Simple Harmonic Motion

1. Simple Harmonic Motion Things that vibrate § 13.1–13.3

2. Hooke’s Law • Force Law: F = –kx • F = force exerted by the spring • k = spring constant (characteristic of the particular spring) • x = distance spring is displaced from equilibrium

3. d2x –kx = m dt2 Conditions of Motion • Newton’s second law: F = ma • Force F depends on position by Hooke’s law: F = –kx –kx=ma • tells how motion changes at each position • second-orderordinary differential equation

4. Group Work • Plot the mass’s position vs. time. • Select at least four events on the plot. At each event, draw arrows to indicate the mass’s velocity and acceleration. A Hooke’s law spring is fixed at one end and has a mass attached to the other end. The spring is initially at its neutral position. The mass is pulled, stretching the spring. The mass is then released.

5. CPS Question The net force on a Hooke’s law object is • zero at the top and bottom • maximum at the top and bottom • minimum but not zero at the top and bottom

6. CPS Question The acceleration of a Hooke’s law object is • zero at the top and bottom • maximum at the top and bottom • minimum but not zero at the top and bottom

7. CPS Question The velocity of a Hooke’s law object is • maximum at the equilibrium position • maximum at the top and bottom • maximum midway between equilibrium and top or bottom

8. Simulation Spreadsheet From class web site, right-click on the “Numerical simulation” link. Enter parameters to produce 1–2 cycles. • Describe how position changes with time. • What is the velocity when position is at an extreme? At 0? • What is the force when velocity is at an extreme? At 0? • Where is max/min U? • Where is max/min K?

9. Simulation Spreadsheet From class web site, right-click on the “Numerical simulation” link. Enter parameters to produce 1–2 cycles. • What happens to the period when k changes? • What happens to the period when m changes?

10. CPS Question All other things being equal, if the mass of a Hooke’s law oscillator increases, its period decreases. does not change. increases.

11. CPS Question All other things being equal, if the stiffnessk of a Hooke’s law spring increases, its period decreases. does not change. increases.

12. Group Work • First MasteringPhysics problem, parts A-E.

13. F0 F0 F0 F0 F0 F0 Uniform Circular Motion • Centripetal force F = mv2/r inwards • Constant magnitude F0; direction depends on position q • Force in y-direction is proportional to –y

14. Uniform Circular Motion • Angle changes at a steady rate. • Projection on y-axis has Hooke’s law force. • So, projection on y-axis must have Hooke’s law motion too! • What is the projection of an angle on the y-axis?

15. w =   k/m d2x –kx = m dt2 Group Work Verify that • if x(t) = A cos (wt + f), • where • A and f are any real constants

16. Equation of Motion x(t) = A cos (wt + f), • Amplitude A • Angular frequency w • Initial phase angle f

17. Angular frequencyw • radians per unit time (units: 1/s = Hz) • w = 2pf • w =   k/m Period and Frequency • Period T • time of one cycle (units: s) • Frequency f • cycles per unit time (units: 1/s = Hz) • f = 1/T

18. Another form x(t) = A cos (wt + f) = C cos (wt) + S sin (wt) where C = A cos (f) and S = –A sin (f)

19. Group Work • Fourth MasteringPhysics problem (Exercise 13.4).

20. Group Work • Second MasteringPhysics problem (“Harmonic Oscillator Acceleration”).

21. Group Work • Fifth MasteringPhysics problem (Exercise 13.10), parts A–C.

22. Initial Conditions • Given m, k, x0, and v0, how does one find the equations of motion? • m and k give w. • x0, v0, and w give A. • x0/A and v0/(Aw) give f. • Now you can do MasteringPhysics Exercise 13.26. • Can practice/explore with “SHM parameters” spreadsheet

23. CPS Question The potential energy of an oscillating mass is greatest at its extreme positions. at its equilibrium (middle) position. between the middle and an extreme position.

24. CPS Question The kinetic energy of an oscillating mass is greatest at its extreme positions. at its equilibrium (middle) position. between the middle and an extreme position.

25. Constant Mechanical Energy E = 1/2kx2 + 1/2mv2

26. 1 1 kx2 PE= 2 2 kA2 PE+ KE = constant = Energy • Potential energy of a stretched spring : • Conservation of energy: where A is the oscillation amplitude. (This of course ignores the sullen reality of energy dispersal by friction and drag.)

27. KE = 0 PE = max KE = max PE = 0 KE = 0 PE = max Energy y time

28. Group Work • First MasteringPhysics problem, parts F and G

29. Effect of Gravity • Less than you might expect: • Changes equilibrium position x = 0 • Does not change k

30. gravity Spring + Gravity spring alone force 0 0 position

31. spring alone gravity spring + gravity 0 0 Spring + Gravity force 0 position

32. Spring + Gravity different equilibrium position same k net force 0 0 position