1. Simple Harmonic Motion

2. Periodic Motion defined: motion that repeats at a constant rate
equilibrium position: forces are balanced
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3. Periodic Motion
For the spring example, the mass is pulled down to y = -A and then released.
Two forces are working on the mass: gravity (weight) and the spring.
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4. Periodic Motion for the spring: ΣF = Fw + Fs ΣFy = mgy + (-kΔy)
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5. Periodic Motion
Damping: the effect of friction opposing the restoring force in oscillating systems
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6. Periodic Motion
Restoring force (Fr): the net force on a mass that always tends to restore the mass to its equilibrium position
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7. Simple Harmonic Motion
defined: periodic motion controlled by a restoring force proportional to the system displacement from its equilibrium position
We must realize that we do this—we pre-suppose.
There must be some starting point.

8. Simple Harmonic Motion
The restoring force in SHM is described by:
Fr x = -kΔx
We must realize that we do this—we pre-suppose.
There must be some starting point.
Δx = displacement from equilibrium position

9. Simple Harmonic Motion
Table 12-1 describes relationships throughout one oscillation
We must realize that we do this—we pre-suppose.
There must be some starting point.

10. Simple Harmonic Motion
Amplitude: maximum displacement in SHM
Cycle: one complete set of motions
We must realize that we do this—we pre-suppose.
There must be some starting point.

11. Simple Harmonic Motion
Period: the time taken to complete one cycle
Frequency: cycles per unit of time
1 Hz = 1 cycle/s = s-1
We must realize that we do this—we pre-suppose.
There must be some starting point.

12. Simple Harmonic Motion
Frequency (f) and period (T) are reciprocal quantities.
We must realize that we do this—we pre-suppose.
There must be some starting point.
f = T 1
T = f 1

13. Reference Circle Circular motion has many similarities to SHM.
Their motions can be synchronized and similarly described.
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14. Reference Circle
The period (T) for the spring-mass system can be derived using equations of circular motion:
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T = 2π k m

15. Reference Circle This equation is used for Example 12-1.
The reciprocal of T gives the frequency.
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Are they important?
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T = 2π k m

16. Periodic Motion and the Pendulum

17. Overview
Galileo was among the first to scientifically study pendulums.

18. Overview
The periods of both pendulums and spring-mass systems in SHM are independent of the amplitudes of their initial displacements.

19. Pendulum Motion
An ideal pendulum has a mass suspended from an ideal spring or massless rod called the pendulum arm.
The mass is said to reside at a single point.

20. Pendulum Motion
l = distance from the pendulum’s pivot point and its center of mass
center of mass travels in a circular arc with radius l.

21. Pendulum Motion forces on a pendulum at rest: weight (mg)
tension in pendulum arm (Tp)
at equilibrium when at rest

22. Restoring Force Fr = Tp + mg
When the pendulum is not at its equilibrium position, the sum of the weight and tension force vectors moves it back toward the equilibrium position.
Fr = Tp + mg

23. Restoring Force Tp = Tw΄+ Fc , where: Tw΄ = Tw = |mg|cos θ Fc = mvt²/r
Centripetal force adds to the tension (Tp):
Tp = Tw΄+ Fc , where:
Tw΄ = Tw = |mg|cos θ
Fc = mvt²/r

24. Restoring Force
Total acceleration (atotal) is the sum of the tangential acceleration vector (at) and the centripetal acceleration.
The restoring forces causes this atotal.

25. Restoring Force
A pendulum’s motion does not exactly conform to SHM, especially when the amplitude is large (larger than π/8 radians, or 22.5°).

26. Small Amplitude
defined as a displacement angle of less than π/8 radians from vertical
SHM is approximated

27. Small Amplitude
For small initial displacement angles:
T = 2π
|g| l

28. Small Amplitude Longer pendulum arms produce longer periods of swing.

29. Small Amplitude
The mass of the pendulum does not affect the period of the swing.
T = 2π
|g| l

30. Small Amplitude
This formula can even be used to approximate g (see Example 12-2).
T = 2π
|g| l

31. Physical Pendulums
mass is distributed to some extent along the length of the pendulum arm
can be an object swinging from a pivot
common in real-world motion

32. Physical Pendulums
The moment of inertia of an object quantifies the distribution of its mass around its rotational center.
Abbreviation: I
A table is found in Appendix F of your book.

33. Physical Pendulums
period of a physical pendulum:
T = 2π
|mg|l I

34. Oscillations in the Real World

35. Damped Oscillations
Resistance within a spring and the drag of air on the mass will slow the motion of the oscillating mass.

36. Damped Oscillations
Damped harmonic oscillators experience forces that slow and eventually stop their oscillations.

37. β is a friction proportionality constant
Damped Oscillations
The magnitude of the force is approximately proportional to the velocity of the system:
fx = -βvx
β is a friction proportionality constant

38. Damped Oscillations
The amplitude of a damped oscillator gradually diminishes until motion stops.

39. Damped Oscillations
An overdamped oscillator moves back to the equilibrium position and no further.

40. Damped Oscillations
A critically damped oscillator barely overshoots the equilibrium position before it comes to a stop.

41. Driven Oscillations
To most efficiently continue, or drive, an oscillation, force should be added at the maximum displacement from the equilibrium position.

42. Driven Oscillations
The frequency at which the force is most effective in increasing the amplitude is called the natural oscillation frequency (f0).

43. Driven Oscillations
The natural oscillation frequency (f0) is the characteristic frequency at which an object vibrates.
also called the resonant frequency

44. Driven Oscillations terminology: in phase pulses driven oscillations
resonance

45. Driven Oscillations A driven oscillator has three forces acting on it:
restoring force
damping resistance
pulsed force applied in same direction as Fr

46. Driven Oscillations
The Tacoma Narrows Bridge demonstrated the catastrophic potential of uncontrolled oscillation in 1940.

47. Waves

48. Waves defined: oscillations of extended bodies
medium: the material through which a wave travels

49. Waves disturbance: an oscillation in the medium
It is the disturbance that travels; the medium does not move very far.

50. Graphs of Waves
Waveform graphs
Vibration graphs

51. Types of Waves
longitudinal wave: disturbance that displaces the medium along its line of travel
example: spring

52. Types of Waves
transverse wave: disturbance that displaces the medium perpendicular to its line of travel
example: the wave along a snapped string

53. Longitudinal Waves Any physical medium can carry a longitudinal wave.
Rarefaction zone: molecules are spread apart and have lower density and pressure
Compression zone: molecules are pushed together and have higher density and pressure

54. Longitudinal Waves travel faster in solids than gases
water waves have both longitudinal and transverse components—a “combination” wave

55. Periodic Waves
carry information and energy from one place to another

56. Periodic Waves
amplitude (A): the greatest distance a wave displaces a particle from its average position
A = ½(ypeak - ytrough)
A = ½(xmax - xmin)

57. Periodic Waves
wavelength (λ): the distance from one peak (or compression zone) to the next, or from one trough (or rarefaction zone) to the next

58. Periodic Waves
A wave completes one cycle as it moves through one wavelength.
A wave’s frequency (f) is the number of cycles completed per unit of time

59. Periodic Waves wave speed (v): the speed of the disturbance
for periodic waves:
λf = v

60. Sound Waves
longitudinal pressure waves that come from a vibrating body and are detected by the ears
cannot travel through a vacuum; must pass through a physical medium

61. Sound Waves
travel faster through solids than liquids, and faster through liquids than gases
have three characteristics:

62. Loudness
the interpretation your hearing gives to the intensity of the wave
intensity (Is): amount of power transported by the wave per unit area
measured in W/m²

63. Loudness
a sound must be ten times as intense to be perceived as twice as loud
sound is measured in decibels (dB)

64. Pitch related to the frequency
high frequency is interpreted as a high pitch
low frequency is interpreted as a low pitch
20 Hz to 20,000 Hz

65. Quality results from combinations of waves of several frequencies
fundamental and harmonics
why a trumpet sounds different than an oboe

66. Sound Waves
All three characteristics affect the way sound is perceived.

67. Doppler Effect
related to the relative velocity of the observer and the sound source
actual sound emitted by the object does not change
an approaching object has a higher pitch than if there were no relative velocity
an object moving away has a lower pitch than if there were no relative velocity

68. Mach Speed
measurement is dependent on the composition and density of the atmosphere
speed of sound changes with altitude