15 • Oscillatory Motion - ECHSPhysics

The inverse of the period is called the frequency f of the motion. Whereas the period
is the time interval per oscillation, the frequency represents the number of oscillations
that the particle undergoes per unit time interval:
(15.11)
The units of f are cycles per second, or hertz (Hz). Rearranging Equation 15.11 gives
(15.12)
We can use Equations 15.9, 15.10, and 15.11 to express the period and frequency of
the motion for the particle–spring system in terms of the characteristics m and k of the
system as
(15.13)
(15.14)
That is, the period and frequency depend only on the mass of the particle and the
force constant of the spring, and not on the parameters of the motion, such as A or .
As we might expect, the frequency is larger for a stiffer spring (larger value of k) and
decreases with increasing mass of the particle.
We can obtain the velocity and acceleration 2 of a particle undergoing simple harmonic
motion from Equations 15.7 and 15.8:
v dx
dt
f 1
T
T 2
f 1
T
1
2f 2
2 √ m
2 √ k
A sin(t )
a d 2 x
dt 2 2 A cos(t )
(15.15)
(15.16)
From Equation 15.15 we see that, because the sine and cosine functions oscillate
between 1, the extreme values of the velocity v are A. Likewise, Equation 15.16
tells us that the extreme values of the acceleration a are 2 A. Therefore, the maximum
values of the magnitudes of the velocity and acceleration are
v max A √ k
m A
a max 2 A k
m A
(15.17)
(15.18)
Figure 15.6a plots position versus time for an arbitrary value of the phase constant.
The associated velocity–time and acceleration–time curves are illustrated in Figures
15.6b and 15.6c. They show that the phase of the velocity differs from the phase of the
position by /2 rad, or 90°. That is, when x is a maximum or a minimum, the velocity
is zero. Likewise, when x is zero, the speed is a maximum. Furthermore, note that the
2 Because the motion of a simple harmonic oscillator takes place in one dimension, we will denote
velocity as v and acceleration as a, with the direction indicated by a positive or negative sign, as in
Chapter 2.
2
T
k
m
SECTION 15.2 • Mathematical Representation of Simple Harmonic Motion 457
▲ PITFALL PREVENTION
15.4 Two Kinds of
Frequency
We identify two kinds of frequency
for a simple harmonic oscillator—f,
called simply the frequency,
is measured in hertz, and
, the angular frequency, is measured
in radians per second. Be
sure that you are clear about
which frequency is being discussed
or requested in a given
problem. Equations 15.11 and
15.12 show the relationship between
the two frequencies.
Period
Frequency
Velocity of an object in simple
harmonic motion
Acceleration of an object in
simple harmonic motion
Maximum magnitudes of speed
and acceleration in simple
harmonic motion