Chapter 11: Vibrations and Waves

1. Chapter 11: Vibrations and Waves

2. Periodic Motion – any repeated motion with regular time intervals

3. Simple Harmonic Motion – vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium

4. SHM • Speed is max at equilibrium • Speed is least at max displacement • Max force is at max displacement • Max acceleration is at max displacement

5. Simple Harmonic Motion

6. Hooke’s Law Felastic = -kx Spring force= -(spring constant x displacement) SI unit for k = N/m (-) signifies the direction of the spring force is always opposite the direction of the mass’s displacement from equilibrium

7. Example • If a mass of 0.55kg attached to a vertical spring stretches the spring 2.0cm from its original equilibrium position, what is the spring constant?

8. Example • Suppose the spring in the previous example is replaced with a spring that stretches 36 cm from its equilibrium position. • What is the spring constant • Is this spring stiffer or less stiff than the one on the original example?

9. Simple Pendulum • Weight (force of gravity) is the restoring force • SHM for small angles only

12. Bellringer 12/3 • A 76N crate is hung from a spring (k=450N/m). How much displacement is caused by the weight of this crate?

13. Amplitude Measuring Simple Harmonic Motion • Amplitude – maximum displacement from equilibrium • Period – (T) – time it takes to complete one cycle • Frequency (f) – number of cycles per unit of time (Hertz = s-1 )

14. Measuring SHM • Frequency and period are inversely related

16. Example • The reading on a metronome indicates the number of oscillations per minute. What are the frequency and period of the metronome’s vibrations when the metronome is set at 180?

17. Pendulum in SHM • Period of a simple pendulum depends on pendulum length and free-fall acceleration • Length is measured from center of mass of the bob and the pivot point Period of a Simple Pendulum in SHM T = 2π L √ ag Period = 2 π x sqrt of (length divided by free fall)

18. Example • You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12s. How all is the tower?

19. Mass/Spring in SHM • Period of a mass spring system depends on mass and spring constant Period of a Mass-Spring System in SHM T = 2π m √ k Period = 2 π x sqrt of (mass divided by spring constant)

20. Example • A child swings on a playground swing with a 2.5m long chain. • What is the period of the child’s motion? • What is the frequency of vibration?

21. Example • The body of a 1275kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153kg. When driven over a pothole in the road, the frame vibrates with a period of 0.840s. For the first seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring.

22. Bellringer 12/4 • A spring of spring constant 30.0N/m is attached to different masses, and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes: • 2.3kg • 15kg • 1.9kg

23. Properties of Waves • Wave – motion of disturbance that transmits energy • Medium – physical environment through which a disturbance can travel • Mechanical waves – requires a medium • Electromagnetic waves – do not require a medium

24. Pulse wave – single traveling pulse Periodic wave – continuous pulse waves Sine wave – source vibrates with SHM

25. Wave Types • Transverse wave – particles vibrate perpendicularly to the direction the wave is traveling • Wavelength (λ)– distance the wave travels along its path during one cycle

26. Wave Types • Longitudinal wave – particles vibrate parallel to the direction the wave is traveling • Crests – coils are compressed (high density and pressure) • Troughs – coils are stretched (low density and pressure)

28. Period and Frequency • Wave frequency – number of waves that pass a given point in a unit of time • Period – time needed for a complete wavelength

29. Wave Speed Wave Speed v = f λ Speed of wave = frequency x wavelength • Speed of a mechanical wave is constant for any given medium

30. Example • A piano string tuned to middle C vibrates with a frequency of 262 Hz. Assuming the speed of sound in air is 343m/s, find the wavelength of the sound waves produced by the string.

31. Example • A piano emits frequencies that range from a low of about 28Hz to a high of about 4200Hz. Find the range of wavelengths in air attained by this instrument when the speed of sound in air is 340m/s.

32. Bellringer 12/5 • A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in air of 1.35m. • What value does this give for the speed of sound in air? • What would be the wavelength of this same sound in water in which sound travels at 1500m/s

33. Constructive interference – two or more waves are added together to form a resultant wave

35. Destructive interference – two or more waves with displacements on opposite sides of equilibrium position are added together to make a resultant wave

37. Reflection • At a free boundary – waves are reflected • At a fixed boundary – waves are reflected and inverted

38. Free End Reflection

39. Fixed End Reflection

40. Standing waves • Standing wave – results when two waves with same frequency, wavelength, and amplitude travel in opposite directions and interfere • Nodes – point in standing wave where it maintains zero displacement • Antinodes – point in standing wave, halfway between two nodes, at which largest displacement occurs

41. Standing Waves

42. Standing Waves

43. Example • A wave of amplitude 0.30m interferes with a second wave of amplitude 0.20m. What is the largest resultant displacement that may occur?