Question

A solid sphere is rolling without slipping on a level surface at a constant speed of $2.0m{s}^{-1}$. How far can it roll up a ${30}^o$ ramp before it stops?

• A. $56cm$
• B. $26cm$
• C. $53cm$
• D. $84$

Answer 1

The correct option is A $56cm$

$\begin{array}{c}wall=\Delta k\\ \Rightarrow -mg\left(l\sin \theta \right)=-\left[\frac{1}{2}m{v}^2+\frac{1}{2}I{w}^2\right]\\ \Rightarrow mg\left(l\sin \theta \right)=\frac{1}{2}m\times v^2\times \frac{1}{2}\times \frac{2}{5}m{R}^2\times \frac{v^2}{R^2}\\ \Rightarrow mgl\sin \theta =\frac{m{v}^2}{2}+\frac{m{v}^2}{5}\\ \Rightarrow mgl\sin \theta =\frac{7m{v}^2}{10}\\ \Rightarrow gl\times \frac{1}{2}=\frac{7\times 4}{10}\\ \Rightarrow l=\frac{7\times 4\times 2}{g\times 10}m\\ =56cm\end{array}$

Hence, the option $A$ is the correct answer.