Question

A solid uniform sphere rotating about its axis with angular velocity ${\omega }_0$ is placed on a rough horizontal plane without any transnational push. Friction co-efficient is not same everywhere on the plane and is different at each point. After some time the sphere begins pure rolling with angular velocity $\omega$ equal to

• A. $\frac{3{\omega }_0}{7}$
• B. $\frac{{\omega }_0}{7}$
• C. $\frac{2{\omega }_0}{7}$
• D. None of these

Answer 1

The correct option is C $\frac{2{\omega }_0}{7}$

$\begin{array}{c}\text{Let the mass of the sphere be}\left(M\right)\text{and}\\ \text{Radius}=R\\ \text{Therefore moment of Inertia}I=\frac{2}{5}M{R}^2\text{-}\\ \text{we have-}\\ \text{Initial angular momentum}\\ L_i=I{\omega }_0=\frac{2}{5}M{R}^2{\omega }_{0}D\end{array}$

$\begin{array}{c}\text{Let after sometime the Sphere starts pure rolling}\\ \text{Then, wet the angular velocity be}{w}^{\prime }\\ \text{angular momentum final -}\\ L_f=M{\omega }^{\prime }{R}^2+\frac{2}{5}M{\omega }^{\prime }{R}^2=\frac{7}{5}m{\omega }^{\prime }{R}^2\end{array}$

$\begin{array}{c}\text{From conservation of angular momentum-}\\ L_i=L_f\text{Foom eq}D\text{and}\left(2\right)\text{we get}-\\ \Rightarrow {\omega }^{\prime }=\frac{2}{7}{w}_0\end{array}$