How high would water rise in the essentially open pipes of a...

How high would water rise in the essentially open pipes of a building if the water pressure gauge shows the pressure at the ground floor to be $270 \mathrm{kPa}$ (about $40 \mathrm{lb} / \mathrm{in}^{2}$.)? Water pressure gauges read the excess pressure just due to the water, that is, the difference between the absolute pressure in the water and the pressure of the atmosphere. The water pressure at the bottom of the highest column that can be supported is $270 \mathrm{kPa}$. Therefore, $P=\rho_{w} p h$ gives $$ h=\frac{P}{\rho_{w} g}=\frac{2.70 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}}{\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)}=27.5 \mathrm{~m} $$

How high would water rise in the essentially open pipes of a building if the water pressure gauge shows the pressure at the ground floor to be $270 \mathrm{kPa}$ (about $40 \mathrm{lb} / \mathrm{in}^{2}$.)?

Water pressure gauges read the excess pressure just due to the water, that is, the difference between the absolute pressure in the water and the pressure of the atmosphere. The water pressure at the bottom of the highest column that can be supported is $270 \mathrm{kPa}$. Therefore, $P=\rho_{w} p h$ gives
$$
h=\frac{P}{\rho_{w} g}=\frac{2.70 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}}{\left(1000 \mathrm{~kg} / \mathrm{m}^{3}\right)\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)}=27.5 \mathrm{~m}
$$

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. This concept plays a crucial role in fluid statics and is directly proportional to the depth of the fluid, the density of the fluid, and the acceleration due to gravity. It explains why pressure increases with depth in a liquid column.

Pressure-Height Relationship (P = ?gh)

The equation P = ?gh establishes the relationship between the pressure at a certain depth in a fluid and the height of the fluid column above that point. It shows that the pressure is a product of the fluid’s density, the acceleration due to gravity, and the height of the fluid, forming the basis for calculating the height of a liquid column from a given pressure reading.

Fluid Density

Fluid density is a measure of mass per unit volume and is a key factor in determining how a fluid behaves under pressure. In the context of hydrostatic pressure, the density of the fluid indicates how much weight a column of that fluid exerts, directly affecting the pressure at the bottom of the column.

Gravitational Acceleration

Gravitational acceleration is the acceleration due to gravity that acts on objects near the Earth's surface. In hydrostatics, it is used to calculate the weight of the fluid column, which in turn determines the pressure that the fluid exerts at a certain depth.

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