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

(N/A) Periodic time: The time taken by an oscillator to complete one full oscillation is called periodic time $(T)$.
Angular frequency: The product of $2\pi$ and the frequency of an oscillator is called angular frequency $(\omega)$.
The displacement of a particle in Simple Harmonic Motion $(SHM)$ with amplitude $A$ and initial phase $\phi = 0$ at time $t$ is given by:
$x(t) = A \sin(\omega t)$ ... $(1)$
Since the motion is periodic with period $T$,the displacement repeats after time $T$:
$x(t) = x(t + T)$
$A \sin(\omega t) = A \sin(\omega(t + T))$
Since the sine function is periodic with a period of $2\pi$,the phase must increase by $2\pi$ for the motion to repeat:
$\omega(t + T) = \omega t + 2\pi$
$\omega t + \omega T = \omega t + 2\pi$
$\omega T = 2\pi$
Therefore,the relation is:
$\omega = \frac{2\pi}{T}$
Since frequency $v = \frac{1}{T}$,we can also write:
$\omega = 2\pi v$