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Q.
What should be the minimum coefficient of static friction between the plane and the cylinder, for the cylinder not to slip on an inclined plane?
System of Particles and Rotational Motion
Solution:
Equation of motion
$M g \sin \theta-f=M a \ldots$ (i)
Also $f R=\tau=I \alpha=M k^{2} \frac{a}{R}\ldots$ (ii)
But $a=\frac{g \sin \theta}{1+\frac{k^{2}}{R^{2}}}\ldots$ (iii)
Putting value of $a$ in equation (ii)
$f=\left(\frac{M k^{2}}{R^{2}}\right) \frac{g \sin \theta}{1+\frac{k^{2}}{R^{2}}}$
For cylinder: $M k^{2}=I=\frac{1}{2} M R^{2}$
$k^{2}=\frac{1}{2} R^{2} ; \quad f=\frac{M\left(\frac{1}{2}\right) g \sin \theta}{\left(1+\frac{1}{2}\right)}=\frac{1}{3} M g \sin \theta$
In case of static friction, $f_{s}=\mu N =\mu M g \cos \theta$
$\frac{1}{3} M g \sin \theta=\mu M g \cos \theta \Rightarrow \mu=\frac{1}{3} \tan \theta$
