Chapter 8: Trigonometric Equations and Applications L8.2 Sine & Cosine Curves: 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 = 2π/ω = 4 → ω = π/2. Consequently the equation of motion is 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.

Ex 2: Given the equation for simple harmonic motion, 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. 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. (b) Frequency = ω/2π = (3π/4) / 2π = ⅜ cycle per second. (a)Max displacement is amplitude, which is 6 cm (c) cm (d) The least positive value: sec.