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Two-cylindrical-hollow-drums-of-radii-R-and-2R-and-of-a-common-height-h-are-rotating-with-angular-velocities-anti-clockwise-and-clockwise-respectively-Their-axes-fixed-are-parallel-and-in




Question Number 24434 by Tinkutara last updated on 17/Nov/17
Two cylindrical hollow drums of radii  R and 2R, and of a common height h,  are rotating with angular velocities ω  (anti-clockwise) and ω (clockwise),  respectively. Their axes, fixed are  parallel and in a horizontal plane  separated by (3R + δ). They are now  brought in contact (δ → 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?
$$\mathrm{Two}\:\mathrm{cylindrical}\:\mathrm{hollow}\:\mathrm{drums}\:\mathrm{of}\:\mathrm{radii} \\ $$$${R}\:\mathrm{and}\:\mathrm{2}{R},\:\mathrm{and}\:\mathrm{of}\:\mathrm{a}\:\mathrm{common}\:\mathrm{height}\:{h}, \\ $$$$\mathrm{are}\:\mathrm{rotating}\:\mathrm{with}\:\mathrm{angular}\:\mathrm{velocities}\:\omega \\ $$$$\left(\mathrm{anti}-\mathrm{clockwise}\right)\:\mathrm{and}\:\omega\:\left(\mathrm{clockwise}\right), \\ $$$$\mathrm{respectively}.\:\mathrm{Their}\:\mathrm{axes},\:\mathrm{fixed}\:\mathrm{are} \\ $$$$\mathrm{parallel}\:\mathrm{and}\:\mathrm{in}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{plane} \\ $$$$\mathrm{separated}\:\mathrm{by}\:\left(\mathrm{3}{R}\:+\:\delta\right).\:\mathrm{They}\:\mathrm{are}\:\mathrm{now} \\ $$$$\mathrm{brought}\:\mathrm{in}\:\mathrm{contact}\:\left(\delta\:\rightarrow\:\mathrm{0}\right). \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Show}\:\mathrm{the}\:\mathrm{frictional}\:\mathrm{forces}\:\mathrm{just}\:\mathrm{after} \\ $$$$\mathrm{contact}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Identify}\:\mathrm{forces}\:\mathrm{and}\:\mathrm{torques}\:\mathrm{external} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{system}\:\mathrm{just}\:\mathrm{after}\:\mathrm{contact}. \\ $$$$\left(\mathrm{c}\right)\:\mathrm{What}\:\mathrm{would}\:\mathrm{be}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{final} \\ $$$$\mathrm{angular}\:\mathrm{velocities}\:\mathrm{when}\:\mathrm{friction}\:\mathrm{ceases}? \\ $$

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