Q13

​Q13. A solid sphere of mass m and radius R is rolling without slipping on a rough horizontal surface with angular acceleration $\alpha$. Centre of mass of sphere lies at a distance $\frac{\mathrm{R}}{\sqrt{2}}$ from the centre of sphere. Find the normal force applied by sphere on the surface at an instant when line joining centre of mass of sphere and centre of sphere makes an angle 45° with vertical and angular velocity of sphere at this instant is $\mathrm{\omega }$.

  $$\begin{gathered} \left(\mathrm{A}\right) \mathrm{mg}+\frac{\mathrm{mR}}{2}\left(\mathrm{\alpha } –{\mathrm{\omega }}^2\right) \left(\mathrm{B}\right) \mathrm{mg}-\frac{\mathrm{mR}}{2}\left(\mathrm{\alpha }+{\mathrm{\omega }}^2\right) \\ \left(\mathrm{C}\right) \mathrm{mg}+\frac{\mathrm{mR}}{2}\left(\mathrm{\alpha } –{\mathrm{\omega }}^2\right) \left(\mathrm{D}\right) \mathrm{mg}+\frac{\mathrm{mR}}{2}\left(\mathrm{\alpha } +{\mathrm{\omega }}^2\right) \end{gathered}$$