Problem 87 - Physics

Question

Concept Simulation 11.1 at reviews the central idea in this problem. In an adjustable nozzle for a garden hose, a cylindrical plug is aligned along the axis of the hose and can be inserted into the hose opening. The purpose of the plug is to change the speed of the water leaving the hose. The speed of the water passing around the plug is to be three times as large as the speed of the water before it encounters the plug. Find the ratio of the plug radius to the inside hose radius.

Step-by-Step Solution

Answer

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Key Concepts

Continuity Equation

The Continuity Equation is a crucial concept in fluid dynamics. It explains how fluid flows in closed systems. This means how the amount of fluid moving through one part of a system must equal the amount moving through another. Mathematically, it's stated as \(A_1 v_1 = A_2 v_2\), where:

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

This equation tells us that if the cross-sectional area of a hose decreases, the speed of the fluid must increase to maintain the same flow rate. This is exactly what happens when the nozzle of a hose is adjusted, focusing the water flow by making the opening smaller. By understanding this, we can determine how various changes in a system affect the fluid's behavior and speed.

Bernoulli's Principle

Bernoulli's Principle is another cornerstone of fluid dynamics. It describes the relationship between the speed, pressure, and potential energy of a fluid in motion. As derived from the conservation of energy, it can be expressed by the following formula: \[ P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \]

• \(P\) is the fluid pressure.
• \(\rho\) is the fluid density.
• \(v\) is the flow velocity.
• \(g\) is the acceleration due to gravity.
• \(h\) is the height above a reference point.

In simple terms, Bernoulli's Principle states that an increase in the fluid's speed results in a decrease in pressure or a decrease in the fluid's potential energy. This principle is apparent when using a hose; as water accelerates through the narrowed nozzle, its pressure drops, explaining the increased speed of the exiting water. Understanding this principle helps us manipulate fluid flow effectively.

Garden Hose Nozzle

A garden hose nozzle is a practical application of fluid dynamics principles. By altering the nozzle's opening size, you control the water's flow rate and velocity. Nozzles work by compressing the water flow into a smaller area, increasing the speed of the water due to the Continuity Equation.

• Adjusting the nozzle can shift the flow from a gentle spray to a powerful jet.
• It operates under the understanding that reducing area increases speed for the same flow volume.

This tool is a daily reminder of fluid dynamics principles in action. It helps gardeners manage water efficiency by focusing the stream onto specific areas, demonstrating how these theories apply in routine, practical scenarios.

Flow Rate Adjustment

Flow rate adjustment is essential for controlling how much fluid passes through a system per unit of time. It involves altering conditions that affect how quickly or slowly a fluid moves.

• It ensures that the right amount of fluid reaches its destination.
• Adjustments can be made by changing the pipe size, nozzle opening, or pump speed.

In our garden hose example, changing the flow rate can be as simple as twisting the nozzle or adjusting the water source. By doing so, we control both speed and pressure, directly affecting how effectively water is delivered. Understanding flow rate adjustments enables efficient resource management, reducing waste and increasing precision in fluid applications.