question_answer

Two identical cylinders 'run a race' starting from rest at the top of an inclined plane, one slides without rolling and other rolls without slipping. Assuming that no mechanical energy is dissipated as heat, which one will win?

Answer:

                When the cylinder slides without rolling,                 \[E=\frac{1}{2}m\upsilon {{'}^{2}}\]         \[\therefore \]  \[\upsilon '=\sqrt{\frac{2E}{m}}\]                 When the cylinder rolls without slipping,                 \[E=\frac{1}{2}m{{\upsilon }^{2}}+\frac{1}{2}I{{\omega }^{2}}=\frac{1}{2}m{{\upsilon }^{2}}+\frac{1}{2}\left( \frac{1}{2}m{{r}^{2}} \right){{\omega }^{2}}\]                 \[=\frac{1}{2}m{{\upsilon }^{2}}+\frac{1}{4}m{{\upsilon }^{2}}=\frac{3}{4}m{{\upsilon }^{2}}\]      \[\therefore \]    \[\upsilon =\sqrt{\frac{4E}{3m}}\] As  \[\upsilon '>\upsilon ,\]therefore sliding cylinder will win the race.