To find the pressure caused by a liquid, use p = ρ × g × h, where ρ (rho) is the density of the liquid, g is the gravitational field strength and h is the vertical depth below the surface. It appears in questions about dams, divers, tanks and submarines.
This lesson follows relating pressure to force and area and sits inside pressure in solids and fluids.
What does each symbol mean?
- p is the pressure from the liquid, in pascals (Pa).
- ρ is the density of the liquid, in kg/m³. Fresh water is about 1000 kg/m³.
- g is the gravitational field strength. Use the value stated in the question, often 10 N/kg.
- h is the vertical depth from the surface to the point, in metres.
The equation works because the liquid above a point has weight, and that weight presses down on each square metre. A deeper point has more liquid above it.
How to work through a question
- Identify the liquid and its density. Convert g/cm³ to kg/m³ by multiplying by 1000 if needed.
- Find the vertical depth. Measure straight down from the surface, not along a slope or a tube.
- Write p = ρgh and substitute with units.
- Decide what is being asked. If the question wants the force on an object, use F = p × A afterwards.
- State the unit and check that it is sensible.
Worked example
A tank holds fresh water to a vertical depth of 1.8 m. Find the pressure caused by the water at the floor. Use ρ = 1000 kg/m³ and g = 10 N/kg. (Invented example data.)
Step 1, equation: p = ρgh.
Step 2, substitute: p = 1000 × 10 × 1.8.
Step 3, calculate: 1000 × 10 = 10 000, and 10 000 × 1.8 = 18 000.
Answer: p = 18 000 Pa, or 1.8 × 10⁴ Pa.
If the floor has a hatch of area 0.050 m², the force on it is F = 18 000 × 0.050 = 900 N.
The mistake to watch for
A common slip is to use the length of a slanting tube instead of the vertical depth.
Mistaken answer: oil of density 800 kg/m³ fills a tilted tube. The oil column along the tube is 1.0 m long, but its top is 0.50 m higher than the bottom. The student uses h = 1.0 m and gets 8000 Pa.
Pressure depends on vertical height only, so the student used the wrong h.
The correction is h = 0.50 m: p = 800 × 10 × 0.50 = 4000 Pa. Always draw a vertical line from the surface to the point and use that length.
Check yourself
Use g = 10 N/kg.
1. Mercury has density 13 600 kg/m³. Find the pressure caused by a mercury column 0.76 m high.
Show answer
p = ρgh = 13 600 × 10 × 0.76 = 13 600 × 7.6 = 103 360 Pa. To 3 significant figures, 1.03 × 10⁵ Pa.
2. The pressure caused by fresh water at a point is 30 000 Pa. How deep is the point? (ρ = 1000 kg/m³)
Show answer
h = p ÷ (ρg) = 30 000 ÷ (1000 × 10) = 30 000 ÷ 10 000 = 3.0 m
3. A narrow tube and a wide lake both hold fresh water. A point in each is 2.0 m below the surface. Compare the pressures.
Show answer
Both are p = 1000 × 10 × 2.0 = 20 000 Pa. They are equal, because the liquid, the depth and g are the same.
Where this leads next
Next, turn the equation into a written argument in explaining a pressure difference with depth. Once you meet a U-tube, see manometer-style diagrams.
Students who can substitute into p = ρgh still lose marks on depth and units under exam conditions. Our teachers look at that kind of pattern in online one-to-one Physics tuition.