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
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.
L T = wR + mv2/L q wR = mg cosq q w = mg m wT = mg sinq Simple Pendulum Force SFT = –wT = –mg sinq
Simple Pendulum Torque SFT= –wT = –mg sinq = LFT = –L mg sinq Restoring torque L q m
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 L q m Lmg mL2 w2 = k/I = = g/L w is independent of mass m (w is not the angular speed of the pendulum)
Board Work 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.
Physical Pendulum Source: Young and Freedman, Figure 13.23.
w = k mgd = I I Physical Pendulum Fnet = –mg sinq tnet = –mgd sinq Approximately Hooke’s law t –mgdq I = Icm + md2
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
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 –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
k – b2 w' = m 4m2 Critical Damping If w' = 0: x(t) = (C1 + C2t) e–at • No oscillation • If displaced, returns directly to equilibrium
k – b2 w' = If w' is imaginary: x(t) =C1 e–at + C2 e–a t m 4m2 1 2 Overdamping • No oscillation • If displaced, returns slowly to equilibrium
= F·v = –bv·v Energy in Damping • Damping force –bv is not conservative • Total mechanical energy decreases over time = –bv2 • Power
Forced Oscillation Periodic driving force F(t) = Fmax cos(wdt)
Forced Oscillation If no damping If wd = w', amplitude increases without bound
Critical or over-damping (b ≥ 2 km): no resonance Resonance If lightly damped: greatest amplitude when wd = w' Source: Young and Freedman, Fig. 13.28