Angular Velocity and its relation with linear velocity

Angular Velocity and its relation with linear velocity:

Typical particle (at a point P) of the rigid body rotating about a fixed axis (taken as the z-axis). The particle describes a circle with a center C on the axis. The radius of the circle is r, the perpendicular distance of the point P from the axis. We also show the linear velocity vector v of the particle at P. it is along the tangent at P to the circle.

Let P' be the position of the particle after an interval of time Δt. The angle PCP' describes the angular displacement Δθ of the particle in time Δt. The average angular velocity of the particle over the interval Δt which is the instantaneous angular velocityof the particle at is. As Δt tends to zero (i.e., takes smaller and smaller values), the ratio approaches a limit the position P. we denote the instantaneous angular velocity by ω (the Greek letter omega).

We know that arc length = radius x angle wept by the particle

By differentiation the above equation we can get

Where v = linear velocity and ω=angular velocity

Where r is the radius of the circle.

We observe that at any given instant the relation v= ωr applies to all particles of the rigid body. Thus for a particle at a perpendicular distance r_t from the fixed axis, the linear velocity at a given instant v_t is given by

                                                         v_t = ω r_t

The index i runs from 1 to n where n is the total number of particles of the body.