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 $\lambda$ be the angle of friction and $\mu$ the coefficient fo static friction, $\lambda \le {\tan }^{-1}\mu$

Answer 1

Let f be applied force
F be frictional force and R be normal reaction

$\therefore u=\tan \lambda =\frac{F}{R}$
When F = \mu R, F is the limiting friction (max friction) when applied force increase, force of friction incrase upto limiting friction $\left(\mu R\right)$
$F<\frac{\mu R}{F}\therefore \tan \lambda =\frac{F}{R}<\mu R\Rightarrow \tan \lambda \le u\Rightarrow \lambda \le {\tan }^{-1}\mu$