Oscillatory :
We shall be studying about Oscillatory and Periodic motion. One special case is Simple Harmonic motion. Basically, this is a repeating motion of an object in which the object continues to observe to and fro motion about a mean position at fixed time interval (under ideal situations). However, if the time interval is not fixed, then the motion may be called as Oscillatory.
The back and forth movements of such an object are called oscillations. We will focus our attention on a special case of periodic motion called simple harmonic motion. It is observed that all periodic motions can be modelled as combinations of simple harmonic motions and hence SHM forms a basic building block for more complicated periodic motion.
Periodic Motion :
Periodic motion is motion of an object that regularly returns to a given position after a fixed time interval. With a little thought, identify several types of periodic motions in everyday life. Your car returns to the driveway each afternoon. You return to the dinner table each night to eat. A bumped chandelier swings back and forth, returning to the same position at a regular rate. The earth returns to the same position in its orbit around the sun each year, resulting in the variation among the four seasons.
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In addition to these everyday examples, numerous other systems exhibit periodic motion. The molecules in a solid oscillate about their equilibrium positions; electromagnetic waves such as light waves, radar and radio waves are characterized by Oscillating electric and magnetic field vectors; in an alternating current electrical circuit, voltage, current and electric charge vary periodically with time.
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If
I
is the period of motion after which it repeats itself, then the frequency ƒ of the periodic motion is the number of cycles performed in 1 s and it is given as
f
= 1/
T
.
-
The units of ƒ are
s
1
or per second. A special name is given to the unit of frequency, hertz (Hz) after the discoverer of radio waves.
-
1 Hz = 1 cycle per second
Oscillations
An oscillation is a special type of periodic motion in which a particle moves to and from about a fixed point called mean position of particle. Oscillations are commonly seen in general life in our surroundings. As discussed, in all types of oscillations, there is always a mean position about which the particle can oscillate. This is the position where the particle is in equilibrium, that 15, net force on the particle at this position is zero. If particle is displaced from the mean position, due to this displacement some forces appear on it which act on the particle in a direction directed toward its equilibrium position, these forces are called restoring forces as these forces tend the particle to move towards its equilibrium position.
Due to restoring forces, particle starts moving toward the mean position and when it reaches the mean position, it gains some
KE
due to work done by the restoring forces and it will overshoot from this point with some velocity in other direction; again restoring forces appear on the particle toward mean position and now the particle is retarded and will stop after travelling some distance. It will return toward the mean position and start accelerating and in such a way motion is continued which we call oscillation. The maximum displacement of particle from its mean position, where it will come to rest or from where it started with zero initial speed, is called as amplitude of oscillations.
An oscillatory motion need not be periodic and need not have fixed extreme positions. For example, motion of pendulum of a wall clock (because the battery of the wall clock wears out with time). The oscillatory motions in which energy is conserved can also be called as periodic. Oscillations in which energy is consumed due to some resistive forces and hence total mechanical energy decreases are called as Damped oscillations. The force/torque (directed towards equilibrium point) acting in oscillatory motion is called restoring force/torque.
Oscillations Periodic Function
If a particle moves along
x
-axis, its position depends upon time
t
. We express this fact mathematically by writing
x
=
f
(
t
) or
x
(
t
)
There are certain motions that are repeated at equal intervals of time. By this we mean that particle is found at the same position moving in the same direction with the same velocity and acceleration, after each period of time. Let
T
be the interval of time in which motion is repeated. Then
x
(
t
) =
x
(
t
+
T
)
where
T
is the minimum change in time. The function that repeats itself is known as a periodic function. During the period, its values may remain finite. Such functions are bound functions. Periodic motion of a particle is also bound because it must not go to infinity and return back in one finite period.
Periodic motions may be oscillatory or non-oscillatory. Uniform circular motion, the motion of a planet around the sun, etc. are periodic but not oscillatory. Also, an oscillatory motion may not repeat its position with the old velocity due to friction and will be non-periodic.
Following are given some general points regarding motion, periodic motion
1.
In general, motion of a body (or its path) depends on two factors:
(i)
the nature of force (or acceleration) of the body and
(ii)
its velocity
Oscillatory Examples
A constant force or constant acceleration always gives a straight line or parabolic path. If initial velocity is zero or parallel (or antiparallel) to constant acceleration then path is straight line. In all other cases, path is a parabola. For small height, acceleration due to gravity (
a
=
g
) is constant. So, path is either straight line or parabola.
If force of constant magnitude is acting on a particle and its direction is always perpendicular to velocity, then path is circular motion in which speed is constant. This is also called uniform circular motion.
Now let us consider a particle free to move along
x
-axis, which is being acted upon by a force given by,
F
= –
kx
n
Here,
k
is a positive constant.
Now, following cases are possible depending on the value of
n
:
(i)
If
n
is an even integer (0, 2, 4, … etc), force is always along negative
x
-axis. If the particle is released from any position on the
x
-axis (except at
x
= 0) a force in negative direction of
x
-axis acts on it and it moves rectilinearly along negative
x
-axis.
(ii)
If
n
is an odd integer (1, 3, 5, … etc), force is along negative
x
-axis for
x
> 0, along positive
x
-axis for
x
< 0 and zero for
x
= 0. Thus, the particle will oscillate about stable equilibrium position (also called the mean position),
x
= 0. The force in this case is called the restoring force. Of these, if
n
= 1, i.e.
F
= –
kx
the motion is said to be SHM.
2.
In every oscillatory motion, there is one mean position (or stable equilibrium position) and two extreme positions.
3.
Distance between mean position and the extreme position is called amplitude of oscillation
A
.
4.
Oscillations does not start by itself. Normally the body has to be displaced from the mean position. In this displacement
F
= –
kx
type force opposes the motion. So, work has to be done against this force which remains stored in the system in the form of mechanical energy. In the absence of any dissipative forces (like friction or viscous force) this mechanical energy remains constant. While moving from extreme positions to mean position potential energy decreases and kinetic energy increases but total mechanical energy remains constant. Similarly, in moving from mean position to extreme positions potential energy increases and kinetic energy decreases.
5.
The more the initial displacement from the mean position, more is the amplitude, more is the initial work done and more is the mechanical energy given for oscillations.
Important Points To Remember
Period is not changed by multiplying or dividing by (or by adding) a constant:
If
x
(
t
) =
x
(
t
+
T
), then
has the same period
T
.
Here,
x
0
and
m
are constant.
Period is reduced α times if
t
be multiplied by α.
If
x
(
t
) =
x
(
t
+
T
) then
x
(α
t
) will have a period
T
/α.
∙
The sum of periodic functions is also periodic.
Illustration
Q. 1 :
What is the time-period of
x
=
A
sin (ω
t
+ α)?
Sol.:
It is known from trigonometry that sin θ = sin (θ + 2π)
Hence
x
=
A
sin (ω
t
+ α + 2π) =
A
sin
x
=
A
sin (ω
t
' + α), where
t
' = t
This shows that the function at
t
coincides with the function at
t
'.
The interval (
t
' –
t
) is the period of
x
.
This period is
Q.2 :
Find the period of the function,
y
= sin ω
t
+ sin 2ω
t
+ sin 3ω
t
Sol :
The given function can be written as
y
=
y
1
+
y
2
+
y
3
Here
y
1
= sin ω
t
,
T
1
= 2π/ω
and
y
3
= sin 3ω
r
,
T
3
= 2π/3ω
T
1
= 2
T
2
and
T
1
= 3
T
3
So, the time period of the given function is
T
1
or 2π/ω.
Because in time
T
= 2π/ω, first function completes one oscillation, the second function two oscillations and the third, three.
