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Physics-

General

Easy

Question

There are two identical small holes of area of cross-section a on the opposite sides of a tank containing a liquid of density $rho$ . The difference in height between the holes is h. Tank is resting on a smooth horizontal surface. Horizontal force which will has to be applied on the tank to keep it in equilibrium is

$g h ρ a$
$\frac{2 g h}{ρ a}$
$2 ρ a g h$
$\frac{ρ g h}{a}$

The correct answer is: $2 rho a g h$

Net force (reaction) = $F equals F subscript B end subscript minus F subscript A end subscript equals fraction numerator d p subscript B end subscript over denominator d t end fraction minus fraction numerator d p subscript A end subscript over denominator d t end fraction$
$equals a v subscript B end subscript rho cross times v subscript B end subscript minus a v subscript A end subscript rho cross times v subscript A end subscript$
\ $F equals a rho open parentheses v subscript B end subscript superscript 2 end superscript minus v subscript A end subscript superscript 2 end superscript close parentheses$ …(i)
According to Bernoulli's theorem
$p subscript A end subscript plus fraction numerator 1 over denominator 2 end fraction rho v subscript A end subscript superscript 2 end superscript plus rho g h equals p subscript B end subscript plus fraction numerator 1 over denominator 2 end fraction rho v subscript B end subscript superscript 2 end superscript plus 0$
Þ $fraction numerator 1 over denominator 2 end fraction rho open parentheses v subscript B end subscript superscript 2 end superscript minus v subscript A end subscript superscript 2 end superscript close parentheses equals rho g h$ Þ $v subscript B end subscript superscript 2 end superscript minus v subscript A end subscript superscript 2 end superscript equals 2 g h$
From equation (i), $F equals 2 a rho g h.$

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