencies (Resonant modes): L = ¼ λ 1 so λ 1 = 4L f n = n f 1 = n v 4L f 1 = v λ 1 = v 4L where n = 1,3,5,

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