Question

a simple harmonic motion is represented by
x(t) = a cos ωt
obtain expressions for velocity and acceleration of the oscillator. also, plot the time variation of displacement, velocity and acceleration of the oscillator.

Answer 1

In SHM the displacement x of a particle from the mean position can be given as: 
 
   x = A Cos ωt
         A = amplitude from the mean position
        ω = angular frequency of vibration
        Ф = 0 = initial phase of SHM, at t = 0
         t = time

velocity = v = dx/dt = - A ω Sin ωt
                   = A ω Cos (ω t + π/2)

   So velocity is π/2 radians ahead of displacement in the phase angle.

acceleration = a = dv/dt = - A ω² Cos ω t
                           = A ω² Sin (ω t + π)

So acceleration is π radians ahead of displacement and π/2 rad ahead of velocity.