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The side wall of a wide vertical vessel ...

The side wall of a wide vertical vessel of height `h=75 cm` has a narrow slit (vertical) running all the way down to the bottom of the vessel.
The length of the slit is `l=50 cm` and the width is `b=1 mm`. With the slit closed, water is filled to the top. Find the resultant reaction force of water coming out as the slit is opened.

Text Solution

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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=rhodAv^2`, as in the previous problem,
`=rho(bdy)2gy`
Integrating `F=rhogbunderset(h-1)overset(h)int2ydy`
`=rhogb[h^2-(h-l)^2]=rhogbl(2h-l)`
(The slit runs from a depth `h-l` to a depth h from the top.)
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