Question

The coefficient of friction of a surface is $\frac{1}{\sqrt{3}}$. what should be the angle of inclination so that a body placed on the surface just begins to slide down?

Answer 1

Step 1: Given

Coefficient of friction, $\mu =\frac{1}{\sqrt{3}}$

Step 2: Formula used

$F=\mu N$, where $F$ is the frictional force, $\mu$ is the coefficient of friction and $N$ is the normal force.

Step 3: Draw the free body diagram

Friction is balanced by the sine component of the weight of the body and the normal reaction R must be balanced by the cosine component.

$$\begin{gathered} F=mg\sin \theta \\ N=mg\cos \theta \end{gathered}$$

Step 4: Determine the angle of inclination

The frictional force is given by,

$$\begin{gathered} F=\mu N \\ mg\sin \theta =\mu mg\cos \theta \\ \tan \theta =\mu \\ \tan \theta =\frac{1}{\sqrt{3}} \\ \theta =30° \end{gathered}$$

Therefore, the angle of inclination is $30°$.