Question

Find the work done to empty the cylendrical tub of radius r and height h filled with fluid of density rho.

Answer 1

Let us consider a cylindrical tub of radius $r$ and height $h$ filled with a fluid of density $\rho$ .

Let us consider a small element of the cylinder of thickness $dx$ at a depth $x$ from the top of the tub.

The volume of this element is

$d V=Area\times height\\ d V=\pi r^2\times dx$

The mass of this element is $dm=volume \times density = \pi r^2 \rho\ dx$

The weight of this element is $\pi r^2 \rho g dx$

The work done to lift this amount of fluid to a height $x$ and thus emptying the cylinder is

$dW=weight\times distance$

$dW=\pi r^2\rho g\ xdx$

To empty the whole cylinder in such way the total work done is

$W=\int _{0}^{h} \pi r^2\rho g \ xdx\\ W=\pi r^2\rho g \int _{0}^{h} xdx\\ W=\pi r^2\rho g \left[x^2/2\right]_0^h\\ W=\frac{1}{2}\pi r^2\rho g h^2$

Answer: The work done to empty the cylindrical tub is $\frac{1}{2}\pi r^2 \rho g h^2$ .