Question

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

Answer 1

Suppose the force F be applied at an angle $\theta$ wih the

horizontal and vertical directions, we have

R + F sin$\theta$ = mg _ _ _ _( 1 )

and nR = f cos $\theta$ _ _ _ _ ( 2 )

form equation 1 and 2

or F = $\frac{nmg}{cos\theta +nsin\theta }$ _ _ _ ( 3 )

For force to be minimum, the denominator should be maximom

i. e. $\theta$ must satisfy the condition.

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

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

or tan$\theta =n$_ _ _ _ _ ( 4 )

Therefore, F will be minimum if $\theta =ta{n}^{-1}\left(n\right)$

Now, using ( 4 )

$sin\theta =\frac{tan\theta }{(1+ta{n}^2\theta {)}^{\frac{1}{2}}}$

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

so, $cos\theta =\sqrt{1-si{n}^2\theta }$

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

Substituting these values in ( 3 ),

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

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