Obtain the expression of displacement from force law for simple harmonic motion.

Explanation

Force law for SHM
`F= -kx (t)`
`ma = -m omega^(2) x(t) " "[therefore F= ma(t), k= m omega^(2)]`
`therefore a(t) = -omega^(2)x (t)`
Integrating w.r.t time t on both sides,
`int a(t) dt = -omega^(2) int x (t) dt`
`= -omega^(2) int A cos (omega t+ phi) dt`
`a(t) xxt = -A omega^(2) int cos (omega t+ phi) dt`
`v(t) = -A omega^(2)xx (sin (omega t+ phi))/(omega)`
`therefore v(t) = -A omega sin (omega t+ phi)`
integrating again w.r.t. time t,
`int v(t) dt= -A omega int sin (omega t+phi)dt`
`therefore v(t)xxt = -A omega [-(cos (omega t+phi))/(omega)]`
`therefore x(t) = -A cos (omega t+phi)`
is the expression of displacement.