RELATION BETWEEN LINEAR VELOCITY AND ANGULAR VELOCITY

RELATION BETWEEN LINEAR VELOCITY AND ANGULAR VELOCITY
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.
Relation between linear velocity and angular velocity
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 = rw
TANGENTIAL 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 = rw
TANGENTIAL ACCELERATION
Suppose 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 be
DVt = r D w
Change 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