Question

A block kept on a rough inclined plane, as shown in the figure remains at rest upto a maximum force $2\text{N}$ down the inclined plane. The maximum external force up the inclined plane that does not move the block is $10\text{N}$. The coefficient of static friction between the block and the plane is
(Take $g=10m/s^2)$

• A. $\frac{2}{3}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{\sqrt{3}}{4}$
• D. $\frac{1}{2}$

Answer 1

The correct option is B $\frac{\sqrt{3}}{2}$

Block does not move upto a maximum applied force of $2\text{N}$ down the inclined plane.

[$f$ is the friction force]
$2+mg\sin \theta =f$
or $2+mg\sin \theta =\mu N=\mu mg\cos \theta$.....(i)
Similarly, block also does not move upto a maximum applied force of $10\text{N}$ up the plane.

Equating forces, we have
$mg\sin \theta +f=10$
$mg\sin \theta +\mu mg\cos \theta =10$ .......(ii)
Now, solving Eqs. (i) and (ii), we get
$mg\sin \theta =4$.....(iii)
and $\mu mg\cos \theta =6$ ........(iv)
Dividing Eqs. (iv) by (iii), we get
$\mu \cot \theta =\frac{3}{2}$
$\Rightarrow \mu =\frac{3\tan \theta }{2}=\frac{3\tan {30}^{\circ }}{2}$
$\Rightarrow \mu =\frac{\sqrt{3}}{2}$