15. Minimum force Required to move a block
(a) Applied force serves two purposes :
(i) It reduces the normal reaction, thus reduces the frictional force.
(ii) It tends to move the block along the surface
(b) Minimum value of F occurs at an angle q at which the normal reaction is reduced so much that the limiting friction becomes just equal to the horizontal component of the applied force.
(c) It happens when the block is pulled at an angle equal to the angle of friction θ = φ
(d) Mathematically
F cos θ = μN = μ (mg – F sin θ)
⇒ F (cos θ + m sin θ) = μ mg

Differentiating the equation with respect to q and putting = 0 for minimum value of F, θ = tan^–1μ