Question

A large open tank has two holes in the wall. One is a square hole of side $L$ at a depth $y$, from the top and the other is a circular hole of a radius $R$ at a depth $4y$ from the top. When the tank is completely filled with water the quantities of water flowing out per second from the holes are the same then the value of $R$ is:
(A) $\dfrac{L}{{\sqrt {2\pi } }}$
(B) $2\pi L$
(C) $L\sqrt {\dfrac{2}{\pi }}$
(D) $\dfrac{L}{{2\pi }}$

Answer 1

The theory of continuity and Bernoulli’s equation is applicable in the solution which is as follows:
${A_1}{v_1} = {A_2}{v_2}$, where ${A_1}$ is the area of cross-section of the first hole and ${v_1}$ is the velocity of the first constriction and ${A_2}$ is the area of cross-section of the second hole and ${v_2}$ is the velocity of the second constriction.

Complete step by step answer:
The theory of continuity represents mass conservation in a given space filled by a fluid.
In elementary fluid mechanics, the simplest well known type of the continuity relationship expresses that the discharge in a pipe for steady flow is constant.
The theory of continuity and Bernoulli’s principle are related.
Bernoulli’s principle states that the flowing fluid’s overall mechanical energy, including the gravitational potential elevation energy, the energy correlated with the fluid friction and the fluid motion’s kinetic energy, remains the same.
For a horizontal unit, the continuity equation indicates that the decrease in diameter would induce an increase in the fluid flow rate for an incompressible fluid. Subsequently, the theory of Bernoulli therefore shows that in the reduced diameter area, there must be a decrease in pressure.
Bernoulli’s velocity of efflux is given by-
$v = \sqrt {2gh}$
where $g =$ acceleration due to gravity and $h =$ height.
Given,
${A_1} = {L^2}$
${v_2} = \sqrt {2gy}$
${A_2} = \pi {R^2}$
${v_2} = \sqrt {2g4y}$
Value of $R$
when the tank is completely filled with water=?
We know that,
${A_1}{v_1} = {A_2}{v_2}$
${L^2}\sqrt {2gy} = \pi {R^2}\sqrt {2g4y} \\\ \Rightarrow {L^2} = 2\pi {R^2} \\\ \Rightarrow R = \dfrac{L} {{\sqrt {2\pi } }} \\\$
Hence, the value of $R$ when the tank is completely filled with water $R = \dfrac{L} {{\sqrt {2\pi } }}$
Therefore, option A is correct.

Note: Here the height is taken as $y$ since, the depth is given as $y$. So, there is not any confusion about this. Also the first hole is in the shape of a square, so we should find the area of the square and the second hole is in the shape of a circle, so we should find the area of a circle. It may be confusing what area to find for the area of cross-section.