Determine the minimum gauge pressure needed in the water pipe leading into a building if water is to come out of a faucet on the fourteenth floor, 44 m above that pipe.

Verified step by step guidance

1. Identify the key concept: The problem involves determining the minimum gauge pressure required to lift water to a height of 44 m. This is a fluid mechanics problem that uses the relationship between pressure, height, and density in a static fluid.
2. Use the hydrostatic pressure formula: The pressure required to lift a fluid to a certain height is given by the equation: $P=\rho gh$, where $\rho$ is the density of water (approximately 1000 kg/m³), $g$ is the acceleration due to gravity (9.8 m/s²), and $h$ is the height (44 m).
3. Substitute the known values into the formula: Replace $\rho$ with 1000 kg/m³, $g$ with 9.8 m/s², and $h$ with 44 m in the equation $P=\rho gh$.
4. Recognize that the result from the formula gives the absolute pressure due to the water column. To find the gauge pressure, note that gauge pressure excludes atmospheric pressure. Since the problem asks for the minimum gauge pressure, the atmospheric pressure is not added to the result.
5. Conclude the calculation: After substituting the values and performing the multiplication, the result will give the minimum gauge pressure in Pascals (Pa). Ensure the units are consistent throughout the calculation.

Key Concepts

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is the pressure measured in a system that excludes the atmospheric pressure acting on the fluid. In this context, it is crucial to determine how much additional pressure is needed in the water pipe to ensure that water can reach the faucet on the fourteenth floor.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid and is calculated using the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. This concept is essential for calculating the pressure needed to lift water to a specific height.

Bernoulli's Principle

Bernoulli's Principle states that in a flowing fluid, an increase in velocity occurs simultaneously with a decrease in pressure or potential energy. This principle helps in understanding how pressure changes in a fluid system, which is important when considering the flow of water from the pipe to the faucet at a higher elevation.