Question

Define angular velocity $\left(\omega \right)$. Derive $v=r\omega$

Answer 1

The rate of change of angular displacement($\theta$) of a body is known as angular velocity($\omega$).

$\omega =\frac{d\theta }{dt}$

Consider a rigid body moving with uniform velocity $v$ along with the circumference of a circle of radius $r$. Assume body covers a linear distance $△x$ in small interval of time $△t$, making an angle $△\theta$ at the center.

Now, length of the arc = radius X angle

$△x=r\times △\theta$

Divide by $△t$ and take limits $△t\to 0$ (small time interval)

${lim}_{t\to 0}\frac{△x}{△t}=r\left({lim}_{t\to 0}\frac{△\theta }{△t}\right)$

But, ${lim}_{t\to 0}\frac{△x}{△t}=v$ and ${lim}_{t\to 0}\frac{△\theta }{△t}=\omega$

Therefore, $v=r\omega$