Explain that simple harmonic motion is the projection of uniform circular motion on a diameter of the reference circle and obtain the velocity and acceleration.

Explanation

Let a particle is moving with a constant angular speed `omega` in an anti-clockwise direction on a circular path having center O and radius A. At time t particle is at P and its phase is `theta= omega t+ phi," where "phi=` initial phase.
The linear velocity of particle at P position `PQ= A omega`
According to figure
In `trianglePNQ`, projection of `vec(PQ)` in X-direction `PN= PQ sin theta`
`therefore vec(PN)= -A omega sin theta`
`therefore P.Q. = -A omega cos (omega t+ phi)`
`therefore v(t)= -A omega cos (omega t+ phi)`
Acceleration : Acceleration of particle at P toward centre `A omega^(2) = PR`
According to figure
In `trianglePMR`, projection of `vec(PR)` on X-direction `MR= PR cos theta`
`P.R.= A omega^(2) cos (omega t+ phi)`
`therefore a(t) = -A omega^(2) cos (omega t+phi)`
Since velocity and acceleration are in negative X-direction, so it.s negative sign is present.