
Linear simple harmonic motion is defined as the linear periodic
oscillatory motion of a body under the influence of a restoring force
which is always directed towards a fixed point in its path
(equilibrium position) and is of magnitude directly proportional to
the magnitude of the displacement of the body from the fixed
position.
Hence, restoring force,

F µ x
F = kx

where, k is a proportionality constant
called the force constant. The negative sign indicates that the
direction of the force is always opposite to the displacement.
Terms Used in the Study of S.H.M.
One Oscillation: One oscillation is defined as a complete to
and fro motion of the oscillating body.
Period (T): The period of S.H.M. is defined as the time taken by
the body to complete one oscillation.
Unit of Period: Second in both SI and CGS systems.
Frequency (n): The frequency of S.H.M. is defined
as the number of oscillations completed by a body in one second.
Unit of Frequency: Hertz in both S.I. and C.G.S. systems.
In time T seconds, a body completes one oscillation.
\ Number of oscillations in one second =

1

= frequency, n


T

\

1

= n


T

Amplitude (A): The maximum displacement of a body in S.H.M.
from its mean position is called the amplitude of S.H.M.
Unit of Amplitude: metre in the SI
system, centimetre in the CGS system.
Path Length or Range: The total length of the path over which
the particle performing SHM moves is called the path length or range
of SHM.
Path Length = 2A, where A is the amplitude.
Unit of Path Length: metre in the SI
system, centimetre in the CGS system.


S.H.M. as a projection of U.C.M.
Consider a particle performing U.C.M. with an angular velocity w along the
circumference of circle of radius A.
At t = 0, the
particle is at P.
At t = t, the
particle is at P'.
Let the angle made by OP with the Y  axis be a m ÐPOP' = wt.
AB is any diameter of the circle.
Let particle M be another particle such that its position always coincides
with the foot of the perpendicular from P to AB. Thus M is the
projection of particle P on the diameter AB.
It is clear that as P moves along the circumference of the circle, M
moves to and fro along diameter AB.
When particle P completes one revolution, particle M completes one
oscillation.
In DOP'M', ÐP'OM' = [90  (wt + a)]
OM'= x = OP' cos
[90  (wt + a)]
\ x = A sin (wt + a)
x is the
displacement of particle M from O.
Velocity is the time rate of change of displacement.
\ Velocity n =

dx

=

d

[A sin (wt + a)]



dt

dt

Acceleration is the time rate of change of velocity.
\ acceleration a =

dn

=

d

[A w cos
(wt + a)]



dt

dt

since x = A sin (wt + a), a = w^{2 }x
w is a
constant.
Hence, from the above relation, a µ x
The acceleration of the particle M is directly proportional to its
displacement and oppositely directed to the displacement
Hence, particle M performs linear S.H.M. M is a projection of a
particle P that performs U.C.M.
Thus, projection of U.C.M. is a linear simple harmonic motion.
Differential Equation of S.H.M.
The restoring force acting on a body performing linear S.H.M. is
given by the relationship F =  kx, where, k is the
force constant and x is the displacement.
From Newton's second
law of motion,
F = ma, where, m is the mass
of the particle and a is its acceleration.
Equating the above equations,
ma =  kx
\ a =

k

.x


m

But a =

dn

=

d

(


t


dx

)=


dt


d^{2}x


dt^{2}



dt

\

d^{2}x

=

k

.x



dt^{2}

m

or

d^{2}x

+

k

x = O



dt^{2}

m

This expression is called the second order differential equation of
linear S.H.M.
