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.

Explain that simple harmonic motion is the projection of uniform circu

1.	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.
Answer» Solution :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.