In a five story building in New York City the water is supplied by a large tank

Question

In a five story building in New York City the water is supplied by a large tank of water on the roof which is open to the atmosphere. In the basement, 29 meters below the water level in the tank there is a 1 cm diameter pipe with a spigot (a valve) which can be turned on and off.
(a) If the spigot is turned off (no water is flowing) what is the absolute pressure in the 1 cm pipe?
Pa

(b) If you open the spigot, what is the velocity of the water coming out of the pipe?
m/s

(c) How long would it take for you to fill up a 20 liter bucket (about five gallons)?
s

Explanation / Answer

a)
P = Po + gh

Atmosperic pressure (Po) is 1.01325*10^5 Pa and the density of water () is about 1000kg/m^3. So;

P = 1.01325*10^5 Pa + (1000kg/m^3)(9.8m/s^2)(29m) = 3.855*10^5 Pa

b) If the spigot is opened to atmospheric pressure then at the spigot (by bernoulli's eq'n)

P1 + gy1 + 1/2v1^2 = P2 + gy2 + 1/2v2^2

We want to find v2. P1=P2=1atm. We'll assume the water in the tank does not move very quickly (v10). So;

gy1 = gy2 + 1/2v2^2

same liquid in both cases so cancels

gy1 = gy2 + 1/2v2^2

2g(y1-y2) = v2^2

2(9.8m/s^2)(29m-0m) = v2^2

v2 = 23.84m/s

c) The flow rate at the spigot is A2v2

flow rate = *(0.01m)^2*(23.84m/s) = 7.49*10^-3 m^3/s

We want to fill 20L which is 0.02m^3

(0.02m^2)/(7.49*10^3 m^3/s) = 2.67s

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