Question

establish a relationship between linear and angular velocity?

Answer 1

Consider a particle “P” in an object (in XY-plane) moving along a circular paths of radius “r” about an
axis through “O” , perpendicular to plane of the figure i.e. z-axis. Suppose the particles moves through
an angle Dq in time Dt sec.If DS is its distance for rotating through angle Dq then, 
Dq = DS / r
Dividing both sides by Dt, we get Dq / Dt = (DS / r. Dt)
r Dq / Dt = DS/Dt
If time interval Dt is very small , then the angle through which the particle moves is also very 
small and therefore the ratio Dq /Dt gives the instantaneous angular speed wins. 
i.e.
V = rwTANGENTIAL VELOCITY
If a particle “P” is moving in a circle of radius “r”, then its linear velocity at any instant is equal to 
tangential velocity which is :V = rwTANGENTIAL ACCELERATIONSuppose an object rotating about a fixed axis changes its angular velocity by Dw in time Dt sec, 
then the change in tangential velocity DVt at the end of this interval will beDVt = r D wChange in velocity in unit time is given by:DV / Dt = r. Dw / Dt
if Dt approaches to zero then DV/Dt will be instantaneous tangential acceleration and Dw/Dt 
will be instantaneous angular acceleration “a “.at = ra