Obtain the force law for SHM from the displacement of SHM particle.

Explanation

Displacement of SHM particle at time t
`x(t)= A cos (omega t+ phi)" ""……."(1)`
where A= amplitude, `omega` = angular frequency and `phi`= initiall phase.
Differentiating equation (1) w.r.t. time t
`v(t)= (d[x(t)])/(dt)= (d)/(dt)[ A cos (omega t+phi)]`
`v(t)= -A omega sin (omega t+phi)`
Differentiating again w.r.t., time t
`a(t) = (d[v(t)])/(dt)= (d)/(dt) [-A omega sin (omega t + phi)]`
`therefore a(t) = -A omega^(2) cos (omega t + phi)`
but `A cos (omega t+phi) = x(t)`
`therefore a(t) = -omega^(2)x (t)`
multiply by mass m on both sides
`ma (t) = -m omega^(2) x(t)`
`therefore F= -kx(t)" where "ma= F, m omega^(2) = k`
`therefore F propto -x(t)`.