Question

show that linear speed of particle performing circular motion is product of radius of circle and angular speed of particle

Answer 1

Let us consider the linear speed of the particle is v.

Let r  denotes the radius of the circle and ω as the angular velocity of the particle.

Let us consider the position of the particle at any instant of time t . Let the angular displacement of the particle is $\theta$.

Let the particle reached at any other  point at instant t'. The angular displacement corresponding to that instant is $\theta '$

The angular velocity is calculated as -

                                         $\omega=\frac{\theta '-\theta}{t'-t}$

We know that linear displacement is the product of radius with angular displacement .

Mathematically linear displacement [s] = $r\theta$

                                               $=\frac{s'/r-s/r}{t'-t}$

                                               $=\frac{1}{r}[ \frac{s'-s}{t'-t} ]$

                                               $=\frac{v}{r}$

                                           ⇒ $\ v\ = \omega r$    [proved]

Here, s and s' are the the linear displacements at instants  t and t'.