Question

A solid cylinder is rolling down on an inclined plane of angle $\theta$. The coefficient of static friction between the plane and the cylinder is ${\mu }_s$. The condition for the cylinder not to slip is

• A. $\tan \theta \ge 3{\mu }_s$
• B. $\tan \theta >3{\mu }_s$
• C. $\tan \theta \le 3{\mu }_s$
• D. $\tan \theta <3{\mu }_s$

Answer 1

The correct option is C $\tan \theta \le 3{\mu }_s$

Torque about point O ${\tau }_o=fR$
where $f={\mu }_{s}mg\cos \theta$
Using ${\tau }_o=I\alpha$
$\therefore$ $fR=\frac{1}{M{R}^2}\alpha$
$⟹f=\frac{M}{2}R\alpha =\frac{1}{2}Ma$ (for rolling $a=R\alpha$)
For Cylinder not to slip,
$Ma\ge Mg\sin \theta -f$
Or $2f\ge Mgsin\theta -f$
Or $3f\ge Mg\sin \theta$
Or $3{\mu }_{s}Mg\cos \theta \ge Mg\sin \theta$
$⟹\tan \theta \le 3{\mu }_s$