Hook scale 0 gPlatform 0 kgWater level 0 cmBuoyancy 0 NSubmerged 0 %loadingFPS 0

Drag to orbit, wheel or pinch to zoom. Arrows: white weight, blue buoyancy, green string tension, red drag.

Archimedes Crown Test: Hydrostatic Weighing, Sinking, Floating and Stokes Drag Lab

Hang a body from a spring balance, lower it into a liquid and watch the reading drop by exactly the weight of the displaced liquid. Cut the string and the same body sinks, rises or floats.

Body (changing any of these resets the run)

Material

Shape

Liquid and world

Liquid

Gravity

Spring balance and hook

Hook speed

Balance resolution

Live readouts

Balance reading against immersion depth (dots: flown, line: closed form)

Forces on the body (last 10 s, newtons)

Experiment: how much gold is in the crown?

A 1.000 kg crown of gold and silver is weighed in air and in the chosen liquid by the same hanging-balance model, seven alloys from pure gold to 40 % gold. The density follows from m/(m - m_w) times the liquid density, and the gold fraction from the density. Try a coarser balance resolution: the 10 g balance cannot tell pure gold from a 90 % alloy reliably.

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom.
With the string attached, move the hook slider or press Lower/Raise. The balance reads the string tension; a floating body slackens the string and the reading falls to zero.
Detach the string to let the body go. Changing material, shape, size, liquid, gravity or water level starts a new run.
Hook speed and balance resolution apply immediately.

The science

Archimedes' principle. A body immersed in a fluid is pushed up by a force equal to the weight of the fluid it displaces: F_b = ρ_f g V_sub. A hanging balance therefore reads T = m g - ρ_f g V_sub. Weighing in air and in liquid gives the density without measuring the volume: ρ_body = ρ_f W_air / (W_air - W_liq). This is the method attributed to Archimedes for the crown of Hiero (the story is a later legend).

Alloy. For gold mass fraction x and silver for the rest, with volumes assumed additive, 1/ρ = x/ρ_Au + (1 - x)/ρ_Ag, ρ_Au = 19 320 and ρ_Ag = 10 490 kg/m³. Real alloys deviate by under about 1 %, and a hollow or lead-cored crown would fool a density test, which is why measuring resolution matters.

Immersion. The submerged volume of a vertical prism is area times depth; for a sphere, the cap V = π d² (3R - d)/3. Displaced liquid raises the tank level by V_sub/A_tank (solved implicitly each step) and overflows beyond 0.50 m. The platform balance under the tank rises by F_b/g: the body pushes down on the liquid with the buoyancy it receives (Newton's third law); this is a quasi-static balance, not a dynamic force reading.

Dynamics. The vertical motion obeys (m + C_a ρ_f V_sub) z'' = -m g + F_b + T + F_d integrated with RK4 at 2 kHz. The spring balance has stiffness 3000 N/m and the string only pulls. The drag is F_d = -½ ρ_f C_D(Re) A |v| v times the submerged fraction, with the sphere law C_D = 24/Re (1 + 0.15 Re^0.687) + 0.42/(1 + 42500 Re^-1.16) and a fixed shape multiplier (cube 2.2, cylinder 1.8, boat 2.5). In the creeping-flow limit this gives Stokes' law, terminal speed 2 (ρ_s - ρ_f) g r² / (9 μ). A floating prism has heave period 2π √((m + C_a ρ_f V_sub)/(ρ_f g A / (1 - A/A_tank))); the factor in A/A_tank is the level rise of the tank.

Simplifications and typical values. Densities and viscosities are typical room-temperature values (check against a table before using them for work). Added-mass coefficients and drag multipliers are rough, the free-surface slam and waves are cosmetic, bodies stay upright (no metacentric stability), the hollow boat is a box with a chosen average density that sinks without flooding detail, and the crown is treated as an equivalent cylinder for partial immersion (full immersion depends only on its volume). Mercury is included as a textbook liquid only. Models built with Blender, scene by Three.js.