RELATION BETWEEN LINEAR VELOCITY AND ANGULAR VELOCITY




Consider a particle “P” in an object (in XYplane) moving along a circular paths of radius “r” about an axis through “O” , perpendicular to plane of the figure i.e. zaxis. 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 w_{ins}. 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 DV_{t} at the end of this interval will be 

DV_{t} = 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 “. 

a_{t} = ra
