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 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. Restoring Force • The push or pull that brings the mass back towards equilibrium • for a pendulum it is a component of the bob’s weight. • for a mass-spring system it is tension from the stretch (or compression) of the spring as described by Hooke’s Law F = -kx

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

8. 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

9. Measuring Simple Harmonic Motion Chapter 11 Section 2

10. Amplitude • The maximum displacement from equilibrium.

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

12. 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)

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

14. 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

15. 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.

16. 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.

17. Properties of Waves Chapter 11 Section 3

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

19. 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

20. 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.

21. 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.

22. 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

23. 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

24. 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.

25. Wave speed • Wave speed is determined completely by the characteristics of the medium • For an unchanging medium, wave speed is constant • Calculate speed of a wave by multiplying wavelength by frequency. • v = f x λ

26. 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

27. 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

28. 11.3 Problems • Page 387 1-4

29. Wave Interactions Chapter 11 Section 4

30. Interference • The combination of two or more waves in a medium at the same time. • 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

33. Constructive Interference • Waves are on the same side of equilibrium. • Waves meet, combine according to the superposition principle, and pass through unchanged. • Amplitude larger than originals

34. Destructive Interference • pulses on opposite sides of equilibrium. • Waves meet, combine according to the superposition principle, and pass through unchanged. • Amplitude smaller than at least one original wave

35. Complete Destructive Interference

36. Interference patterns • Interference patterns result from continuous interference. • Check it out!

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

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

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

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

41. Standing wave parts • Node – point that maintains zero displacement • Antinode – point at which largest displacement occurs

42. Standing waves • Only certain frequencies produce standing wave patterns.

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