Density and upthrustEdexcel A-Level Physics: Revision notes

Density is the mass per unit volume of a substance:

$\rho = \frac{m}{V}$

where ρ is in kg m⁻³, m in kg and V in m³. Density depends on the material and not on the size of the sample, so cutting an object in half halves both m and V and leaves ρ unchanged.

Unit conversions matter: 1 g cm⁻³ = 1000 kg m⁻³, and 1 cm³ = 10⁻⁶ m³. Water has a density of about 1000 kg m⁻³.

Worked example: a block of 0.135 kg and volume 5.0 × 10⁻⁵ m³ has ρ = 0.135 / 5.0 × 10⁻⁵ = 2700 kg m⁻³.

When an object is placed in a fluid (a liquid or a gas), the fluid exerts an upward force on it called upthrust. It arises because pressure in a fluid increases with depth, so the fluid pushes up on the lower surface of the object with a greater force than it pushes down on the upper surface. The resultant of these forces is directed upwards.

The size of the upthrust is given by Archimedes' principle: the upthrust on an object equals the weight of fluid it displaces.

$U = \rho_{fluid} V_{displaced}\, g$

The fluid displaced has volume equal to the submerged volume of the object. A fully submerged object displaces its whole volume; a floating object displaces only the part below the surface.

Worked example: a sphere weighs 3.60 N in air and reads 2.25 N when fully in water. The upthrust is 3.60 − 2.25 = 1.35 N, so V = 1.35 / (1000 × 9.81) = 1.4 × 10⁻⁴ m³. Its density is (3.60 / 9.81) / 1.38 × 10⁻⁴ ≈ 2.7 × 10³ kg m⁻³.

The motion of an object in a fluid depends on the balance between its weight W and the upthrust U:

• U = W: equilibrium. The object floats at rest on the surface, or hovers at constant depth if fully submerged.
• W > U when fully submerged: a downward resultant force, so the object sinks.
• U > W when fully submerged: an upward resultant force, so the object rises until it floats.

A floating object displaces just enough fluid to have a weight equal to its own weight. An object denser than the fluid cannot float, because even fully submerged the upthrust is smaller than its weight. A submarine changes its weight by flooding or emptying ballast tanks, while its volume, and so its maximum upthrust, stays fixed.