Question

Water is filled in a rectangular tank of size $3m\times 2m\times 1m.$.
(a) Find the total force exerted by the water on the bottom surface of the tank.
(b) Consider a vertical side of area $2m\times 1m$. Take a horizontal strip of width $\delta xmeter$ in this side, situated at a depth of $xmeter$ from the surface of the water. Find the force by the water on this strip.
(c) Find the torque of the force calculated in part (b) about the bottom edge of this side.
(d) Find the total force by the water on this side.
(e) Find the total torque by the water on the side about the bottom edge. Neglect the atmospheric pressure and take $g=10m{s}^{-2}$.

Answer 1

Force on bottom

$F_B=Pressure\times Area$

$F_B=(\rho gh+{\rho }_0)\left(lw\right)$

$F_B=({10}^3\times 10\times 1+10)(2\times 3)$

$F_B=6\times {10}^{4}N$

Here, atmospheric pressure $\left({\rho }_0\right)$ is neglected

$\left(b\right)$ Force on strip

$df=Pressure\times Area$

$df=\left(\rho gx\right)\left[2dx\right]$

$df=2\times 104\left(xd\infty \right)$

$\left(d\right)$ Torque about $AB$

$dt=df(1-x)$

$t={\int }_0^{I}dt={\int }_0^{I}2\infty {10}^4\times (1-10)dx$

$t=2\times {10}^4\int (x-x^2)dx$

$=2\times {10}^4{[\frac{x^2}{2}-\frac{x^3}{3}]}_0^I$

$=2\infty {10}^4\times \frac{1}{6}=3.33\times {10}^{3}Nm$

$(-)$ torque about the force on $\left(b\right)$ point is simply $dt=df(1-\infty )$

$dt=2\times {10}^4\times (1-\infty )dx$

$dt=2\times {10}^4[x-x^2].dx$