Question

The side wall of a wide vertical cylindrical vessel of height $h=75cm$ has a narrow vertical slit running all the way down to the bottom of the vessel. The length of the slit is $l=50cm$ and the width $b=1.0mm$. With the slit closed, the vessel is filled with water. Find the resultant force of reaction (in $N$) of the water flowing out of the vessel immediately after the slit is opened.

Answer 1

Consider an element of height $dy$ at a distance $y$ from the top. The velocity of the fluid coming out of the element is
$v=\sqrt{2gy}$
The force of reaction $dF$ due to this is $dF=\rho dA{v}^2=\rho \left(bdy\right)2gy$
Integrating $F=\rho gb{\int }_{h-1}^{h}2ydy$
$=\rho gb[h^2-{(h-l)}^2)]=\rho gbl(2h-l)$
(The slit runs from a depth $h-l$ to a depth $h$ from the top.)