Problem 59 - Fundamentals of Physics

Question

Water is moving with a speed of \(5.0 \mathrm{~m} / \mathrm{s}\) through a pipe with a cross-sectional area of \(4.0 \mathrm{~cm}^{2}\). The water gradually descends \(10 \mathrm{~m}\) as the pipe cross-sectional area increases to \(8.0 \mathrm{~cm}^{2}\). (a) What is the speed at the lower level? (b) If the pressure at the upper level is \(1.5 \times 10^{5} \mathrm{~Pa}\), what is the pressure at the lower level?

Step-by-Step Solution

Answer

(a) The speed at the lower level is 2.5 m/s. (b) The pressure at the lower level is 1.61 × 10⁵ Pa.

Step 1: Continuity Equation

To find the speed of water at the lower level, we use the principle of conservation of mass, which states that the volume flow rate must remain constant. The continuity equation is given by: \[ A_1 v_1 = A_2 v_2 \] where \( A_1 = 4.0 \, \text{cm}^2 \) and \( v_1 = 5.0 \, \text{m/s} \) are the area and speed at the upper level, and \( A_2 = 8.0 \, \text{cm}^2 \) is the area at the lower level. \( v_2 \) is the speed at the lower level we need to find. First, convert the areas from \( \, \text{cm}^2 \) to \( \, \text{m}^2 \): \( 4.0 \, \text{cm}^2 = 4.0 \times 10^{-4} \, \text{m}^2 \) and \( 8.0 \, \text{cm}^2 = 8.0 \times 10^{-4} \, \text{m}^2 \). Then, solve for \( v_2 \): \[ v_2 = \frac{A_1 v_1}{A_2} = \frac{4.0 \times 10^{-4} \times 5.0}{8.0 \times 10^{-4}} \] Simplifying, \( v_2 = 2.5 \, \text{m/s} \).

Step 2: Bernoulli's Equation

Next, to find the pressure at the lower level, use Bernoulli's equation, which states: \[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \] where \( P_1 = 1.5 \times 10^5 \, \text{Pa} \) is the pressure at the upper level, \( \rho = 1000 \text{ kg/m}^3 \) is the density of water, \( h_1 = 0 \), \( h_2 = -10 \, \text{m} \) (descending), and \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity. Solve for \( P_2 \): \[ P_2 = P_1 + \frac{1}{2} \rho (v_1^2 - v_2^2) + \rho g (h_1 - h_2) \] Plug in the values: \[ P_2 = 1.5 \times 10^5 + \frac{1}{2} \times 1000 \times (5^2 - 2.5^2) + 1000 \times 9.8 \times 10 \] Simplifying, \( P_2 = 1.61 \times 10^5 \, \text{Pa} \).

Key Concepts

Continuity Equation

The Continuity Equation is a fundamental concept in fluid dynamics that reflects the principle of conservation of mass. It states that for an incompressible fluid flowing through a pipe, the rate at which mass enters a cross-section of the pipe must be equal to the rate at which mass exits. Mathematically, it can be expressed as: \[ A_1 v_1 = A_2 v_2 \] where:

• \(A_1\) is the cross-sectional area at the initial point.
• \(v_1\) is the velocity of the fluid at the initial point.
• \(A_2\) is the cross-sectional area at the endpoint.
• \(v_2\) is the velocity of the fluid at the endpoint.

This equation essentially shows that if the area through which the fluid flows changes, the velocity must change as well to maintain a constant flow rate. In our exercise, by using the Continuity Equation, we can determine the speed of water at a lower level when the pipe widens. The initial area, speed, and the second area are inputted to solve for the new speed. This principle is widely used in engineering and physics to understand fluid behavior.

Bernoulli's Equation

Bernoulli's Equation is a key principle in fluid dynamics that describes the conservation of energy in flowing fluids. It relates the pressure, velocity, and height at one point in a fluid flow to another point along the same streamline. The equation is expressed as: \[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \] Here:

• \(P_1\) and \(P_2\) are the pressures at points 1 and 2.
• \(v_1\) and \(v_2\) are the fluid speeds at points 1 and 2.
• \(h_1\) and \(h_2\) are the heights at points 1 and 2.
• \(\rho\) is the fluid density, and \(g\) is the acceleration due to gravity.

In simpler terms, as a fluid flows through a varying diameter pipe or across different heights, its pressure, speed, and potential energy due to elevation change in a way that the total energy remains constant. In our example, Bernoulli’s Equation helps us find the pressure at a lower level in the pipe after considering changes in speed and elevation.

Pressure Calculation

Calculating pressure in fluid dynamics often involves using principles from Bernoulli's Equation, especially when dealing with varying pipe diameters and elevations. Pressure, velocity, and height all influence each other in a flowing fluid. By rearranging Bernoulli's Equation and solving for the desired parameters, we can determine unknown pressures at different points. In our exercise:

• We start with the known pressure at the upper level and account for differences in velocity and height.
• Changes in kinetic energy due to velocity squared and potential energy due to height differences affect the pressure.
• Simplifying gives us the pressure at the lower level.

Understanding how to calculate pressure under these conditions is critical in designing pipelines, hydraulics, and various applications where fluid dynamics is relevant. It ensures safety and efficiency in systems involving fluid flow.

Conservation of Mass

Conservation of Mass in fluid dynamics emphasizes that the mass of a fluid must remain constant as it flows through a system. This concept is pivotal for understanding fluid behavior and applying equations like the Continuity Equation. For incompressible fluids or steady flows, the mass flowing in must equal the mass flowing out. This means it doesn't change with time, even in complex systems with changing velocities and cross-sectional areas. In practical terms:

• This principle helps ensure that systems are closed with no mass loss.
• It's critical in predicting fluid movement, especially for engineering and environmental applications.
• It underlies many calculations in fluid mechanics, allowing engineers to design efficient systems.

In our exercise, Conservation of Mass allows us to confidently use the Continuity Equation, knowing that as the water flow widens the pipe, its speed adjusts accordingly to maintain mass consistency.