A uniformly dense solid cylinder of mass M and radius R is released from rest on an inclined plane and starts performing pure rolling. During its downward journey along the incline the cylinder moves distance l, the angle of inclination from horizontal is $α$ and the coefficient of friction is given as $μ .$ [The acceleration due to gravity is g downwards. Air resistance is not present.]

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The acceleration of center of mass of cylinder is $\frac{2 g \sin α}{3}$

The final angular speed of cylinder is $\sqrt{\frac{4}{3} \frac{g l \sin α}{R^{2}}}$

The minimum coefficient of friction required so that there is no slipping is $\frac{\tan α}{3}$

The final angular speed of cylinder is $\sqrt{\frac{2}{3} \frac{g l \sin α}{R^{2}}}$

answer is A, B, C.

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Detailed Solution

For [A]:

$\begin{array}{l} a = \frac{g \sin α}{1 + \frac{K^{2}}{R^{2}}} = \frac{2}{3} g \sin α \\ F o r [B] : \\ V^{2} = 0 + 2 \times \frac{2}{3} g \sin α l \\ \Rightarrow V = \sqrt{\frac{4}{3}} g l \sin α \Rightarrow ω = \frac{V}{R} = \sqrt{\frac{4 g l \sin α}{3 R^{2}}} \\ F o r [C] : \end{array}$

For pure rolling

$\begin{array}{l} a = α R \Rightarrow \frac{M g \sin α - f}{M} = \frac{f \times R^{2}}{\frac{M R^{2}}{2}} \\ \Rightarrow M g \sin α - f = 2 f \Rightarrow f = \frac{M g \sin α}{3} \\ \Rightarrow F o r minimum μ, μ m g \cos α = \frac{M g \sin α}{3} \Rightarrow μ_{\min} = \frac{\tan α}{3} \end{array}$