question_answer 57)

                  Two cylindrical hollow drums of radii R and 2R, and of a common height h, are rotating with angular velocities \[\omega \] (anti clockwise) and \[\omega \] (clockwise), respectively. Their axes, fixed are parallel and in a horizontal plane separated by \[(3R+\delta )\]. They are now brought in contact \[(\delta \,\to 0)\].                 (a) Show the frictional forces just after contact.                 (b) Identify forces and torques external to the system just after contact.                 (c) What would be the ratio of final angular velocities when friction ceases?                

Answer:

                  (a)                 Frictional force on drum of radius R (F)                 (b) \[\Sigma \,F=\,O,\,\,{{F}_{1}}\,=\,{{F}_{2}}=F\]                 \[{{\tau }_{ext}}\,=F\times \,3R\] (c) When force of friction cases                 \[{{\upsilon }_{1}}\,={{\upsilon }_{2}}\] or \[R{{\omega }_{1}}\,=2R\,{{\omega }_{2}}\]                 \[\therefore \] \[\frac{{{\omega }_{1}}}{{{\omega }_{2}}}=2\].