Many physical phenomena can be modeled with simple harmonic motion.

the swinging of a pendulum

radio and television waves

light and sound waves water waves

a spring-mass system

An object that moves on a coordinate

axis so that its distance from its rest

position at time is

simple harm

given by either

cos or sin

where and 0 are constants,

moves wit onic motionh

d

t

d a t d a t

a

.

The amplitude is |a|. 2The period is

The frequency is the number of oscillations per unit time. It is the reciprocal of the period.

2

2f

cosd a tSimple harmonic motion can be expressed with either sine or cosine but since this object starts at 4 when t = 0, cosine makes sense. If you used sine, you'd have to use a phase shift to have a different value than 0 at t = 0.

4cosd t2

2T

2

2

4cosd t

x

y

x

An object attached to a coiled spring is pushed up a distance 4 units from its rest position and then released. Assuming that the motion is simple harmonic with period 2 seconds, write an equation that relates the displacement d of the object from its rest potion after t seconds. Assume that the positive direction of the motion is up.

If we knew the equation of motion for an object, we could determine some things about it.

The equation modeling the motion of an object is:

This negative tells us that after t = 0, it will be stretched down first

x

Since we know , we can find the period

2 2

4 2T

3sin 4d t

This tells us that the motion will repeat

every seconds. 1.62

x

1.6

What is the frequency?

It is the reciprocal of the period so 2

.64

This means in one second, it has completed .64 oscillations.

x

.64

Often the motion of an object is affected by friction or other resistive forces. These slow the object or damp its motion.

The equation modeling an oscillating object with damping is:

damping factor

mass of the object

2

222

cos4

bt

m bd t ae t

m

2

222

cos4

bt

m bd t ae t

m

What does the graph of look like? 2

bt

me

b and m are positive so this is a graph of e to the negative something t

t

d

here is te

tehere is

So as time increases, this function decreases and approaches zero

2

222

cos4

bt

m bd t e a t

m

What does the graph of look like? 2

22

cos4

ba t

m

This is a basic cosine graph with amplitude a and

period 22

2

2

4bm

t

2

222

cos4

bt

m bd t e a t

m

so this part wants to oscillate

but this part which multiplies the amplitude is going to zero

t

--- and this is what happens!

t

Notice the amplitude decreasing exponentially

When you push a button on your phone, the tone made is actually two tones of different frequencies added together.

To obtain the graph of the sum of two or more functions, we can simply

use the function property (f + g)(x) = f(x) + g(x)

So we'll find the value of each function at x and then add the values together

x x

Let's find the graph of f(x) = 2 sin x + sin 2x

x 2sin x sin 2x

0 2(0) 0 0

2sin sin 2x x

0

32 3

2

2 3sin

3 2

33 2.6

2

2 1 2 sin 0 2

32 3

2

4 3sin

3 2

33 0.9

2

3

2

2

3

2(0) 0 0 04

3

32 3

2

8 3sin

3 2

33 0.9

2

3

2

2( 1) 2 sin 3 0 2

5

3

32 3

2

10 3

sin3 2

33 2.6

2

x

Plot these points

The purple is 2 sin x and the red is sin 2x. See how adding the values makes the blue graph.

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au