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Chapter 6: Problem 4

As air rushes steadily into the narrow opening of a vacuum cleaner attachment, it accelerates to high speed and its pressure drops well below atmospheric pressure. From where does the air’s newfound kinetic energy come?

Short Answer

Expert verified
The kinetic energy comes from the work done by the pressure difference between the atmosphere and the low pressure inside the attachment.

Step by step solution

01

Understanding Pressure and Kinetic Energy

Kinetic energy in fluids is often converted from other forms of energy. In this problem, the air’s velocity increases, implying an increase in kinetic energy.
02

Application of Bernoulli’s Principle

Bernoulli's principle states that in a steady flow, the sum of all forms of energy in a fluid (potential energy, kinetic energy, and pressure energy) remains constant. For a streamline flow, it can be expressed as: \[P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2\,\] where \(P\) is the pressure, \(\rho\) is the density, \(v\) is the velocity, and \(g\) is the gravitational constant. In this scenario, the gravitational term can be ignored as the height difference is minimal.
03

Setting Up the Bernoulli's Equation for This Problem

For the vacuum cleaner attachment, assuming the height change is negligible, the Bernoulli equation simplifies to: \[P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2\,\] where \(P_1\) is the atmospheric pressure and \(P_2\) is the pressure inside the attachment. The air initially at rest has \(v_1 = 0\).
04

Solving for Kinetic Energy

Rewriting the equation: \[P_1 = P_2 + \frac{1}{2} \rho v_2^2\,\] can be rearranged to solve for \(v_2\): \[ \frac{1}{2} \rho v_2^2 = P_1 - P_2\,\] giving \[ v_2 = \sqrt{\frac{2(P_1 - P_2)}{\rho}}\,\] indicating that the increased kinetic energy comes from the pressure difference.
05

Conclusion

The air’s increased kinetic energy inside the vacuum cleaner attachment comes from the work done by the pressure difference between atmospheric pressure outside the attachment and the much lower pressure inside the attachment.

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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 the behavior of liquids and gases in motion. It encompasses principles and equations that describe how fluids move, the forces acting on them, and energy transformations within the fluid. One of the foundational principles in fluid dynamics is Bernoulli's principle, which links pressure, velocity, and height in a moving fluid.

When fluid flows through a constriction, like a narrow vacuum cleaner attachment, its speed increases, causing a drop in pressure. This is a direct application of Bernoulli's principle, which states that an increase in fluid velocity comes with a decrease in pressure. Understanding fluid dynamics helps in realizing that the motion and pressure changes in fluids are interrelated and can be predicted using mathematical models.
Pressure Difference
The pressure difference is a crucial concept in fluid dynamics. It refers to the variation in pressure from one point in a fluid to another. This difference in pressure is the driving force that moves the fluid from regions of high pressure to regions of low pressure.

In the case of a vacuum cleaner attachment, the low pressure inside the attachment compared to the atmospheric pressure outside creates a pressure difference that accelerates the air into the attachment. As the air accelerates, its speed increases, leading to a corresponding drop in pressure. This behavior aligns perfectly with Bernoulli's principle and is key in understanding how devices like vacuum cleaners work efficiently.
  • Higher pressure outside the attachment
  • Lower pressure inside the attachment
  • Airflow is driven by this pressure difference
Kinetic Energy Conversion
Kinetic energy conversion refers to the transformation of energy into motion. In fluids, this often involves converting potential or pressure energy into kinetic energy, which manifests as the fluid's velocity.

In our vacuum cleaner example, Bernoulli's equation : -Before starting: -

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Most popular questions from this chapter

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