Short on time? This page is the recap. For the derivations, bar charts and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A pressure difference between two places makes a fluid flow.
- In a full pipe, an incompressible fluid leaves at the same rate it enters. That is conservation of mass.
- Volume flow rate V/t = Av comes from the cylinder of fluid (area A, length vt) that passes a line in time t.
- Bernoulli’s equation is conservation of energy for a small volume of ideal fluid: work done by pressure differences changes its kinetic and gravitational potential energy.
- Torricelli’s result is Bernoulli between an open surface and an open hole: v = √(2gh), like free fall through h.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| V/t = Av | m³/s | find the volume flow rate |
| mass flow rate = ρAv | kg/s | find mass per second |
| A₁v₁ = A₂v₂ | A in m², v in m/s | compare speeds in two sections |
| A = πr² = πd²/4 | m² | turn a diameter into an area |
| P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂² | each term in Pa = J/m³ | compare pressure, height and speed at two points |
| v = √(2gh) | h = depth of opening below the surface | speed of fluid leaving an opening |
| v = 0 in Bernoulli → P = P₀ + ρgh | Pa | check against the depth rule |
Assumptions behind the numbers
- The fluid is ideal: incompressible and with no viscosity, so no mechanical energy is lost.
- Pipes are completely full.
- Flow is steady: the speed at a given point does not change with time.
- For Torricelli: the tank is much wider than the hole, so the surface speed is about zero, and both the surface and the jet are at atmospheric pressure.
- g = 9.8 m/s²; water 1000 kg/m³; 1 atm = 1.0 × 10⁵ Pa.
Mistakes to avoid
- Faster means higher pressure. In a level pipe, faster means lower pressure.
- Diameter ratio used as area ratio. Square it.
- Mixing gauge and absolute pressure. Use the same kind at both points.
- Dropping the ρgy term when the points are at different heights.
- Units: cm² to m² is × 10⁻⁴; litres to m³ is × 10⁻³.
- Density in Torricelli. It cancels; the exit speed depends only on h.
- Unequal bar-chart totals. For an ideal fluid, the total P + ρgy + ½ρv² must match at every point.
Quick self-check
- Water flows into a pipe section whose diameter is three times larger. By what factor does its speed change? (It becomes 1/9 of its earlier value.)
- A small hole is 1.25 m below the open surface of a wide tank. What is the speed of the water leaving it? (4.9 m/s)
- In a level pipe, water speeds up from 2.0 m/s to 4.0 m/s. By how much does its pressure fall? (6000 Pa = 6.0 kPa)
Next: practice questions.