This document discusses simple harmonic motion and modeling it using sinusoidal functions. Simple harmonic motion is motion that repeats, such as a buoy bobbing up and down in waves or a piano string vibrating. It can be described using a sinusoidal function of the form f(t) = a cos(bt + c) + d, where a controls amplitude, b controls period, c controls phase shift, and d controls vertical shift. The document provides examples of writing sinusoidal functions to model real-world situations involving simple harmonic motion, such as a bobbing buoy or vibrating piano string. It also discusses using the functions to make predictions about the motion.

1. 8.4: Simple Harmonic
Motion & Modeling
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Write a sinusoidal
function to represent
simple harmonic motion.
•Find a sinusoidal model
and use it to make
predictions.

2. Definition
The standard forms for sine
and cosine functions are:
( ) sin( )f t a bt c d= + +
( ) cos( )g t a bt c d= + +
where a,b,c and d are
constants.

3. Important Idea
In the standard form:
( ) sin( )f t a bt c d= + +
( ) cos( )g t a bt c d= + +
•a controls amplitude
•b controls period
•c controls phase shift
•d controls vertical shift
Sketchpad

4. Definitions
Amplitude= a
Period =
2
b
π
Phase Shift=
c
b
−
Vertical Shift = d
From
Lesson
7.4

5. Write a sine function and a
cosine function for the
sinusoidal graph.
Example
2π− 2π−

6. Write a sine function and a
cosine function for the
sinusoidal graph.
Try This
2π− 2π
-7
1

7. Solution
( ) 4sin(2 ) 3f t t= −
( ) 4cos 2 3
2
g t t
π 
= − − ÷
 

8. Definition
2π− 2π−
The wave
shape of
these
graphs are
called
sinusoids
and their
functions
are called
sinusoidal
functions.
3
( ) 3sin 2 1
2
f t t
π 
= − + ÷
 
Sinusoidal
Function
Sinusoid

9. Definition
( ) sin( )f t a bt c d= + +
Motion that can be
described by a function of
the form:
( ) cos( )g t a bt c d= + +
is called simple harmonic
motion. Simple harmonic
motion is motion that repeats.
or

10. Try This
What are examples of
harmonic motion?

11. Example
A buoy in a lake bobs up and
down as waves move past.
The buoy moves 6 feet from
its high point to its low point
every 10 seconds. At t=0 the
buoy is at its high point.
Write an equation using
cosine to describe its motion.

12. Procedure
1.Find the mid-point of the
motion. This value is d.
2.A is the distance from the
midpoint to the highpoint.
3.Find b and c using period.
4.Substitute using a standard
equation, usually cosine.
5.Make predictions using the
equation.

13. Example
When a pianist plays middle
C, the piano string vibrates at
a frequency of 264 cycles per
second. Write an equation of
simple harmonic motion of
the string when A is 1mm.

14. Example
The population of foxes in a
certain forest vary with time.
Records started being kept at
t=0 when a minimum number
of 200 appeared. The next
maximum, 800 foxes,
occurred at t=5.1 years.
Predict the population when
t=7,8,9 &10 years.

15. Try This
The Coast Guard observes a
raft at the bottom of a wave
bobbing up and down a total
distance of 10 feet.The raft
completes a full cycle every
12 seconds. Write an
equation describing the
motion. Where is the raft
after 18 seconds?

16. Solution
5cos
6
t
h
π 
= −  ÷
 
At 18 sec, h=5 ft.

17. Example
a. Find a sinusoidal function to
represent the motion of the
moving weight.
b. Sketch a graph of the function
you found in part a.
c. What is the height of the
weight after 3 sec.
d. When will the height of the
weight be 6 cm. below the
equilibium.

18. Example
Suppose that a weight
hanging from a spring is set
in motion by an upward push.
It takes 5 sec. for it to
complete 1 cycle of moving
from its equilibrium position
to 8 cm. above, then
dropping to 8 cm. below, and
finally returning to its
equilibrium position.

19. Lesson Close
Harmonic motion problems
occur in medicine,
economics, science and
engineering.