A block of weight W rests on horizontal floor with coefficient of static friction $\mathrm{\mu }$ . It is desired to make the block move by applying minimum amount of force. The angle $\theta$ from the horizontal at which the force should be applied and the magnitude F of the force is

Normal force,

N = mg − Fsinθ

If  F is the minimum to move the block, then;

f = frictional force (static)

f = μN= μ(mg−Fsinθ)

For the block to just move,

Fcosθ = f = μ(mg−Fsinθ)

F(cosθ+μsinθ) = μmg

F = μmg​/(cosθ+μsinθ)
 

If F is the minimum, then
 

let y=cosθ+μsinθ   ...........(1) should be maximum

dy​/dθ= −sinθ + μcosθ = 0

sinθ = μcosθ

θ = tan⁻¹μ
 

For the value of θ=tan⁻¹μ, the force F will be minimum,

and

y = cosθ+μsinθ

=cosθ+μ²cosθ

=cosθ(1+μ²)

=$\sqrt{(1+{\mu }^2)}$
 

hence, minimum force,

F=μmg/y​ 

$\Rightarrow F=\frac{\mu W}{\sqrt{1+{\mu }^2}}$