Oscillations

The displacement of an oscillating particle is given by x = a sin ( ω t ) + b cos ( ω t ), where a and b are constant show that the motion of particle is S. H. M.

Solution:

S . H . M .  ( Simple Harmonic Motion )
Definition:

S. H. M. can be defined as the periodic motion of body in which the restoring force ( or acceleration)  is directed towards the mean position and its magnitude is directly proportional to the displacement from mean position.

The displacement of particle is given by

x = a sin ( ω t ) + b cos ( ω t )

- - - - - - - - - ( 1 )

We know that the velocity is rate if change of displacement.

Therefore,
v = dx / dt

v = d / dt ( a sin ω t + b cos ω t)
v = a ω cos ω t - b ω sin ω t

But acceleration is rate of change if velocity, the above equation can be written as
a = dv / dt
a = d / dt ( a ω cos ω t - b ω sin ω t )
a = - a ω ² sin ω t + b ω ² cos ω t
a = - ω ² [ a sin ω t + b cos ω t ]

From equation ( 1 )
a=  - ω ² x
as ω ² is constant

a α - x

Since acceleration is directly proportional to the displacement, the motion of particle is S. H. M.