A block of mass M is kept on a rough horizontal surface. The coefficient of static friction between the block and the surface is μ. The block is to be pulled by applying a force to it. What minimum force is needed to slide the block? In which direction should this force act?

Question

Philoid · Accepted Answer

Let us consider the force F is applied on the body which is making angle θ with horizontal.

From the free body diagram,
R + F sin θ = mg
⇒ R = −F sin θ + mg (1)
μR = F cos θ (2)

From Equation (1),
μ(mg − Fsin θ)−F cos θ = 0
⇒ μmg = μF sin θ +F cos θ

The applied force F should be minimum, when μ sin θ + cos θ is maximum.
Again, μ sin θ + cos θ is maximum when its derivative is zero:

d/dθ (μsin θ+ cos θ) =0

⇒ μ cos θ − sin θ = 0
θ = tan⁻¹ μ
So,

Dividing numerator and denominator by cos θ, we get

Therefore, the minimum force required is