MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Simple Harmonic Motion Page 3

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Phase of S.H.M.

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 w2 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 -Aw2 0 Aw2 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 = -Aw2 sin (wt + a) = -Aw2 cos (wt + p )= -A w2 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 -Aw2 0 Aw2 0 -Aw2

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 = kx2

\ P.E. = kx2

 Kinetic energy of the particle = mn2 = m w2(A2- x2) = k(A2 - x2)

\ Total energy, T.E. = P.E. + K.E. = kx2 + k(A2 - x2)
\ T.E. = kA2
Since
k and A are constants, T.E. is a constant.

 Since k = mw2 = m . (2pn)2 = m . ( 2p ) 2 T

 T.E. = mw2A2 = 2p2mn2A2 = 2p2 mA2 T2

\ T.E. µ A2 and T.E. µ n2

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
x1 = A sin (wt + a) and x2 = 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 = x1 + x2 = 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.

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