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Q.

A uniform solid cylinder rolling without slipping along a horizontal plane with angular speed $ω_{1}$ suddenly contacts a plane inclined at an angle $θ$ as shown in fig.

The value of $θ$ which could bring the cylinder immediately to rest on impact is $y \times 30 º$ find y

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answer is 4.

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Detailed Solution

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Taking angular momentum about contact point p
$I w_{1} + m v_{1} (R \cos θ) = I w_{2} + m v_{2} R$
$\frac{M R^{2}}{2} w_{1} + m (w_{1} R) (R \cos θ) = \frac{m R^{2}}{2} w^{2} + m (R w_{2}) R$
$\Rightarrow w_{2} = w_{1} (\frac{1}{3} + \frac{2}{3} \cos θ)$
$O = w_{1} [\frac{1}{2} + \frac{2}{3} \cos θ]$ $(∴ w_{2} = 0)$
$\cos θ = - \frac{1}{2}$
$θ = 120^{0}$

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