Question

Consider a rectangular tank of size $l\times b$ filled with a liquid of density $\rho$ to a height $H$ as shown in the figure. Find the ratio of the force on the base and the vertical wall of the tank.

• A. $\frac{l}{H}$
• B. $\frac{2l}{H}$
• C. $\frac{4l}{H}$
• D. $\frac{3l}{H}$

Answer 1

The correct option is B $\frac{2l}{H}$

Whenever liquid comes in contact with solid boundaries it exerts a force on it. The force on the boundary may be obtained by integrating the pressure over the entire area of the boundary. The variations of liquid pressure acting at the base and at the wall are shown in figure $\left(a\right)$ and $\left(b\right)$, respectively.



$\left(i\right)$ $\text{Force at the base}$

Since the pressure is uniform at the base, force acting at the base is given by,

$F_b=p\times$ (area of the base)

Since, $p=\rho gH$

Therefore, $F_b=\rho gH\left(lb\right)=\rho g\left(lbH\right)$


$\left(ii\right)$ $\text{Force acting on the vertical wall}$

Pressure acting on the vertical wall is not uniform but increases linearly with depth. Pressure at a depth $h$ from the free surface is $p=\rho gh.$


Since pressure is increasing linearly, so we can take average pressure directly.

$p_w=\frac{\rho gH}{2}$

So, force acting on wall will be,

$F_w=\frac{\rho gH}{2}\times bH=\frac{\rho gb{H}^2}{2}$

$\therefore \frac{F_b}{F_w}=\frac{\rho g\left(lbH\right)}{\rho gb{H}^2/2}=\frac{2l}{H}$

Hence, option $\left(\text{B}\right)$ is correct.