A block of mass $\mathrm{m}$ 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.

Hint We know that friction is a force between two surfaces that are sliding, or trying to slide, across each other. For example, when you try to push a book along the floor, friction makes this difficult. Friction always works in the direction opposite to the direction in which the object is moving, or trying to move. It is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact.Based on this concept we have to solve this question.

Complete step by step answer

From the given data, we can derive that,

$\text{R}+\text{F}\sin \theta =\text{mg}\ldots \ldots \text{Equation (i)}$

$\text{ }\!\!\mu\!\!\text{ R}=\text{F}\cos \theta \ldots \ldots \text{Equation (ii)}$

or $\quad \text{R}=\dfrac{\text{F}\cos \theta }{\text{ }\!\!\mu\!\!\text{ }}$

using this in equation (i), we get

$\dfrac{\text{F}\cos \theta }{\text{ }\!\!\mu\!\!\text{ }}+\text{F}\sin \theta =\text{mg}$

or $\quad \text{F}=\dfrac{\text{ }\!\!\mu\!\!\text{ mg}}{\cos \theta +\text{ }\!\!\mu\!\!\text{

}\sin \theta }\ldots \ldots \text{Equation (iii)}$

It must satisfy the condition,

$\dfrac{\text{d}}{\text{dq}}(\cos \theta +\text{ }\!\!\mu\!\!\text{ }\sin \theta )=0$

or $-\sin \theta +\text{ }\!\!\mu\!\!\text{ }cos\theta =0$

$\tan \theta =\text{ }\!\!\mu\!\!\text{ }$

$\mathrm{F}$ is minimum $=\theta {{\tan }^{-1}}(\text{ }\!\!\mu\!\!\text{ })$

$\sin \theta =\dfrac{\tan \theta }{{{\left( 1+{{\tan }^{2}}\theta \right)}^{1/2}}}$

$=\dfrac{\text{ }\!\!\mu\!\!\text{ }}{{{\left( 1+{{\text{ }\!\!\mu\!\!\text{ }}^{2}} \right)}^{1/2}}}$

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

$=\dfrac{1}{{{\left( 1+{{\text{ }\!\!\mu\!\!\text{ }}^{2}} \right)}^{1/2}}}$

${{\text{F}}_{\min }}=\dfrac{\text{ }\!\!\mu\!\!\text{ mg}}{\left( \dfrac{1}{{{\left( 1+{{\text{

}\!\!\mu\!\!\text{ }}^{2}} \right)}^{1/2}}+\dfrac{{{\text{ }\!\!\mu\!\!\text{ }}^{2}}}{{{\left(

1+{{\text{ }\!\!\mu\!\!\text{ }}^{2}} \right)}^{1/2}}}} \right)}$

${{\text{F}}_{\min }}=\dfrac{\text{ }\!\!\mu\!\!\text{ mg}}{{{\left( 1+{{\text{ }\!\!\mu\!\!\text{

}}^{2}} \right)}^{1/2}}}$

Therefore, the minimum possible force applied to move the body with a mass m and the acceleration due to gravity m, and the coefficient of static friction $\text{ }\!\!\mu\!\!\text{ }$ is $\dfrac{\text{ }\!\!\mu\!\!\text{ mg}}{{{\left( 1+{{\text{ }\!\!\mu\!\!\text{ }}^{2}} \right)}^{1/2}}}$.

Note We know that static friction is a force that keeps an object at rest. Static friction definition can be written as the friction experienced when individuals try to move a stationary object on a surface, without actually triggering any relative motion between the body and the surface which it is on.It hinders the movement of an object moving along the path. When two fabrics slide over each other, this friction occurs. There's friction all around us. When we walk, for instance, our feet are in touch with the floor. Static friction is caused by adhesion, light chemical attraction between two surfaces. And friction, in general, is caused by the imperfections in every surface gripping together and overlapping.
It should also be known to us that kinetic friction (also referred to as dynamic friction) is the force that resists the relative movement of the surfaces once they're in motion.