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Problem 2.101: A cylinder of radius $R$ is kept embedded along the wall of a container of water of density $\rho$. Length of the cylinder is $L$. Determine the vertical force exerted by water on the cylinder.

Solution:

The cylinder is completely immersed in water. The pressure at the lower point $A$ is greater than that at the upper point $B$.

This difference in pressure creates an upward force on the cylinder which is called the buoyant force.

So the vertical force mentioned in the problem is the buoyant force.

Buoyant force on a body immersed in a fluid is equal to the weight of the liquid displaced.

How much volume of water is displaced by the cylinder?

As the cylinder is fully inside water, volume of water displaced will be equal to the volume of the cylinder.

This will be equal to $\pi\:r^2\:L$.

Mass of this water will be $M\:=\:\pi\:r^2\:L\:\rho$.

Weight of this water will be $Mg\:=\:\pi\:r^2\:L\:\rho\:g$.

So  the vertical force exerted by water on the cylinder is

$\pi\:r^2\:L\:\rho\:g$.