Example of Equilibrium of body on the rough inclined:

Magnitude of minimum force 'p' that is required to move the body up the plane. At the time when 'p' is acted with the angle of φ.

Resolving the forces which are Parallel to Plane OA:

Pcosφ  - µR - W.sinα  = 0                                                                                                                ...(i)

Resolving all the forces Perpendicular to Plane OA:

R + Psinφ - W.cos  α = 0                                                                                                                ...(i)

Putting value of 'R' from (ii) in equation (i) we get

P = W.[(µ.cosα + sinα )/( µ.sinφ + cosφ )]

Now putting µ = tanφ , on solving

P = W.[sin(α   + φ  )/cos(φ-  θ )]                                                               ...(iii)

Now P is minimum at cos(φ - θ ) is max

i.e., cos(φ   -  θ ) = 1 orφ     -θ   = 0     i.e., φ    =θ

P_min = W.sin( θ +φ )