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Introduction Oscillations : motions that repeat themselves

XII. Periodic Motion. Introduction Oscillations : motions that repeat themselves a) Swinging chandeliers, boats bobbing at anchor, oscillating guitar strings, pistons in car engines

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Introduction Oscillations : motions that repeat themselves

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  1. XII. Periodic Motion • Introduction • Oscillations: motions that repeat themselves a) Swinging chandeliers, boats bobbing at anchor, oscillating guitar strings, pistons in car engines • 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

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

  3. XII.B. Simple Harmonic Motion (SHM) 2. Displacement x(t): x(t) = xmcos(wt + f), where (XII.B.2) xm = Amplitudeof the motion (wt + f) = Phase of the motion f = Phase constant (or phase angle) : depends on the initial displacement and velocity w = Angular frequency = 2p/T = 2pf (rad/s) (XII.B.3) 3. Simple harmonic motion = periodic motion is a sinusoidal function of time (represented by sine or cosine function)

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

  5. XII.C. Force Law for SHM • From Newton’s 2nd Law: F = ma = –mw2x (XII.C.1) • 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 = mw2 (XII.C.2) (XII.C.4) (XII.C.3)

  6. XII.D. Energy in SHM 1. Elastic potential energy U = 1/2kx2 = 1/2kxm2cos2(wt + f) (XII.D.1) 2. Kinetic energy K = 1/2mv2 = 1/2kxm2sin2(wt + f) (XII.D.2) • Total mechanical energy = E = U + K E = 1/2kxm2cos2(wt + f) + 1/2 kxm2sin2(wt + f); = 1/2kxm2 (XII.D.3)

  7. XII.E. Pendula • 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 • Consider the Forces acting on the bob: Fq = –mgsinq= mg(s/L); with sinq = s/L (XII.E.1) • If q is small ( 150 or so) then sinq  q: • Fq ~ –mgq = –mgs/L. (XII.E.2) q L s ^ q q W = mg

  8. XII.E. Pendula • A simple pendulumconsists 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 = –w2 x), so: w = (mg/L/ m)1/2 = (g/L)1/2 and (XII.E.3) T = 2p(L/g)1/2 (XII.E.4)

  9. 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. xm = 0.02 m; vm = 0.10 m/s

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