Chapter 13 Simple Harmonic Motion
“We are to admit no more causes of natural things
than such as are both true and sufficient to explain
their appearances.” Isaac Newton
13.1 Introduction to Periodic Motion
Periodic motion is any motion that repeats itself in equal intervals of time. The
uniformly rotating earth represents a periodic motion that repeats itself every 24
hours. The motion of the earth around the sun is periodic, repeating itself every 12
months. A vibrating spring and a pendulum also exhibit periodic motion. The period
of the motion is defined as the time for the motion to repeat itself. A special type of
periodic motion is simple harmonic motion and we now proceed to investigate it.
13.2 Simple Harmonic Motion
An example of simple harmonic motion is the vibration of a mass m, attached to a
spring of negligible mass, as the mass slides on a frictionless surface, as shown in
figure 13.1. We say that the mass, in the unstretched position, figure 13.1(a), is in
its equilibrium position. If an applied force FA acts on the mass, the mass will be
displaced to the right of its equilibrium position a distance x, figure 13.1(b). The
distance that the spring stretches, obtained from Hooke’s law, is
FA = kx
The displacement x is defined as the distance the body moves from its equilibrium
position. Because FA is a force that pulls the mass to the right, it is also the force
that pulls the spring to the right. By Newton’s third law there is an equal but
opposite elastic force exerted by the spring on the mass pulling the mass toward the
left. Since this force tends to restore the mass to its original position, we call it the
restoring force FR. Because the restoring force is opposite to the applied force, it is
given by
FR = −FA = −kx (13.1)
When the applied force FA is removed, the elastic restoring force FR is then the only
force acting on the mass m, figure 13.1(c), and it tries to restore m to its equilibrium
position. We can then find the acceleration of the mass from Newton’s second law as
ma = FR
= −kx
Thus,
a=−k x (13.2)
m
13-1
, Chapter 13 Simple Harmonic Motion
Figure 13.1 The vibrating spring.
Equation 13.2 is the defining equation for simple harmonic motion. Simple
harmonic motion is motion in which the acceleration of a body is directly
proportional to its displacement from the equilibrium position but in the opposite
direction. A vibrating system that executes simple harmonic motion is sometimes
called a harmonic oscillator. Because the acceleration is directly proportional to the
displacement x in simple harmonic motion, the acceleration of the system is not
constant but varies with x. At large displacements, the acceleration is large, at small
displacements the acceleration is small. Describing the vibratory motion of the mass
m requires some new techniques and we will do so in section 13.3. However, let us
first look at the motion from a physical point of view. The mass m in figure 13.2(a)
is pulled a distance x = A to the right, and is then released. The maximum restoring
force on m acts to the left at this position because
FRmax = −kxmax = −kA
The maximum displacement A is called the amplitude of the motion. At this
position the mass experiences its maximum acceleration to the left. From equation
13-2
, Chapter 13 Simple Harmonic Motion
Figure 13.2 Detailed motion of the vibrating spring.
13.2 we obtain
a=− kA
m
The mass continues to move toward the left while the acceleration
continuously decreases. At the equilibrium position, figure 13.2(b), x = 0 and hence,
from equation 13.2, the acceleration is also zero. However, because the mass has
inertia it moves past the equilibrium position to negative values of x, thereby
compressing the spring. The restoring force FR now points to the right, since for
negative values of x,
FR = −k(−x) = kx
The force acting toward the right causes the mass to slow down, eventually coming
to rest at x = −A. At this point, figure 13.2(c), there is a maximum restoring force
pointing toward the right
FRmax = −k(−A)max = kA
13-3
, Chapter 13 Simple Harmonic Motion
and hence a maximum acceleration
amax = − k (−A) = k A
m m
also to the right. The mass moves to the right while the force FR and the
acceleration a decreases with x until x is again equal to zero, figure 13.2(d). Then FR
and a are also zero. Because of the inertia of the mass, it moves past the
equilibrium position to positive values of x. The restoring force again acts toward
the left, slowing down the mass. When the displacement x is equal to A, figure
13.2(e), the mass momentarily comes to rest and then the motion repeats itself. One
complete motion from x = A and back to x = A is called a cycle or an oscillation.
The period T is the time for one complete oscillation, and the frequency f is the
number of complete oscillations or cycles made in unit time. The period and the
frequency are reciprocal to each other, that is,
f= 1 (13.3)
T
The unit for a period is the second, while the unit for frequency, called a hertz, is
one cycle per second. The hertz is abbreviated, Hz. Also note that a cycle is a
number not a dimensional quantity and can be dropped from the computations
whenever doing so is useful.
13.3 Analysis of Simple Harmonic Motion -- The
Reference Circle
As pointed out in section 13.2, the acceleration of the mass in the vibrating spring
system is given by equation 13.2 as
a=−k x
m
Since the acceleration a = d2x/dt2, equation 13.2 can be written as
d2x = − k x (13.4)
dt2 m
Equation 13.4 is a second-order differential equation that completely describes the
simple harmonic motion of the mass m. Unfortunately, the solution of such
differential equations is beyond the scope of this course. We also can not use the
kinematic equations derived in chapter 2 because they were based on the
assumption that the acceleration of the system was a constant. As we can see from
13-4
“We are to admit no more causes of natural things
than such as are both true and sufficient to explain
their appearances.” Isaac Newton
13.1 Introduction to Periodic Motion
Periodic motion is any motion that repeats itself in equal intervals of time. The
uniformly rotating earth represents a periodic motion that repeats itself every 24
hours. The motion of the earth around the sun is periodic, repeating itself every 12
months. A vibrating spring and a pendulum also exhibit periodic motion. The period
of the motion is defined as the time for the motion to repeat itself. A special type of
periodic motion is simple harmonic motion and we now proceed to investigate it.
13.2 Simple Harmonic Motion
An example of simple harmonic motion is the vibration of a mass m, attached to a
spring of negligible mass, as the mass slides on a frictionless surface, as shown in
figure 13.1. We say that the mass, in the unstretched position, figure 13.1(a), is in
its equilibrium position. If an applied force FA acts on the mass, the mass will be
displaced to the right of its equilibrium position a distance x, figure 13.1(b). The
distance that the spring stretches, obtained from Hooke’s law, is
FA = kx
The displacement x is defined as the distance the body moves from its equilibrium
position. Because FA is a force that pulls the mass to the right, it is also the force
that pulls the spring to the right. By Newton’s third law there is an equal but
opposite elastic force exerted by the spring on the mass pulling the mass toward the
left. Since this force tends to restore the mass to its original position, we call it the
restoring force FR. Because the restoring force is opposite to the applied force, it is
given by
FR = −FA = −kx (13.1)
When the applied force FA is removed, the elastic restoring force FR is then the only
force acting on the mass m, figure 13.1(c), and it tries to restore m to its equilibrium
position. We can then find the acceleration of the mass from Newton’s second law as
ma = FR
= −kx
Thus,
a=−k x (13.2)
m
13-1
, Chapter 13 Simple Harmonic Motion
Figure 13.1 The vibrating spring.
Equation 13.2 is the defining equation for simple harmonic motion. Simple
harmonic motion is motion in which the acceleration of a body is directly
proportional to its displacement from the equilibrium position but in the opposite
direction. A vibrating system that executes simple harmonic motion is sometimes
called a harmonic oscillator. Because the acceleration is directly proportional to the
displacement x in simple harmonic motion, the acceleration of the system is not
constant but varies with x. At large displacements, the acceleration is large, at small
displacements the acceleration is small. Describing the vibratory motion of the mass
m requires some new techniques and we will do so in section 13.3. However, let us
first look at the motion from a physical point of view. The mass m in figure 13.2(a)
is pulled a distance x = A to the right, and is then released. The maximum restoring
force on m acts to the left at this position because
FRmax = −kxmax = −kA
The maximum displacement A is called the amplitude of the motion. At this
position the mass experiences its maximum acceleration to the left. From equation
13-2
, Chapter 13 Simple Harmonic Motion
Figure 13.2 Detailed motion of the vibrating spring.
13.2 we obtain
a=− kA
m
The mass continues to move toward the left while the acceleration
continuously decreases. At the equilibrium position, figure 13.2(b), x = 0 and hence,
from equation 13.2, the acceleration is also zero. However, because the mass has
inertia it moves past the equilibrium position to negative values of x, thereby
compressing the spring. The restoring force FR now points to the right, since for
negative values of x,
FR = −k(−x) = kx
The force acting toward the right causes the mass to slow down, eventually coming
to rest at x = −A. At this point, figure 13.2(c), there is a maximum restoring force
pointing toward the right
FRmax = −k(−A)max = kA
13-3
, Chapter 13 Simple Harmonic Motion
and hence a maximum acceleration
amax = − k (−A) = k A
m m
also to the right. The mass moves to the right while the force FR and the
acceleration a decreases with x until x is again equal to zero, figure 13.2(d). Then FR
and a are also zero. Because of the inertia of the mass, it moves past the
equilibrium position to positive values of x. The restoring force again acts toward
the left, slowing down the mass. When the displacement x is equal to A, figure
13.2(e), the mass momentarily comes to rest and then the motion repeats itself. One
complete motion from x = A and back to x = A is called a cycle or an oscillation.
The period T is the time for one complete oscillation, and the frequency f is the
number of complete oscillations or cycles made in unit time. The period and the
frequency are reciprocal to each other, that is,
f= 1 (13.3)
T
The unit for a period is the second, while the unit for frequency, called a hertz, is
one cycle per second. The hertz is abbreviated, Hz. Also note that a cycle is a
number not a dimensional quantity and can be dropped from the computations
whenever doing so is useful.
13.3 Analysis of Simple Harmonic Motion -- The
Reference Circle
As pointed out in section 13.2, the acceleration of the mass in the vibrating spring
system is given by equation 13.2 as
a=−k x
m
Since the acceleration a = d2x/dt2, equation 13.2 can be written as
d2x = − k x (13.4)
dt2 m
Equation 13.4 is a second-order differential equation that completely describes the
simple harmonic motion of the mass m. Unfortunately, the solution of such
differential equations is beyond the scope of this course. We also can not use the
kinematic equations derived in chapter 2 because they were based on the
assumption that the acceleration of the system was a constant. As we can see from
13-4
