SCI 340 L27 Simple Pendulum Question The period of a simple harmonic oscillator depends on: A. The spring constant k. B. The mass m. C. The maximum amplitude A. D. The gravitational field g.

SCI 340 L27 Simple Pendulum almost follow Hooke’s law Pendulums almost follow Hooke’s law § 10.4

Torsional Oscillator Angular Hooke’s law: t = –kq SCI 340 L27 Simple Pendulum Torsional Oscillator 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 SCI 340 L27 Simple Pendulum Simple Pendulum q L m Massless, inextensible string/rod Point-mass bob 4

Question The period of a simple pendulum depends on: A. The length L. B. The mass m. C. The maximum amplitude A. D. The gravitational field g.

Simple Pendulum Force SFT = –wT = –mg sinq L T = wR + mv2/L q SCI 340 L27 Simple Pendulum Simple Pendulum Force q L m T = wR + mv2/L w = mg q wT = mg sinq wR = mg cosq SFT = –wT = –mg sinq

Simple Pendulum Torque SCI 340 L27 Simple Pendulum Simple Pendulum Torque q L m SFT = –wT = –mg sinq = LFT = –L mg sinq Restoring torque 7

Small-Angle Approximation SCI 340 L27 Simple Pendulum 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 SCI 340 L27 Simple Pendulum Simple Pendulum q L m t = –L mg sinq t  –L mg q = –kq k = Lmg I = mL2 Lmg mL2 w2 = k/I = = g/L w is independent of mass m (w is not the angular speed of the pendulum) 9

Board Work About how long is the pendulum of a grandfather clock? SCI 340 L27 Simple 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?