While taking a shower, you notice that the shower head is made up of 44 small ro

Question

While taking a shower, you notice that the shower head is made up of 44 small round openings, each with a radius of 2.00 mm. You also determine that it takes 4.00 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 5.70 m below the level of the shower head. The pump maintains an absolute pressure of 1.50 atm. Use g = 10 m/s2, and assume that 1 atmosphere is 1.0 105 Pa.

(a) At what speed does the water emerge from the shower head? The answer is .452 m/s which I understand.

(b) What is the speed of the water in the pipe connected to the pump?

(c) What is the cross-sectional area of the pipe connected to the pump?

Explanation / Answer

a)

let the speed of water from the shower head is v1

Now , using equation of continuity

v1 * A1 * N = volume flow rate

44 * pi * (0.002)^2 * v1 = 1 *10^-3/4

v1 = 0.452m/s

the speed of water from shower head is 0.497 m/s

b)

let the speed of water is pump is v2

Using bernoull's equation

1.5 * 1.01 *10^5 - 1.01 *10^5 - 1000 * 9.8 * 5.7 = 0.5 * 1000 *(0.452^2 - v2^2)

solving for v2

v2 = 3.31 m/s

the speed of water from the pump is 3.31 m/s

c)

using equation of continuity

A2 * v2 = volume flow rate

3.31 * A2 = 1 *10^-3/4

A2 = 7.553 *10^-5 m^2

the area of new cross section is 7.553 *10^-5 m^2

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