Question

# The coefficient of friction between two surfaces is 0.2. The angle of friction is$\text{A}\text{. }{{\sin }^{-1}}(0.2)$\text{B}\text{. co}{{\text{s}}^{-1}}(0.2)$\text{C}\text{. ta}{{\text{n}}^{-1}}(0.1)$$\text{D}\text{. co}{{\text{t}}^{-1}}(5)$

Hint: The angle of friction ($\theta$) is defined as the angle made by the resultant of the normal reaction and the maximum frictional force with the normal reaction. In this case, $\tan \theta =\dfrac{{{f}_{\max }}}{N}$ and ${{f}_{\max }}=\mu N$.

Formula used:
${{f}_{\max }}=\mu N$

Consider a block of mass m resting on a horizontal surface. Let the coefficient of friction between the surface of the block and the horizontal surface be $\mu$.
Friction is a force that opposes the relative motion between two surfaces. Therefore, if we try to move this block with a force, the frictional force between the block and horizontal surface will oppose the applied force. The direction of frictional force is always parallel to the surfaces in contact and opposite to the relative motion or force applied.
Force of friction is a variable force. The maximum value of the frictional force on the block is given by ${{f}_{\max }}=\mu N$, where N is the normal reaction on the block by the horizontal surface. In this case, the normal reaction on the block is in upward direction.
The angle between the resultant of the maximum force of friction and the normal reaction and the normal reaction is called angle of friction.

Let us find the value of the angle of friction. Let the angle be $\theta$.
If you see the given figure, $\tan \theta =\dfrac{{{f}_{\max }}}{N}$ …… (i).
And we know that ${{f}_{\max }}=\mu N$. Substitute this value in equation (i).
$\Rightarrow \tan \theta =\dfrac{\mu N}{N}$.
$\Rightarrow \tan \theta =\mu$
This means the tangent of the angle of friction between two surfaces is equal to the coefficient of friction between the two surfaces.
In the given question, $\mu$=0.2.
Hence, $\Rightarrow \tan \theta =0.2$.
The reciprocal of $\tan \theta$ is $\cot \theta$.
$\Rightarrow \cot \theta =\dfrac{1}{\tan \theta }=\dfrac{1}{0.2}$.
$\Rightarrow \cot \theta =\dfrac{1}{\tan \theta }=\dfrac{1}{0.2}=5$
$\Rightarrow \theta ={{\cot }^{-1}}(5)$
This means the angle of friction between the given two surfaces is ${{\cot }^{-1}}(5)$.
Hence, the correct answer is option D.

Note:
Students may confuse between angle of friction and angle of repose.
We already know what angle of friction is.
Angle of repose is the minimum angle of inclination of a surface on which a block begins to slide under the gravitational force.