Question

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

Answer 1

Explanation:

Let 'p' be the force applied too at an angle θ

From the free body diagram

R+Psinθ−mg=0

R+Psinθ+mg...(i)

μR−pcosθ=0...(ii)

Equn.(i) is μ(mg−Psinθ)−Pcosθ=0

⇒μmg=μPsinθ−Pcosθ⇒p=

μsinθ+cosθ

μmg

Applied force P should be minimum, when μsinθ+cosθ is maximum

Again, μsinθ+cosθ is maximum when its derivative is zero

∴d/dθ(μsinθ+cosθ)=0

⇒μcosθ−sinθ=0⇒θ=tan

1μ

So, P=μsinθ+cosθμmg= cosθ

μsinθ

+

cosθ

cosθ

μmg/cosθ

=

1+μtanθ

μmgsecθ

=

1+tan

2

θ

μmgsecθ

=

secθ

μmg

=

(1+tan 2θ)μmg

=

1+μ

2

μmg

Minimum force is

1+μ

2

μmg

at an angle θ=tan

−1μ