Simple Harmonic Motion
shows a sequence of “snapshots” of a simple oscillating system. a particle moving repeatedly back and forth about the origin of an x axis. I this section we simply describe the motion. Later, we shall discuss how to attain such motion. One important property of oscillatory motion is its requency, or number of oscillations that are completed each second. The symbol for frequency is f, and its SI unit is the hertz (abbreviated Hz), where I hertz = I Hz = I oscillation per second = I s -I. (16-1 ) Related to the frequency is the period T of the motion. which is the time for one complete oscillation (or cycle); that is. (16-2) Any motion that repeats itself at regular intervals is called periodic motion or harmonic motion. We are interested here in otion that repeats itself in a particular way-namely. like that in Fig. 16-10. For such motion the displacement x of theA sequence of “snapshots” (taken at equal time intervals) showing the position of a particle as it oscillates back and forth about the origin along an x axis, between the limits +x••and -x . The vector arrows are scaled )0 indicate the speed of the particle. The speed is maximum when the .particle is at the origin and zero when it is at :tx••.If the time I is chosen to be zero when the particle is at +x.,. then the particle returns to +~m at I = T. where T is the period of the motion. The motion is then repeated. (b) A graph of x as a function of time for the motion of (0). panicle from the origin is given as a function of time by
x(t) = x”, cos(wt + c/J) (displacement). (16-3)
in which x m’ IAJ. and c/Jare constants. This motion is called simple harmonic motion
(SHM), a term that means the periodic motion is a sinusoidal function of time.
Equation 16-3, in which the sinusoidal function is a cosine function. is graphed in
. (You can get that graph by rotating ‘counterclockwise by 90′
and then connecting the successive locations of the particle with a curve.) The quantities
that detennine the shape of the graph are displayed in Fig. 16-2 with their
names. We now shall define those quantities. .
Tbe quantity xm’ called the amplitude of the motion. is a positive constant
whose value depends on how the motion was started. The subscript m stands fOI
maximum because the amplitude is the magnitude of the maximum displacement of
the particle in “either direction. The cosine function in Eq. 16-3 varies between the
limits ❗ 1. so the displacement x(t) varies between the limits :!:xm’
The time-varying quantity (WI + c/J) in Eq. 16-3 is called the phase of the
motion, and the constant c/Jis called the phase constant (or phase angle). The value
of c/Jdepends on the displacement and velocity of the particle at time t = O. For the
x(t) plots of Fig. 16-3a, the phase constant c/Jis zero.
To interpret the constant w, called the angular frequency of the motion, we
first note that the displacement x(t) must return to its initial value after one period l’
of the motion; that is, X(I) must equal xCt + T) for all t. To simplify this analysis.
let us put c/J= 0 in Eq, 16-3. From that equation we then can write
Xnt cos ~ = Xm cos.w(t + T~. (16-4)
The cosine function first repeats itself when its argument (the phase) has increase
by 217″rad, so Eq. 16-4 gives us
wet + T) = WI + 217″
Thus, from Eq. 16-2 the angular frequency is
wT = 217″.
In each case. the blue curve is obtained from Eq. 16-3
with •• = 9. (a)”1’bered curse diffe•••from the blue curve only .
in that its amplitlldex';’ is areater (the red-curve extremes.of
dirpacelRSt are higher and lower). (b) The red curve diffen
.from the blue’curve only in that its period is T’ = Tf}. (the red
curve is compressed horizhntaJly).(e) The reelcurve differs
from the blue curve only in that ~ •• – .,,/4 Tadt1IdIer than zero
• (the negative value of ~ shifts the red curve to the right).
The SI unit of angular -iRqu, ency is the radian per second. (To be consistent,
then, t/J must-he-in radians.) Figu re 16-3 compares x(t) for two simple hannonic
motions that differ either in arnplitu de, in period (and thus in frequency and angular
frequency), or in phase constant. nHECKPOINT 1: A particle undergoing si~ple harmonic oscillation of periof Ttlike /’!..
that in Fig. 16-1} is at -x'” at time 1= O. Lilt at -x”” at +x••’ at 0, between – 0,
or between 0 and +x., when (a) 1= 2.00T, ‘b) 1= 3.S0T, and (c) 1= S.25T.