Question

A body of mass M is kept on a rough horizontal surface (friction coefficient = μ). A person is trying to pull the body by applying a horizontal force but the body is not moving. The force by the surface on A is F where

(a) F = Mg

(b) F = μMg

(c) Mg < F < Mg√(1 + μ2)

(d) Mg > F > Mg√(1 + μ2).

Answer 1

(c) Mg < F < Mg√(1 + μ2).

Explanation:

Weight of the body Mg = Normal force on the body by the surface Let pull by the person be P. Then the value of P ≤ µMg. Force by the surface on A = F = √{(P)²+(Mg)²} For Maximum F, P = µMg

So, Fmax = √{(µMg)²+(Mg)²} = Mg√(1+µ²) i.e. F≤ Fmax → F≤ Mg√(1+µ²) ..................... (i)

Now F = Mg.secΘ [Where Θ is the angle between F and vertical, See figure below]

Since secΘ≥1

→Mg.secΘ≥Mg

→F≥Mg

→Mg≤F ..........................................(ii)

Combine (i) and (ii) and we get,

Mg≤F≤ Mg√(1+µ²)