Question

The angle between the resultant contact force and the normal force exerted by a body on the other is called the angle of friction. Show that, if λ be the angle of friction and μ the coefficient of static friction λ ≤ tan⁻¹ μ.

Answer 1

Let
f be the applied force,
R be the normal reaction force and
F be the frictional force.



The coefficient of static friction is given by
$u=\tan \mathrm{\lambda }=\frac{\mathit{F}}{\mathit{R}}$
(where λ is the angle of friction)

When F = μR, F is the limiting friction (maximum friction). When applied force increases and the body still remains still static then the force of friction increases up to its maximum value equal to limiting friction (μR).

$$\begin{gathered} F\mathit{<}\mu R \\ \therefore \tan \mathrm{\lambda }=\frac{\mathit{F}}{\mathit{R}}\mathit{\le }\frac{\mathit{\mu }\mathit{R}}{\mathit{R}} \end{gathered}$$
⇒ tan λ ≤ μ
⇒ λ ≤ tan⁻¹ μ