Vibrations and Waves

1. Vibrations and Waves Chapter 11

2. Simple Harmonic Motion Chapter 11 Section 1

3. Periodic Motion • Any repetitive, or cyclical, types of motion • Examples? • Simple Harmonic Motion(SHM) is a specialized form of periodic motion

4. Simple Harmonic Motion • Periodic vibration about an equilibrium position • Restoring force must be • proportional to displacement from equilibrium • in the direction of equilibrium

5. Simple Harmonic Motion • Common examples include: • mass-spring system • pendulum for small angles

6. Mass on a Spring When a spring is stretched, the restoring force from the tension in The spring is described by Hooke’s Law… F = kx The force acting on the mass is proportional to its displacement from equilibrium and in a direction towards equilibrium, thus SHM

7. The Pendulum • A simple pendulum consists of a mass called a bob, which is attached to a fixed string. Effectively, all the mass is in the bob. • The x component of the weight (Fg sin q) is the restoring force.

8. The Pendulum • The magnitude of the restoring force (Fgsin q) is proportional to sin q. • When the angle of displacement q is relatively small, sin q is approximately equal to q in radians… sin 0 = 0 • So, for small angles, the restoring force is very nearly proportional to the displacement, and the pendulum’s motion is an excellent approximation ofsimple harmonic motion.

10. Virtual Simple Harmonic Motion • http://phet.colorado.edu/simulations/sims.php?sim=Pendulum_Lab • http://phet.colorado.edu/simulations/sims.php?sim=Masses_and_Springs

11. Measuring Simple Harmonic Motion Chapter 11 Section 2

12. Amplitude • The maximum displacement from equilibrium.

13. Period • The time it takes for one complete cycle of motion. • Represented by the symbol T • Unit of seconds

14. Frequency • The number of cycles completed in a unit of time (usually seconds) • Represented by the symbol f • Unit of s-1 (also known as Hertz)

15. Period and Frequency • Period and frequency are inversely related. • f = 1/T and T = 1/f

16. A mass-spring system vibrates exactly 10 times each second. What is its period and frequency? f = 10 cycles per second = 10 Hz T = 1/f = 1/10 s = 0.1 s

17. Factors Affecting Pendulums • For small amplitudes, the period of a pendulum does not depend on the mass or amplitude. • Length and acceleration due to gravity do affect the period of a pendulum.

18. Factors Affecting Mass-Spring Systems • The heavier the mass, the longer the period (more inertia) • The stiffer the spring, the less time it will take to complete one cycle.

19. Properties of Waves Chapter 11 Section 3

20. What is a wave? • A wave is an means by which energy is transferred from one place to another via periodic disturbances

21. Some general terminology… • Pulse – a single disturbance, single cycle • Periodic wave – continuous, repeated disturbances • Sine wave – a wave whose source vibrates with simple harmonic motion • Medium – whatever the wave is traveling through

22. Mechanical Waves • Waves that require a physical medium to travel through. • Examples: sound, disturbance in a slinky • Examples of physical media are water, air, string, slinky.

23. Electromagnetic waves • Waves that do not require a physical medium. • Comprised of oscillating electric and magnetic fields • Examples include x-rays, visible light, radio waves, etc.

24. Transverse Waves • Particles of the medium move perpendicular to the direction of energy transfer • You should be able to identify crests, troughs, wavelength (distance traveled during one full cycle), and amplitude Crest Trough

25. Longitudinal Waves • Particles of the medium move parallel to the direction of energy transfer (slinky demo) • Be able to Identify compressions, rarefactions, wavelengths Compressions Rarefactions

26. Waves transfer energy • Note that, while energy is transferred from point A to point B, the particles in the medium do not move from A to B. • Individual particles of the medium merely vibrate back and forth in simple harmonic motion • The rate of energy transfer is proportional to the square of the amplitude • When amplitude is doubled, the energy carried increases by a factor of 4.

27. Wave speed • Wave speed is determined completely by the characteristics of the medium • For an unchanging medium, wave speed is constant • The speed of a wave can be calculated by multiplying wavelength by frequency. v = f x λ

28. Practice #1 • Q: Microwaves travel at the speed of light, 3.00108 m/s. When the frequency of microwaves is 9.00 109 Hz, what is their wavelength? • A: 0.0300 m

29. Practice #2 • Q: The piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string. • A: 1.30 m

30. 11.3 Problems • Page 387 1-4

31. Wave Interactions Chapter 11 Section 4

32. 5 behaviors common to all waves: • Reflection • Interference • Rectilinear Propagation • Refraction • Diffraction

33. 1. Reflection • The bouncing of a wave when it encounters the boundary between two different media

34. Fixed End Reflection • At a fixed boundary, waves are inverted as they are reflected.

35. Free End Reflection • At a free boundary, waves are reflected on the same side of equilibrium

36. 2. Interference • The combination of two or more waves in a medium at the same time. • Physical matter cannot occupy the same space at the same time, but energy can. • The Superposition Principle describes what happens when waves interfere… • Waves (energy) pass through each other completely unaffected • The medium will be displaced an amount equal to the vector sum of what the waves would have done individually

37. Constructive Interference • Pulses on the same side of equilibrium. • Waves meet, combine according to the superposition principle, and pass through unchanged. • Displacement of medium greater than originals

38. Destructive Interference • pulses on opposite sides of equilibrium. • Waves meet, combine according to the superposition principle, and pass through unchanged. • Displacement of medium less than at least one original

39. Complete Destructive Interference

40. Interference patterns • Interference patterns result from continuous interference. • http://phet.colorado.edu/en/simulation/wave-interference

41. Standing Waves • An interference pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.

42. Standing wave parts • Node – point that maintains zero displacement, complete destructive interference • Antinode – point at which largest displacement occurs, constructive interference

43. Standing waves • Only specific frequency-wavelength combinations will produce standing wave patterns in a given medium.

44. If a string is 4.0 m long, what are three wavelengths that will produce standing waves on this string?

45. 3. Rectilinear Propagation • Waves travel in straight lines • The direction of travel is perpendicular to the wavefront. Wavefront - The set of points in space reached by a wave at the same instant as the wave travels through a medium.

46. Parallel Wavefronts: Circular Wavefronts: Direction of a single wave Direction of a single wave

47. 4. Refraction The bending of the path of a wave as it enters a new medium of different wave speed.

48. 5. Diffraction • The spreading of wave energy around the edges of barriers and obstacles