question_answer

Equal torques are applied on a cylinder and a hollow sphere. Both have same mass and radius. The cylinder rotates about its axis and the sphere rotates about one of its diameters. Which will acquire greater speed? Explain.

Answer:

                                From     \[\tau =I\alpha \]                             or            \[\alpha =\frac{\tau }{I}\]                             As \[\tau =cons\tan t\], \[\alpha \propto \frac{1}{I}\]                     \[\therefore \]  \[\frac{{{\alpha }_{c}}}{{{\alpha }_{s}}}=\frac{{{I}_{s}}}{{{I}_{c}}}\] Now moment of inertia of cylinder about its axis is \[{{I}_{c}}=\frac{1}{2}M{{R}^{2}}\] and moment of inertia of hollow sphere about its diameter is \[{{I}_{s}}=\frac{2}{3}M{{R}^{2}}\] \[\therefore \]  \[\frac{{{\alpha }_{c}}}{{{\alpha }_{s}}}=\frac{2/3\,M{{R}^{2}}}{1/2M{{R}^{2}}}=\frac{4}{3}\]                        \[\therefore \]  \[{{\alpha }_{c}}>{{\alpha }_{s}}\] Hence, the cylinder will acquire greater speed compared to the hollow sphere.