09 · 16 min
Density and pressure
Density, pressure on a surface, and pressure in a liquid.
P = F / A
The idea
Density is mass per unit volume, ρ = m / V. Water is 1 g/cm³, which is 1 000 kg/m³. To convert, multiply g/cm³ by 1 000. A 60 cm³ volume is 60 × 10⁻⁶ m³, because 1 m³ = 10⁶ cm³.
Pressure is force per unit area, P = F / A, in pascals. One pascal is one newton per square metre. The same force on a smaller area gives a larger pressure, which is why a drawing pin is easy to push in point-first.
In a still liquid the extra pressure due to depth is P = hρg. It acts in all directions and does not depend on the width of the container. This is the pressure on top of atmospheric pressure, which is about 100 kPa and is often left out when a question says “extra pressure”.
Formulas
Density
ρ = m / V
Keep mass and volume in a matching pair: kg with m³, or g with cm³.
- ρ
- density
- kg/m³
- m
- mass
- kg
- V
- volume
- m³
Pressure
P = F / A
Force perpendicular to the area. A larger area means a smaller pressure for the same force.
- P
- pressure
- Pa
- F
- force
- N
- A
- area
- m²
Pressure in a liquid
P = hρg
Extra pressure at depth h, not including the atmosphere, unless the question adds it.
- P
- pressure due to the liquid
- Pa
- h
- depth
- m
- ρ
- density
- kg/m³
- g
- gravitational field strength
- N/kg
Worked examples
Density of a stone
A stone has mass 0.15 kg and volume 60 cm³. Find the density in g/cm³ and in kg/m³.
- Mass
- 0.15 kg = 150 g
- Volume
- 60 cm³
Try it on paper first. Then open the working one step at a time.
Pressure under a block
A block weighs 40 N. Its base is 20 cm by 10 cm. Find the pressure on the bench.
- Force
- 40 N
- Base
- 0.20 m by 0.10 m
Try it on paper first. Then open the working one step at a time.
Checks
A sample has mass 250 g and volume 100 cm³. What is its density, in g/cm³?
A force of 500 N acts on an area of 0.25 m². What is the pressure, in pascals?
A diver is 8.0 m underwater. Take the density of water as 1 000 kg/m³ and g = 10 N/kg. What extra pressure does the water exert, in pascals?
Pressure at a point in a still liquid
Easy places to lose a mark
- Mixing g/cm³ with m³, or cm with m, in the area.
- Using the diameter where the area needs the radius, or forgetting A = length × width for a rectangle.
- Adding atmospheric pressure when the question asks only for the pressure due to the liquid.
Practical
Pressure pad
Compare pressure from a force on an area with pressure from a depth of liquid.