Density, Pressure and Upthrust

Density — mass per unit volume:

ρ = m / V

Measured in kg m⁻³. Water is 1000 kg m⁻³, which makes it a useful reference for checking whether an answer is sensible.

Pressure — force per unit area, acting normally to the surface:

p = F / A

Measured in pascals, where 1 Pa = 1 N m⁻². Pressure explains why the same force can be harmless or destructive: a drawing pin has a tiny tip area, so a modest force produces enormous pressure.

Pressure in a fluid — a column of fluid of height h exerts a pressure at its base of

p = ρgh

Note what this does not contain: the area of the column, or its shape. The pressure at a given depth depends only on the depth, the density and g, which is why the water pressure at the bottom of a narrow pipe and a wide lake of the same depth is identical.

Example: at 3.0 m depth in water, p = 1000 × 9.81 × 3.0 = 2.9 × 10⁴ Pa. That is on top of atmospheric pressure, about 1.0 × 10⁵ Pa, so the total pressure is roughly 1.3 × 10⁵ Pa.

Upthrust — the net upward force on a body immersed in a fluid. It arises because pressure increases with depth, so the fluid pushes up on the bottom of the body harder than it pushes down on the top.

Archimedes' principle — the upthrust equals the weight of the fluid displaced by the body.

Floating and sinking — an object floats if it can displace a weight of fluid equal to its own weight before it is fully submerged. A steel ship floats not despite being steel but because its hull encloses a large volume of air, so its average density is well below that of water. Solid steel of the same mass would sink immediately.