The ratio of the accelerations for a solid sphere (mass 'm' and radius 'R') rolling down an incline of angle…
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:
5: 7
2: 3
2: 5
7: 5
Solution
For solid sphere rolling without slipping on inclined plane, acceleration
$
a_{1}=\frac{g \sin \theta}{1+\frac{K^{2}}{R^{2}}}
$
For solid sphere slipping on inclined plane without rolling, acceleration
$
a_{2}=g \sin \theta
$
Therefore required ratio $=\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}$
$
=\frac{1}{1+\frac{K^{2}}{R^{2}}}=\frac{1}{1+\frac{2}{5}}=\frac{5}{7}
$