Question

A body of mass M is kept on a rough horizontal surface (friction coefficient =$\mu$ ). A person is trying to pull the body by applying a horizontal force but the body is not moving. The force by the surface on A is F where

• A. $F=Mg$
• B. $F=\mu Mg$
• C. $Mg\le F\le Mg\sqrt{1+{\mu }^2}$
• D. $Mg\ge F\ge Mg\sqrt{1-{\mu }^2}$

Answer 1

The correct option is C $Mg\le F\le Mg\sqrt{1+{\mu }^2}$

Force applied by any surface is resultant of friction force and Normal force.
0 $\le$ Friction force (f) $\le$ $\mu \times N$
where, N is Normal force applied by surface.
Normal force $=$ Mg (perpendicular to surface)
$\therefore$ Resultant force (F) $=$ $\sqrt{{\left(N\right)}^2+f^2}$
$F_{min}$ $=$ N $=$ Mg when f $=$ 0
$F_{max}$ $=$ = $\sqrt{N^2+f^2}$ when f = $\mu$mg
$=$ $\sqrt{{\left(Mg\right)}^2+{\left(\mu Mg\right)}^2}$
$=$ $Mg\sqrt{1+{\mu }^2}$
$\therefore$ $Mg\le F\le Mg\sqrt{1+{\mu }^2}$