A cylindrical vessel is filled with a liquid such that the thrust on the bottom of the vessel and that on its side wall become equal. Prove that the height of the liquid column in the cylinder is numerically equal to the radius of the vessel.

Explanation

Let the height of the liquid column in the vessel = h, the radius of the cylindrical vessel = r and the density of the liquid = `rho`.
`therefore" "` The weight of the liquid in the cylinder = `pir^(2)hxxrhoxxg`
`therefore" "` Thrust on the base of the cylinder = weight of the liquid in the cylinder
=`pir^(2)hrhog" "...(1)`
Now, the average pressure on the wall of the cylinder = `1/2xxhrhog`,
area of the curved surface of the cylinder = `2pirh`.
`therefore` Total thrust on the wall of the cylinder
`=1/2hrhogxx2pirh=pirh^(2)rhog" "...(2)`
According to the problem, from equations (1) and (2), we get
`pir^(2)hrhog=pirh^(2)rhogor,r=h`
i.e., the radius of the cylinder will be equal to the height of the liquid column.