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A sphere rolls without slipping on an incline of inclination $θ$ . The minimum coefficient of static friction to support pure rolling is to be

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$\frac{2}{3} Tan θ$

$\frac{2}{5} Tan θ$

$\frac{2}{7} Tan θ$

$\frac{3}{7} Tan θ$

answer is C.

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Detailed Solution

$μ_{s} = \frac{\tan θ}{(1 + \frac{r^{2}}{k^{2}})}$

For solid sphere

$I = \frac{2}{5} {mr}^{2} \Rightarrow \frac{2}{5} {mr}^{2} = {mk}^{2}; \Rightarrow \frac{r^{2}}{k^{2}} = \frac{5}{2};$

$∴ μ_{s} = \frac{\tan θ}{1 + \frac{5}{2}} = \frac{2 \tan θ}{7}$

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