A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is where x = __________. [2025]

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Solution

Q.
A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{5}}$ where x = __________. [2025]

Ans.
(4)

Applying Mechanical Energy conservation:

$k_{i} + U_{i} = k_{f} + U_{f}$

$\Rightarrow 0 + m g h = \frac{1}{2} m v^{2} (1 + \frac{k^{2}}{R^{2}}) + 0$

$\Rightarrow V = \sqrt{\frac{2 g h}{1 + \frac{k^{2}}{R^{2}}}}$

So, Ratio of velocities

$\frac{V_{Ring}}{V_{Solid sphere}} = \sqrt{\frac{1 + \frac{2}{5}}{1 + 1}} = \sqrt{\frac{7}{10}}$

x = 3.5 $\Rightarrow$ Rounding off x = 4.

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