Vector relation between linear velocity and angular velocity :
When a rigid body rotates on its stable axis, then every particle of the body moves in a circular path. Every circle is situated on the rotational plane which is vertical to rotational axis and is centered on the axis. In pure rotational motion every particle of the rigid body moves with the same angular velocity. The value of the speed of the particle is directly proportional to its angular velocity. In the diagram, a rigid body particle P is moving in a circular path of radius r. If at any instant its angular displacement is ‘θ’ radian, then
θ = \(\frac{\mathrm{arc}}{\text { radius }}=\frac{s}{r}\) ⇒ s = rθ
Differentiating the above equation with respect to time taking r constant, we get \(\frac{d s}{d t}=r \frac{d \theta}{d t}\)
Taking absolute value at both the ends;
From equation(2), it is clear that the particle that is away from the axis of rotation, will move with more linear velocity.
To represent the above relationship in vector form, assume that the particle P has position vector \(\vec{R}=\overrightarrow{O P}\) at time t = 0. Therefore, from the figure,
\(\frac{r}{R}\) = sinϕ
From equation (2)
o = ωRsinϕ ………….. (3)
or \(\vec{v}=\vec{\omega} \times \vec{R}\) ……… (4)
This, is the vector relation between linear velocity and angular velocity.