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

Pendulums almost follow Hooke’s law § 14.4–14.6

Angular Oscillators Angular Hooke’s law:  = –  Angular Newton’s second law:  = I So –  = I General Solution:  =  cos(  t +  ) where  2 =  / I ;  and  are constants

Simple Pendulum  L 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 . 8.The gravitational field g.

Simple Pendulum Force  F T = –w T = –mg sin   L m T = w R + mv 2 /L w = mg  w T = mg sin  w R = mg cos 

Simple Pendulum Torque  F T = –w T = –mg sin   = LF T = –L mg sin  Restoring torque  L m

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

Simple Pendulum  = –L mg sin   –L mg  = –   = Lmg I = mL 2  L m  2 =  / I = = g/L Lmg mL 2  is independent of mass m (  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 A.greater than B.the same as C.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 A.longer than B.the same as C.shorter than The period of a simple pendulum of length L.

Physical Pendulum Source: Young and Freedman, Figure

Physical Pendulum F net = –mg sin   net = –mgd sin  Approximately Hooke’s law   –mgd   =  I mgd I = I = I cm + md 2

Example: Suspended Rod Mass M, center of mass at L/2 I = ML L L 2 Physical pendulumSimple pendulum L 2 harder to turneasier to turn

Physical Pendulum What is the period when d >> R?