Transcript Sullivan Algebra and Trigonometry: Section 9.5

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
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.
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.
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
d  a cost
or
d  a sin t
where a and  > 0 are constants, moves with
simple harmonic motion. The motion has
amplitude a and period 2  .
The frequency f of an object in simple
harmonic motion is the number of
oscillations per unit of time. Thus,
 1
f 
=
2 T
 0
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.
d  a cost
2
Amplitude =  7  7
2 


Period =
4
4 2


d  7 cos t
2
Suppose that the distance d (in
centimeters) an object travels in time t (in
seconds) satisfies the equation
d  15 sin 4t
(a) Describe the motion of the object.
Simple harmonic
(b) What is the maximum displacement
from its resting position?
A   15  15 centimeters
Suppose that the distance d (in
centimeters) an object travels in time t (in
seconds) satisfies the equation
d  15 sin 4t
(c) What is the time required for one
oscillation?
2 
Period  T 
 seconds
4 2
(d) What is the frequency?
1 2
frequency  f   oscillations per second
T 
Damped Motion
The displacement d of an oscillating object
from its at rest position at time t is given by
2


b
bt 2 m
2
d  ae
cos   2 t 
4m 

where b is a damping factor (damping
coefficient) and m is the mass of the
oscillating object.
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.
m  8; b  0.7, a  15
2

4


2
m  8; b  0.7, a  15


2
2


b
bt 2 m
2
d  ae
cos   2 t 
4m 

2
2


0.7

 0.7 t 2 (8)
d  15e
cos   
2


4(8)
 2
2

 0.49 
 0.7 t 16
d  15e
cos

t
256 
 4

t 

(b) Using a graphing utility, graph the
function.
2

 0.49 
 0.7 t 16
d  15e
cos

t
256 
 4
(c) Determine the maximum
displacement of the bob after the first
oscillation.