almost follow Hooke’s law Pendulums almost follow Hooke’s law § 13.6

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

Simple Pendulum L q m Massless, inextensible string/rod Point-mass bob 3

Poll 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.

Simple Pendulum Force SFT = –wT = –mg sinq L T = wR + mv2/L q w = mg q wT = mg sinq wR = mg cosq SFT = –wT = –mg sinq

Simple Pendulum Torque SFT = –wT = –mg sinq = LFT = –L mg sinq Restoring torque 6

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

Simple Pendulum t = –L mg sinq t  –L mg q = –kq k = Lmg I = mL2 Lmg w2 = k/I = = g/L w is independent of mass m (w is not the angular speed of the pendulum) 8

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

Think Question An extended object with its center of mass a distance L from the pivot, has a moment of inertia greater than the same as less than a point mass a distance L from the pivot.

Poll 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. 11

Physical Pendulum Source: Young and Freedman, Figure 13.23.

Physical Pendulum k mgd = w = Fnet = –mg sinq tnet = –mgd sinq Approximately Hooke’s law t  –mgdq w = k I mgd I = I = Icm + md 2

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

Damped and Forced Oscillations Introducing non-conservative forces § 13.7–13.8

Damping Force Such as viscous drag v Drag opposes motion: F = –bv

Poll Question How does damping affect the oscillation frequency? Damping increases the frequency. Damping does not affect the frequency. Damping decreases the frequency.

Light Damping x(t) = Ae cos(w't + f) – w' = If w' > 0: Oscillates –bt 2m x(t) = Ae cos(w't + f) k – b2 w' = m 4m2 If w' > 0: Oscillates Frequency slower than undamped case Amplitude decreases over time

Critical Damping – w' = If w' = 0: x(t) = (C1 + C2t) e–at k m 4m2 b2 – If w' = 0: x(t) = (C1 + C2t) e–at No oscillation If displaced, returns directly to equilibrium

Overdamping – w' = If w' is imaginary: x(t) = C1 e–a t + C2 e–a t k m 4m2 b2 – If w' is imaginary: x(t) = C1 e–a t + C2 e–a t 1 2 No oscillation If displaced, returns slowly to equilibrium

Energy in Damping Damping force –bv is not conservative Total mechanical energy decreases over time Power = F·v = –bv·v = –bv2

Forced Oscillation Periodic driving force F(t) = Fmax cos(wdt)

Forced Oscillation If no damping If wd = w', amplitude increases without bound

Resonance If lightly damped: greatest amplitude when wd = w' Critical or over-damping (b ≥ 2 km): no resonance Source: Young and Freedman, Fig. 13.28