Question

An equilateral uniform prism of mass $m$ rests on a rough horizontal surface with coefficient of friction $\mu$ and a horizontal force $F$ is applied. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, then the minimum force required to topple the prism is:

• A. $\frac{mg}{\sqrt{3}}$
• B. $\frac{mg}{4}$
• C. $\frac{\mu mg}{\sqrt{3}}$
• D. $\frac{\mu mg}{\sqrt{4}}$

Answer 1

The correct option is A $\frac{mg}{\sqrt{3}}$

From the figure,


$AP=a\sin {60}^o=\frac{a\sqrt{3}}{2}$
For just topple condition, normal reaction will shift to $\text{O}$.
Since friction act through horizontal line passing through $\text{O}$, there is no torque due to it.
Net torque about point $\text{O}=0$,
$F\cdot \frac{a\sqrt{3}}{2}=mg\times \left(\frac{a}{2}\right)$
$F=\frac{mg}{\sqrt{3}}$