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Race of Rollers

Question

If a sphere is rolling on an inclined plane with a velocity 'v' without slipping, the vertical height of the incline in terms of velocity will be

$\frac{7 v}{10 g}$
$\frac{3 v}{5 g}$
$\frac{7 v^{2}}{10 g}$
$\frac{2 v^{2}}{5 g}$

A

\frac{7 v}{10 g}

B

\frac{7 v^{2}}{10 g}

C

\frac{3 v}{5 g}

D

\frac{2 v^{2}}{5 g}

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Solution

Verified by Toppr

The velocity of solid sphere on the bottom of inclined plane $v = \sqrt{\frac{2 g h}{1 + \frac{I}{M R^{2}}}}$
The moment of inertia of solid sphere about its diameter
$I = \frac{2}{5} M R^{2}$
$v = \sqrt{\frac{2 g h}{(1 + \frac{2}{5})}} = \sqrt{\frac{10}{7}} g h$
$h = \frac{7 v^{2}}{10 g}$

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