A cylindrical container with small holes drilled vertically is filled with water, as shown in the figure. The problem asks about the application of Bernoulli's principle to the water jets coming out of these holes. The pressure at the top of the water surface is atmospheric pressure, which is given as $\pi$ cmHg. The density of the water is $\rho$ kg/m³. The specific questions are: (a) State the conditions necessary for Bernoulli's equation to be valid. (b) Write down Bernoulli's equation and identify each term. Show that the equation is dimensionally correct. (c) Derive an expression for the velocity ($V_A$) of the water exiting hole A using Bernoulli's principle. (d) Derive an expression for the horizontal distance ($x_A$) the water jet from hole A travels before hitting the ground, in terms of $H$, $h$, and $g$. (e) Derive an expression for the horizontal distance ($x_C$) the water jet from hole C travels before hitting the ground, in terms of $H$, $h$, and $g$. (f) Given $h = 25$ cm, $H = 100$ cm, $\rho = 1000$ kg/m³, and $g = 10$ m/s², determine A and C's distances. (g) Find the distance the water jet from hole B (located in the middle of the container) travels before hitting the ground. (h) Sketch a diagram showing how the distance traveled by the water jet changes with depth.

1. Problem Description

A cylindrical container with small holes drilled vertically is filled with water, as shown in the figure. The problem asks about the application of Bernoulli's principle to the water jets coming out of these holes. The pressure at the top of the water surface is atmospheric pressure, which is given as $\pi$ cmHg. The density of the water is $\rho$ kg/m³.

The specific questions are:

(a) State the conditions necessary for Bernoulli's equation to be valid.

(b) Write down Bernoulli's equation and identify each term. Show that the equation is dimensionally correct.

(c) Derive an expression for the velocity ($V_A$) of the water exiting hole A using Bernoulli's principle.

(d) Derive an expression for the horizontal distance ($x_A$) the water jet from hole A travels before hitting the ground, in terms of $H$, $h$, and $g$.

(e) Derive an expression for the horizontal distance ($x_C$) the water jet from hole C travels before hitting the ground, in terms of $H$, $h$, and $g$.

(f) Given $h = 25$ cm, $H = 100$ cm, $\rho = 1000$ kg/m³, and $g = 10$ m/s², determine A and C's distances.

(g) Find the distance the water jet from hole B (located in the middle of the container) travels before hitting the ground.

(h) Sketch a diagram showing how the distance traveled by the water jet changes with depth.

2. Solution Steps

(a) Conditions for Bernoulli's Equation:

Bernoulli's equation is valid under the following conditions:

1. The fluid is incompressible.

2. The flow is steady (time-independent).

3. The flow is inviscid (no viscosity).

4. The flow is along a streamline.

(b) Bernoulli's Equation:

$P + \frac{1}{2}\rho v^2 + \rho g h = constant$

Where:

- $P$ is the pressure.

- $\rho$ is the density of the fluid.

- $v$ is the velocity of the fluid.

- $g$ is the acceleration due to gravity.

- $h$ is the height above a reference point.

Dimensional Analysis:

$[P] = ML^{-1}T^{-2}$

$[\frac{1}{2}\rho v^2] = ML^{-3}(LT^{-1})^2 = ML^{-1}T^{-2}$

$[\rho g h] = ML^{-3}(LT^{-2})(L) = ML^{-1}T^{-2}$

All terms have the same dimensions, so the equation is dimensionally correct.

(c) Velocity at hole A ($V_A$):

Let's apply Bernoulli's equation at the surface of the water in the tank (point 1) and at hole A (point 2). Let the height of hole A be $h$ from the bottom of the tank.

$P_1 + \frac{1}{2}\rho v_1^2 + \rho g H = P_2 + \frac{1}{2}\rho v_A^2 + \rho g h$

Where:

- $P_1 = P_2 = P_{atm}$ (atmospheric pressure)

- $v_1 \approx 0$ (velocity at the surface is negligible)

So, $P_{atm} + 0 + \rho g H = P_{atm} + \frac{1}{2}\rho v_A^2 + \rho g h$

$\rho g H = \frac{1}{2}\rho v_A^2 + \rho g h$

$gH = \frac{1}{2}v_A^2 + gh$

$v_A^2 = 2g(H - h)$

$V_A = \sqrt{2g(H - h)}$

(d) Distance $x_A$ for hole A:

The water jet from hole A is launched horizontally. The time it takes to fall a distance $h$ is given by:

$h = \frac{1}{2}gt^2$

$t = \sqrt{\frac{2h}{g}}$

The horizontal distance $x_A$ is then:

$x_A = V_A \cdot t = \sqrt{2g(H - h)} \cdot \sqrt{\frac{2h}{g}} = \sqrt{4(H - h)h}$

$x_A = 2\sqrt{(H - h)h}$

(e) Distance $x_C$ for hole C:

Let the height of hole C be $h_C$ from the bottom. Using the same logic as above:

$V_C = \sqrt{2g(H-h_C)}$

$t = \sqrt{\frac{2h_C}{g}}$

$x_C = V_C \cdot t = \sqrt{2g(H - h_C)} \cdot \sqrt{\frac{2h_C}{g}}$

$x_C = 2\sqrt{(H - h_C)h_C}$

(f) Given Values

We have $H = 100$ cm = 1 m, $h = 25$ cm = 0.25 m

Hole A's distance from bottom is h=0.25m

Hole C is located at the bottom so $h_C$=

0.

Since the diagram does not explicitly show the placement of A & C, the question is unclear. Presuming hole A is at 25cm from the bottom & hole C is at 0cm (bottom), we have:

$x_A = 2\sqrt{(1-0.25)(0.25)} = 2\sqrt{(0.75)(0.25)} = 2\sqrt{0.1875} \approx 2(0.433) = 0.866$ m

$x_C = 2\sqrt{(1-0)(0)} = 0$ m

(g) Distance $x_B$ for hole B:

Hole B is located in the middle, so $h_B = H/2 = 1/2 = 0.5$ m.

$x_B = 2\sqrt{(H - h_B)h_B} = 2\sqrt{(1 - 0.5)(0.5)} = 2\sqrt{(0.5)(0.5)} = 2(0.5) = 1$ m

(h) Sketch:

The horizontal distance $x = 2\sqrt{(H - h)h}$ is a function of $h$. This is a parabola opening downwards. $x = 0$ when $h = 0$ or $h = H$. The maximum distance occurs when $h = H/2$.

3. Final Answer

(a) Conditions: Incompressible, steady, inviscid flow along a streamline.

(b) Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho g h = constant$. Dimensionally correct.

(c) $V_A = \sqrt{2g(H - h)}$

(d) $x_A = 2\sqrt{(H - h)h}$

(e) $x_C = 2\sqrt{(H - h_C)h_C}$

(f) $x_A \approx 0.866$ m (if h=0.25 m from bottom), $x_C = 0$ m (if at bottom).

(g) $x_B = 1$ m

(h) The distance is maximum at $h = H/2$ and zero at $h=0$ and $h=H$. The sketch is a parabola with maximum x at h=H/

2.