1. Chapter 13 Periodic Motion

2. Special Case: Simple Harmonic Motion (SHM)

3. Only valid for small oscillation amplitude But SHM approximates a wide class of periodic motion, from vibrating atoms to vibrating tuning forks...

4. Starting Model for SHM: mass m attached to a spring Demonstration

5. Simple Harmonic Motion (SHM) x = displacement of mass m from equilibrium Choose coordinate x so that x = 0 is the equilibrium position If we displace the mass m, a restoring force F acts on m to return it to equilibrium (x=0)

6. Simple Harmonic Motion (SHM) By ‘SHM’ we mean Hooke’s Law holds: for small displacement x (from equilibrium), F = – k x ma = – k x negative sign: F is a ‘restoring’ force (a and x have opposite directions) Demonstration: spring with force meter

8. What is x(t) for SHM? We’ll explore this using two methods The ‘reference circle’: x(t) = projection of certain circular motion A little math: Solve Hooke’s Law

9. The ‘Reference Circle’ P = mass on spring: x(t) Q = point on reference circle P = projection of Q onto the screen

10. The ‘Reference Circle’ P = mass on spring: x(t) Q = point on reference circle A = amplitude of x(t) (motion of P) A = radius of reference circle (motion of Q)

11. The ‘Reference Circle’ P = mass on spring: x(t) Q = point on reference circle f = oscillation frequency of P = 1/T (cycles/sec)  = angular speed of Q = 2  /T (radians/sec)  = 2  f

12. What is x(t) for SHM? P = projection of Q onto screen. We conclude the motion of P is: See additional notes or Fig. 13-4 for 

13. Alternative: A Little Math Solve Hooke’s Law: Find a basic solution: Solve for x(t)

14. v = dx/dt v = 0 at x = A |v| = max at x = 0 a = dv/dt |a| = max at x = A a = 0 at x = 0 See notes on x(t), v(t), a(t)

15. Show expression for 

16. going from 1 to 3, increase one of A, m, k (a) change A : same T (b) larger m : larger T (c) larger k : shorter T Do demonstrations illustrating (a), (b), (c)

17. Summary of SHM for an oscillator of mass m A = amplitude of motion,  = ‘phase angle’ A,  can be found from the values of x and dx/dt at (say) t = 0

18. Energy in SHM As the body oscillates, E is continuously transformed from K to U and back again See notes on v max

19. E = K + U = constant Do Exercise 13-17

20. Summary of SHM x = displacement from equilibrium (x = 0) T = period of oscillation definitions of x and  depend on the SHM

21. Different Types of SHM horizontal (have been discussing so far) vertical (will see: acts like horizontal) swinging (pendulum) twisting (torsion pendulum) radial (example: atomic vibrations)

22. Horizontal SHM

23. Now show: a vertical spring acts the same, if we define x properly.

24. Vertical SHM Show SHM occurs with x defined as shown Do Exercise 13-25

25. ‘Swinging’ SHM: Simple Pendulum Derive  for small x Do Pendulum Demonstrations

26. ‘Swinging’ SHM: Physical Pendulum Derive  for small  Do Exercises 13-39, 13-38

27. Angular SHM: Torsion Pendulum (fiber-disk)

28. Application: Cavendish experiment (measures gravitational constant G). The fiber twists when blue masses gravitate toward red masses

29. Angular SHM: Torsion Pendulum (coil-wheel) Derive  for small 

30. Radial SHM: Atomic Vibrations Show SHM results for small x (where r = R 0 +x)

31. Announcements Homework Sets 1 and 2 (Ch. 10 and 11): returned at front Homework Set 5 (Ch. 14): available at front, or on course webpages Recent changes to classweb access: see HW 5 sheet at front, or course webpages

32. Damped Simple Harmonic Motion See transparency on damped block-spring

33. SHM: Ideal vs. Damped Ideal SHM: We have only treated the restoring force: F restoring = – kx More realistic SHM: We should add some ‘damping’ force: F damping = – bv Demonstration of damped block-spring

34. Damping Force this is the simplest model: damping force proportional to velocity b = ‘damping constant’ (characterizes strength of damping)

35. SHM: Ideal vs. Damped In ideal SHM, oscillator energy is constant: E = K + U, dE/dt = 0 In damped SHM, the oscillator’s energy decreases with time: E(t) = K + U, dE/dt < 0

36. Energy Dissipation in Damped SHM Rate of energy loss due to damping:

37. What is x(t) for damped SHM? We get a new equation of motion for x(t): We won’t solve it, just present the solutions.

38. Three Classes of Damping, b small (‘underdamping’) intermediate (‘critical’ damping) large (‘overdamping’)

39. ‘underdamped’ SHM

40. ‘underdamped’ SHM: damped oscillation, frequency  ´

41. ‘underdamping’ vs. no damping underdamping: no damping (b=0):

42. ‘critical damping’: decay to x = 0, no oscillation can also view this ‘critical’ value of b as resulting from oscillation ‘disappearing’: See sketch of x(t) for critical damping

43. ‘overdamping’: slower decay to x = 0, no oscillation See sketch of x(t) for overdamping

44. Application Shock absorbers: want critically damped (no oscillations) not overdamped (would have a slow response time)

45. Forced Oscillations (Forced SHM)

46. Forced SHM We have considered the presence of a ‘damping’ force acting on an oscillator: F damping = – bv Now consider applying an external force: F driving = F max cos  d t

47. Forced SHM Every simple harmonic oscillator has a natural oscillation frequency (  if undamped,  ´ if underdamped) By appling F driving = F max cos  d t we force the oscillator to oscillate at the frequency  d (can be anything, not necessarily  or  ´)

48. What is x(t) for forced SHM? We get a new equation of motion for x(t): We won’t solve it, just present the solution.

49. x(t) for Forced SHM If you solve the differential equation, you find the solution (at late times, t >> 2m/b)

50. Amplitude A(  d ) Shown (for  = 0): A(  d ) for different b larger b: smaller A max Resonance: A max occurs at  R, near the natural frequency,  = (k/m) 1/2 Do Resonance Demonstrations

51. Resonance Frequency (  R ) A max occurs at  d =  R (where dA/d  d =0):

52. natural, underdamped, forced:  >  ´ >  R natural frequency: underdamped frequency: resonance frequency:

53. Introduction to LRC Circuits (Electromagnetic Oscillations) See transparency on LRC circuit

54. Electric Quantity Counterpart charge Q(t) x(t) current I = dQ/dt v = dx/dt (moving charge) (generates a magnetic field, B)

55. Electrical Concepts electric charge: Q current (moving charge): I = dQ/dt resistance (Q collides with atoms): R voltage (pushes Q through wire): V = RI

56. Voltage (moves charges) resistance R causes charge Q to lose energy: V = RI (voltage = potential energy per unit charge) C and L also cause energy (voltage) changes

57. Circuit Element  (Voltage) R = resistanceV R = RI (Q collides with atoms) C = capacitanceV C = Q/C (capacity to store Q on plate) L = inductanceV L = L(dI/dt) (inertia towards changes in I)

58. Change in Voltage = Change in Energy voltage = potential energy per unit charge recall, around a closed loop:

59. Which looks like:

60. Circuit Element Counterpart 1/C = 1/capacitance k L = inductance m R = resistance b (Extra Credit: Exercise 31-35) Use this table to write our damped SHM as damped electromagnetic oscillations

61. In the LRC circuit, Q(t) acts just like x(t)! underdamped, critically damped, overdamped

62. Driven (and resonance): V driving = V max cos  d t