Water is filled in a cylindrical container to a height of $3m. $ The ratio of the cross-sectional area of the orifice and the beaker is $ 0.1. $ The square of the speed of the liquid coming out from the orifice is ....... $m^2/s^2$ ($g = 10 m/s^2$)
A$50 $
B$50.5$
C$51 $
D$52$
IIT 2004, Diffcult
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A$50 $
a (a)Let $A = $ cross-section of tank
$a = $ cross-section hole
$V =$ velocity with which level decreases
$v = $ velocity of efflux
From equation of continuity $av = AV \Rightarrow V = \frac{{av}}{A}$
By using Bernoulli's theorem for energy per unit volume
Energy per unit volume at point A
= Energy per unit volume at point B
$P + \rho gh + \frac{1}{2}\rho {V^2} = P + 0 + \frac{1}{2}\rho {v^2}$
==> ${v^2} = \frac{{2gh}}{{1 - {{\left( {\frac{a}{A}} \right)}^2}}} = \frac{{2 \times 10 \times (3 - 0.525)}}{{1 - {{(0.1)}^2}}} = 50{(m/\sec )^2}$
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