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. 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-order ordinary differential equation –kx = m d2xd2x dt 2

4. Group Work a.Plot the mass’s position vs. time. b.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 A.zero at the top and bottom B.maximum at the top and bottom C.minimum but not zero at the top and bottom

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

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

8. Simulation Spreadsheet 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? From class web site, right-click on the “Numerical simulation” link. Enter parameters to produce 1–2 cycles.

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

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

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

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

13. Uniform Circular Motion Centripetal force F = mv 2 /r inwards Constant magnitude F 0 ; direction depends on position  F0F0 F0F0 F0F0 F0F0 F0F0 F0F0 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. Group Work Verify that if x(t) = A cos (  t +  ), where A and  are any real constants –kx = m d2xd2x dt 2  = k/m

16. Equation of Motion x(t) = A cos (  t +  ), Amplitude A Angular frequency  Initial phase angle 

17. Period and Frequency Period T –time of one cycle (units: s) Frequency f –cycles per unit time (units: 1/s = Hz) –f = 1/T Angular frequency  – radians per unit time (units: 1/s = Hz) –  = 2  f –  = k/m

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

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, x 0, and v 0, how does one find the equations of motion? –m and k give . –x 0, v 0, and  give A. –x 0 /A and v 0 /(A  ) give . 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 A.at its extreme positions. B.at its equilibrium (middle) position. C.between the middle and an extreme position.

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

25. Constant Mechanical Energy E = 1/2 kx 2 + 1/2 mv 2

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

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

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. Spring + Gravity position force 0 0 spring alone gravity

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

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