Relate liquid pressure to depth and density

Compare depth before container size

Imagine pressure sensors at the same vertical depth in two differently shaped containers of the same liquid. Both free surfaces experience the same atmospheric pressure. Predict whether the wider container necessarily gives the larger sensor reading. More total liquid does not automatically mean greater pressure at a particular depth.

The investigation assumes a uniform liquid at rest, constant density and teaching gravitational acceleration g = 10 m/s². Its free surface is held at an absolute pressure of 100 kPa. You choose depth from 0 to 4 m and density from 500 to 1500 kg/m³. These are ideal density values, not identifications of particular liquids. Temperature, compression, surface tension and fluid motion are omitted.

Connect density, weight and pressure

Density ρ, pronounced rho, is mass divided by volume in kg/m³. A column with horizontal area A and depth h has volume Ah, mass ρAh and weight ρAhg. Here A is in m², h in m and g in m/s². A density of 1000 kg/m³ means a cubic metre contains 1000 kg; it does not specify the total mass of a container of unknown volume.

Pressure is normal force divided by area. One pascal, Pa, is one N/m². One kilopascal, kPa, is 1000 Pa. The supporting pressure difference across a stationary liquid column must balance its weight. Dividing the weight by area gives Δp = ρgh. The area cancels: a wider column weighs more but spreads that weight over a larger area.

The bottom pressure also includes the pressure already acting at the top. Thus p = p_surface + ρgh, where all pressure terms must use the same units. Pressure at a depth supports the liquid column’s weight in addition to the surface-pressure contribution. OpenStax fluids, density and pressure develops these quantities from force balance.

Work through the default measurement

At 2 m depth in density 1000 kg/m³, the pressure increment is 1000 × 10 × 2 = 20000 Pa = 20 kPa. Adding the specified surface pressure gives 120 kPa absolute. The gauge pressure relative to the 100 kPa surface reference is 20 kPa.

At 4 m the increment becomes 40 kPa and absolute pressure 140 kPa. Doubling depth doubles the gauge pressure in this setup, but it does not double absolute pressure. The surface contribution remains 100 kPa. Capture both depths and explain the two different proportionality claims.

The graph uses depth on the horizontal axis and pressure on the vertical axis. The solid line shows absolute pressure; the dashed line shows gauge pressure. Both have the same slope, and their separation stays 100 kPa. The selected-depth markers move along these lines; changing depth does not change their slope.

Change density and interpret the slope

Keep depth at 2 m. At density 500 kg/m³, gauge pressure is 10 kPa and absolute pressure 110 kPa. At density 1500 kg/m³, those values become 30 and 130 kPa. Tripling density triples the column weight per unit area and therefore the pressure increment, while the surface pressure stays fixed.

The slope is ρg, expressed as pascals per metre or converted to kilopascals per metre. For density 1000 kg/m³ it is 10 kPa/m. If measured pressure differences did not grow linearly with depth, possible explanations would include density changing with depth, fluid acceleration, an incorrect depth scale or a pressure-sensor calibration problem. A nonlinearity needs investigation rather than automatic deletion of discrepant measurements.

State the boundaries of the model

In a connected uniform liquid at rest with the same surface reference, equal depths have equal pressure even if container widths differ. The container walls can support forces; the force on a particular base need not equal the entire liquid weight for every container shape. Do not substitute a whole-container weight argument for local hydrostatic equilibrium.

Depth is vertical distance below the free surface, not distance along a sloping wall. Pressure at a point is a scalar; the force it exerts on a small surface is perpendicular to that surface. The fact that pressure increases downward does not mean pressure forces act only downward.

At depth zero, the hydrostatic increment is zero while absolute pressure is 100 kPa. This is the surface limit of the model. A zero gauge reading is not a vacuum. The panel schematic introduces a separate gas compartment; its force account is developed in the next lesson.

Transfer to another surface pressure

A stationary liquid has density 800 kg/m³ and surface pressure 90 kPa. Using g = 10 m/s², predict absolute pressure 3 m below the surface and the gauge pressure relative to that surface. Then compare two sensors separated vertically by 1.5 m in the same liquid.

Check the depth transfer

The increment at 3 m is 800 × 10 × 3 = 24000 Pa = 24 kPa. Absolute pressure is 90 + 24 = 114 kPa; gauge pressure relative to the surface is 24 kPa. The deeper of two sensors separated by 1.5 m reads 800 × 10 × 1.5 = 12000 Pa, or 12 kPa, more than the shallower sensor. Their absolute offset cancels from this difference.