A house at the bottom of a hill is fed by a full tank of water 5.0 m deep and connected to the house by a pipe that is 110 m long at an angle of 58° from the horizontal (Fig. 13–48). (a) Determine the water gauge pressure at the house.
(b) How high could the water shoot if it came vertically out of a broken pipe in front of the house?
a)188 Nm^2
b)77m
To determine the water gauge pressure at the house, we need to calculate the hydrostatic pressure at the bottom of the tank and take into account the height difference and the length of the pipe.
(a) To begin, we need to find the pressure at the bottom of the tank due to the weight of the water. The pressure at a certain depth in a fluid can be calculated using the equation:
Pressure = density x gravity x depth
In this case, since the tank is full, the depth is equal to 5.0 m. The density of water is approximately 1000 kg/m³, and the acceleration due to gravity is 9.8 m/s². Substituting these values into the equation, we get:
Pressure = 1000 kg/m³ x 9.8 m/s² x 5.0 m = 49000 Pa
Thus, the pressure at the bottom of the tank is 49000 Pa (or 49 kPa).
Next, we need to calculate the pressure at the house by taking into account the height difference and the length of the pipe. We can use the following equation:
Pressure = Pressure at bottom of tank + (density x gravity x height difference) - (density x gravity x length x sin(angle))
In this equation, the height difference is equal to the vertical distance between the bottom of the tank and the house, and the angle is the angle of the pipe from the horizontal (58°). The density of water, the acceleration due to gravity, and the length of the pipe (110 m) are the same as before.
Since the house is at the bottom of the hill, the height difference is 0. Therefore, the equation simplifies to:
Pressure = Pressure at bottom of tank - (density x gravity x length x sin(angle))
Substituting the known values, we get:
Pressure = 49000 Pa - (1000 kg/m³ x 9.8 m/s² x 110 m x sin(58°))
Using a calculator, we can compute the value of sin(58°) ≈ 0.8480 and simplify the equation further:
Pressure ≈ 49000 Pa - (1000 kg/m³ x 9.8 m/s² x 110 m x 0.8480)
≈ 49000 Pa - 89656 Pa ≈ -40656 Pa
The negative pressure value indicates that the water gauge pressure at the house is below atmospheric pressure. To convert it to positive gauge pressure, we take the absolute value:
Water gauge pressure at the house ≈ |-40656 Pa| ≈ 40656 Pa
Therefore, the water gauge pressure at the house is approximately 40656 Pa (or 40.7 kPa).
(b) To determine how high the water could shoot if it came vertically out of a broken pipe in front of the house, we can use the concept of potential energy. The potential energy of an object at a certain height is given by the equation:
Potential Energy = mass x gravity x height
Since the water shooting out of the broken pipe is not confined, we can neglect the mass of the water, and hence the equation simplifies to:
Potential Energy = gravity x height
To find the maximum height, we can rearrange the equation to solve for height:
Height = Potential Energy / gravity
Substituting the known values, we get:
Height = (49000 Pa - atmospheric pressure) / (9.8 m/s²)
Assuming atmospheric pressure is approximately 101325 Pa, we can calculate:
Height = (49000 Pa - 101325 Pa) / (9.8 m/s²) = -52325 Pa / (9.8 m/s²)
Simplifying, we find the height to be approximately:
Height ≈ -5346.9 m
The negative value indicates that the water cannot shoot above the ground level. Therefore, if the water came vertically out of a broken pipe in front of the house, the maximum height it would reach is approximately 0 m.