Question

A shower head has 20 circular openings, each with radius 1.0 mm. The shower head is connected to a pipe with radius 0.80 cm. If the speed of water in the pipe is 3.0 m/s, what is its speed as it exits the shower-head openings?

Short answer

The speed of water exiting the openings is 9.6 m/s.

Step-by-step solution

Understanding the Problem

We need to find the speed of water as it exits the 20 openings of the shower head. We can use the principle of conservation of mass (continuity equation) for incompressible fluids, which states that the flow rate must remain constant between the pipe and the shower head openings.

Calculate Area of Pipe

The radius of the pipe is given as 0.80 cm, which is 8.0 mm. The area, \( A_\text{pipe} \), of the pipe's cross-section can be calculated using the formula for the area of a circle: \( A = \pi r^2 \). Therefore, \( A_\text{pipe} = \pi (8.0 \text{ mm})^2 = 64\pi \text{ mm}^2 \).

Calculate Total Area of Shower Head Openings

Each of the 20 openings has a radius of 1.0 mm. So, the area of one opening \( A_\text{opening} = \pi (1.0 \text{ mm})^2 = \pi \text{ mm}^2 \). Since there are 20 openings, the total area \( A_\text{total openings} = 20 \times \pi \text{ mm}^2 = 20\pi \text{ mm}^2 \).

Apply Continuity Equation

The continuity equation states that \( A_\text{pipe} \cdot v_\text{pipe} = A_\text{total openings} \cdot v_\text{openings} \), where \( v_\text{pipe} \) is the velocity of water in the pipe, and \( v_\text{openings} \) is the velocity of water exiting the openings. Substituting known values: \( 64\pi \cdot 3.0 = 20\pi \cdot v_\text{openings} \).

Solve for Velocity at Openings

Cancel \( \pi \) from both sides, and solve for \( v_\text{openings} \): \( 64 \cdot 3.0 = 20 \cdot v_\text{openings} \). Thus, \( v_\text{openings} = \frac{64 \times 3.0}{20} = 9.6 \text{ m/s} \).

Key concepts

Incompressible Fluids

Incompressible fluids are fluids in which the density remains constant regardless of changes in pressure. This assumption simplifies the analysis of fluid motion because the mass flowing into a system will equal the mass flowing out of it.
For incompressible fluids, the density does not vary from point to point, making mathematical modeling more straightforward. Water is often considered incompressible in many practical applications due to its minimal change in density for usual pressure variations.
This concept is crucial when applying the Continuity Equation, as it allows us to make predictions about fluid behavior based on the assumption that material properties (like density) do not change during the flow process.

Conservation of Mass

The conservation of mass, often referred to in the context of fluids as the Continuity Equation, is a fundamental principle asserting that mass cannot be created or destroyed in an isolated system. In fluid dynamics, this means that what goes into a system must come out, assuming no accumulation within the system.
For a pipe that narrows or widens, the product of the cross-sectional area and velocity at one point must equal the product of these values at another point, given an incompressible fluid.
Therefore, for a flow through varying cross-sectional areas, \( A_1v_1 = A_2v_2 \). This relationship is essential for calculating changes in fluid speed when cross-sectional dimensions change.

Flow Rate

Flow rate describes the volume of a fluid passing through a given surface per unit time and is often denoted as \( Q \). It is influenced by both the velocity of the fluid and the cross-sectional area it moves through. Flow rate is mathematically expressed as \( Q = Av \), where \( A \) is the cross-sectional area and \( v \) is the velocity.

• In the context of the problem, the flow rate through the initial pipe matches the flow rate of the water exiting all shower head openings.
• This constancy is due to the Conservation of Mass, ensuring that the mass flow (or volumetric flow rate, for incompressible fluids) is maintained.

Understanding flow rate allows for the calculation of necessary changes in velocity due to area changes in fluid systems.

Velocity Calculation

Calculating the velocity of a fluid in different sections of a system involves applying the Continuity Equation. When dealing with a system like a pipe connected to a shower head, the velocity of the fluid as it exits can be determined by ensuring that the product of the flow area and velocity in the larger inlet pipe is equal to that at the smaller outlets of the shower head.

• The given problem demonstrates this with calculations showing how the area and inlet velocity determine the outlet velocity.
• In the exercise: Knowing \( A_\text{pipe} \) and \( v_\text{pipe} \), as well as \( A_\text{total openings} \), we rearrange the equation \( A_\text{pipe} \, v_\text{pipe} = A_\text{total openings} \, v_\text{openings} \) to solve for \( v_\text{openings} \).

This application helps predict outputs in systems where fluid must transition between spaces of varying dimensions.