# A Cylinder Having A Radius Of 0.4m, Initially Rotating (Att=0) With Ω0 = 54rad/S Is Placed On A Rough Inclined Plane With Θ=37° Having Friction Coefficient Μ=0.5. The Time Taken By The Cylinder To Start Pure Rolling Is (G=10m/S 2)

$a = (\mu \;g \;cos \Theta ) + (g \;sin \Theta) \\\Rightarrow 0.5 * 10 * 0.8 + 10 * 0.6 = 10m/s^{2} \\\Rightarrow \alpha = \frac{(\mu\; mg \;cos\Theta) R}{\frac{1}{2}\;mR^{2}} = \frac{2\mu\; g\; cos\Theta}{R} \\\Rightarrow \frac{2 * 0.5 * 10 * 0.8}{0.4} = 20 rad/s^{2}$

Pure rolling will start when, v = RΩ or at = R (Ω 0 – at)

Therefore, 10 t = 0.4 (54 – 20 t)

t = 1.2 seconds.

The time taken by the cylinder to start pure rolling is 1.2 seconds.

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