Pressure in a liquid: p = hρg
Pressure due to a column of liquid = height of column × density of liquid × gravitational field strength. In symbols, p = h ρ g. p is in pascals (Pa), h is in metres (m), ρ is in kg/m³ and g is in N/kg. AQA, Edexcel and both OCR physics courses give it on the exam sheet as a Higher tier equation. The AQA and Edexcel combined science courses leave it out.
Equation
p = h ρ g
| Symbol | Quantity | Unit |
|---|---|---|
p | pressure | Pa |
h | height | m |
ρ | density | kg/m³ |
g | gravitational field strength | N/kg |
What each specification says
| Exam board | Status | Equation |
|---|---|---|
| AQA Physics (8463) | Given on the sheet · Higher tier only | p = h ρ g |
| AQA Combined Science (8464) | Not in this specification | |
| Edexcel Physics (1PH0) | Given on the sheet · Higher tier only | P = h × ρ × g |
| Edexcel Combined Science (1SC0) | Not in this specification | |
| OCR Gateway Physics A (J249) | Given on the sheet · Higher tier only | p = h ρ g |
| OCR Twenty First Century Physics B (J259) | Given on the sheet · Higher tier only | p = ρ g h |
Calculator: p = h ρ g
Why pressure grows with depth
The deeper you go in water, the more liquid sits above you. All that liquid has weight, and it presses down on everything below. So the pressure grows with depth.
You can feel it in your ears at the bottom of a swimming pool. Dams are built thicker at the base than at the top for the same reason: the water pushes hardest at the bottom.
A denser liquid gives more pressure at the same depth. A metre of sea water pushes a little harder than a metre of fresh water, and a metre of mercury pushes far harder. The shape and width of the container do not matter; only the depth, the density and g do.
Which courses include it
- AQA Physics (8463): p = h ρ g. Higher tier. Combined Science Trilogy (8464) does not have it.
- OCR Gateway Physics A (J249): p = h ρ g. Higher tier.
- OCR Twenty First Century Physics B (J259): p = ρ g h, with depth last. Higher tier.
- Edexcel Physics (1PH0): given on the sheet, Higher tier only (point 15.14P). Edexcel Combined Science (1SC0): not included.
Where it appears, it is on the list of equations you select and apply. AQA and OCR put it on the sheet in the exam, along with every other physics equation in their courses. The OCR Gateway list and the AQA list show it in context.
The examples below use g = 9.8 N/kg. Your exam question states the value it wants, so read it before you start.
Related equations
The answer from p = h ρ g is the extra pressure caused by the liquid. The air above the water pushes down too, so the total pressure at the bottom of a lake is this value plus the pressure of the atmosphere.
If a question gives the mass and volume of the liquid instead of its density, find ρ first with the density equation. To turn a pressure into a force on a window or a hatch, use p = F/A.
Worked examples
p
- p = h ρ g
- p = 2.5 × 1000 × 9.8
- p = 24 500 Pa
24 500 Pa
p
- Change the depth to metres: 30 cm = 0.3 m
- p = h ρ g
- p = 0.3 × 920 × 9.8
- p = 2704.8 Pa
- p = 2700 Pa (3 s.f.)
2700 Pa
h
- Change the pressure to pascals: 147 kPa = 147 000 Pa
- h = p ÷ (ρ g)
- h = 147 000 ÷ (1000 × 9.8)
- h = 147 000 ÷ 9800
- h = 15 m
15 m
Common mistakes
- Measuring h from the bottom of the tank. The depth is measured down from the surface of the liquid to the point you care about.
- Using a depth in centimetres. A 30 cm column entered as 30 gives a pressure 100 times too big.
- Using a density in g/cm³. Water is 1 g/cm³, which is 1000 kg/m³, and the equation needs the second one.
- Adding in air pressure when the question asks only for the pressure due to the liquid.
Solve p = hρg for h or ρ in the Equation Rearranger.
Questions people ask
Does a wide tank give more pressure than a narrow one?
No. Two tanks filled to the same depth with the same liquid have the same pressure at the bottom. The wide one holds more water, but it spreads that weight over a bigger floor.
Why does J259 write ρgh and the others hρg?
The order of the letters is different, but multiplication gives the same answer in any order. Write it the way your own board prints it.