question_answer

A uniform circular disc of radius R is rolling on a horizontal surface. Determine the tangential velocity (i) at the upper most point, (ii) at the centre of mass and (Hi) at the point of contact.

Answer:

                Rolling can be regarded as the combination of pure rotation and pure translation. As shown in Fig. (a), in case of pure rotation the, velocity of CM is zero and the tangential velocity at points A and B is \[{{\upsilon }_{A}}={{\upsilon }_{B}}=R\omega .\]. As shown in Fig. (b), in case of pure translation, \[{{\upsilon }_{A}}={{\upsilon }_{B}}={{\upsilon }_{CM}}=R\omega =\upsilon \](say)   As shown in Fig. (c), in case of rolling, the tangential velocity at any given point is the vector sum of the velocities in (a) and (b) at that point. Hence \[{{\upsilon }_{A}}2{{\upsilon }_{CM}},{{\upsilon }_{CM}}=\upsilon =R\omega \] and \[{{\upsilon }_{B}}=0\] It may be noted that in case of rolling the instantaneous velocity at the point of contact B with surface is zero, because the body is not slipping.