4.7 Simple Harmonic Motion

Many physical periodic happenings can be represented as a sinusoidal function

* t is time

* is amplitude

* is period

* is frequency

a (maximum displacement)

(time for 1 cycle)

( ) sin ( ) or ( ) cos ( )f x a b t c d f x a b t c d

2

b

2

b

(# of cycles per unit of time)

Which do we choose? sine or cosine?- Whichever seems the most appropriate!

Some Examples (but not all the various kinds!)

Ex 1) An alternating current can be described by a sinusoidal function where c(t) is the current measured in amperes at t seconds. Find the amplitude, period, & frequency.

a b

( ) 18cos 1202

1( ) 18cos120

240

c t t

c t t

c

amp =

Per = 2 1

120 60

frequency = 60

18

Ex 2) The amplitude of a sound wave produced by the note E above middle C is 0.8 and the frequency is 330 cycles per second. Determine the sinusoidal function representing this sound.

a = 0.8

Per =

1330

2Per =

2 2 3302 660

per 1

b

b

frequency = 330 1

330

0.8sin 660y t

Ex 3) A weight hanging from the end of a spring is pulled down 12 cm below its resting place & released. It takes 2.4 seconds to complete one cycle. Determine an equation of motion for the weight.

a = 12

b = ?

2 2 5Per 2.4

2.4 6b

b

(12 down from rest)

512cos

6y t

Use the cosine functionWhy cosine? Starts low… not in middleBut it’s LOW instead of high How to fix this?Flip over x-axisMake equation negative

12 cm

Ex 4) A Ferris wheel is 80 ft. in diameter & rises 86 ft from the ground. Each revolution of the wheel takes 28 seconds. Express the height of a rider as a function of time t if the rider is at the bottom when t = 0.

• start @ bottom …• bottom not at 0 at 6• d = 6 + 40

Per 28 sec

2 228

28 14b

b

g40cos 46

14y t

40 ft

40 ft

diam = 80 ft radius = 40 ft86 ft

6 ft

86 6 8040

2 2a

6

86

neg. cos

= 46 (middle)

Homework

#408 Pg 233 #1, 7, 9, 13, 15, 17, 19, 27, 30, 33, 37, 39