Question

Prove that linear velocity v=r w

Answer 1

Solution:-

=> Linear velocity is as usual ,

$\rm \: \to \: v = \dfrac{ds}{dt} \: \: or \: \: \dfrac{dr}{dt}$

=> Magnitude of linear velocity is called linear speed v . thus

$\rm \to \: v = |v| = \bigg | \dfrac{ds}{dt} \: or \: \dfrac{dr}{dt} \bigg |$

Relation between linear speed and angular speed

=> In the figure, linear distance PP' travelled by the particle in time ∆t is

$\rm \to \: \Delta s = r \Delta \theta$

$\rm \to \: \displaystyle\lim_ { \rm\Delta t \to0} \dfrac{ \rm\Delta s}{ \Delta t} = \rm r\displaystyle\lim_ { \rm\Delta t \to0}\dfrac{ \rm\Delta \theta}{ \Delta t}$

$\rm \to \: \dfrac{ds}{dt} = r \dfrac{d \theta}{dt}$

where

$\omega = \displaystyle\lim_ { \rm\Delta t \to0}\dfrac{ \rm\Delta \theta}{ \Delta t} = \rm \frac{d \theta}{dt}$

We get

$\to \boxed{\rm \: v = r \omega}$