Question

In figure, a crate slides down an inclined right-angled trough. The coefficient of kinetic friction between the crate and the trough is ${\mu }_k.$ What is the acceleration of the crate in terms of ${\mu }_k,\theta$ and g?

• A. $g(\sin \theta -\sqrt{2}{\mu }_k\cos \theta )$
• B. $g(\sin \theta -\frac{{\mu }_k\cos \theta }{\sqrt{2}})$
• C. $g(\sin \theta +\sqrt{2}{\mu }_k\cos \theta )$
• D. $g(\sin \theta +\frac{{\mu }_k\cos \theta }{\sqrt{2}})$

Answer 1

The correct option is A $g(\sin \theta -\sqrt{2}{\mu }_k\cos \theta )$

The force of gravity is acting downwards
Resolving gravitation force in inclination plane
along incline plane $mgsin\theta -f=ma$..(i)
perpendicular to incline plane $\frac{N_1+N_2}{\sqrt{2}}=mgcos\theta$...(ii)
The kinetic friction is given by $f=\sqrt{2}{\mu }_{k}mgcos\theta$...(iii)
Substituting value of $f$ in equation (i)
$a=g(sin\theta -\sqrt{2}{\mu }_{k}cos\theta )$