Q: A heavy body of mass 25 kg is to be dragged along a horizontal plane (μ =1/√3). The least force required is

(a) 250 N

(b) 25 N

(c) 125 N

(d) 62.5 N

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N + F sin θ = mg

⇒ N = mg – F sin θ …(i)

The block is about to move if

F cos θ = f_{max} = µN

F cos θ = µ (mg – F sin θ) From(i)

F(cos θ + µ sin θ ) = µ mg

$ \displaystyle F = \frac{\mu mg}{cos\theta + \mu sin\theta} $ …(ii)

For F to be minimum, denominator y should be maximum.

$ \displaystyle y = cos\theta + \mu sin\theta $

$\displaystyle \frac{dy}{d\theta} = 0 $

( Where y = cos θ + µ sin θ )

$ \displaystyle -sin\theta + \mu cos\theta = 0 $

$ \displaystyle tan\theta = \mu $

$\displaystyle sin\theta = \frac{\mu}{\sqrt{1+\mu^2}} \; , cos\theta = \frac{1}{\sqrt{1+\mu^2}} $

On putting the value of sinθ and cosθ in (ii)

$\displaystyle F_{min} = \frac{\mu mg}{\sqrt{1+\mu^2}} $

Since m= 25 kg , μ =1/√3

F_{min} = 125 N