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.

DefinitionThe 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.

Important IdeaIn 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

DefinitionsAmplitude=a

Period =2

b

Phase Shift=c

b

Vertical Shift =

d

From

Lesson

7.4

Write a sine function and a cosine function for the sinusoidal graph.

Example

2 2

Write a sine function and a cosine function for the sinusoidal graph.

Try This

2 2

-7

1

Solution

( ) 4sin(2 ) 3f t t

( ) 4cos 2 32

g t t

Definition

2 2

The wave shape of these graphs are called sinusoids and their functions are called sinusoidal functions.

3( ) 3sin 2 1

2f t t

Sinusoid

al Function

Sinusoi

d

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

Try ThisWhat are examples of harmonic motion?

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.

Procedure1.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.

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.

ExampleThe 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.

Try ThisThe 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?

Solution

5cos6

th

At 18 sec, h=5 ft.

Examplea. 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.

ExampleSuppose 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.

Lesson Close

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