A box of mass m slides down an inclined plane that makes an angle of φ with the horizontal. if the coefficient of kinetic friction is μk, find an expression for the acceleration of the box down the slope. the acceleration due to gravity is g.

In a force diagram set-up, we name the angle of inclination theta, g as the acceleration due to gravity. In this case, the forces acting on the box going down is the weight itself impeded by the friction between the box and the inclined plane.

The weight of the box is expressed as mg sin theta

The frictional force is expressed as the normal force times the coefficient of friction that is expressed as mu g cos theta.

By Newton's second law of motion, F = ma = mg sin theta - mu g cos theta

Thus, a = g (sin theta - u cos theta