question_answer

The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle \[\theta \] without slipping and slipping down the incline without rolling is                     [NEET 2014]

A)       5 : 7

B) 2 : 3

C) 2 : 5

D) 7 : 5

Correct Answer: A

Solution :

A solid sphere rolling without slipping down an inclined plane

In this case, \[{{a}_{1}}=\frac{g\,\sin \theta }{1+\frac{{{k}^{2}}}{{{R}^{2}}}}=\frac{g\,sin\theta }{1+\frac{(2/5){{R}^{2}}}{{{R}^{2}}}}\]

\[\left[ \therefore \text{for}\,\text{solid}\,\text{sphere},{{K}^{2}}=\frac{2}{5}{{R}^{2}} \right]\]

\[=\frac{g\sin \theta }{7/5}\]

\[\Rightarrow \]   \[{{a}_{1}}=\frac{5}{7}g\,\sin \theta \]

For a sphere slipping down an inclined plane

\[\Rightarrow \]   \[{{a}_{2}}=g\sin \theta \]

\[\Rightarrow \]   \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{5/7g\,\sin \theta }{g\,\sin \theta }\]

\[\Rightarrow \]   \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{5}{7}\]