4.3.3Dynamics, forces and pressure
Derivation of Pressure Difference (Delta p = rho g Delta h) – A Level Physics 9702
5 min read 9 parts Lesson 20 of 24 1 quiz
Lesson summary
4.3.3 Derivation of Pressure Difference (Δp = ρgΔh) — AS PHYSICS 9702 notes: Derive Δp = ρgΔh and use it for hydrostatic pressure, total pressure and U-tubes.Consider a vertical column of liquid at rest with uniform cross-sectional area A, height Δh and constant density ρ.The vol…
- The pressure difference depends only on density, field strength and vertical height
- The pressure at the bottom of a volume is greater than at the top
- Total pressure at depth h
- Atmospheric pressure (barometric pressure) is 101 325 Pa
- The fluid is incompressible
- Use the vertical depth
Section 1 of 9Deriving Δp = ρgΔh
1
Consider a vertical column of liquid at rest with uniform cross-sectional area A, height Δh and constant density ρ.
2
The volume of the column is V = AΔh.
3
Its mass is m = ρV = ρAΔh, using density = mass ÷ volume.
4
Its weight is W = mg = ρAΔhg, the extra downward force on the area A at the bottom.
5
Pressure is force per unit area, so Δp = F ÷ A = ρAΔhg ÷ A.
6
The area cancels, giving Δp = ρgΔh.
Section 2 of 9Meaning of the result
1Hydrostatic pressure
Hydrostatic pressure the pressure at a point in a fluid exerted by the weight of the fluid above that point.
2
The pressure difference depends only on density, field strength and vertical height
- not on container shape or cross-sectional area.
3
Points at the same depth in the same continuous liquid are at the same pressure.
4
The pressure acts equally in all directions at a point in a fluid at rest.
5The pressure at the bottom of a volume is greater than at the top
- The pressure at the bottom of a volume is greater than at the top more fluid lies above it.
6Units
- ρ in kg m-3, g in N kg-1 and Δh in m give Δp in Pa.
Section 3 of 9Total pressure
Total pressure
Total pressure at depth h
Total pressure at depth h atmospheric pressure + hydrostatic pressure: p = patm + ρgh.
The pressure depends on the density of the liquid, the depth and the gravitational field strength.
Atmospheric pressure (barometric pressure) is 101 325 Pa
- Atmospheric pressure (barometric pressure) is 101 325 Pa about 1.0 × 105 Pa.
Include atmospheric pressure whenever total or absolute pressure is asked for.
Convert kPa and MPa to Pa before calculating.
Section 4 of 9Assumptions
Assumptions
The fluid is incompressible
- The fluid is incompressible with constant density.
The gravitational field strength g is uniform.
Use the vertical depth
- Use the vertical depth not the length of a pipe.
Section 5 of 9U-tube manometer
A manometer measures pressure differences using a liquid in a U-tube.
Equal levels
- Equal levels the pressure on both sides is the same, atmospheric pressure.
If the pressure on one side rises
- If the pressure on one side rises the liquid is forced down on that side and rises in the other limb.
The difference between the two levels
- The difference between the two levels gives the pressure difference between the ends of the tube.
With two immiscible liquids
- With two immiscible liquids the denser liquid stands at the smaller height,
- because pressures at the same level in the same liquid are equal: ρ1gh1 = ρ2gh2.
Section 6 of 9Upthrust from the pressure difference
1
A submerged object has greater pressure on its lower surface than on its upper surface
- giving a net upward force.
2Upthrust is a force proportional to the pressure difference
- F = (pbottom − ptop)A.
3It is much larger in liquids than gases
- It is much larger in liquids than gases liquids are denser.
4
Upthrust depends on the pressure difference and the fluid, not on the density of the object.
Section 7 of 9More from the specification
More from the specification
Derivation of hydrostatic pressure
- Considering a fluid cylinder of area A and height h, downward force F = mg = (rho * A * h) * g, yielding pressure difference delta p = F / A = rho * g * delta h.
Archimedes' principle and buoyancy
- The buoyant upthrust on a submerged body equals the weight of the displaced fluid, deriving directly from hydrostatic pressure differences.
Depth and density dependence
- Fluid pressure depends solely on depth, fluid density,
- and gravitational field strength, acting equally in all spatial directions at a given depth.
Newton
- Unit of force, kg m s-2.
One newton is the force that gives a mass of 1 kg an acceleration of 1 m s-2.
What the symbols mean:
Weight is a force,
- so it is measured in newtons: W = mg.What the symbols mean: W = weight (N) · m = mass (kg) · g = gravitational field strength (9.81 N kg−1 on Earth)
Pressure is force per unit area, so 1 pascal = 1 newton per square metre (1 Pa = 1 N m-2).
Section 8 of 9Key terms
| Term | Meaning |
|---|---|
| Hydrostatic pressure | The pressure in a fluid due to the weight of the fluid above. |
| Pressure difference | The change in pressure between two depths, Δp = ρgΔh. |
| Atmospheric pressure | The pressure due to the weight of the air above, 101 325 Pa. |
| Total pressure | Atmospheric pressure plus hydrostatic pressure. |
| Manometer | A U-tube instrument that measures pressure differences. |
| Incompressible fluid | A fluid whose density does not change with pressure. |
| Immiscible | Unable to mix, as two liquids that form separate layers. |
| Depth | The vertical distance below the free surface of a liquid. |
Section 9 of 9Exam technique
1
Using the length of the pipe instead of the vertical depth in ρgh.
2
Forgetting atmospheric pressure when asked for total or absolute pressure.
3
Thinking pressure depends on the shape or width of the container.
4
Thinking upthrust depends on the density of the object rather than the fluid displaced.