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A uniform cylinder of mass M and radius R is placed on a rough horizontal board, which in turn is placed on a smooth surface. The coefficient of friction between the board and the cylinder is $μ$ . If the board starts accelerating with constant acceleration a, as shown in the figure, then

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for pure rolling motion of the cylinder, direction of frictional force is forward and its magnitude is Ma/3

the maximum value of a, so that the cylinder performs pure rolling is 3 $μ$ g

the acceleration of the centre of mass of the cylinder under pure rolling condition for the given a is a/3

the acceleration of the centre of mass of the cylinder under pure rolling condition for the given a is a/8.

answer is A, B, C.

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Detailed Solution

Let the frictional force be in the forward direction, then

$f = {Ma}_{0}$

and $fR = \frac{{MR}^{2}}{2} α_{0}$

$a_{0} = \frac{f}{M} and α_{0} = \frac{2 f}{MR}$

For pure rolling, $a_{0} + {Rα}_{0} = a$

$\frac{f}{M} + \frac{2 f}{M} = a \Rightarrow f = \frac{Ma}{3}$

For pure rolling, $f < f_{L} = μMg$

$\frac{Ma}{3} < μMg \Rightarrow a < 3 μg$

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