Question

A Cylindrical vessel of cross-sectional area $1000c{m}^2$, is fitted with a frictionless piston of mass 10 kg, and filled with water completely. A small hole of cross-sectional area $10m{m}^2$ is opened at a point 50 cm deep from the lower surface of the piston. The velocity of effluent from the hole will be

• A. 10.5 m/s
• B. 3.4 m/s
• C. 0.8 m/s
• D. 0.2 m/s

Answer 1

The correct option is B 3.4 m/s

Given that
$A=1000c{m}^3$
$m=10Kg=10000g$
$A_h=10m{m}^2=0.1m^2$
$(H-h)=50cm$
Here, the force on piston is
$F=mg$
Hence, Increase in pressure on the liquid in the wider tube is
$P=\frac{F}{A}=\frac{mg}{A}$
Let, H is the level of water to which piston will move, thus
$P=H\rho g$
$H=\frac{P}{\rho g}$
$H=\frac{mg}{A\rho g}=\frac{m}{A\rho }$
$H=\frac{10000}{1000}$.......($\because \rho =1$)
$\therefore H=10cm$ from surface of tube.
Since, (H-h) = 50cm, h = 60cm = 0.6 m
The velocity of effluent from the hole is
$v=\sqrt{h}\rho g$
$v=\left(\sqrt{2}gh\right)$
$v=\sqrt{(}2\left)\right(9.8\left)\right(0.6)=3.4m{s}^{-1}$