Problem 87 - Master Resource Book in JEE Main Physics

Question

Water flowing out of the mouth of a tap and falling vertically in streamline flow forms a tapering column, Le., the area of cross-section of the liquid column decreases as it moves down. Which of the following is the most accurate explanation for this? (a) Falling water tries to reach a terminal velocity and hence, reduces the area of cross-section to balance upwand and downward forces (b) As the water moves down, its speed increases and hence, its pressure decreases. It is then compressed by atmosphere (c) The surface tension causes the exposed surface area of the liquid to decrease continuously (d) The mass of water flowing out per second through any cross-section must remain constant. As the water is almost incompressible, so the volume of water flowing out per second must remain constant. As this is equal to velocity \(x\) area, the area decreases as velocity increases

Step-by-Step Solution

Answer

(d) The area decreases as velocity increases to maintain constant flow rate.

Step 1: Analyze the Question

The question is asking why the cross-section of a liquid column of water flowing out of a tap decreases as it falls. We need to analyze the given options and relate them to principles of fluid dynamics to find the correct explanation.

Step 2: Understand Principle of Continuity Equation

According to the principle of continuity, for an incompressible fluid, the mass flow rate must remain constant. This implies that the product of cross-sectional area and velocity must be constant while the fluid flows.

Step 3: Relate Mass Conservation to Decreasing Area

As the water falls, the gravitational force acts on it, causing it to accelerate and thus increasing its velocity. To maintain a constant mass flow rate, as per the continuity equation, the cross-sectional area must decrease if the velocity increases.

Step 4: Evaluate the Options

Option (d) states that the mass of water per second must remain constant, and as velocity increases, area decreases to keep the product of velocity and area constant. This matches the principle of continuity. Other options are either incorrectly stating the effects or mixing concepts.

Step 5: Select the Correct Explanation

Based on the analysis, option (d) is consistent with the principles of continuity and fluid dynamics, indicating that the cross-sectional area decreases to maintain a constant flow rate as velocity increases.

Key Concepts

Principle of Continuity

In fluid dynamics, the principle of continuity is a fundamental concept that helps us understand how fluids behave as they move through different spaces. This principle states that for an incompressible fluid (like water), the mass flow rate must remain constant as it flows. This means that if a fluid moves into a narrow channel, its speed has to increase so the same amount of fluid passes through per unit of time. Mathematically, this can be represented by the equation:\[ A_1 \times v_1 = A_2 \times v_2 \]where \( A \) is the cross-sectional area and \( v \) is the velocity of the fluid at different points along its path. The equation shows that if the area (\( A \)) decreases, the velocity (\( v \)) must increase to keep the product \( A \times v \) constant. This relationship is central to understanding how fluids flow, especially in pipes and nozzles.

Incompressible Flow

Incompressible flow refers to a situation where the density of a fluid remains constant throughout its motion. Most liquids, like water, are considered incompressible because their density doesn’t change significantly with pressure changes. When analyzing fluid dynamics problems involving incompressible flow, we assume the fluid doesn't expand or contract. This simplifies calculations and helps us use principles like the continuity equation effectively.

• Density changes are negligible, so volume flow rate remains constant.
• Helps predict how a fluid’s speed and cross-sectional area will change.

Understanding incompressibility is crucial for engineers and scientists working with systems where liquids flow, allowing them to design efficient and effective piping systems, among other applications.

Mass Flow Rate

Mass flow rate in fluid dynamics refers to the amount of mass passing through a given cross-section of a flow per unit of time. It is a critical concept for understanding how fluids move through different systems.The formula for mass flow rate \( \dot{m} \) is:\[ \dot{m} = \rho \cdot A \cdot v \]where \( \rho \) is the density, \( A \) is the cross-sectional area, and \( v \) is the velocity of the fluid. In scenarios involving incompressible fluids like water, the density \( \rho \) is constant. Therefore, the product of the area and velocity (\( A \cdot v \)) becomes the primary aspect ensuring that the mass flow rate remains unchanged. This understanding allows engineers to efficiently design systems where fluid dynamics is crucial, ensuring constant flow rates through variations in cross-sectional areas or velocities.

Velocity-Area Relationship

The velocity-area relationship is a key component of fluid flow systems that relies on the principle of continuity. This relationship explains how the speed of a fluid changes when it travels through areas of different sizes.As mentioned in the principle of continuity, the equation \( A_1 \times v_1 = A_2 \times v_2 \) illustrates that an increase in velocity must compensate for a decrease in area, and vice versa.

• Smaller cross-sectional areas lead to higher fluid velocities.
• Larger cross-sectional areas result in lower fluid velocities.

Understanding this relationship helps in various applications, from designing efficient water distribution systems to creating aerodynamic car shapes. By controlling the area through which the fluid passes, one can manage its velocity, and thereby its dynamic pressure and kinetic energy, which are vital parameters in fluid dynamics engineering.