Oscillations: motions that repeat themselves

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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

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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 s1
  
• Period time for one complete oscillation (or
  cycle)
• T 1/f (XII.B.1)

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XII.B. Simple Harmonic Motion (SHM)

• 2. Displacement x(t)
• x(t) xmcos(wt f), where (XII.B.2)
• xm Amplitude of 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)

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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)
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XII.C. Force Law for SHM

• From Newtons 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 Hookes Law for a spring
• F kx, where k mw2 (XII.C.2)

(XII.C.4)
(XII.C.3)
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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)

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XII.E. Pendula

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

• 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
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XII.E. Pendula

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 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)

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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