$\underset{\Delta t \rightarrow 0}{Lim}\: \frac{\Delta \theta}{\Delta t}$ = Instantaneous Angular Velocity $(\omega)$
| $v=r\omega$ |
Relation between angular velocity and linear velocity
Relation between angular velocity $(\omega)$ and linear velocity$(v)$:
We know that the angular displacement of the particle is
$\Delta \theta= \frac{\Delta s}{r} \qquad(1)$
Where $r$ = The radius of a circle.
Now divide by $\Delta t$ on both side of equation $(1)$
$\frac{\Delta \theta}{\Delta t}=\frac{1}{r} \frac{\Delta s}{\Delta t} $
If $\Delta t \rightarrow 0$ then the above equation can be written as
$\underset{\Delta t \rightarrow 0}{Lim}\: \frac{\Delta \theta}{\Delta t}=\frac{1}{r}\: \underset{\Delta t \rightarrow 0}{Lim} \: \frac{\Delta s}{\Delta t} \qquad(2)$
Where
$\underset{\Delta t \rightarrow 0}{Lim}\: \frac{\Delta \theta}{\Delta t}$ = Instantaneous Angular Velocity $(\omega)$
$\underset{\Delta t \rightarrow 0}{Lim} \: \frac{\Delta s}{\Delta t}$= Instantaneous Linear Velocity $(v)$
Now equation $(2)$ can be written as
$\omega=\frac{1}{r}v$
This is the relation between linear velocity and angular velocity.
$\underset{\Delta t \rightarrow 0}{Lim}\: \frac{\Delta \theta}{\Delta t}$ = Instantaneous Angular Velocity $(\omega)$
