Chapter 13: Problem 23
If you turn on the faucet in the bathroom sink, you will observe that the stream seems to narrow from the point at which it leaves the spigot to the point at which it hits the bottom of the sink. Why does this occur?
Short Answer
Expert verified
Answer: The stream of water narrows from the faucet to the bottom of the sink because the velocity of the water increases due to gravity. The conservation of mass principle requires that the cross-sectional area of the water stream decreases to compensate for the increase in velocity, thus maintaining the constant mass flow rate in the system.
Step by step solution
01
Identifying the phenomenon
First, let's observe the stream of water as it flows from the faucet to the sink. As it flows down, due to gravity, we can notice that the stream becomes narrower. We will use the concept of conservation of mass to understand this behavior.
02
Introducing the conservation of mass
The conservation of mass states that the mass of a system remains constant over time, as long as no matter enters or leaves the system. In our case of the water stream, we can consider the mass of water flowing through a section of the stream remains constant over time.
03
Applying the conservation of mass to the water stream
Let's consider a small section of the water stream with a cross-sectional area A₁ at the top near the faucet and A₂ at the bottom near the sink. The stream flows at a velocity v₁ at the top and v₂ at the bottom. According to the conservation of mass, the mass flow rate in the system should remain constant, which means:
m₁ = m₂
ρA₁v₁ = ρA₂v₂
where ρ is the density of the water (which remains constant), A₁ and A₂ are the cross-sectional areas, and v₁ and v₂ are the velocities of the water stream at the top and bottom, respectively.
04
Analyzing velocity and cross-sectional area relationship
From the above equation, we can deduce the relationship between the velocities and the cross-sectional areas:
A₁v₁ = A₂v₂
As the water falls, due to gravity, its velocity increases (v₂ > v₁). According to the equation, in order for the mass flow rate to remain constant, the cross-sectional area must decrease (A₂ < A₁).
05
Conclusion
To summarize, the stream of water narrows from the faucet to the bottom of the sink because the velocity of the water increases due to gravity. The conservation of mass principle requires that the cross-sectional area of the water stream decreases to compensate for the increase in velocity, thus maintaining the constant mass flow rate in the system.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mass Flow Rate
The mass flow rate is a measure of the amount of mass moving through a particular point in a system per unit time. It's a central concept when analyzing systems involving fluid flow, such as the water from a faucet. Defined mathematically, the mass flow rate, often denoted by the symbol \( m \), is calculated by the equation \( m = \rho A v \) where \( \rho \) represents the fluid density, \( A \) is the cross-sectional area through which the fluid passes, and \( v \) is the velocity of the fluid at that point.
Understanding mass flow rate is crucial because it helps us comprehend how fluids behave under different conditions, such as varying velocities or cross-sectional areas. In the faucet example, the mass flow rate must remain constant from the point where the water exits the faucet to the point where it hits the sink. This concept directly relates to the conservation of mass, as the quantity of water does not change, only its velocity and cross-sectional area do as it falls under gravity.
Understanding mass flow rate is crucial because it helps us comprehend how fluids behave under different conditions, such as varying velocities or cross-sectional areas. In the faucet example, the mass flow rate must remain constant from the point where the water exits the faucet to the point where it hits the sink. This concept directly relates to the conservation of mass, as the quantity of water does not change, only its velocity and cross-sectional area do as it falls under gravity.
Cross-sectional Area
Cross-sectional area, often symbolized as \( A \), refers to the size of a cut through an object or fluid stream, perpendicular to the flow direction. Imagine slicing through a water pipe or stream; the area of this slice is what we call the cross-sectional area. This concept is vital in fluid dynamics because it impacts the flow characteristics, such as velocity and pressure.
The relationship between cross-sectional area and fluid velocity is inversely proportional, assuming a steady, incompressible flow and no external forces except gravity. A decrease in the cross-sectional area results in an increase in velocity, which is what we observe with the stream of water narrowing as it falls from the faucet to the sink. This phenomenon occurs because the fluid, with a given mass flow rate, must move faster as the area it flows through becomes smaller.
The relationship between cross-sectional area and fluid velocity is inversely proportional, assuming a steady, incompressible flow and no external forces except gravity. A decrease in the cross-sectional area results in an increase in velocity, which is what we observe with the stream of water narrowing as it falls from the faucet to the sink. This phenomenon occurs because the fluid, with a given mass flow rate, must move faster as the area it flows through becomes smaller.
Fluid Dynamics
Fluid dynamics is the study of the behavior of fluids (liquids and gases) in motion. It involves the analysis of fluid flow, the forces that develop as a result, and the resulting effects on surrounding environments. The equations governing fluid dynamics are the foundations of understanding phenomena such as the narrowing stream of water from a faucet.
In the context of our example, fluid dynamics allows us to calculate changes in velocity and cross-sectional area of the water stream by applying the principles of conservation of mass and momentum. The Navier-Stokes equations, for instance, describe how the velocity field of a fluid evolves over time. They might be complex, but for simple cases like this, we can use more straightforward approaches that still derive from these fundamental principles in fluid dynamics.
In the context of our example, fluid dynamics allows us to calculate changes in velocity and cross-sectional area of the water stream by applying the principles of conservation of mass and momentum. The Navier-Stokes equations, for instance, describe how the velocity field of a fluid evolves over time. They might be complex, but for simple cases like this, we can use more straightforward approaches that still derive from these fundamental principles in fluid dynamics.
Gravity's Effect on Velocity
Gravity profoundly influences the behavior of fluids by accelerating them towards the Earth. This acceleration affects the velocity of a fluid in a free fall, such as water leaving a faucet. According to the principles of physics, specifically kinematics, the velocity of an object increases linearly with time as it falls under the influence of gravity, in a vacuum and without air resistance.
In our fluid example, the water exiting the faucet accelerates due to gravity, causing an increase in velocity as it falls toward the sink. Because of the conservation of mass, as the velocity of the stream increases, the cross-sectional area must decrease to maintain a constant mass flow rate. This continuity equation, combined with gravity's acceleration, results in the observable narrowing of the water stream from the faucet to the sink.
In our fluid example, the water exiting the faucet accelerates due to gravity, causing an increase in velocity as it falls toward the sink. Because of the conservation of mass, as the velocity of the stream increases, the cross-sectional area must decrease to maintain a constant mass flow rate. This continuity equation, combined with gravity's acceleration, results in the observable narrowing of the water stream from the faucet to the sink.