Question

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?

• A. $\frac{1}{3}\sin \theta$
• B. $\frac{1}{3}\tan \theta$
• C. $\frac{2}{3}\sin \theta$
• D. $\frac{2}{3}\tan \theta$

Answer 1

Answer: (B)

Equation of motion
$Mg\sin \theta -f=Ma\dots$ (i)
Also $fR=\tau =I\alpha =M{k}^2\frac{a}{R}\dots$ (ii)
But $a=\frac{g\sin \theta }{1+\frac{k^2}{R^2}}\dots$ (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: $M{k}^2=I=\frac{1}{2}M{R}^2$
$k^2=\frac{1}{2}{R}^2;f=\frac{M\left(\frac{1}{2}\right)g\sin \theta }{\left(1+\frac{1}{2}\right)}=\frac{1}{3}Mg\sin \theta$
In case of static friction, $f_s=\mu N=\mu Mg\cos \theta$
$\frac{1}{3}Mg\sin \theta =\mu Mg\cos \theta \Rightarrow \mu =\frac{1}{3}\tan \theta$