A spherical reservoir that contains 4.55 105 kg of water when full. (h = 7.70 m.) The reservoir is vented to the atmosphere at the top. For a full reservoir, find the gauge pressure that the water has at the faucet in each house. Ignore the diameter of the delivery pipes.
(The height of the tower is 15m)
To find the gauge pressure that the water has at the faucet in each house, we need to consider the pressure at the bottom of the water column due to its height. The pressure at a certain depth in a fluid is given by the equation:
P = P₀ + ρgh
Where:
P is the pressure at the desired depth,
P₀ is the atmospheric pressure (which we assume to be 1 atm),
ρ is the density of the fluid,
g is the acceleration due to gravity, and
h is the height of the fluid column.
In this case, the height of the water column (h) is 7.70 m. The density of water (ρ) is approximately 1000 kg/m³. Acceleration due to gravity (g) is approximately 9.8 m/s².
Now let's calculate the gauge pressure at the bottom of the water column:
P = 1 atm + (1000 kg/m³) * (9.8 m/s²) * (7.70 m)
P = 1 atm + 75460 N/m²
P = 1.07546 x 10⁵ N/m²
Since 1 N/m² is equal to 1 Pascal (Pa), we can also express this pressure as:
P ≈ 1.07546 x 10⁵ Pa
So, the gauge pressure at the faucet in each house when the reservoir is full is approximately 1.07546 x 10⁵ Pa.