A block of mass m is lying on a horizontal surface of coefficient of friction μ. A force F is applied to the block at an angle θ with the horizontal. The block will move with a minimum force F if

Refer to diagram, Vertical component of F is F sin θ and the horizontal component is F cos θ. Thus              R+F sin ⁡θ=mg  or   R=mg−F sin⁡ θFrictional force  μR=μ(mg-F sin θ). Also μ(mg-F sin θ)=F cos θor    F=μmg(μ sin⁡θ+cos ⁡θ)         (i)F will be minimum if the denominator is maximum, i.e., if ddθ(μ sin⁡ θ+cos ⁡θ)=0or μcos ⁡θ−sin ⁡θ=0or μ=tan ⁡θ