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C
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Oscillations
15-1 SIMPLE HARMONIC MOTION
After reading this module, you should be able to . . .
15.01 Distinguish simple harmonic motion from other types of
periodic motion.
15.02 For a simple harmonic oscillator, apply the relationship
between position x and time t to calculate either if given a
value for the other.
15.03 Relate period T, frequency f, and angular frequency v.
15.04 Identify (displacement) amplitude x m , phase constant
(or phase angle) f, and phase vt ϩ f.
15.05 Sketch a graph of the oscillator’s position x versus time
t, identifying amplitude x m and period T.
15.06 From a graph of position versus time, velocity versus
time, or acceleration versus time, determine the amplitude
of the plot and the value of the phase constant f.
15.07 On a graph of position x versus time t describe the effects of changing period T, frequency f, amplitude x m , or
phase constant f.
15.08 Identify the phase constant f that corresponds to the
starting time (t ϭ 0) being set when a particle in SHM is
at an extreme point or passing through the center point.
15.09 Given an oscillator’s position x(t) as a function of time,
find its velocity v(t) as a function of time, identify the velocity amplitude v m in the result, and calculate the velocity at
any given time.
15.10 Sketch a graph of an oscillator’s velocity v versus time t,
identifying the velocity amplitude v m .
15.11 Apply the relationship between velocity amplitude v m ,
angular frequency v, and (displacement) amplitude x m .
15.12 Given an oscillator’s velocity v(t) as a function of time,
calculate its acceleration a(t) as a function of time, identify
the acceleration amplitude a m in the result, and calculate
the acceleration at any given time.
15.13 Sketch a graph of an oscillator’s acceleration a versus
time t, identifying the acceleration amplitude a m .
15.14 Identify that for a simple harmonic oscillator the acceleration a at any instant is always given by the product of a
negative constant and the displacement x just then.
15.15 For any given instant in an oscillation, apply the relationship between acceleration a, angular frequency v, and displacement x.
15.16 Given data about the position x and velocity v at one
instant, determine the phase vt ϩ f and phase constant f.
15.17 For a spring–block oscillator, apply the relationships between spring constant k and mass m and either period T or
angular frequency v.
15.18 Apply Hooke’s law to relate the force F on a simple harmonic oscillator at any instant to the displacement x of the
oscillator at that instant.
● The frequency f of periodic, or oscillatory, motion is the
number of oscillations per second. In the SI system, it is
measured in hertz: 1 Hz ϭ 1 s
Ϫ1
.
● The period T is the time required for one complete oscillation, or cycle. It is related to the frequency by T ϭ 1/f.
● In simple harmonic motion (SHM), the displacement x(t) of a
particle from its equilibrium position is described by the equation
x ϭ x m cos(vt ϩ f) (displacement),
in which x m is the amplitude of the displacement, vt ϩ f is
the phase of the motion, and f is the phase constant. The
angular frequency v is related to the period and frequency of
the motion by v ϭ 2p/T ϭ 2pf.
● Differentiating x(t) leads to equations for the particle’s
SHM velocity and acceleration as functions of time:
v ϭ Ϫvx m sin(vt ϩ f) (velocity)
and
a ϭ Ϫv
2
x m cos(vt ϩ f) (acceleration).
In the velocity function, the positive quantity vx m is the velocity amplitude v m . In the acceleration function, the positive
quantity v
2
x m is the acceleration amplitude a m .
● A particle with mass m that moves under the influence of a
Hooke’s law restoring force given by F ϭ Ϫkx is a linear simple harmonic oscillator with
(angular frequency)
and
(period).
T ϭ 2p A
m
k
v ϭ A
k
m
Key Ideas
Learning Objectives
C
H
A
P
T
E
R
1
5
Oscillations
15-1 SIMPLE HARMONIC MOTION
After reading this module, you should be able to . . .
15.01 Distinguish simple harmonic motion from other types of
periodic motion.
15.02 For a simple harmonic oscillator, apply the relationship
between position x and time t to calculate either if given a
value for the other.
15.03 Relate period T, frequency f, and angular frequency v.
15.04 Identify (displacement) amplitude x m , phase constant
(or phase angle) f, and phase vt ϩ f.
15.05 Sketch a graph of the oscillator’s position x versus time
t, identifying amplitude x m and period T.
15.06 From a graph of position versus time, velocity versus
time, or acceleration versus time, determine the amplitude
of the plot and the value of the phase constant f.
15.07 On a graph of position x versus time t describe the effects of changing period T, frequency f, amplitude x m , or
phase constant f.
15.08 Identify the phase constant f that corresponds to the
starting time (t ϭ 0) being set when a particle in SHM is
at an extreme point or passing through the center point.
15.09 Given an oscillator’s position x(t) as a function of time,
find its velocity v(t) as a function of time, identify the velocity amplitude v m in the result, and calculate the velocity at
any given time.
15.10 Sketch a graph of an oscillator’s velocity v versus time t,
identifying the velocity amplitude v m .
15.11 Apply the relationship between velocity amplitude v m ,
angular frequency v, and (displacement) amplitude x m .
15.12 Given an oscillator’s velocity v(t) as a function of time,
calculate its acceleration a(t) as a function of time, identify
the acceleration amplitude a m in the result, and calculate
the acceleration at any given time.
15.13 Sketch a graph of an oscillator’s acceleration a versus
time t, identifying the acceleration amplitude a m .
15.14 Identify that for a simple harmonic oscillator the acceleration a at any instant is always given by the product of a
negative constant and the displacement x just then.
15.15 For any given instant in an oscillation, apply the relationship between acceleration a, angular frequency v, and displacement x.
15.16 Given data about the position x and velocity v at one
instant, determine the phase vt ϩ f and phase constant f.
15.17 For a spring–block oscillator, apply the relationships between spring constant k and mass m and either period T or
angular frequency v.
15.18 Apply Hooke’s law to relate the force F on a simple harmonic oscillator at any instant to the displacement x of the
oscillator at that instant.
● The frequency f of periodic, or oscillatory, motion is the
number of oscillations per second. In the SI system, it is
measured in hertz: 1 Hz ϭ 1 s
Ϫ1
.
● The period T is the time required for one complete oscillation, or cycle. It is related to the frequency by T ϭ 1/f.
● In simple harmonic motion (SHM), the displacement x(t) of a
particle from its equilibrium position is described by the equation
x ϭ x m cos(vt ϩ f) (displacement),
in which x m is the amplitude of the displacement, vt ϩ f is
the phase of the motion, and f is the phase constant. The
angular frequency v is related to the period and frequency of
the motion by v ϭ 2p/T ϭ 2pf.
● Differentiating x(t) leads to equations for the particle’s
SHM velocity and acceleration as functions of time:
v ϭ Ϫvx m sin(vt ϩ f) (velocity)
and
a ϭ Ϫv
2
x m cos(vt ϩ f) (acceleration).
In the velocity function, the positive quantity vx m is the velocity amplitude v m . In the acceleration function, the positive
quantity v
2
x m is the acceleration amplitude a m .
● A particle with mass m that moves under the influence of a
Hooke’s law restoring force given by F ϭ Ϫkx is a linear simple harmonic oscillator with
(angular frequency)
and
(period).
T ϭ 2p A
m
k
v ϭ A
k
m
Key Ideas
Learning Objectives