Question

A thin circular ring of mass M and radius r​ is rotating about its axis with constant angular velocity omega. Two objects each of mass m are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with angular velocity?

Answer 1

Dear Student,

$$\begin{gathered} {\mathrm{I}}_1{\mathrm{\omega }}_1={\mathrm{I}}_2{\mathrm{\omega }}_2............\left(1\right) \\ {\mathrm{I}}_1={\mathrm{MR}}^2...................\left(2\right) \\ \mathrm{and}{\mathrm{I}}_2={\mathrm{MR}}^2+2{\mathrm{mR}}^2...................\left(3\right) \\ \mathrm{from}\left(1\right),\left(2\right)\mathrm{and}\left(3\right),\mathrm{we}\mathrm{get}, \\ {\mathrm{\omega }}_2=\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}\mathrm{\omega } \\ {\mathrm{\omega }}_2=\frac{{\mathrm{MR}}^2}{{\mathrm{MR}}^2+2{\mathrm{mR}}^2}\times \mathrm{\omega } \\ {\mathrm{\omega }}_2=\frac{\mathrm{M\omega }}{\mathrm{M}+2\mathrm{m}} \end{gathered}$$