While taking a shower, you notice that the shower head is made up of 42 small ro

Question

While taking a shower, you notice that the shower head is made up of 42 small round openings, each with a radius of 2.00 mm. You also determine that it takes 2.00 s for the shower to completely fill a 1.00-liter container you hold in the water stream. The water for the shower is pumped by a pump that is 5.30 m below the level of the shower head. The pump maintains an absolute pressure of 1.50 atm. Use g = 10 m/s, and assume that 1 atmosphere is 1.0 × 105 Pa (a) At what speed does the water emerge from the shower head? m/s (b) What is the speed of the water in the pipe connected to the pump? m/s (c) What is the cross-sectional area of the pipe connected to the pump? m2

Explanation / Answer

part A :

Area of the opening A = pir ^2

A = 3.14* 2*10^-3 * 2*10^-3

A = 1.256 *10^-7 m^2

flow rate Q/t = 1L/2 s = 0.5 L/s

Q/t = 0.5 *10^-3 m^3/s

from flow rate, flow Q = Av for 1 opening

for 42 openings

Q = 42 * 1.256 *10^-7 * v = 0.5*10^-3

v = 0.947 m/s

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part B:

using Bernouliis theorem

P1 + 0.5 rho v1^2 + rho g h1 = P2 + 0.5 rho v2^2 + rho g h2

(1.5*10^5) + (0.5 * 1000 * v1^2) + (1000* 10* 0) = (1*10^5) + (0.5* 1000*0.947^2) + (1000* 10 * 5.3)

v1 = 2.626 m/s

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part C:

use the equation of Continuity as A1 v1 = A2 V2

2.626* A1 = 0.947 * 42 * 1.256 *10^-7

A1 = 190.23 *10^-6 m^2

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