Oscillations & Waves

1. Simple Harmonic Motion Forced Oscillations & Resonance Oscillations & Waves IB Physics WAVES

2. Make sure to read page 99

3. Simple Harmonic Motion • Oscillation 4. Physics. a. an effect expressible as a quantity that repeatedly and regularly fluctuates above and below some mean value, as the pressure of a sound wave or the voltage of an alternating current. b. a single fluctuation between maximum and minimum values in such an effect. From: http://dictionary.reference.com

4. Simple Harmonic Motion • Terms • Displacement(x,Θ) • Amplitude (xo,Θo) • Period (T) • Frequency (f) • Phase Difference {There’s a nice succinct explanation of the Radian on p.101. Check it out.}

5. Table 13-1Typical Periods and Frequencies

6. Simple Harmonic Motion • Definition • Oscillators that are perfectly isochronous & whose amplitude does not change in time • Real World Approximations • Pendulum (Θ0 < 40o) • Weight on a spring (limited Amplitude)

7. Simple Harmonic Motion • Angular Frequency • In terms of linear frequency: ω = 2πf • There is a connection between angular frequency and angular speed of a particle moving in a circle with a constant speed.

8. 13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:

9. Here, the object in circular motion has an angular speed of where T is the period of motion of the object in simple harmonic motion. 13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion

10. Figure 13-5Position versus time in simple harmonic motion

11. Figure 13-6Velocity versus time in simple harmonic motion

12. Figure 13-7Acceleration versus time in simple harmonic motion

13. Figure 13-2Displaying position versus time for simple harmonic motion

14. Figure 13-3Simple harmonic motion as a sine or a cosine

15. Simple Harmonic Motion • Mathematical Definition a is directly proportional to x a = - ω2x

16. Simple Harmonic Motion • What does this mean about force? F = - kx • Apply 2nd Law ma = - kx

17. Simple Harmonic Motion • Acceleration not constant • Force-accel relation: 2nd order diff eq x= P cos ω t + Q sin ω t • P & Q constants • ω = √(k/m) • Compare T calculation for spring vs. pendulum

18. 13-4 The Period of a Mass on a Spring Therefore, the period is

19. 13-6 The Pendulum A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass). The angle it makes with the vertical varies with time as a sine or cosine.

20. 13-6 The Pendulum Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ,whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

21. 13-6 The Pendulum However, for small angles, sin θ and θ are approximately equal.

22. 13-6 The Pendulum Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:

23. Solutions of the SHM equation

24. SHM Equation Solutions x = xocosωt x = xosinωt v = vocosωt v = -vosinωt v = ±ω√(xo2 - x2)

25. Boundary Conditions • x = xo when t=0 • x = 0 when t=0 • Solutions differ in phase by π 2

26. Energy Changes • Kinetic • Potential • Total

27. Figure 13-10Energy as a function of position in simple harmonic motion

28. Figure 13-11Energy as a function of time in simple harmonic motion

29. Forced Oscillations & Resonance • Damped Oscillations – decr w/ time • Heavily – decr very quickly • Critically – no/barely • Damping Force • Opposite in direction to motion of oscillating particle • Dissipative

30. Forced Oscillations & Resonance • Natural Frequency • Frequency at which system oscillates when not being driven • Forced (driven) Oscillations • Added energy to prevent damping

31. Forced Oscillations & Resonance • Driver Frequency = Natural Frequency • Max E from driver when @ max amplitude • Max amplitude of oscillation • Resonance • re: A vs. f graphs

32. WAVES

33. Waves • A means by which energy is transferred between two points in a medium • No net transfer of the medium • Single: “pulse” • Continuous: “wave train” • Mechanical waves need a medium. • Example: sound & water • Radiant energy does not need a medium. • Example: light

34. Transverse • Vibratory is perpendicular to the direction of energy transfer. • Examples: water & light Wavelength Crest Amplitude Equilibrium Height Trough

35. Longitudinal • Vibratory motion is parallel to the direction of energy transfer. • Compressional or Pressure wave ////// / / / / / ////// / / / / / • Example: sound Compression Wavelength Rarefaction

36. Waves in 2 Dimensions • Previous representations are cut-aways, showing length & amplitude • Wave Fronts use parallel lines to represent crests, showing width & length • Rays are often drawn perpendicular to fronts to indicate the direction of travel of the wave λ Fronts RAY

37. Wave Characteristics Crest – highest point - (max displacement ) Trough – lowest point - (max displacement ) Compression – particles are closest - (max displacement  ) Rarefaction – particles are farthest apart - (max displacement)

38. Wave Characteristics Amplitude (A,a) – maximum displacement from equilibrium position Period (T) – time for one complete oscillation Frequency (f) – oscillations per second Wavelength (λ) – distance between two successive particles that have the same displacement Wave Speed (v,c) – speed energy moved through medium by the wave Intensity (I) – energy per unit time transported across a unit area of medium

39. Wave Characteristics • Wave Speed depends on nature & properties of medium • Water waves travel faster in deep water • Frequency of wave depends upon frequency of source • Will not change if wave enters a different medium or the properties of the medium change • intensity ∞ amplitude2

40. Wave Characteristics • Relationships: f = 1 T v = fλ • Waves are periodic in both time and space.

41. Wave Graphs Displacement Distance Equilibrium Position Displacement-Position Graph

42. Wave Graphs Displacement Time Displacement-Time Graph

43. Electromagnetic Waves • Electric & Magnetic Fields oscillating at right angles to each other • Same speed in free space • Know spectrum p.117