A cubical box of side a sitting on a rough table-top is pushed ho… - Vidyadip

MCQ

A cubical box of side $a$ sitting on a rough table-top is pushed horizontally with a gradually increasing force until the box moves. If the force is applied at a height from the table top which is greater than a critical height $H$, the box topples first. If it is applied at a height less than $H$, the box starts sliding first. Then, the coefficient of friction between the box and the table top is

✓
$\frac{a}{2 H}$
B
$\frac{2 H}{a}$
C
$\frac{a}{H}$
D
$\frac{H}{a}$

✓

Answer

Correct option: A.

$\frac{a}{2 H}$

a
(a)

When the block started to topple, normal reaction passes through the edge along which block tends to turn, as shown below.

For toppling, torque of applicd force $F$ must be greater than torque of weight about $O$.

$\Rightarrow \quad F \times H \geq m g \frac{a}{2}$

and block begins to slide when

$F \geq \text { friction }$

$\Rightarrow F \geq \mu m g \quad \dots(i)$

Equating limiting values from Eqs. $(i)$ and $(ii)$, we have

$\mu m g H=m g \frac{a}{2}$

$\text { or } \mu=\frac{a}{2 H}$

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