A cylindrical tank of radius $2\text{m}$ and height $5\text{m}$ is filled with water. Water is pumped out through a pipe at the top of the tank. Find the total work required to empty the tank. Take water density $\rho =1000{\text{kg/m}}^3$ and $g=9.8{\text{m/s}}^2$. Give your answer in kilojoules (kJ).

Step-by-step solution

1. Consider a thin horizontal slice of water at a depth $y$ metres from the bottom; its thickness is $dy$.
2. The volume of the slice is $dV=\pi r^{2}dy=\pi (2{)}^{2}dy=4\pi dy$.
3. Mass of the slice: $dm=\rho dV=1000\times 4\pi dy=4000\pi dy$ kg.
4. Weight of the slice: $dW=dmg=4000\pi \times 9.8dy=39200\pi dy$ N.
5. The slice must be lifted a distance equal to the remaining height $5-y$ to reach the top pipe.
6. Work done on the slice: $dU=dW(5-y)=39200\pi (5-y)dy$.
7. Total work $U={\int }_0^{5}39200\pi (5-y)dy$.
8. Integrate: $U=39200\pi [5y-\frac{y^2}{2}{]}_0^5=39200\pi \left(25-12.5\right)=39200\pi \times 12.5$.
9. Calculate: $U=490000\pi \text{J}$.
10. Convert to kilojoules: $U=\frac{490000\pi }{1000}\approx 1539.4\text{kJ}\approx 1.54\times 10^3\text{kJ}$.

Concept tested: Work done in pumping fluid

Published 14 September 2026 · How this solution was prepared and checked