Derive the expressions for displacement velocity and acceleration of a particle executes S.H.M.

Explanation

Displacement of a body in S.H.M. `X=A cos(omegat+phi)`.
i) Displacement(x): At `t=0` displacement `x=A` i.e., extreme position when `omegat=phi=90^(@)` displacement `x=0` at mean position at any point `x=Acos(omegat+phi)`.
ii) Velocity (V): Velocity of a body in S.H.M. is `V=(dx)/(dt)=(d)/(dt){Aomegacos(omegat+phi)}`
`therefore V=-A omegasin(omegat+phi) or V=-omegasqrt(A^(2)-x^(2))`
When `(omegat=phi)=0` then velocity `v=0`. For points where `(omegat+phi)=90^(@)`
Velocity `V=-Aomega`i.e., velocity is maximum.
iii)Acceleration (a): Acceleration of a body in S.H.M. is `a=(dy)/(dx)`
`=d/dt(-Aomegasin(omegat+phi)=-Aomega^(2)cos(omegat+phi)=-omega^(2)x)`
`a_(max)=-omega^(2)A.`