Precalculus 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves
Objective: able to build a model for and analyze an object in simple harmonic motion; analyze an object in damped motion;
graph the sum of two functions.
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>0.
The swinging of a pendulum, the vibrations of a tuning fork, and the bobbing of a weight attached to a coiled
spring are examples of simple harmonic motion.
The amplitude of vibration is the distance from the object’s rest position to its point of greatest displacement.
The period of a vibrating object is the time required to complete one vibration.
Note: If friction is present, the amplitude will decrease with time to 0. This type of motion is an example of
damped motion.
Simple harmonic motion is related to circular motion.
Consider a circle centered at the origin with radius, a.
http://www.animations.physics.unsw.edu.au/jw/SHM.htm#projection
An object that moves on a coordinate axis so that its distance, d, from the origin at time, t, with angular
d = a cos(ωt ) or d = a sin (ωt ) where a and ω > 0 are constants, moves with
speed, ω, is given by either
simple harmonic motion.
This motion has an amplitude
2π
a and a period of
ω
The frequency, f, of an object in simple harmonic motion is the number of oscillations per unit time.
Frequency is the reciprocal of the period, so
f =
ω
2π
,ω > 0
1. An object attached to a coiled spring is pulled down a distance of 10 cm from its rest position and then
released. Assuming that the motion is simple harmonic motion with period 3 seconds, write an equation that
relates the distance, d, of the object from its rest position after t seconds. Also assume that the positive
direction of the motion is up.
2. The distance, d (in meters), that an object travels in time, t (in seconds), is given by d=4sin(2t)
a. Describe the motion of the object.
b. What is the maximum displacement from its resting point?
c. What is the time required for one oscillation?
d. What is the frequency?
Most physical phenomena are affected by friction and other resistive forces. These forces remove energy from
a moving system and thereby damp its motion. The amplitude will decrease with time to zero.
A function that describes this phenomenon maintains a sinusoidal component, but the amplitude of this
component will decrease with time in order to account for the damping effect.
Damped Motion:
The displacement, d, of an oscillating object from its rest position at time, t, is given by
− bt

b2
d = ae 2 m cos t ω 2 −

4m 2





where
b is a damping factor or damping coefficient and m is the mass of the
oscillating object. This motion has a displacement of
a
2π
at
t = 0, and a period of ω
under simple harmonic
motion (no damping effect).
Notice: when b = 0 (zero damping) we have the formula for simple harmonic motion with amplitude
2π
a
and a
period of ω
3. Suppose that a simple pendulum with a bob of mass 10 grams and a damping factor of 0.8 grams/second
is pulled 20 centimeters from its at-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
b. Use a graphing calculator to graph the function found in part a)
c. Determine the maximum displacement of the bob after the first oscillation
d. What happens to the displacement of the bob as time increases without bound?
Many physical and biological applications require the graph of the sum of two functions. For example, if two
tones are emitted, the sound produced is the sum of the waves produced by the two tones. On a touch-tone
phone, each button produces a unique sound. The sound produced is the sum of two tones. To graph the
sum of two (or more) functions, we can use the method of adding y-coordinates of the functions.
4. Use the method of adding y-coordinates to graph the function f(x) = x + cos x
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