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  4. A sphere and circular disc of same mass and radius are allowed to roll down an inclined plane from the same height without slipping. Find the ratio of times taken by these two to come to the bottom of incline :

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Q. A sphere and circular disc of same mass and radius are allowed to roll down an inclined plane from the same height without slipping. Find the ratio of times taken by these two to come to the bottom of incline :

System of Particles and Rotational Motion

Solution:

Acceleration of a rolling body on inclined plane is
$a=\frac{g \sin \theta}{1+\frac{k^{2}}{R^{2}}}$
For two bodies time of rolling same length is
$\frac{t_{1}}{t_{2}}=\sqrt{\frac{a_{2}}{a_{1}}}$
$=\sqrt{\frac{1+k_{1}^{2} / R^{2}}{1+k_{2}^{2} / R^{2}}}$
For a solid sphere $ k_{1}^{2}=\frac{2}{5} R^{2}$
and for disc $k_{2}^{2}=\frac{1}{2} R^{2}$
$\Rightarrow \frac{t_{1}}{t_{2}}=\sqrt{\frac{1+2 / 5}{1+1 / 2}}$
$=\sqrt{\frac{14}{15}}$