Question

A uniform cylinder of height $h$ and radius $r$ is placed with its circular face on a rough inclined plane and the inclination of the plane to the horizontal is gradually increased. If $\mu$ is the coefficient of friction, then under what condition the cylinder will slide before toppling.

• A. $\mu <\frac{2r}{h}$
  
• B. $\mu >\frac{2r}{h}$
  
• C. $\mu <\frac{r}{h}$
  
• D. $\mu >\frac{r}{h}$
  

Answer 1

The correct option is A $\mu <\frac{2r}{h}$


Given,
Height of cylinder$=h$,
Radius of cylinder$=r$,

The free body diagram of the problem is shown below. Let $f$ is the friction force acting on the cylinder.


The cylinder will slide if
$mg\sin \theta >f$

$\Rightarrow mg\sin \theta >\mu N$

$\Rightarrow mg\sin \theta >\mu mg\cos \theta$

$\Rightarrow \tan \theta >\mu ....\left(1\right)$

Condition for toppling:
The cylinder will topple if anticlockwise moment of force $mg\sin \theta$ will be greater than the clockwise moment of the force $mg\cos \theta$ about point $\text{O}$ as shown in figure.

$\left(mg\sin \theta \right)\frac{h}{2}>\left(mg\cos \theta \right)r$

$\Rightarrow \tan \theta >\frac{2r}{h}....\left(2\right)$

Thus, the condition of sliding is $\tan \theta >\mu$ and condition of toppling is $\tan \theta >\frac{2r}{h}$.

So, the cylinder will slide before toppling if
$\mu <\frac{2r}{h}$

Hence, Option (a) is the correct answer.