What gauge pressure (in atm) in the water mains is necessary if a firehose is to spray water to a height of 6.0 m?
To determine the necessary gauge pressure in the water mains for a firehose to spray water to a height of 6.0 m, we can use the concept of hydraulic head and Bernoulli's equation.
Step 1: Convert the height into meters.
Given: Height (h) = 6.0 m
Step 2: Determine the acceleration due to gravity.
The acceleration due to gravity (g) is approximately 9.8 m/s^2.
Step 3: Apply Bernoulli's equation.
Bernoulli's equation states that the total energy of a fluid remains constant along a streamline. In this case, we can equate the pressure energy and gravitational potential energy.
P₁ + ρgh₁ = P₂ + ρgh₂
Where:
P₁ = gauge pressure at the water mains
ρ = density of water
g = acceleration due to gravity
h₁ = height from the hose nozzle to the water level in the mains (in this case, 0 since the water is at the same level)
P₂ = gauge pressure required to lift the water to a height of 6.0 m
h₂ = height to which the water is lifted
Step 4: Rearrange the equation and substitute known values.
P₂ = P₁ + ρgh₂
Given:
h₁ = 0 m
h₂ = 6.0 m
ρ (density of water) = 1000 kg/m³
Substituting these values into the equation:
P₂ = P₁ + (1000 kg/m³)(9.8 m/s²)(6.0 m)
Step 5: Calculate the gauge pressure.
P₂ = P₁ + 58800 kg·m²/s²
Since the pressure is usually measured in pascals (Pa), which is equivalent to N/m², we divide the above result by the conversion factor 1 atm = 101325 Pa.
P₂ = P₁ + 58800 kg·m²/s² / (101325 Pa/atm)
Step 6: Convert the pressure to atm.
P₂ ≈ P₁ + 0.58 atm
Therefore, the gauge pressure in the water mains necessary for a firehose to spray water to a height of 6.0 m is approximately 0.58 atm.
To determine the gauge pressure required in the water mains for the firehose to spray water to a height of 6.0 meters, we can use the principles of fluid mechanics and the concept of pressure.
Step 1: Determine the height of the water above the firehose nozzle (h).
Given that the water sprays to a height of 6.0 meters, the height (h) is 6.0 meters.
Step 2: Convert the height into the pressure equivalent.
The pressure at the base of the firehose is atmospheric pressure, which is 1 atm. We need to calculate the additional pressure required to spray water to a height of 6.0 meters.
Using the concept of pressure, we know that the pressure (P) at a certain height (h) can be calculated using the equation: P = ρgh, where ρ is the density of water (1000 kg/m³) and g is the acceleration due to gravity (9.8 m/s²).
P = ρgh
P = (1000 kg/m³) * (9.8 m/s²) * (6.0 m)
Step 3: Convert the pressure to atmospheres (atm).
Since the given pressure is in atm, we need to convert the pressure obtained in step 2 to atm. 1 atm is equivalent to 101325 Pa.
P_atm = P / (101325 Pa/atm)
Substituting the value of P from step 2:
P_atm = [(1000 kg/m³) * (9.8 m/s²) * (6.0 m)] / (101325 Pa/atm)
Step 4: Calculate the gauge pressure.
Gauge pressure is the pressure above atmospheric pressure. To obtain the gauge pressure, we subtract 1 atm from the calculated pressure in step 3.
Gauge pressure = P_atm - 1 atm
Finally, substitute the value of P_atm from step 3 into this equation to find the gauge pressure.
The resulting value will provide the gauge pressure (in atm) required in the water mains for the firehose to spray water to a height of 6.0 meters.
Use the Bernoulli equation with height change, and the velocity equal to zero at both the top of the stream and in high-pressure water mains.
(1/2)(density)(g)(height) = pressure difference = gauge pressure