Define periodic time and angular frequency and obtain the relation between them.

Explanation

Periodic time : ..The time required to complete one rotation is called periodic time (T)...
Angular frequency : `2pi` time the frequency of an oscillator is called angular frequency `(omega)`
`therefore omega = (2pi)/(T)`
The displacement of SHM particle with amplitude A and initial phase `phi=0` at time t is `x(t)= A sin omega t" ""........."(1)`
Since the motion has a period T, hence
`x(t)= x(t+T)" That is "A cos omega t= A cos (omega t + phi)" ""........."(2)`
Now the cosine function is periodic with period `2pi` means phase increase by `2pi` and it motion repeared.
`therefore omega t+2pi = omega(t+T)`
`therefore omega t+ 2pi = omega t+ omega T`
`therefore 2pi = omegaT`
`therefore omega = (2pi)/(T)`
but `(1)/(T)=v` (frequency)
`therefore omega= 2pi v`
So `pi" is "2pi` time the frequency of oscillation `v(1/T)`.