Chapter 5
Simple Harmonic Motion
We begin our study of waves with the study ofsimple harmonic motion. Simple harmonic motion is the
motion that a particle exhibits when under the influence of a force of the form given byHooke’s law(named
for the 17th century English scientist Robert Hooke):
FDkx: (5.1)A force of this form describes, for example, the force on a mass attached to a horizontal spring with spring
constantk, wherekis a measure of the stiffness of the spring. In this caseFis the force exerted by the spring,
andxis the distance of the mass from itsequilibrium position—that is, the “resting” position at which the
mass can be left where it will not oscillate.
It can be shown using the calculus that when the particle is displaced from the equilibrium position and
released, then this force results in an oscillating motion of the particle about the equilibrium position that
varies sinusoidally with timet:
x.t/DAcos.!tCı/: (5.2)Here!is called theangular frequencyof the motion, and measures how fast the particle oscillates back and
forth. The constantAis called theamplitudeof the motion, and is the maximum distance the particle travels
from its equilibrium position,xD 0. The constantıcalled thephase constant, and determines where in its
cycle the particle is at timetD 0. A plot ofx.t/is shown in Fig. 5.1.
Since the sine and cosine function differ only by a phase shift (sincos.=2/; cossin.C
=2/), we could replace the cosine function in Eq. (5.2) with a sine by simply adding an extra=2to the
phase constantı. So either the sine or the cosine can be used equally well to describe simple harmonic motion
(Fig. 5.2); here we will choose to use the cosine function.
The calculus may also be used to find the velocity of the particle at any timet; the result is
v.t/DA!sin.!tCı/: (5.3)Further, it can be shown that the acceleration at any timetis
a.t/DA!^2 cos.!tCı/ (5.4)
D!^2 x.t/: (5.5)Multiplying Eq. (5.5) by the particle massm,wefind
ma.t/DF.t/Dm!^2 x.t/: (5.6)