Consider about the figure shown below. Particle P is describing a uniform circular motion with constant angular velocity ( \omega ) in anti-clockwise direction in a circle of radius ( A ) . Suppose –
At time ( t = 0 ) , the particle is at point A and ( \angle { XOA } = \phi _{ 0 } )
After time ( t = t ) , the particle reaches at the point P and ( \angle { AOP } = \omega t ) .
A perpendicular PN is drawn on fixed diameter XX' .
When the particle moves on the circumference of the circular path, point N will move “to & fro” on the diameter XX' .
Hence, motion of the foot of perpendicular N will be a simple harmonic motion.
Displacement in a Simple Harmonic Motion
Displacement of point N from the mean position O in time ( t ) will be –
x = ON
In right angle triangle \triangle {ONP} –
\angle { PON } = ( \omega t + \phi _ { 0 } )
Therefore, \quad \left ( \frac { ON }{ OP } \right ) = [ \cos ( \omega t + \phi _{ 0 } ) ]
Or, \quad \left ( \frac { x }{ A } \right ) = [ \cos ( \omega t + \phi _{ 0 } ) ]
Or, \quad x = [ A \cos ( \omega t + \phi _ { 0 } ) ] …….. (1)
This equation is called simple harmonic motion equation.
060201 SIMPLE HARMONIC MOTION EQUATION
This equation gives displacement of a particle executing SHM at any instant after time ( t ) from initial position.
Quantity ( \omega t + \phi _ { 0 } ) is called phase of the particle and ( \phi _{ 0 } ) is called initial phase or phase constant.
Quantity ( A ) is called amplitude of the motion. When the particle is at extreme position, the maximum displacement is ( \pm A ) .
When a particle execute a simple harmonic motion, a force always acts on particle which has a tendency to bring the particle in its mean position. This force is called restoring force.
For a simple harmonic motion –
F = - k x
Restoring force is proportional to displacement of the particle from its mean position.
060204 RESTORING FORCE & DISPLACEMENT IN SIMPLE HARMONIC MOTION EQUATION
Therefore, force constant of simple harmonic equation is –
k = m \omega ^ 2
Also, \quad \omega = \sqrt { \left ( \frac { k }{ m } \right ) }
A graph for variation of restoring force verses displacement is shown in figure.
Different forms of Simple Harmonic Equations
Depending upon the initial conditions of particle, general equations for simple harmonic motion can be expressed in different forms. These are illustrated in following figures.
060205 FORMS OF SIMPLE HARMONIC MOTION EQUATION
(1) Case – A ( Initially particle is at extreme right & start moving in negative x direction )
Consider about the figure in Case -A
Initial position of particle is at extreme right position. This means, when ( t = 0 ) , position of particle is at ( x = + A ) .
Thus, the particle moves in “negative x direction” i.e. in direction from ( x = + A ) to ( x = 0 )
Then, general form of this simple harmonic equation will be –
x = A \cos ( \omega t + \phi _ { 0 } ) ……… (1A)
(2) Case – B ( Initially particle is at extreme left & start moving in positive x direction)
Consider about the figure in Case -B
Initial position of particle is at extreme left position. This means, when ( t = 0 ) , position of particle is at ( x = - A )
Thus, the particle moves in “positive x direction” i.e. in direction from ( x = - A ) to ( x = 0 )
Then, general form of this simple harmonic equation will be –
x = - A \cos \left ( \omega t + \phi _ { 0 } \right ) …….. (1B)
(3) Case – C (Initially particle is at mean position & start moving in negative x direction)
Consider about the figure in Case -C
The initial position of particle is at mean position. This means, when ( t = 0 ) , position of particle is at ( x = 0 )
Thus, the particle moves in “negative x direction” i.e. in direction from ( x = 0 ) to ( x = - A )
Then, general form of this simple harmonic equation will be –
x = - A \sin ( \omega t + \phi _ { 0 } ) ………. (1C)
(4) Case – D (Initially particle is at mean position & start moving in positive x direction)
Consider about the figure in Case -D
The initial position of particle is at mean position. This means, when ( t = 0 ) , position of particle is at ( x = 0 )
Thus, the particle moves in “positive x direction” i.e. in direction from ( x = 0 ) to ( x = + A )
Then, general form of this simple harmonic equation will be –