A. Introduction 1. Oscillations: motions that repeat themselves a)Swinging chandeliers, boats bobbing at anchor, oscillating guitar strings, pistons in car engines 2. Understanding periodic motion essential for later study of waves, sound, alternating electric currents, and light 3.An object in periodic motion experiences restoring forces or torques that bring it back toward an equilibrium position 4.Those same forces cause the object to “overshoot” the equilibrium position XII. Periodic Motion

1.Definitions a) Frequency (f) = number of oscillations that are completed each second [f] = hertz = Hz = 1 oscillation per sec = 1 s –1 b) Period = time for one complete oscillation (or cycle): T = 1/f(XII.B.1) XII.B. Simple Harmonic Motion (SHM)

2.Displacement x(t): x(t) = x m cos(  t +  ), where(XII.B.2) x m = Amplitude of the motion (  t +  ) = Phase of the motion )  = Phase constant (or phase angle) : depends on the initial displacement and velocity  = Angular frequency = 2  T = 2  f (rad/s)(XII.B.3) 3.Simple harmonic motion = periodic motion is a sinusoidal function of time (represented by sine or cosine function) XII.B. Simple Harmonic Motion (SHM)

4. velocity of a particle moving with SHM: 5.The acceleration for SHM: XII.B. Simple Harmonic Motion (SHM) (XII.B.4) (XII.B.5)

1. From Newton’s 2 nd Law: F = ma = –m  2 x(XII.C.1) 2. This result (a restoring force that is proportional to the displacement but opposite in sign) is the same as Hooke’s Law for a spring: F = –kx, where k = m  2 (XII.C.2) XII.C. Force Law for SHM (XII.C.3) (XII.C.4)

1.Elastic potential energy U = 1/2kx 2 = 1/2kx m 2 cos 2 (  t +  )(XII.D.1) 2.Kinetic energy K = 1/2mv 2 = 1/2kx m 2 sin 2 (  t +  )(XII.D.2) 3. Total mechanical energy = E = U + K E = 1/2kx m 2 cos 2 (  t +  ) + 1/2 kx m 2 sin 2 (  t +  ); = 1/2kx m 2 (XII.D.3) XII.D. Energy in SHM

1. A simple pendulum consists of a particle of mass m (bob) suspended from one end of an unstretchable, massless string of length L that is fixed at the other end a)Consider the Forces acting on the bob: F  = –mgsin  = mg(s/L); with sin  = s/L  XII.E.1  b)If  is small (  15 0 or so) then sin    :  F  ~ –mg  = –mgs/L.  XII.E.2  XII.E. Pendula W = mg  ^   L s

simple pendulum 1. A simple pendulum consists of a particle of mass m (bob) suspended from one end of an unstretchable, massless string of length L that is fixed at the other end c)This equation is the angular equivalent of the condition for SHM (a = –  2 x), so:  = (mg/L/ m) 1/2 = (g/L) 1/2 and (XII.E.3) T = 2  (L/g) 1/2 (XII.E.4) XII.E. Pendula

Example Problem #12 A pendulum bob swings a total distance of 4.0 cm from end to end and reaches a speed of 10.0 cm/s at the midpoint. Find the period of oscillation. x m = 0.02 m; v m = 0.10 m/s