Question

A uniform cube of side '$b$' and mass $M$ rest on a rough horizontal table. A horizontal force $F$ is applied normal to one of the face at a point, at a height $3b/4$ above the base. What should be the coefficient of friction $\left(\mu \right)$ between cube and table so that it will tip about an edge before it starts slipping?

• A. $\mu >\frac{2}{3}$
• B. $\mu >\frac{1}{3}$
• C. $\mu >\frac{3}{2}$
• D. none

Answer 1

The correct option is A $\mu >\frac{2}{3}$

For equilibrium in y direction we have
$N=Mg$
For equilibrium in x direction we have
$F=f$(here f is the frictional force)
or
$F=\mu N=\mu Mg$
For equilibrium of torque about center we have
$N\times x=f\times \frac{b}{2}+F\times \frac{b}{4}$
or
$Mg\times \frac{b}{2}=\mu Mg\times \frac{b}{2}+\mu Mg\times \frac{b}{4}$(as $x=\frac{b}{2}$for toppling)
or
$\mu =\frac{2}{3}$
Thus for tipping we have
$\mu >\frac{2}{3}$