Simple Harmonic Motion :
We will learn that When a body repeats its motion after regular time intervals we say that it is in
harmonic motion
or
periodic motion
. The time interval after which the motion is repeated is called the
time period
. If a body moves to and fro on the same path, it is said to perform
oscillations
.
Simple harmonic motion
(SHM) is a special type of oscillation in which the particle oscillates on a straight line, the acceleration of the particle is always directed towards a fixed point on the line and its magnitude is proportional to the displacement of the particle from this point. This fixed point is called the
centre of
oscillation
. Taking this point as the origin and the line of motion as the X-axis, we can write the defining equation of a simple harmonic motion as
a
= −ω
2
x
where ω
2
is a positive constant. If
x
is positive,
a
is negative and if
x
is negative,
a
is positive. This means that the acceleration is always directed towards the centre of oscillation.
SHM is a special type of oscillatory motion in which the restoring force is proportional to the displacement from the mean position (for small displacement from mean position). It is the simplest (easy to analyse) form of oscillatory motion. The amplitude of oscillations of the particle is very small. At any instant the displacement of a particle in oscillatory motion can be expressed in terms of sinusoidal functions (sine and cosine functions).
These functions are called harmonic functions. That is why an oscillatory motion is also called a harmonic motion. A simple harmonic motion can be expressed in terms of one single sine or cosine function or a linear combination of sine and cosine functions. If a particle is moving to and fro about an equilibrium point
M
(called as the mean position), along a straight line as shown in Figure, then we can call its motion to be SHM, where
A
and
B
are extreme positions and
AM
=
MB
= Amplitude (
a
)
However, when a body or particle is free to rotate about a given axis executing angular oscillations, then this type of SHM can be regarded as angular SHM.
Dynamics of Simple Harmonic Motion :
The restoring force for small displacement
x
is given by
F
= –
k
x
, where
k
is called the force constant of the system. Force constant is defined as the restoring force per unit displacement. Its SI unit is Nm
−1
. If m is mass of the body and
a
is its acceleration, then for simple harmonic motion.
∙
This is the differential equation of SHM.
The general solution to this equation is
Where,
a
is the amplitude,
is the phase (also called as instantaneous phase),
is phase constant or initial phase angle (in radian) also called as epoch.
Consider a particle to execute SHM and assume that the path of the particle is on a straight line, then
x
is the displacement of particle from the mean position at that instant. The displacement of a particle executing SHM at time
t
is given by
x
=
a
sin (ω
t
+ ϕ)
Also, note that the displacement of a particle executing SHM at time
t
can be given by
y
=
a
sin (ω
t
+ ϕ)
Amplitude,
a
or
A
, is the maximum value of displacement of the particle from its equilibrium position i.e., mean position. So, amplitude of SHM is half the separation between the extreme points of SHM. It depends upon the energy of the system.
Angular frequency
, ω of SHM is
and its SI unit is rads
–1
.
Frequency,
f
or
v
is the number of oscillations completed in unit time interval, so we have
Its unit is sec
–1
or Hz.
Time period,
T
is the smallest time interval after which oscillatory motion gets repeated, so time period
T
given by
Mathematically
at
t
=
T
, the displacement
x
of SHM must satisfy
x
(
t
+
T
) =
x
(
t
). For example,
and
Velocity And Acceleration In Shm :
We can write the equation of motion of a particle performing SHM is
y
=
A
sin (ω
x
+ ϕ)
…(i)
Differentiating
y
with respect to ‘
t
’ given velocity of the particle
…(ii)
From Eq. (i), we can write sin (ω
t
+ ϕ) =
and we know that
Substituting the value of cos (ω
t
+ ϕ) in Eq. (i), we get
…(iii)
Differentiating Eq. (ii) again w.r.t. time.
Equation (iii) can be rearranged in the form
∙
This shows that velocity-position graph is an ellipse. This is also true for momentum-position graph. The momentum-position graph is known as phase-space graph of the particle.
(a)
Displacement-time graphs:
The displacement of the particle is always measured from the mean position
P
wheareas, time
t
can be calculated from any position of the oscillating particle.
|
Situations
|
Graphs
|
|
Particle starts SHM from mean position.
Displacement equation
y
=
A
sin ω
t
|
|
|
Particle starts SHM from amplitude position.
Displacement equation
y
=
A
cos ω
t
|
|
|
Particle starts SHM from any general position.
Displacement equation
y
=
A
sin (ω
t
+
|
|
(b)
Velocity-time and acceleration-time graphs:
If particle starts SHM from mean position, then
Displacement equation,
y
=
A
sin ω
t
…(i)
Differentiation of equation (i) gives velocity of the particle.
…(ii)
Differentiation of Eq. (ii) again gives acceleration of the particle.
|
Velocity-time graph
|
Acceleration-time graph
|
|
|
|
Potential Energy Of A Particle In Simple Harmonic Motion :
The potential energy
U
at a displacement
x
is the work done against the restoring force in moving the body from the mean position to this position.
∙
The maximum value of potential energy is
at the extreme position i.e., at
x
= ±
A
and minimum value of potential energy is
U
min
= 0 at the mean position i.e., at
x
= 0.
Kinetic Energy Of A Particle In Simple Harmonic Motion
Kinetic Energy Of A Particle In Simple Harmonic Motion :
If
x
=
A
sin (ω
t
), then
v
=
A
ω cos (ω
t
), so kinetic energy
K
of a particle executing SHM is given by
∙
The maximum value of kinetic energy is
at the mean position i.e., at
x
= 0 and minimum value of kinetic energy is
K
min
= 0 at the extreme position i.e., at
x
= ±
A
.
Mechanical Energy Of A Particle In Simple Harmonic Motion
Mechanical Energy Of A Particle In Simple Harmonic Motion :
The mechanical energy
E
of a particle executing SHM is
So, we observe that total mechanical energy is a constant. The variations of
U
,
K
and
E
with time are shown in Figure.
The variations of
U
,
K
and
E
with displacement are shown in Figure.
At
x
= 0,
U
= 0 and the energy is purely kinetic i.e.,
At extreme points or turning points of the SHM, kinetic energy is zero and the energy is purely potential i.e.,
Note that the frequency of vibration of the kinetic and potential energy is twice that of the frequency of oscillation.
The instantaneous total mechanical energy of the spring-mass system may be written as
Differentiating it w.r.t. time, we get
Since,
and
therefore,
This is the differential equation of SHM.
