Question

A body of mass M is kept on a rough horizontal surface (friction coefficient = μ). 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 = μ Mg
(c) Mg ≤ F ≤ Mg $\sqrt{1+{\mathrm{\mu }}^2}$
(d) Mg ≥ F ≥ Mg $\sqrt{1-{\mathrm{\mu }}^2}$

Answer 1

(c) Mg ≤ F ≤ Mg $\sqrt{1+{\mathrm{\mu }}^2}$

Let T be the force applied on an object of mass M.

If T = 0, F_min = Mg.

If T is acting in the horizontal direction, then the body is not moving.
∴ $T=\mathrm{\mu }\left(mg\right)$

$$\begin{gathered} F_{\max }=\sqrt{(Mg{)}^2+(T{)}^2} \\ =\sqrt{(Mg{)}^2+(\mathrm{\mu }Mg{)}^2} \end{gathered}$$

Thus, we have:
$Mg\le F\le Mg\sqrt{1+(\mathrm{\mu }{)}^2}$