Question

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

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

Answer 1

Answer: (B)

$3.4m/s$

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, $(H-h)=50cm,H=60cm=0.6m$
The velocity of effluent from the hole is
$v=\sqrt{h\rho g}$
$v=\left(\sqrt{2gh}\right)$
$v=\sqrt{\left(2\right)\left(9.8\right)\left(0.6\right)}=3.4m{s}^{-1}$