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Obtain instantaneous acceleration of a S...

Obtain instantaneous acceleration of a SHM particle with the help of reference circle.

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Acceleration of a particlel executing uniform circular motion with angular speed `omega` and amplitude A is `(v^2)/(A)" or "A omega^(2)` and direction of acceleration is toward centre so it is negative.
The acceleration of projected particle in uniform circular motionl at time t `a(t)= -omega^(2) x(t)`
This acceleration is towards the centre :

The centripetal force at point P is `omega^(2)A`, its projection on X-axis is `A omega^(2) cos (omega t+phi)` and its direction is opposite to +X-axis hence instantaneous acceleration of projected particle at time t is
`a(t)= -A omega^(2) cos (omega t+ phi)`
`therefore a(t) = -omega^(2) x(t)" "[therefore A cos (omega t+ phi)= x(t)]`
is the acceleration of SHM particle.
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