A worker is pushing a box of mass M up a rough incline by applying a horizontal pushing force of constant magnitude P. The incline makes an angle θ with respect to the vertical, and the coefficient of kinetic friction between the slope and the box is μ. Derive an expression for the acceleration of the box

The incline makes an angle θ with respect to the vertical,

component of weight along the inclined plane, downwards = mgcosθ

component of weight perpendicular to inclined surface, downwards = mg sinθ

Component of force P along the surface, upwards = Psinθ

component of force P perpendicular to surface = Pcosθ

Reaction force of surface

R = Pcosθ + mgsinθ

Force of friction

= μ. (Pcosθ + mgsinθ) acting downwards

Net force in upward direction

= P sinθ - mgcosθ - μ. (Pcosθ + mgsinθ)

acceleration

a = [P sinθ - mgcosθ - μ. (Pcosθ + mgsinθ) ] / m