Question

The speed of water in a shower pipe having a radius of 2.00 cm is about 1.20 m/s. If it comes out of the shower head with a speed of 0.530 m/s and each small nozzle in the shower head has a radius of 0.850 mm, how many small nozzles are there in this showerhead?

Answer 1

There are approximately 100 small nozzles in this showerhead.

Step-by-step explanation:

To determine the number of small nozzles in the showerhead, we can utilize the principle of conservation of mass for an incompressible fluid flowing through pipes. The formula relating the velocities and cross-sectional areas of the pipe and nozzles is A₁v₁ = A₂v₂, where A represents the cross-sectional area and v represents the velocity.

Given the radius of the shower pipe is 2.00 cm (0.020 m) and its speed is 1.20 m/s, we calculate the cross-sectional area of the pipe using the formula A = πr². Substituting the values, the cross-sectional area of the pipe (A₁) can be computed.

Next, given the speed of water coming out of the showerhead is 0.530 m/s and the radius of each small nozzle is 0.850 mm (8.50 × 10^-4 m), we compute the cross-sectional area of a single nozzle (A₂) using the formula for the area of a circle.

Using the principle of conservation of mass, A₁v₁ = total number of nozzles × A₂v₂, we can solve for the total number of nozzles. Rearranging the formula to solve for the number of nozzles gives us: total number of nozzles = (A₁v₁) / (A₂v₂).

After performing the calculations with the respective values for cross-sectional areas and velocities, the total number of small nozzles in the showerhead is approximately 100, assuming the flow of water remains steady and there are no other losses in the system.

Answer 2

To find the number of nozzles in the showerhead, apply the conservation of mass using the continuity equation, calculate the total cross-sectional area required for the nozzles, find the area of one nozzle, and divide the total required area by the area of one nozzle.

Step-by-step explanation:

The student's question involves applying the principle of conservation of mass in the context of fluid dynamics, specifically using the continuity equation to compute the number of nozzles in a shower head based on the speed of water in the pipe and the speed of water coming out of the nozzles.

The continuity equation states that the product of the cross-sectional area (A) and the velocity (v) of a fluid must be constant throughout its flow if the fluid is incompressible, which is the assumption we make for water in this scenario. Therefore, we can express this as A1 * v1 = A2 * v2, where A1 and v1 refer to the area and velocity in the shower pipe, and A2 and v2 refer to the area and velocity in one of the small nozzles.

First, we calculate the cross-sectional area of the shower pipe, A1 = π * (2.00 cm)^2, and then use the given velocities to find the total cross-sectional area required for the nozzles, A2_total = A1 * (v1 / v2). After that, we compute the area of one small nozzle A2_single = π * (0.850 mm)^2, and then determine the number of small nozzles by dividing the total area for the nozzles by the area of one nozzle.