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The spool shown in the figure is placed on a rough horizontal surface has inner radius r and outer radius R. The angle $θ$ between the applied force F (tangential to the circle of radius r) and the horizontal can be varied. The critical angle $(θ)$ for which the spool does not roll and remains stationary is $\frac{π}{n}$ , then n is___(if $R = 2 r$ )

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Detailed Solution

If spool is not to translate $F \cos θ = f (1)$
If spool is not to rotate Fr = fR.. (2)
From eq. (1) and (2) we get static friction $\frac{f R}{r} \cos θ = f$
or $\cos θ = \frac{r}{R}$ or $θ = \cos^{- 1} (\frac{r}{R})$
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