A solid sphere of mass $M$ and radius $R$ rolls without slipping on a horizontal surface with a linear velocity $v$. What is the total kinetic energy of the sphere?

Solution

1. The total kinetic energy is given by $$\begin{array}{rl}K& =K_{\text{trans}}+K_{\text{rot}}\\ & =\frac{1}{2}M{v}^2+\frac{1}{2}I{\omega }^2\end{array}$$.
2. For a solid sphere, the moment of inertia is $I=\frac{2}{5}M{R}^2$ and the angular velocity for rolling without slipping is $\omega =\frac{v}{R}$.
3. Substituting these values into the equation gives $$\begin{array}{rl}K& =\frac{1}{2}M{v}^2+\frac{1}{2}(\frac{2}{5}M{R}^2)(\frac{v}{R}{)}^2\\ & =\frac{1}{2}M{v}^2+\frac{1}{5}M{v}^2\end{array}$$.
4. Simplifying the expression results in $$\begin{array}{rl}K& =0.5M{v}^2+0.2M{v}^2\\ & =0.7M{v}^2\end{array}$$.
5. Hence the answer is (B).