Theory — Pressure and Flow in Fluids
Pressure Increases with Depth
In a fluid at rest, the pressure at a point comes from the weight of everything above it: the column of fluid plus the atmosphere pressing on the surface. The deeper you go, the more fluid sits above you, so the pressure rises in direct proportion to depth.
P_atm = atmospheric pressure (≈ 101.3 kPa at sea level)
ρ = fluid density (kg/m³), g = gravity (m/s²), h = depth (m)
Why Doubling Density Does Not Double the Reading
The pressure gauge reads the total pressure, P_atm + ρgh. Only the ρgh part depends on the fluid. If you double the density, you double ρgh, but the atmospheric part stays the same — so the total reading goes up, but it does not double. Turn the atmosphere off (place the tank in vacuum) and P_atm = 0, so the reading is just ρgh; then doubling the density really does double the reading.
Pascal's Principle
Pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid. In a connected (enclosed) tank, pressing down on the fluid on one side raises the pressure everywhere by the same amount, so gauges at the same height read the same value — this is the basis of the hydraulic press.
Continuity — Flow Through a Pipe
For an incompressible fluid, the volume flow rate is the same everywhere along a pipe. Where the pipe is narrow, the area A is small, so the speed v must be large to carry the same flow; where it is wide, the fluid slows down.
Bernoulli's Equation and the Water Tower
For an ideal fluid, the sum of pressure energy, kinetic energy, and gravitational energy per unit volume is constant along a streamline. For a tank open to the atmosphere, this gives Torricelli's result: fluid leaving an opening a depth h below the surface comes out at v = √(2gh). A stream directed straight up would, in the absence of losses, rise back to the height of the water surface in the tank.
Exit speed from depth h: v = √(2 · g · h)
Deeper or denser
More fluid weight above → higher gauge pressure (ρgh grows).
Narrower pipe
Smaller area → faster flow, since A·v stays constant.
Lower water level
Smaller depth h → slower exit speed and a shorter stream.
| Quantity | Equation | Notes |
|---|---|---|
| Pressure at depth | P = P_atm + ρgh | gauge reads total pressure |
| Gauge (fluid) pressure | ρgh | doubles if density doubles |
| Pascal's principle | ΔP transmitted equally | enclosed fluid, hydraulics |
| Continuity | A₁v₁ = A₂v₂ | flow rate constant |
| Exit speed (Torricelli) | v = √(2gh) | opening depth h below surface |
Instructions — Running the Virtual Experiment
The simulation has three tabs — Pressure, Flow, and Water Tower — matching the three parts of the worksheet. Record every reading in your worksheet. Pressures are shown in kilopascals (kPa).
Simulation — Fluid Pressure and Flow
Tank & fluid
Enclosed tank — add weight
Pipe & fluid
Water tower
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide if your instructor asks for a full report; otherwise complete the worksheet.
Fluid Pressure and Flow
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To investigate how pressure in a fluid depends on depth, density, gravity, and atmospheric pressure; to confirm that an enclosed fluid transmits pressure equally (Pascal's principle); to observe the continuity relation A·v = constant for flow in a pipe; and to study the stream from a water tower using Torricelli's and Bernoulli's relations.
Theory
Pressure at depth is P = P_atm + ρgh. The gauge reads the total pressure, so the fluid contribution ρgh doubles when the density doubles, but the total reading does not (the atmospheric part is unchanged). With the atmosphere off, the reading is ρgh alone and doubles with density. An enclosed fluid transmits an applied pressure equally everywhere. For flow, A₁v₁ = A₂v₂. For a tank open to air, the exit speed at depth h is v = √(2gh).
Calculations — Sample values
Gauge pressure (water, h = 2 m): ρgh = 1000 × 9.8 × 2 = 19 600 Pa = 19.6 kPa; total = 19.6 + 101.3 = 120.9 kPa
Density effect (vacuum): gasoline ρgh = 700 × 9.8 × 2 = 13.7 kPa; honey = 1400 × 9.8 × 2 = 27.4 kPa — exactly doubled
Gravity effect: at g = 12, ρgh = 1000 × 12 × 2 = 24.0 kPa (up from 19.6 kPa); the gauge reading rises in proportion to g
Water tower (h = 3 m): v = √(2 × 9.8 × 3) = 7.67 m/s
Results Table — Pressure (h = 2 m, g = 9.8 m/s²)
| Fluid | ρ (kg/m³) | ρgh (kPa) | Atmosphere on (kPa) | Atmosphere off (kPa) |
|---|---|---|---|---|
| Gasoline | 700 | 13.7 | 115.0 | 13.7 |
| Honey | 1400 | 27.4 | 128.8 | 27.4 |
With the atmosphere on, the reading rose but did not double; with the atmosphere off, doubling the density doubled the reading.
Discussion
The pressure rose with depth and density as predicted by ρgh. With the atmosphere on, doubling the density did not double the reading because the gauge also includes the fixed atmospheric pressure; in vacuum the reading was just ρgh and doubled exactly. Increasing gravity raised the gauge pressure proportionally. In the enclosed tank, adding weight on one side raised the pressure equally on both gauges, confirming Pascal's principle. In the pipe, narrowing the middle increased the speed while the flow rate stayed constant (A·v = constant), and the water-tower stream slowed and fell shorter as the level dropped, consistent with v = √(2gh) and Bernoulli's equation.
Conclusion
Pressure in a fluid follows P = P_atm + ρgh; the gauge (fluid) part scales with density and gravity, while the atmospheric part is fixed. Enclosed fluids obey Pascal's principle, flowing fluids obey continuity, and the water-tower stream obeys Torricelli's/Bernoulli's relations. Experimental readings agreed with theory throughout.
Practice Questions
Show all work and include units. Use P_atm = 101.3 kPa and g = 9.8 m/s² unless told otherwise.