A uniform disc of mass m and radius $\mathrm{R}=\frac{80}{23{\mathrm{\pi }}^2}\mathrm{m}$ is pivoted smoothly at P. If a uniform ring of mass m and radius R is welded at the lowest point of the disc, find the period of SHM of the system (disc + ring). (in seconds) 

The time period of a physical pendulum is 

           $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{I}}_{\mathrm{P}}}{\mathrm{Mgr}}}$

Here we have three quantities I_P, m and r.

Let us calculate the quantities one by one as follows: 

Finding ${\mathrm{I}}_{\mathrm{P}}:{\mathrm{I}}_{\mathrm{P}}={\left({\mathrm{I}}_{\mathrm{P}}\right)}_{\text{disc }}+{\left({\mathrm{I}}_{\mathrm{P}}\right)}_{\text{ring }}$

where ${\left({\mathrm{I}}_{\mathrm{P}}\right)}_{\mathrm{disc}}={\mathrm{I}}_{\mathrm{A}}+\mathrm{m}(\mathrm{PA}{)}^2=\frac{{\mathrm{mR}}^2}{2}+{\mathrm{mR}}^2=\frac{3{\mathrm{mR}}^2}{2}$

and ${\left({\mathrm{I}}_{\mathrm{P}}\right)}_{\text{ring }}={\mathrm{I}}_{\mathrm{B}}+\mathrm{m}(\mathrm{PB}{)}^2={\mathrm{mR}}^2+\mathrm{m}(3\mathrm{R}{)}^2=10{\mathrm{mR}}^2$

Then, we have ${\mathrm{I}}_{\mathrm{P}}=\frac{3}{2}{\mathrm{mR}}^2+10{\mathrm{mR}}^2=\frac{23}{2}{\mathrm{mR}}^2$

Finding r: 

        $\mathrm{r}={\overrightarrow{\mathrm{r}}}_{\mathrm{C}}=\frac{{\mathrm{m}}_1\overrightarrow{{\mathrm{r}}_1}+{\mathrm{m}}_2\overrightarrow{{\mathrm{r}}_2}}{{\mathrm{m}}_1+{\mathrm{m}}_2}$

where ${\mathrm{m}}_1=\mathrm{m}, {\mathrm{m}}_2=\mathrm{m}$

      $\overrightarrow{{\mathrm{r}}_{1\mathrm{C}}}=-\mathrm{R}\text{ and }{\mathrm{r}}_{2\mathrm{C}}=-3\mathrm{R}$

This gives ${\mathrm{r}}_{\mathrm{C}}=2\mathrm{R}$

Finding M: 

      $\mathrm{M}=(\mathrm{m}{)}_{\mathrm{disc}}+(\mathrm{m}{)}_{\mathrm{ring}}=\mathrm{m}+\mathrm{m}=2\mathrm{m}$

Substituting ${\mathrm{I}}_{\mathrm{P}}=23/2{\mathrm{mR}}^2, \mathrm{M}=2\mathrm{m}\text{ and }\mathrm{r}=2\mathrm{R}$ in the expression

      $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{I}}_{\mathrm{P}}}{\mathrm{Mgr}}}\text{ we have }\mathrm{T}=\mathrm{\pi }\sqrt{\frac{23\mathrm{R}}{2\mathrm{g}}}$

After substituting the values we get t = 2 s.