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Consider the arrangement shown in the figure. The pulley is light and the spool has mass $M$ . Table is sufficiently rough for pure rolling. String is light and inextensible. Center of mass of spool is at geometrical center and moment of inertia of spool about centroidal axis is $\frac{5}{2} M R^{2}$ .

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Acceleration of the spool is $\frac{4 F}{9 M}$

Acceleration of the spool is $\frac{4 F}{13 M}$

Minimum value of $μ$ for pure rolling is $\frac{9 F}{13 M g}$

Minimum value of $μ$ for pure rolling is $\frac{5 F}{9 M g}$

answer is B, C.

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Detailed Solution

$F - f = M a ... (i)$

$f \cdot 2 R - F R = l \cdot α ... (i i)$

$a = α \cdot 2 R ... (i i i)$

$\Rightarrow a = \frac{4 F}{13 M}$
$\Rightarrow f = \frac{9 F}{13}$

but $f \leq μ M g \Rightarrow μ \geq \frac{f}{M g} or μ_{\min} = \frac{9 F}{13 M g}$

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