Question

A block of mass m is placed on a horizontal surface. The coefficient of friction between them is μ. The block has to be moved by applying a single external force on it. The force may be applied in any direction. The minimum value of this force must be

(a) mg, applied vertically upward, if μ > 1

(b) μmg, applied horizontally, if μ < 1

(c) μmg/√(μ2 + 1) for all values of μ

(d) μ2mg/μ2 + 1 for all values of μ

Answer 1

Correct Answer is: (c) μmg/√(μ2 + 1) for all values of μ

mg = N + P sin θ

or N = mg - P sin θ.

Pcos θ = Flim = μN = μmg - μP sin θ

or P[cos θ + μ sin θ] = μmg

or P = μmg/cos θ + μ sinθ

For P to be minimum, d/dθ (cos θ + μ sin θ) = 0

or - sin θ + μ cos θ = 0

or tan θ = μ.