Sullivan Algebra and Trigonometry: Section 9.5

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Sullivan Algebra and Trigonometry Section 9.5

• Objectives of this Section
• Find an Equation for an Object in Simple
  Harmonic Motion
• Analyze Simple Harmonic Motion
• Analyze an Object in Damped Motion
• Graph the Sum of Two Functions

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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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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 an object is attached to a pendulum and
is pulled a distance 7 meters from its rest
position and then released. If the time for one
oscillation is 4 seconds, write an equation that
relates the distance d of the object from its
rest position after time t (in seconds). Assume
no friction.
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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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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 a simple pendulum with a bob of mass 8
grams and a damping factor of 0.7 grams/second is
pulled 15 centimeters to the right of its rest
position and released. The period of the
pendulum without the damping effect is 4 seconds.
(a) Find an equation that describes the position
of the pendulum bob.
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(b) Using a graphing utility, graph the function.
(c) Determine the maximum displacement of the bob
after the first oscillation.