Question

Consider the bottom of the cubic tank of edge $x$ as reference level, determine the gravitational potential energy of water completely filled in the tank. (Take the density of water as $\rho$ and acceleration due to gravity as $g$)

• A. $\rho g{x}^4$
• B. $\frac{1}{4}\rho g{x}^4$
• C. $\frac{1}{2}\rho g{x}^4$
• D. $\frac{1}{3}\rho g{x}^4$

Answer 1

The correct option is C $\frac{1}{2}\rho g{x}^4$

Taking an elementary strip of mass $dm$ at a height $h$, it has gravitational potential energy:
$dU=\left(dm\right)gh$ ..........(i)

where : $dm=\rho x^{2}dh=\left(\rho dV\right)$

From (i) :-

$dU=\left(\rho x^{2}dh\right).gh$

$U=\rho g{x}^2{\int }_0^{x}hdh=\rho g{x}^2.\frac{x^2}{2}$

$U=\frac{1}{2}\rho g{x}^4$

Correct Option - (C)