Chapter 13

1. Chapter 13 Periodic Motion

2. Special Case: Simple Harmonic Motion (SHM)

3. Simple Harmonic Motion (SHM) • 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) w = angular speed of Q = 2p /T (radians/sec) w = 2p 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 q

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

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

15. Show expression for f

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 SHMfor an oscillator of mass m • A = amplitude of motion, f = ‘phase angle’ • A,f 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 vmax

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 w 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. Horizontal SHM • 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 w for small x Do Pendulum Demonstrations

26. ‘Swinging’ SHM: Physical Pendulum Derive w for small q 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 w for small q

30. Radial SHM:Atomic Vibrations Show SHM results for small x (where r = R0+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: • Frestoring = – kx • More realistic SHM: • We should add some ‘damping’ force: • Fdamping = – 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 w´

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 aslow response time)

45. Forced Oscillations (Forced SHM)

46. Forced SHM • We have considered the presence of a ‘damping’ force acting on an oscillator:Fdamping = – bv • Now consider applying an external force: Fdriving = Fmax coswdt

47. Forced SHM • Every simple harmonic oscillator has a natural oscillation frequency • (w if undamped, w´ if underdamped) • By appling Fdriving = Fmax coswdt we force the oscillator to oscillate at the frequency wd (can be anything, not necessarily w or w´)

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(wd) • Shown (for f = 0):A(wd) for different b • larger b: smaller Amax • Resonance:Amax occurs at wR, near the natural frequency,w = (k/m)1/2 Do Resonance Demonstrations