Harmonic Motion
Lesson 2.8
The Spring Has Sprung
Consider a weight on a spring
that is bouncing up and down
It moves alternately above
and below an equilibrium
point
The movement can be modeled by
y a cos 2 f t or y a sin 2 f t
2
Simple Harmonic Motion
For the functions
y a cos 2 f t or y a sin 2 f t
t is time
f is the frequency
1/f is the period
|a| is the amplitude
3
Try It Out
For each of the following, find the
Amplitude
Frequency
Period 2 t
y cos
3 3
t
y 2sin
4
y 5cos 2 t
4
Try It the Other Way
Given
Frequency = .8 cps
Amplitude = 4
Write the function
What if
Amplitude = 3.5
Assume maximum
Period = 0.5 sec displacement occurs
when t = 0
5
Spring Constants
For a particular spring system
When mass = m
When spring constant = k
The frequency is calculated
1 k
f
2 m
See exercise
Given k and m, substitute into function 172
28, page
y a cos 2 f t
6
Damped Harmonic Motion
What if the a is not a constant
Rather it is a function
As time, t increases, the motion is
lessened by a dampening influence
y enx sin 2 f t
Experiment with spreadsheet
Where is dampening important on an
automobile? 7
Damp Your Motion
.06 x
Given y 12e cos t
How many complete oscillations
during time interval 0 ≤ t ≤10
How long until the absolute value of
the displacement is always less than
0.01
Hint: use calculator
8
Damp Your Motion
Count oscillations for 0 ≤ t ≤10
.06 x
y 12e cos t
For when movement is less than .01
zoom in
draw lines at y = ± .01
9
Damp Your Motion
Double check values at the peak
y 12e.06 x cos t
Ask calculator for intersections
10
Assignment
Lesson 2.8
Page 208
Exercises 1 – 35 odd
11