A cyclindrical vessel of cross-sectional area $1000$ $cm^2$, is fitted with a frictionless piston of mass $10$ $kg$, and filled with water completely.A small hole of cross-sectional area $10$ $mm^2$ is opened at a point $50$ $cm$ deep from the lower surface of the piston. The velocity of efflux from the hole will be ........... $m/s$
Diffcult
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Given that
$A = 1000 cm^3$
$m = 10 kg =10000 g$
$A_h = 10 mm^2 = 0.1 m^2$
$(H-h) = 50 cm$
Here, the force on piston is
$F = mg$
$A = 1000 cm^3$
$m = 10 kg =10000 g$
$A_h = 10 mm^2 = 0.1 m^2$
$(H-h) = 50 cm$
Here, the force on piston is
$F = mg$
Hence, Increase in pressure on the liquid in the wider tube is
$P = \dfrac{F}{A} = \dfrac{mg}{A}$
Let, $H$ is the level of water to which piston will move, thus
$P = H\rho g$
$H = \dfrac{P}{\rho g}$
$H = \dfrac{mg}{A\rho g} = \dfrac{m}{A\rho}$
$H = \dfrac{10000}{1000}$.......($\because \rho = 1$)
$\therefore H = 10 cm$ from surface of tube.
Since, $(H-h) = 50cm, H = 60cm = 0.6 m$
The velocity of effluent from the hole is
$v = \sqrt {h\rho g}$
$v = (\sqrt {2gh}) $
$v = \sqrt {(2)(9.8)(0.6) } = 3.4 ms^{-1}$
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