Question

A block of mass m rests on a horizontal floor with which it has a coefficient of static friction µ. It is desired to make the body move by applying the minimum possible force F. Find the magnitude of F and direction in which it has to be applied.

Answer 1

Let F be applied at angle θ as shown in figure. Normal reaction in this case will be

N = mg – F sin θ

The limiting friction is therefore

f_L = µN = µ(mg – F sin θ)

For the block to move,

F cos θ = f_L = µ(mg – F sin θ)

or F = \(\frac{\mu mg}{\cos \theta + \mu \sin \theta}\)    .....(i) 

For F to be minimum, denominator should be maximum.

or \(\frac d{d \theta}\) (cos θ + µ sin θ) = 0

or – sin θ + µ cos θ = 0

or tan θ = µ or θ = tan^–1(µ)

Substituting this value of θ in Eq. (i), we get

F_min = mg sinθ

Answer 2

Magnitude of F and direction mg sin\(\theta\)