1. CPS Question The period of a simple harmonic oscillator depends on: (Add together the numbers for all correct choices and enter the sum.) 1. The spring constant k. 2. The mass m. 4. The maximum amplitude A. 8. The gravitational field g.

2. Pendulums almost follow Hooke’s law § 13.4–13.6

3. Angular Oscillations • Angular Hooke’s law: t = –kq • Angular Newton’s second law: t = Ia • So –kq =Ia • General Solution: q = Qcos(wt + f) • where w2 = k/I; Q and f are constants

4. Group Work • MasteringPhysics problem 13.34

5. Simple Pendulum L q m

6. CPS Question The period of a simple pendulum depends on: (Add together the numbers for all correct choices and enter the sum.) 1. The length L. 2. The mass m. 4. The maximum amplitude Q. 8. The gravitational field g.

7. L T = wR q wR = mg cosq q w = mg m wT = mg sinq Simple Pendulum Fnet = –wT = –mg sinq

8. Small-Angle Approximation For small q (in radians) q sin q  tan q

9. Simple Pendulum Fnet= –wT = –mg sinq t = –L mg sinq t  –L mg q = –kq k = Lmg I = mL2 L q m Lmg mL2 w2 = k/I = = g/L w is independent of mass m

10. Group Work Find the length of a simple pendulum whose period is 2 s. About how long is the pendulum of a grandfather clock?

11. Group Work • Do the lab.

12. CPS Question If a pendulum is an extended object with its center of mass a distance L from the pivot, its period is • longer than • the same as • shorter than The period of a simple pendulum of length L.

13. Physical Pendulum Source: Young and Freedman, Figure 13.23.

14. w = k mgd = I I Physical Pendulum Fnet = –mg sinq tnet = –mgd sinq Approximately Hooke’s law t –mgdq

15. New Powers • Now you can do MasteringPhysics problem 13.52

16. Example: Suspended Rod Mass M, center of mass at L/2 Physical pendulum Simple pendulum L L L 2 2 1 1 3 4 I =  ML2 I =  ML2 harder to turn easier to turn

17. Example Problem A pendulum is constructed of a circular disk pivoting at a point midway between its center of mass and its edge. What must its radius be for it to have a period of 1 s?

18. Physical Pendulum • What is the period whend >> R?