A body of mass $m$ is kept on a rough horizontal surface (coefficient of friction $=\mu$ ). Horizontal force is applied on the body, but it does not move. The resultant of normal reaction and the frictional force acting on the object is given $\mathrm{F}$, where $\mathrm{F}$ is

• A

  $|\mathrm{F}|=m g+\mu m g$
• B

  $|\mathrm{F}|=\mu m g$
• Correct answer
  C

  $|\mathrm{F}| \leq m g \sqrt{1+\mu^2}$
• D

  $|\mathrm{F}|=m g$

Solution

$$ \text { The situation can be drawn as } $$
The frictional force, $\mathrm{f}=\mu \mathrm{N}=\mu \mathrm{mg} \quad[\because \mathrm{N}=\mathrm{mg}]$ From Free body diagram (FBD), the resultant force is $$ \begin{aligned} |F| & =\sqrt{\mathrm{N}^2+\mathrm{f}^2} \\ & =\sqrt{(\mathrm{mg})^2+(\mu \mathrm{mg})^2}=\mathrm{mg} \sqrt{1+\mu^2} \end{aligned} $$ This is the minimum force required to move the object. But as the body is not moving $$ \therefore \quad|\mathrm{F}| \leq \mathrm{mg} \sqrt{1-\mu^2} $$

Hence, the correct answer is option: 'C'.