To what height should a cylindrical vessel be filled with a homogeneous liquid such that the force exerted by the liquid on the wall of the vessel be equal to the force exerted by the liquid on the bottom of the vessel?

To what height should a cylindrical vessel be filled with a homogeneou

1.	To what height should a cylindrical vessel be filled with a homogeneous liquid such that the force exerted by the liquid on the wall of the vessel be equal to the force exerted by the liquid on the bottom of the vessel?
Answer» Solution :Let h be the height of the liquid of density `rho` in the cylinder of radius r. The pressure of the liquid COLUMN at the bottom of the cylinder is `hrhog`. THEREFORE, the FORCE exerted by the liquid column on the bottom of the cylinder is `F_(1)="pressure"xx"area of the bottom"` `=hrhogxxpir^(2)` The pressure of the liquid column on the wall of the vessel VARIES from zero at the TOP of the column to `hrhog` at the bottom. Thus, the average pressure on the wall is `(0+hrhog)/2=1/2hrhog` Therefore, the force exerted by the liquid column on the wall of the cylinder is `F_(2)="pressure"xx"area of wall"=1/2hrhogxx2pirh` For `F_(2)=F_(1)` we should have `1/2hrhogxx2pirh=hrhogxxpir^(2)` or, h = r i.e., the height of the liquid column in the cylinder should be equal to the radius of the cylinder.