Chapter 15 Oscillations

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Chapter 15 Oscillations

• 15.1. What is Physics?      
• 15.2. Simple Harmonic Motion      
• 15.3. The Force Law for Simple Harmonic
  Motion      
• 15.4. Energy in Simple Harmonic Motion      
• 15.6. Pendulums      
• 15.7. Simple Harmonic Motion and Uniform Circular
  Motion       
• 15.9. Forced Oscillations and Resonance

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What is Physics?    

• Any measurable quantity that repeats itself at
  regular time intervals is defined as undergoing
  periodic motion.
• If the periodic variation of a physical quantity
  over time has the shape of a sine (or cosine)
  function, we call it a sinusoidal oscillation or
  simple harmonic motion.
• Any periodic motion is superposition of simple
  harmonic motions.

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Simple Harmonic Motion
The maximum excursion from equilibrium is the
amplitude A of the motion
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• The weight of an object on a vertical spring
  stretches the spring by an amount d 0. Simple
  harmonic motion of amplitude A occurs with
  respect to the equilibrium position of the object
  on the stretched spring.

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Displacement
    




                                                                                                    

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Some Observables for Simple Harmonic Motion

• The period T is the time required for one
  complete motional cycle.
• The frequency f of the motion is the number of
  cycles of the motion per second (unit is 1
  cycle/second1 Hz).
• Frequency and period are related according to

    

• Angular frequency ?

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VELOCITY

• The velocity in simple harmonic motion is given
  by

• The maximum magnitude of velocity is

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ACCELERATION

• The acceleration in simple harmonic motion is

• The maximum magnitude of the acceleration is

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Force on an object in Simple Harmonic Motion
Where Km?2 is a constant
Any object under a force of
will be in simple harmonic motion. This force is
called restoring force.
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Km?2 is spring constant
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Check Your Understanding 2

• The drawing shows plots of the displacement x
  versus the time t for three objects undergoing
  simple harmonic motion. Which object, I, II, or
  III, has the greatest maximum velocity?

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Example 1

• The diaphragm of a loudspeaker moves back and
  forth in simple harmonic motion to create sound,
  as in Figure. The frequency of the motion is
  f1.0 kHz and the amplitude is A0.20 mm. (a)
  What is the maximum speed of the diaphragm? (b)
  Where in the motion does this maximum speed
  occur? (c) What is the maximum acceleration of
  the diaphragm, and (d) where does this maximum
  acceleration occur?

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Example 2

• An 0.80-kg object is attached to one end of a
  spring, as in Figure, and the system is set into
  simple harmonic motion. The displacement x of the
  object as a function of time is shown in the
  drawing. With the aid of these data, determine
  (a) the amplitude A of the motion, (b) the
  angular frequency w, (d) the speed of the object
  at t1.0 s, and (e) the magnitude of the objects
  acceleration at t1.0 s.

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Energy in Simple Harmonic Motion
Km?2 is spring constant, then mk/ ?2
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The Simple Pendulum
we can write this restoring torque as
If ? is small, then sin??,
For simple pendulum, ImL2
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The Physical Pendulum
   



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Sample Problem

• In Fig. 15-11a, a meter stick swings about a
  pivot point at one end, at distance h from the
  sticks center of mass.
• (a) What is the period of oscillation T?
• (b) What is the distance L0 between the pivot
  point O of the stick and the center of
  oscillation of the stick?

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Simple Harmonic Motion and Uniform Circular
Motion

• Simple harmonic motion is the projection of
  uniform circular motion on a diameter of the
  circle in which the circular motion occurs.

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Forced Oscillations and Resonance

• Two angular frequencies are associated with a
  system undergoing driven oscillations
• the natural angular frequency ? of the system,
  which is the angular frequency at which it would
  oscillate if it were suddenly disturbed and then
  left to oscillate freely, and
• (2) the angular frequency ?d of the external
  driving force causing the driven oscillations.

How large the displacement amplitude xm is
depends on a complicated function of ?d and ?.
The velocity amplitude vm of the oscillations is
easier to describe it is greatest when
a condition called resonance.