Question

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

Answer 1

$R+Fsin\theta =mg⟶eq1$

$nR=Fcos\theta ⟶eq2$

$orR=\frac{Fcos\theta }{n}$

using this in equation 1

$\frac{Fcos\theta }{n}+Fsin\theta =mg$

$orF=\frac{nmg}{cos\theta +nsin\theta }⟶eq3$

it must satisfy the condition,

$\frac{d}{dq}(cos\theta +nsin\theta )=0$

$or-sin\theta +ncos\theta =0$

$tan\theta =n$

$Fisminimum=\theta {tan}^{-1}\left(n\right)$

$sin\theta =\frac{tan\vartheta }{{(1+{tan}^2\theta )}^{1/2}}$

$=\frac{n}{{(1+n^2)}^{1/2}}$

$cos\theta =\sqrt{1-{sin}^2\theta }$

$=\frac{1}{{(1+n^2)}^{1/2}}$

$F_{min}=\frac{nmg}{\left(\frac{1}{{(1+n^2)}^{1/2}+\frac{n^2}{{(1+n^2)}^{1/2}}}\right)}$

$F_{min}=\frac{nmg}{{(1+n^2)}^{1/2}}$