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We have two unknowns and will need two equations. Using Bernoulli's Equation with \(h_{1}=h_{2}\), we have $$P_{1}+\frac{1}{2} \rho v_{1}^{2}=P_{2}+\frac{1}{2} \rho v_{2}^{2} \quad \text { or } \quad P_{1}+\frac{1}{2} \rho\left(v_{1}^{2}-v_{2}^{2}\right)=P_{2}$$ Furthermore, \(v_{1}=2.0 \mathrm{~m} / \mathrm{s}\), and the equation of continuity tells us that $$v_{2}=v_{1} \frac{A_{1}}{A_{2}}=(2.0 \mathrm{~m} / \mathrm{s})\left(\frac{r_{1}}{r_{2}}\right)^{2}=(2.0 \mathrm{~m} / \mathrm{s})(9.0)=18 \mathrm{~m} / \mathrm{s}$$ Substituting then gives $$1.80 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}+\frac{1}{2}\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)\left[(2.0 \mathrm{~m} / \mathrm{s})^{2}-\left(18 \mathrm{~m} / \mathrm{s}^{2}\right)\right]=P_{2}$$ from which \(P_{2}=0.20 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}=20 \mathrm{kPa}\). Since \(P_{\text {out }}=P_{A}\), this is the gauge pressure inside the hose. How could you obtain this latter equation directly from Torricelli's Theorem?

Step by step solution

01

Understanding Bernoulli's Equation

Bernoulli's Equation describes the relationship between pressure, velocity, and height of a fluid flow: \[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]. In this problem, since \(h_{1} = h_{2}\), we focus only on pressure and velocity differences.

02

Using the Continuity Equation

The continuity equation helps relate the velocities at two points in a flow system: \(v_{2} = v_{1} \frac{A_{1}}{A_{2}}\). Given \(v_{1} = 2.0\; \text{m/s}\) and \( \left(\frac{r_{1}}{r_{2}}\right)^{2} = 9.0 \), we find that \(v_{2} = 18\; \text{m/s}\).

03

Substitute into Bernoulli's Equation

Substitute known values into the rearranged Bernoulli's equation: \[ P_1 + \frac{1}{2} \rho (v_1^2 - v_2^2) = P_2 \]. Use \( \rho = 1000 \; \text{kg/m}^3 \), \(v_1 = 2.0\; \text{m/s}\), and \(v_2 = 18\; \text{m/s}\) to compute \( P_2 = 0.20 \times 10^{5} \; \text{N/m}^2 \).

04

Applying Torricelli's Theorem

Torricelli's Theorem states \( v = \sqrt{2gh} \) for an ideal fluid leaving an orifice. For this problem, if viewed in a similar experimental setup, rearranging terms in Bernoulli’s equation effectively computes velocity differences leading to the pressure \(P_2\), showing equivalence with the Torricelli’s principle when adjustments account for pressure changes in the fluid flow.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Dynamics

Fluid dynamics is the study of fluids (liquids and gases) in motion. It deals with the behavior and movement of fluids and the forces acting on them. In many real-world applications, understanding fluid dynamics is crucial, such as designing pipelines, predicting weather patterns, and even understanding blood circulation in biology.
One of the key principles in fluid dynamics is Bernoulli's Equation, which relates the pressure, velocity, and height (or elevation) of a fluid moving along a streamline. This equation helps predict how a fluid behaves under varying conditions.
Some essential considerations in fluid dynamics include:

• Streamlines: Imaginary lines that represent the flow of a fluid; the tangent at any point on a streamline indicates the fluid velocity at that point.
• Incompressibility: For many problems, especially with liquids, we assume fluids are incompressible — meaning their density doesn’t change with pressure variations.
• Viscosity: A measure of a fluid's resistance to deformation. While higher viscosity means thicker fluid and a slower flow, lower viscosity indicates a thinner and faster flow.

Understanding these concepts will aid in solving problems using equations like Bernoulli's Equation and the Continuity Equation.

Pressure and Velocity

Pressure and velocity are two critical elements in fluid dynamics that often interplay within a system. According to Bernoulli’s Equation, there is an inverse relationship between the pressure and velocity of a fluid: when one increases, the other decreases, assuming the fluid density remains constant.
In the context of the given problem, the exercise illustrates how the pressure changes when fluid moves between two points with differing velocities.
Key factors to keep in mind include:

• Static Pressure: The pressure exerted by a fluid at rest.
• Dynamic Pressure: The kinetic energy per unit volume of a fluid particle (linked to fluid velocity).
• Total Pressure: The sum of static and dynamic pressures, often constant along a streamline.

The exercise demonstrates that when the fluid sped up (velocity increased), its pressure decreased as a consequence, which can be quantified using Bernoulli’s Equation.

Continuity Equation

The Continuity Equation is a fundamental principle in fluid dynamics that expresses the conservation of mass in a fluid flow system. It states that the amount of fluid entering a system must equal the amount exiting, assuming there is no accumulation within the system.
Mathematically, the equation for an incompressible fluid can be expressed as:

• \[ A_1 v_1 = A_2 v_2 \]

where:

• \( A_1 \) and \( A_2 \) are the cross-sectional areas at two points in the flow.
• \( v_1 \) and \( v_2 \) are the fluid velocities at these points.

The exercise uses this principle to find the velocity of the fluid at the second point \(v_2\), given \(v_1\) and the ratio of the radii \((r_1/r_2)^2\). By applying this, we can relate the changes in velocity to the changes in the cross-sectional area of the fluid flow.
This concept underscores the importance of maintaining a consistent flow in various engineering applications, allowing for efficient design and management of systems transporting fluids.