Simple Harmonic Motion

Report
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 = 2π/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
Ex 2: Given the equation for simple harmonic motion d  6 cos t ,
4
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.
3
 4  6 cos(3 )  6  1  6 cm
4
3
3
3
 3 5
(d) d (t )  6 cos t  0  cos t  0  t  , , ,...
4
4
4
2 2 2
(c) d (4)  6 cos
The least positive value:
3

2
t   t  sec.
4
2
3

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