Question

A force $F$ is applied on the top of a cube as shown in figure. The coefficient of friction between the cube and the ground is $\mu$. If $F$ is gradually increased, the cube will topple before sliding if

• A. $\mu >\frac{1}{4}$
• B. $\mu <\frac{1}{2}$
• C. $\mu >\frac{1}{2}$
• D. $\mu <1$

Answer 1

The correct option is C $\mu >\frac{1}{2}$

Equilibrium of an object is broken when the object is on the point of toppling,
hence we draw the FBD and write the force and torque equation.
Let $m$ be the mass of the cube and $a$ the side of cube. The cube will slide if :
$F>\mu mg...\left(i\right)$ (By applying the force balancing equation).

It will topple if torque of $F$ about $P$, is greater than torque of $mg$ about $P$ i.e.,
$Fa>\left(\frac{a}{2}\right)mg$
or $F>\frac{1}{2}mg...\left(ii\right)$ (By applying the torque balancing equation).

From Eqs. $\left(i\right)$ and $\left(ii\right)$, we see that cube will topple before sliding if

$\mu mg>\frac{1}{2}mg$
$\mu >\frac{1}{2}$

Hence option C is the correct answer