Chapter 6.6, Problem 38E

To determine

To Calculate: The hydrostatic force on a parabola shaped end of a tank filled with gasoline.

Answer to Problem 38E

The total hydrostatic force on the end of the tank is $136.5\overline{3}lb$ .

Explanation of Solution

Given: The tank contains gasoline with density $42lb/f{t}^3$ .

The end face of the tank is in the shape of the region between the curves $y=\frac{1}{2}{x}^2$ and $y=12$ , measured in $ft$ .

Concept Used:

• The hydrostatic pressure of a liquid of density $\rho$ at a depth $d$ from the surface of the liquid is $P=\rho gd$ .
• The force on a surface due to pressure $P$ exerted is $F=PA$ , where $A$ is the area of the surface being considered.

The equation of the curve defining the shape of the end face is $y=\frac{1}{2}{x}^2$ , that is

  $height=\frac{1}{2}{\left(\frac{width}{2}\right)}^2=\frac{widt{h}^2}{8}$

or

  $h=\frac{w^2}{8}$

  $\Rightarrow w=\sqrt{8h}$

At a height $h$ from the bottom of the tank, the distance from the surface of water is $8-h$ .

So, the pressure at height $h$ from the bottom of the tank is

  $P_h=\text{ρ}g\left(8-h\right)$

The area of aninfinitesimally thin horizontal strip of thickness $dh$ on the end face of the tanklocated at height $h$

  $d{A}_h=width\times thickness=\sqrt{8h}\cdot dh$

Therefore, the force on this thin strip is

  $\begin{array}{l}dF=P\times A=P_{h}d{A}_h=\rho g(8-h)\cdot \sqrt{8h}dh\\ =42\left(8^{\frac{3}{2}}{h}^{\frac{1}{2}}-8^{\frac{1}{2}}{h}^{\frac{3}{2}}\right)dh\end{array}$

Now the total force on the end surface is the integral of $dF$ from $h=0$ (bottom) to $h=8ft$ (surface of gasoline). So

  $F={{\displaystyle \int }}_0^{8}dF={{\displaystyle \int }}_0^{8}42\left(8^{\frac{3}{2}}{h}^{\frac{1}{2}}-8^{\frac{1}{2}}{h}^{\frac{3}{2}}\right)dh$

  $={42\times \left(\frac{2}{3}\times 8^{\frac{3}{2}}{h}^{\frac{3}{2}}-\frac{2}{5}\times 8^{\frac{1}{2}}{h}^{\frac{5}{2}}\right)]}_0^8$

  $=42\times (341.\overline{3}-204.8)$

  $=136.5\overline{3}$

Therefore, the total hydrostatic force on the end of the tank is $136.5\overline{3}lb$ .