Question

The ratio of radius of gyration of a circular disc about a tangential axis in the plane of the disc and a circular ring of the same radius about a tangential axis in the plane of the ring is

Answer 1

$$\begin{gathered} Disc \\ Momentofinertiaaboutthediameter=I_d=\frac{m{r}^2}{4} \\ momentofinertiaaboutthe\tan gentialaxisI=I_d+m{r}^2 \\ {I}_{disc}=\frac{m{r}^2}{4}+m{r}^2=\frac{5m{r}^2}{4} \\ {K}_{disc}=radiusofgyration \\ \frac{5m{r}^2}{4}=m{k_{disc}}^2 \\ {k}_{disc}=\sqrt{\frac{5}{4}}r \\ Ring \\ Momentofinertiaaboutthediameter=I_{dring}=\frac{m{r}^2}{2} \\ momentofinertiaaboutthe\tan gentialaxisI=I_{dring}+m{r}^2 \\ {I}_{ring}=\frac{m{r}^2}{2}+m{r}^2=\frac{3m{r}^2}{2} \\ {k}_{ring}=radiusofgyrationofring \\ m{k_{ring}}^2=\frac{3m{r}^2}{2} \\ {k}_{ring}=\sqrt{\frac{3}{2}}r \\ \frac{k_{disc}}{k_{ring}}=\frac{\sqrt{\frac{5}{4}}r}{\sqrt{\frac{3}{2}}r}=\sqrt{\frac{5}{6}} \end{gathered}$$