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Physics Question on Inclined planes

A solid cylinder is rolling down on an inclined plane of angle θ,\theta , The coefficient of static friction between the plane and cylinder is μs \mu _{s} . Then condition for the cylinder not to slip is

A

tan⁡ θ≥ 3μs \tan \,\theta \ge \,3\mu_{s}

B

tan⁡ θ>3μs \tan \,\theta >3\mu _{s}

C

tan⁡ θ≤3μs \tan \,\theta \le 3\mu _{s}

D

tan⁡ θ<3μs \tan \,\theta < 3\mu_{s}

Answer

tan⁡ θ≤3μs \tan \,\theta \le 3\mu _{s}

Explanation

Solution

For the rolling of a solid cylinder acceleration
a=g3sin⁡θa=\frac{g}{3} \sin \theta
∴\therefore The condition for the cylinder to remain in equilibrium
ma≤μsRm a \leq \mu_{s} R
⇒13Mgsin⁡θ≤Mgcos⁡θ⋅μs\Rightarrow \frac{1}{3} M g \sin \theta \leq M g \cos \theta \cdot \mu_{s}
or μs≥13tan⁡θ\mu_{s} \geq \frac{1}{3} \tan \theta
or tan⁡θ≤3 μs\tan \theta \leq 3\, \mu_{s}