Question

To what height (in $cm$ ) should a cylindrical vessel of radius $7\, cm$ be filled with a homogeneous liquid to make the force with which the liquid presses on the sides of the vessel to be equal to the force exerted by the liquid on the bottom of the vessel?

Answer 1

Consider a cylindrical vessel of radius $r$ containing a liquid of density $\rho$.
The liquid needs to be filled to a height $h$, so that force on its cylindrical wall is equal to the force on its circular base.
$\therefore $ Weight of the liquid in the cylinder $=\pi r^{2} h \rho g$ ....(i)
Force on the wall of the vessel is average pressure on the wall
$\therefore $ Average pressure on the wall
$=\frac{1}{2}$ (pressure at bottom + pressure at top)
$=\frac{1}{2}( h \rho g+0)=\frac{1}{2} h \rho g$
Since area of the cylindrical wall in contact with the water is $2 \pi rh$,
Force on the cylindrical wall $=2 \pi rh \times \frac{1}{2} h \rho g$
$=\pi rh ^{2} \rho g$ ...(ii)
Equating (i) and (ii),
$\pi rh ^{2} \rho g =\pi r ^{2} h \rho g$
$\therefore h = r =7\, cm$