Question

A cylindrical vessel of height $500mm$ has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height H. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being $200mm$. Find the fall in height (in mm) of water level due to opening of the orifice. (Take atmospheric pressure $=1.0\times {10}^{5}N/m^2$, density of water $=1000kg/m^3$ and $g=10m/s^2$. Neglect any effect of surface tension.)

Answer 1

Pressure due to falling water level at $200mm$ is

$P+\rho gh=P_0$

$P=P_0-\rho gh$

or $P={10}^5-\left(1000\right)\left(10\right)\left(0.2\right)=98\times {10}^{3}N/m^2$

now, $P_0{V}_0=PV$

or ${10}^5\left[A\right(0.5-H\left)\right]=98\times {10}^3\left[A\right(0.5-0.2\left)\right]$

where $A=$ cross-sectional area of a vessel.

$0.5-H=0.294$

$\Rightarrow H=0.206m=206mm$

The fall in height (in mm) of water level $=206mm-200mm=6mm$