Question

A cubical block of mass $m$ slides in an inclined right angled trough as shown in the figure. If the coefficients of kinetic friction between block and material composing the trough is ${\mu }_k$, find the acceleration of the block. (Both walls of the trough make an angle of 45

^∘

with the inclined plane)

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

Answer 1

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

Equation of motion of the block is
$ma=mg\sin \theta -2{\mu }_{k}mg\cos \theta \cos {45}^{\circ }.$
(Net force = Gravitational force + Frictional force)
In the second term on the right side of above equation factor $2$ arises due to the fact that frictional force arises from two walls of the trough, and term $\cos {45}^{\circ }$ takes care of the fact that the walls are tilted at ${45}^{\circ }$ to the axis of channel, and the normal reaction is accordingly reduced.
$\therefore a=g(\sin \theta -\sqrt{2}{\mu }_k\cos \theta )$
Hence, the correct answer is option (a).