question_answer

If two circular discs of the same mass and thickness are made from metals of different densities, which disc will have the larger moment of inertia about its central axis ? Explain.

Answer:

                \[{{I}_{1}}=\frac{1}{2}mr_{1}^{2}\]  and \[{{I}_{2}}=\frac{1}{2}mr_{2}^{2}\]                 \[\therefore \]  \[\frac{{{I}_{1}}}{{{I}_{2}}}=\frac{r_{1}^{2}}{r_{2}^{2}}\]                 Also,      \[m=\pi r_{1}^{2}t{{\rho }_{1}}=\pi r_{2}^{2}t{{\rho }_{2}}\]                 \[\therefore \]  \[\frac{r_{1}^{2}}{r_{2}^{2}}=\frac{{{\rho }_{2}}}{{{\rho }_{1}}}\]                 Hence   \[\frac{{{I}_{1}}}{{{I}_{2}}}=\frac{{{\rho }_{2}}}{{{\rho }_{1}}}\] i.e.,          \[I\propto \frac{1}{\rho }\] Thus the disc with greater density will have less moment of inertia.