Density and Upthrust

Density is mass per unit volume, $\rho =\frac{m}{V}$, in $\mathrm{kg}{\mathrm{m}}^{-3}$. It is a property of the material, independent of the size of the sample.

Unit conversion is the usual trap. $1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}=1000\mathrm{kg}{\mathrm{m}}^{-3}$, because a cubic metre is a million cubic centimetres while a kilogram is only a thousand grams. Water is about $1000\mathrm{kg}{\mathrm{m}}^{-3}$, steel about $7800$, air about $1.2$.

For a regularly shaped solid, measure dimensions and mass. For an irregular one, measure the volume of water it displaces — which is exactly the idea behind upthrust.

Pressure in a fluid increases with depth, because the fluid above has weight: $\Delta p=\rho g\Delta h$.

Consider a block of height $h$ and base area $A$ fully immersed. The fluid pushes down on its top and up on its bottom. The bottom is deeper by $h$, so the pressure there is greater by ${\rho }_{f}gh$, and the net upward force is that pressure difference times the area: ${\rho }_{f}ghA$.

But $hA$ is the block’s volume, and ${\rho }_{f}V$ is the mass of fluid that volume would contain. So the upthrust equals ${\rho }_{f}Vg$ — the weight of the fluid displaced. The sideways pressures cancel in pairs and contribute nothing.

$\text{Archimedes’ principle}$ states that the upthrust on an object in a fluid equals the weight of fluid it displaces. The derivation above shows why for a simple block, and the result holds for any shape.

For a fully submerged object the displaced volume is its whole volume. For a floating object it is only the submerged part.

A spring balance reading drops when an object is lowered into water, and the drop is exactly the upthrust — which is how the principle is checked experimentally.

An object floats when it can displace a weight of fluid equal to its own weight before becoming fully submerged. In equilibrium, upthrust equals weight.

Setting ${\rho }_f{V}_{\text{sub}}g={\rho }_{o}Vg$ gives the fraction submerged directly: it equals the ratio of the object’s density to the fluid’s. An object less dense than the fluid floats with that fraction below the surface; denser than the fluid, the fraction would exceed 1, so it sinks.

That is how a steel ship floats: its hull encloses a large volume of air, so its $\text{average}$ density — total mass over total enclosed volume — is below that of water.