Definition: Phase of S.H.M. means the state of oscillation of
the particle performing S.H.M. The state of oscillation is described
by the displacement and direction of motion of the particle.
In the expression x = A sin(wt + a) the angle
(wt + a) specifies
the state of oscillation of the particle.
Thus, this angle is called the phase angle.
Phase Constant or Epoch
The initial state of oscillation or phase of the particle at t = 0 is
specified by a. a is called the phase constant or epoch of S.H.M.
Graphs
Graphs of displacement (x), velocity (n),
acceleration (a) versus wt
Case I: Particle starting at the mean position and moving to
the right.
Here, x = 0 at t = 0, a = 0
Hence x = A sin wt, n = A w cos wt, a = A w^{2} sin wt
The values of x, n and a for different
values of t and wt are shown
in the table.
t

0

T/4

T/2

3 T/4

T

wt

0

p/2

p

3 p/2

2p

x

0

A

0

A

0

n

Aw

0

Aw

0

Aw

a

0

Aw^{2}

0

Aw^{2}

0

Variation of x vs. wt
Variation
of n vs. wt
Variation
of a vs. wt
Case
II: Particle starting from the positive extremity and
moving to the left.
Here, x = A at t = 0; a = 90° or

p

radian


2

\ x = A sin (wt + a) = A sin
(wt +

p

) = A cos wt


2

n = A w cos
(wt + a) = Aw cos (wt +

p

) = A w sin wt


2

a = Aw^{2} sin (wt + a) = Aw^{2} cos (wt +

p

)= A w^{2} cos wt


2

The values of x, n and a for different
values of t and wt are given
in the table.
t

0

T/4

T/2

3 T/4

T

wt

0

p/2

p

3 p/2

2p

x

A

0

A

0

A

n

0

Aw

0

Aw

0

a

Aw^{2}

0

Aw^{2}

0

Aw^{2}

Potential
Energy, Kinetic Energy and Total Energy of a Particle in S.H.M
Potential energy: Definition
Potential energy of a particle in S.H.M. is the work done by an
external agent against the restoring force in displacing the particle
from the mean position to a given point in its path.
Let AOB be the path of a particle performing linear S.H.M. Let it be
displaced through an infinitesimal distance dx. Work done
by the external agent against the restoring force = dw = Fdx = kxdx ( F = kx)
\ Total work
done in displacing the particle from its mean position through a
distance x
W =dw = kx dx = kx^{2}
\ P.E. = kx^{2}
Kinetic energy of the particle =

mn^{2}

=

m w^{2}(A^{2} x^{2}) = k(A^{2 } x^{2})

\ Total
energy, T.E. = P.E. + K.E. = kx^{2} + k(A^{2}  x^{2})
\ T.E. = kA^{2}
Since k and A are
constants, T.E. is a constant.
Since k = mw^{2} = m . (2pn)^{2} = m .

(

2p

)

2


T

T.E. = mw^{2}A^{2}
= 2p^{2}mn^{2}A^{2}
=

2p^{2} mA^{2}


T^{2}

\ T.E. µ A^{2}
and T.E. µ n^{2}
Graphs of K.E., P.E. and T.E. Against Displacement
Composition
of Two SHMs having Same Period and Parallel
to Each Other (Analytic Method)
Let the two parallel S.H.Ms of same period
and different amplitudes and initial phases be represented by the
equation
x_{1} = A sin (wt + a) and x_{2} = B sin (wt + b), where a and b are the
respective epochs.
By the principle of superposition, the resultant displacement (x) when these
two S.H.Ms act on a particle is given by
x = x_{1} + x_{2} = A sin (wt + a) + B sin (wt + b)
On simplifying and putting A cos a + B cos b = C cos d
and A sin a + B sin b = C sin d
We get x = C sin (wt + d)
This shows that the resultant motion is also simple harmonic in
nature. The resultant amplitude C is given by
C =
The initial phase of the resultant SHM is given by
d = tan^{1}

[

A sin a + B sin b

]


A cos a + B cos
b

Case I:

If the phase difference between the two individual SHMs is zero, i.e. if a = b,
C = A + B

Case II:

If the phase difference between the two individual, SHMs is 90°, i.e. if a  b = 90°,
C = \ C > A
and C > B

Case III:

If the phase difference between the two SHMs is 180°, i.e. if a  b = 180°
C = A  B
If A = B, C = 0

Application:
Simple Pendulum
Ideal Simple Pendulum:
An
ideal simple pendulum is defined as a heavy point mass suspended from
a rigid support by a weightless, inextensible, twistless,
flexible string and allowed to oscillate in a vertical plane under
gravity.
Practically:

(1) A heavy metallic sphere is used as the bob.


(2) A light string is used.

Length of a Simple Pendulum: The distance between the point of
suspension and the centre of gravity of the bob is called the length
of the simple pendulum.
