Friction revisited: rough surfaces, μ and the angle of friction

Horizontal plane, slanted pull. A pull P at angle θ above the horizontal splits into P cos θ along the floor and P sin θ upwards. The upward part helps to carry the weight, so the normal reaction becomes smaller: N = W − P sin θ. Horizontally the friction balances the other part: F = P cos θ. If the body is about to slide, F = μN, so P cos θ = μ(W − P sin θ), which gives P = μW / (cos θ + μ sin θ). A push at an angle below the horizontal does the opposite: N = W + P sin θ. Inclined plane. Take a body at rest on a rough plane tilted at θ. Only two forces act on it, the weight W and the resultant reaction R′, so R′ equals W, vertically upwards. Splitting W along the plane and across it: the friction up the line of greatest slope is F = W sin θ and the normal reaction is N = W cos θ. As the plane is tilted more, the friction needed grows like tan θ. The body is about to slide when F = μN, that is tan θ = μ, so θ = λ: the body rests as long as θ is not more than the angle of friction.