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CHAPTER 9
OSCILLATIONS
Some of the motions we encounter are of repetitive nature, these motions are called
oscillations. Some examples of these motions are : the swing of a pendulum, the
vibrations of a guitar string or the diaphragms of speaker systems. Some other forms
of oscillations which are less obvious or evident are those of air molecules
transmitting sounds, the atoms of solids conveying temperature or the electrons in
antennas. We are now going to study some of the properties of such periodic
oscillations, often called Harmonic Motions.
The frequency
A simple oscillating system consists of a particle moving repeatedly back and forth
about a reference origin. an important property of this kind of systems is its
frequency, which is the number of completed oscillations ( or cycles) per second.
We usually use the symbol f and the unit of the frequency is called the Hertz
(abbreviated Hz), we then have
1 hertz = 1 Hz = 1 oscillation per second = 1 s 1 .
The period
Another important property of an oscillatory motion is its period T, which is the time
for one complete cycle (or oscillation). The period is related to the frequency via the
formula
T
1
.
f
Obviously, the unit for the period is the second s.
Simple harmonic motion
Any motion that repeats itself at regular intervals of time is called periodic motion or
harmonic motion. The simplest presentation of the displacement of a particle from
the origin, for such a harmonic motion, is given as function of time by
x(t )  x m cos(t   ).
In this formula, the values x m ,  and  are constants. This motion is known as a
simple harmonic motion (SHM), which means that the periodic motion is a
sinusoidal function of time, as shown in Fig. 9.1. The three parameters in this
formula are the amplitude, the phase and the angular frequency.
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The amplitude
The amplitude of a sinusoidal motion, like the one described in the previous formula,
is given by x m . The subscript m stands for maximum because the amplitude is the
magnitude of the maximum displacement of the particle in either direction about the
origin. Since the cosine function oscillates between -1 and +1, then the displacement
x(t) varies between - x m and + x m , as shown in Fig. 9.1.
Displacement
xm
T
Time
0
-x m
Fig. 9.1 Simple harmonic motion.
The phase
The equation of the displacement contains the term (t   ) which is called the phase
of the motion, and the constant  is called the phase constant (or phase angle). The
value of the phase constant depends on the displacement and the velocity of the
particle at the time t=0, and its SI unit is the radian. The plot of x(t) in Fig. 9.1 has a
phase constant   0 .
The angular frequency
The angular frequency is the constant  . The displacement x(t) must return to its
initial value after one period T (or any integer number of the period) of the motion. In
the case of   0 , we can write
 (t  T )  t  2 ,
so then the angular frequency can be written as
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
2
 2f .
T
The SI unit of the angular frequency is the radian per second.
The velocity
If we differentiate the equation of the displacement with respect to time, we can find
the velocity of the particle moving with the simple harmonic motion. That is,
v(t ) 
dx
 x m sin(t   ).
dt
Similarly to the displacement, the positive quantity x m is called the velocity
amplitude v m . So then the velocity of the particle can vary between the values of - v m
and + v m .
The acceleration
We know that the acceleration of a particle is simply the derivative, with respect to
time, of the velocity of that particle. Knowing the velocity v(t) for simple harmonic
motion, we can calculate the acceleration as
a(t ) 
dv
  2 x m cos(t   ).
dt
Again the positive quantity  2 x m is called the acceleration amplitude am . That is,
the acceleration of the particle can vary between the values - am and + am .
We can see that if we combine the displacement and acceleration we obtain
a(t )   2 x(t ) .
This means that the acceleration is proportional to the displacement but opposite in
sign, and the two quantities are related by the square of the angular frequency.
The force law
If we want to know what force must act on the particle, we need to know its
acceleration and how it varies with time. Using Newton’s second law and the
formula for the acceleration of a particle in a simple harmonic motion, we get
F  ma  (m 2 ) x ,
which is very similar to Hooke’s law for a spring which gives
F  kx .
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We can say then that the spring constant is
k  m 2 .
Now we can have an alternative definition of the simple harmonic motion, taking
into account the equation relating the force and the displacement. This states
Simple harmonic motion is the motion executed by a particle of mass m
subject to a force that is proportional to the displacement of the particle
but opposite in sign.
Linear simple harmonic oscillator
Let us consider a block-spring system, like the one shown in Fig. 9.2.
k
m
x
-xm
x=0
xm
Fig. 9.2 A simple harmonic oscillator.
This system forms a linear simple harmonic oscillator, where linear indicates that the
force F is proportional to the displacement x rather than to some other power of x. As
we have seen before, the angular frequency  of the simple harmonic motion of the
block is related to the spring constant k and the mass m of the block via the formula
k  m 2 , which can be written as

k
.
m
The period of the linear oscillator can be obtained then as
m
.
k
T  2
Energy considerations
The potential energy of a linear oscillator is associated entirely with the spring. Its
value depends on how much the spring is stretched or compressed. It is
U (t ) 
1 2 1 2
kx  kx m cos 2 (t   ).
2
2
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The kinetic energy of the same system is associated entirely with the block, because
its value depends on how fast the block is moving. It is given by
K (t ) 
1 2 1
mv  m(x m ) 2 sin 2 (t   )
2
2
but we know that k  m 2 , so then the kinetic energy becomes
K (t ) 
1 2 1 2
mv  kx m sin 2 (t   ).
2
2
The mechanical energy E(t) is given by E = U + K, which is
E (t ) 
1 2
1
kx m cos 2 (t   )  kx m2 sin 2 (t   ).
2
2
We know that cos 2   sin 2   1 , then the energy can be simplified to
E UK 
1 2
kx m ,
2
which is a constant, independent of time.
Simple pendulum
A simple pendulum consists of a particle of mass m suspend from an unstretchable,
massless string of length L, as the one shown in Fig. 9.3. The mass is free to swing
back and forth.
L

Tension
m
mg
Fig. 9.3 A simple pendulum.
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For small oscillations (   10  ), we can work out a relation between m, g, L and k,
and this relation is k  mg L . Substituting this equation in the period equation
obtained previously, we obtain the period of a simple pendulum as
T  2
L
.
g
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