Question

In the figure shown, the velocity and pressure of the liquid at the cross-section (2) are given by (If $P_o$ is the atmospheric pressure).

• A. $\sqrt{2hg},P_0+\frac{\rho hg}{2}$
• B. $\sqrt{hg},P_0+\frac{\rho hg}{2}$
• C. $\sqrt{\frac{hg}{2}},P_0+\frac{3\rho hg}{4}$
• D. $\frac{\sqrt{hg}}{2},P_0+\frac{3\rho hg}{4}$

Answer 1

The correct option is C $\sqrt{\frac{hg}{2}},P_0+\frac{3\rho hg}{4}$


Velocity of the liquid just after coming out of section $\left(3\right)$ is
$V_3=\sqrt{2gh}......\left(i\right)$

Using continuity equation, across section ${}^{\prime }{2}^{\prime }$ and section${}^{\prime }{3}^{\prime }$

$\frac{A}{2}\times V_2=\frac{A}{4}\times V_3$

$V_2=\frac{1}{2}\times V_3=\sqrt{\frac{hg}{2}}$

Now using Bernoulli's theorem across sections $2$ and $3$

$P_0+\frac{1}{2}\rho V_3^2=P_2+\frac{1}{2}\rho V_2^2$

$\Rightarrow P_2=P_0+\frac{1}{2}\rho (V_3^2-V_2^2)$

$=P_0+\frac{1}{2}\rho (2gh-\frac{gh}{2})$

$\therefore P_2=P_0+\frac{3\rho gh}{4}$

Hence, $\left(C\right)$ is the correct answer.

Why this Q? Bernoulli's theorem: Bernoulli's theorem states that the total energy (pressure energy, kinetic energy, gravitational potential energy) per unit volume is constant at all points along a streamline. $P+\frac{1}{2}\rho v^2+\rho gh=$constant or $\frac{P}{\rho g}+\frac{1}{2}\frac{v^2}{g}+h=$constant Principle of continuity The principle of continuity represents the conservation of mass. According to this principle. $av$ =constant $a$ =area of cross section, $v$ =velocity of fluid flow $\Rightarrow a_1{v}_1=a_2{v}_2$