Problem 58 - College Physics

Question

(a) If the radius of a pipe narrows to half of its original size, will the flow speed in the narrow section (1) increase by a factor of \(2,\) (2) increase by a factor of 4 , (3) decrease by a factor of \(2,\) or (4) decrease by a factor of \(4 ?\) Why? (b) If the radius widens to three times its original size, what is the ratio of the flow speed in the wider section to that in the narrow section?

Step-by-Step Solution

Answer

(a) The flow speed increases by a factor of 4. (b) The ratio is 1:36.

Step 1: Understand the Problem

We need to use the principle of conservation of mass, specifically the equation for the continuity of fluid flow, which states that for an incompressible fluid, the product of the cross-sectional area and the velocity of the fluid is constant across any two points in the pipe.

Step 2: Use the Continuity Equation

The continuity equation is given by: \(A_1 v_1 = A_2 v_2\), where \(A\) is the cross-sectional area and \(v\) is the flow speed. The cross-sectional area \(A\) of a pipe is given by \(\pi r^2\), where \(r\) is the radius.

Step 3: Calculate the Area Change When Radius Narrows

If the radius narrows to half of its original size: \(A_2 = \pi (\frac{r}{2})^2 = \frac{1}{4} \pi r^2.\)

Step 4: Relate Areas to Velocities for Narrowing

Using the continuity equation: \(\pi r^2 v_1 = \frac{1}{4} \pi r^2 v_2.\) Simplify to find \(v_2 = 4v_1.\) The flow speed in the narrow section increases by a factor of 4.

Step 5: Calculate the Area Change When Radius Widens

If the radius widens to three times its original size: \(A_3 = \pi (3r)^2 = 9\pi r^2.\)

Step 6: Relate Areas to Velocities for Widening

Using the continuity equation for widened section compared to the original: \(\pi r^2 v_1 = 9\pi r^2 v_3,\) solve for \(v_3\) to get \(v_3 = \frac{v_1}{9}.\)

Step 7: Find Ratio of Flow Speeds in Wide to Narrow Section

Using the results \(v_2 = 4v_1\) and \(v_3 = \frac{v_1}{9}\), the ratio \(\frac{v_3}{v_2} = \frac{\frac{v_1}{9}}{4v_1} = \frac{1}{36}.\)

Key Concepts

Continuity Equation

In fluid dynamics, the continuity equation is a fundamental principle that ensures mass conservation in fluid flow regimes. This principle is particularly useful when dealing with incompressible fluids, where the density of the fluid remains constant. This means that the amount of fluid moving into a certain part of a system must equal the amount of fluid moving out. Mathematically, this is expressed as:

• \(A_1 v_1 = A_2 v_2\)

Here, \(A\) represents the cross-sectional area, and \(v\) symbolizes the flow speed. By applying this equation, we can predict how the speed of the fluid changes as it moves through varying pipe cross-sections.
For example, if a pipe is constricted (or narrows), the fluid speed must increase to maintain the flow rate, assuming no fluid is added or removed from the system. This lays the foundation for understanding why flow speed changes when the pipe's radius changes.

Flow Speed

Flow speed refers to the velocity at which a fluid element travels along the pipeline. In our exercise scenario, this is especially relevant when considering how a narrowing or widening of the pipe affects the speed. Using the continuity equation, we determine the effect of such changes on fluid motion. When a pipe narrows, the flow speed increases, as observed in the exercise where the radius halves:

• Narrowed radius results in \(v_2 = 4v_1\).

This highlights that the flow speed quadruples - a direct consequence of reducing the cross-sectional area to a quarter of its former size, thus multiplying the speed by four to maintain a constant flow rate.
Conversely, when the pipe widens, as in the case of expanding to three times the initial radius, the flow speed drops. This is due to the significant increase in the cross-sectional area:

• Widened radius results in \(v_3 = \frac{v_1}{9}\).

Hence, the speed decreases ninefold, emphasizing the inverse relationship between flow speed and cross-sectional area.

Pipe Radius

The radius of a pipe is crucial in determining the behavior of fluid flow within it. It directly influences the cross-sectional area, calculated as \(A = \pi r^2\). This area is key in applying the continuity equation to fluid dynamics problems since a change in radius affects the area.
In scenarios where the pipe narrows:

• The area shrinks by the square of the change in radius.
• This results in a calculated area of \(\frac{1}{4}\) of the original when the radius halves.

As the radius becomes larger, the opposite occurs. Consider widening the pipe to three times the original radius:

• The cross-sectional area increases to \(9\pi r^2\), nine times larger than the initial size.

Analyzing these changes allows us to quantify adjustments that occur in flow speed using the continuity equation. This understanding facilitates effective design and operation in real-life fluid systems, ensuring efficiency and preventing issues related to pressure and flow rate distributions.