Understanding Simple Harmonic Motion and Pendulum Dynamics

1. Simple Harmonic Motion

2. Periodic Motion • defined: motion that repeats at a constant rate • equilibrium position: forces are balanced

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.

4. Periodic Motion • for the spring: ΣF = Fw + Fs ΣFy = mgy + (-kΔy)

5. Periodic Motion • Damping: the effect of friction opposing the restoring force in oscillating systems

6. Periodic Motion • Restoring force (Fr): the net force on a mass that always tends to restore the mass to its equilibrium position

7. Simple Harmonic Motion • defined: periodic motion controlled by a restoring force proportional to the system displacement from its equilibrium position

8. Simple Harmonic Motion • The restoring force in SHM is described by: Fr x = -kΔx • Δx = displacement from equilibrium position

9. Simple Harmonic Motion • Table 12-1 describes relationships throughout one oscillation

10. Simple Harmonic Motion • Amplitude: maximum displacement in SHM • Cycle: one complete set of motions

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

12. 1 1 f = T = T f Simple Harmonic Motion • Frequency (f) and period (T) are reciprocal quantities.

13. Reference Circle • Circular motion has many similarities to SHM. • Their motions can be synchronized and similarly described.

14. m T = 2π k Reference Circle • The period (T) for the spring-mass system can be derived using equations of circular motion:

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

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 • 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 • Centripetal force adds to the tension (Tp): Tp = Tw΄+ Fc , where: Tw΄ = Tw = |mg|cosθ 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. l T = 2π |g| Small Amplitude • For small initial displacement angles:

28. l T = 2π |g| Small Amplitude • Longer pendulum arms produce longer periods of swing.

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

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

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

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. 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