Question

A solid cube is placed on a horizontal surface. The coefficient of friction between them is $\mu$, where $\mu <1/2$. A variable horizontal force is applied on the cube's upper face, perpendicular to one edge and passing through the mid-point of the edge, as shown in figure. The maximum acceleration with which it can move without toppling is

• A. $g(1-2\mu )$
• B. $g(1+2\mu )$
• C. $\frac{g}{2}(1-2\mu )$
• D. $\frac{g}{2}(1+2\mu )$

Answer 1

The correct option is C $\frac{g}{2}(1-2\mu )$

FBD of the block given in the question, viewed from the side is shown below.


By using principle of moments, we balance clockwise and anticlockwise torque about $\text{O}$.
Thus, we get
$P\times a=\left(mg\right)\times \frac{a}{2}$ (where, $a\to$ side of cube)
$\Rightarrow P=\frac{mg}{2}$ ... (1) [for just toppling the block]
Now, applying Newton's 2nd law on the cube:
$P-\mu N=m{a}_{\text{max}}$
$\Rightarrow \frac{mg}{2}-\mu mg=m{a}_{\text{max}}$
$\Rightarrow a_{\text{max}}=\frac{g}{2}(1-2\mu )$
Thus, option (c) is the correct answer.