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?

Verified step by step guidance

1. Calculate the cross-sectional area of the pipe using the formula for the area of a circle, \( A_{\text{pipe}} = \pi r_{\text{pipe}}^2 \), where \( r_{\text{pipe}} \) is the radius of the pipe.
2. Calculate the cross-sectional area of one shower head opening using the formula for the area of a circle, \( A_{\text{opening}} = \pi r_{\text{opening}}^2 \), where \( r_{\text{opening}} \) is the radius of one opening.
3. Multiply the area of one opening by the number of openings to find the total area of all the shower head openings, \( A_{\text{total openings}} = 20 \times A_{\text{opening}} \).
4. Apply the principle of conservation of mass, which states that the mass flow rate must be constant throughout the system, implying \( A_{\text{pipe}} \times v_{\text{pipe}} = A_{\text{total openings}} \times v_{\text{openings}} \), where \( v_{\text{pipe}} \) is the speed of water in the pipe and \( v_{\text{openings}} \) is the speed of water exiting the openings.
5. Solve for \( v_{\text{openings}} \) using the equation \( v_{\text{openings}} = \frac{A_{\text{pipe}} \times v_{\text{pipe}}}{A_{\text{total openings}}} \).

Key Concepts

Continuity Equation

The Continuity Equation in fluid dynamics states that for an incompressible fluid, the mass flow rate must remain constant from one cross-section of a pipe to another. This means that the product of the cross-sectional area and the fluid velocity is constant. In this case, the speed of water in the shower head openings can be determined by comparing the area of the pipe to the total area of the openings.

Cross-Sectional Area

The cross-sectional area of a circular opening is calculated using the formula A = πr², where r is the radius. For the shower head, the total area is the sum of the areas of all openings. Understanding how to calculate the cross-sectional area is essential for applying the Continuity Equation to find the speed of water exiting the shower head.

Fluid Velocity

Fluid velocity refers to the speed at which a fluid flows through a given area. In this problem, the velocity of water in the pipe is given, and we need to find the velocity as it exits the shower head. The relationship between the velocities in different sections of the flow is governed by the Continuity Equation, which allows us to solve for the unknown exit velocity.