Forces on a rough inclined plane: Effort parallel to the plane to move a body upward A body of weight W is to be moved up a rough plane inclined at angle α to the horizontal by a force applied parallel to the plane. The coefficient of friction is μ = tanφ. Choose the correct expression for the required effort P (just impending upward motion).

Given
Incline angle α, weight W, coefficient of friction μ, effort P applied parallel to plane to move body up the plane.

Free-body equations (limiting equilibrium upward)
Along plane: P − W sinα − F = 0 Normal: R = W cosα At impending upward motion, friction acts down the plane with magnitude F = μ R = μ W cosα.

Compute P
P = W sinα + μ W cosα = W (sinα + μ cosα)

Alternative angle of friction form
Using μ = tanφ, the same result can be expressed via trigonometric identities if needed, but the simplest correct form is above.

Final Answer
P = W (sinα + μ cosα)