Section 4.5

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Section 4.5 Simple Harmonic Motion Damped
Motion Combining Waves
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Equilibrium is the location where the object is
at rest
A
C
B equilibrium
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The amplitude of vibration is the distance from
the equilibrium position to its point of greatest
displacement (A or C).
The period of a vibrating object is the time
required to complete one vibration - that is, the
time required to go from point A through B to C
and back to A.
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Simple harmonic motion is a special kind of
vibrational motion in which the acceleration a of
the object is directly proportional to the
negative of its displacement d from its rest
position. That is, a -kd, k gt 0.
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Theorem Simple Harmonic Motion
An object that moves on a coordinate axis so that
its distance d from the origin at time t is given
by either
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The frequency f of an object in simple harmonic
motion is the number of oscillations per unit of
time. Thus,
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Suppose that the distance d (in centimeters) an
object travels in time t (in seconds) satisfies
the equation
(a) Describe the motion of the object.
Simple harmonic
(b) What is the maximum displacement from its
resting position?
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Suppose that the distance d (in centimeters) an
object travels in time t (in seconds) satisfies
the equation
(c) What is the time required for one oscillation?
(d) What is the frequency?
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The simple harmonic motion on the previous slides
assumes that there is no resistance to the
motionno friction. The motion of such systems
will continue indefinitely In most physical
systems, this is not the case. Friction or
resistive forces remove energy from the system,
dampening its motion.
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Theorem Damped Motion
The displacement d of an oscillating object from
its at rest position at time t is given by
where b is a damping factor (damping coefficient)
and m is the mass of the oscillating object.
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Suppose that a simple pendulum with a bob of
mass 10 grams and a damping factor of .8
grams/sec is pulled 20 centimeters from its at
rest position and released. The period of the
pendulum without the damping effect is 4 seconds.
20 cm
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1. Find the equation that describes the position of
   the pendulum bob.
2. Using a graphing utility, graph the function
   found in part (a).
3. Determine the maximum displacement of the bob
   after the first oscillation.
4. What happens to the displacement of the bob as
   time increases without bound?

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1. Find the equation that describes the position of
   the pendulum bob.

m 10, a 20, b 0.8 The period under simple
harmonic motion is 4 seconds.
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b. Using a graphing utility, graph the function
found in part (a).
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c. Determine the maximum displacement of the bob
after the first oscillation.
About 17 cm
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d. What happens to the displacement of the bob as
time increases without bound?
Notice that the dampened motion formula contains
the factor
What happens to this factor as t grows without
bound, t -gt 8?
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• Read pages 305-307 on Combining Waves
• To graph a function h(x) f(x) g(x)
• Graph f(x) and g(x)
• Select several x values from the domains of the
  f(x) and g(x)
• Calculate f(x) and g(x) for the values of x
  selected in 2.
• The value of h(x) for a given value of x will be
  the sum of f(x) and g(x).

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Example If we pick x 2 and discover that for
some given f(x) and g(x) we have f(2) 4, and
g(2) 7, then the value of h(2) 4 7 11,
and the point (2,11) appears on the graph of
h(x).