Chapter 8: Trigonometric Equations and Applications

L8.2 Sine & Cosine Curves: Simple Harmonic Motion

Simple Harmonic Motion

The periodic nature of the trigonometric functions is useful for describing motion of a point on an object that vibrates, oscillates, rotates or is moved by wave motion. For ex, consider a ball that is bobbing up and down on the end of a spring.  10cm is the maximum distance that the ball moves vertically upward or downward from its equilibrium (at rest) position.  It takes 4 seconds for the ball to move from its maximum displacement above zero to its maximum displacement below zero and back again.  With ideal conditions of perfect elasticity and no friction or air resistance, the ball would continue to move up and down in a uniform manner.

 Motion of this nature can be described by a sine or cosine function and is called simple harmonic motion.  For this particular example, the amplitude is 10cm, the period is 4 seconds, and the frequency is ¼ cps (cycles per second).

Simple Harmonic Motion

A point that moves on a coordinate line is in

simple harmonic motion

if its distance d from the origin at time t is given by either d = a sin ωt or d = a cos ωt where a and ω are real numbers such that ω > 0. The motion has

amplitude

|a|,

period

2π/ω and

frequency

ω/2π.

Ex 1: Write the equation for simple harmonic motion of a ball suspended from a spring that moves vertically 8 cm from rest. It takes 4 seconds to go from its maximum displacement to its minimum and back. What is the frequency of the motion?

Since the spring is at equilibrium (d = 0) when t=0, we will use the equation d = a sin ωt. The maximum displacement from 0 is 8 cm and the period is 4 sec so amplitude = |a| = 8, period = 2 π/ω = 4 → ω = π/2.

Consequently the equation of motion is d  8 sin  2 t The frequency = ω/2π = (π/2)/(2π) = ¼ cycle per second.

* Note that ω (lower case omega) is just a stand-in for the coefficient, B.

Since time, unlike an angle, is not measured in π, ω frequently has π in it for cancelation purposes.

Simple Harmonic Motion (cont)

A point that moves on a coordinate line is in

simple harmonic motion

if its distance d from the origin at time t is given by either d = a sin ωt or d = a cos ωt where a and ω are real numbers such that ω > 0. The motion has

amplitude

|a|,

period

2π/ω and

frequency

ω/2π.

3  where d is in cm and t is in seconds, find: (a) the maximum displacement, (b) the frequency, (c) the value when t = 4, and (d) the least positive value for t for which d = 0.

(a) Max displacement is amplitude, which is 6 cm (b) Frequency = ω/2π = (3π/4) / 2π = ⅜ cycle per second. (d) d ( 4 ) d ( t )  6 cos  6 cos 3  3 4  4 t  4  6 cos( 3  )   0  cos   1   6 3  4 t  0  3  t   3  4 2 4 t   2  t  2 , 3  2 , 5  2 ,...