Buoyancy Experiment Calculator - Two-Liquid Density Solver

The buoyancy experiment calculator is a physics tool that back-solves the density of an unknown liquid when a sphere floats across the interface of two stacked liquids. You measure the ball's mass and diameter, record how deeply it sinks into the bottom liquid, and read off the top liquid density using Archimedes' principle and spherical-cap geometry.

• • Classroom buoyancy experiment: Run a saltwater-and-dish-soap demo in a high-school or first-year physics lab and recover the upper liquid's density from the measured cap height.
• • Density of an unknown household liquid: Drop a known ball into two stacked liquids, measure the cap height, and identify the density of syrup, oil, alcohol, or another fluid on hand.
• • Quality check for layered fluids: Verify that a layered sample (oil on water, brine on alcohol) sits at the densities you expect by checking the equilibrium position of a calibrated ball.
• • Homework and lab report cross-check: Compare a hand-solved two-liquid force balance against this calculator to confirm the algebra and unit conversions before turning in the assignment.

Most buoyancy experiments use one liquid, but a two-liquid version lets a single sphere float in a way that one fluid alone cannot. The calculator handles the geometry and force-balance algebra so you can focus on the measurement.

If you only know a liquid in grams per millilitre or pounds per gallon, switch to the density calculator first to convert that density into the kg per cubic metre units this calculator expects.

The calculator combines a spherical-cap volume with a static force balance across two liquids, then solves that force balance for the unknown upper liquid density. The per-liquid buoyant forces and the ball's overall density are reported as intermediates so the result is auditable.

• d: Ball diameter in metres.
• m: Ball mass in kilograms.
• rho_1: Bottom liquid density in kg per cubic metre (known).
• h: Submerged cap height in metres, measured from the bottom of the ball to the liquid interface.
• rho_2: Top liquid density in kg per cubic metre (the unknown the calculator solves for).

The first part of the calculation is geometric: the cap volume V_cap uses the closed-form spherical-cap formula, and the dome volume is V_ball minus V_cap. The second part is a force balance. At equilibrium the ball's weight m * g equals the sum of the buoyant forces from each liquid, and that equation is solved for the unknown rho_2. If the cap height equals the diameter, the top liquid does no work and the tool returns null rather than dividing by zero.

According to Wikipedia (Archimedes' principle), the buoyant force on a body equals the weight of displaced fluid, written F_b = rho * V * g.

To sanity-check the V_ball term in the closed-form spherical-cap force balance, run the diameter through the sphere volume calculator before reading the top-density result here.

Four ideas sit underneath the buoyancy experiment calculator. Read these once and the worked example above falls into place.

If the lower liquid is denser than the upper one, the sphere settles at the interface with part of itself in each. If the upper liquid were denser, the sphere would simply float on top and the two-liquid experiment would reduce to a single-liquid Archimedes solve. The interesting case is the first one, which is what this calculator targets.

For the cap and dome geometry that drives the spherical-cap concept above, the same diameter and cap height can be cross-checked against the surface-area and volume results from the sphere calculator.

Use the tool in five steps; default values reproduce a golf-ball saltwater interface so you can verify the workflow first.

1. 1 Weigh the ball: Measure the ball's mass on a scale to at least three significant figures and enter it in kilograms in the Ball Mass field.
2. 2 Measure the ball diameter: Use callipers or a precision ruler to read the diameter in metres and enter it in the Ball Diameter field.
3. 3 Prepare the two liquids: Pour the denser bottom liquid into a clear container, then gently layer the lighter top liquid on top so the interface stays sharp.
4. 4 Drop the ball and read the cap height: Lower the ball in until it floats at the interface, wait for the meniscus to settle, then measure the submerged cap height from the bottom of the ball to the interface in metres.
5. 5 Set the bottom density and compute: Look up or measure the density of the bottom liquid in kg/m^3, enter it in the Bottom Liquid Density field, and read the solved top liquid density plus the per-liquid buoyant forces.

A classroom demo with a 0.04267 m diameter, 0.04593 kg golf ball sitting in 1145 kg/m^3 saltwater with a measured cap height of 0.0127 m returns a top liquid density of about 1124.8 kg/m^3.

Once the top liquid density is known, the same fluid properties feed a moving-fluid problem through the bernoulli equation calculator if the next lab segment covers streamline energy balances instead of static buoyancy.

Reasons to reach for this tool instead of doing the spherical-cap algebra by hand.

• • One screen for the entire experiment: Ball, bottom liquid, and cap height share a panel with the per-liquid buoyant forces, so the result and the intermediate physics stay visible together.
• • Reproduces a canonical worked example: The default values match a golf ball at a saltwater-dense-liquid interface, so you can verify the workflow against a hand-solved force balance before plugging in your own numbers.
• • Handles the divide-by-zero edge cleanly: When the cap height equals the diameter, or when the inputs force the top density negative, the tool returns null and a clear reason instead of crashing or printing NaN.
• • Reveals the per-liquid buoyancy split: The buoyant force from the bottom liquid and the buoyant force from the top liquid are reported separately in newtons, so you can see how much of the weight each fluid carries.
• • Pairs with classroom fluid mechanics: The same density values flow naturally into the Bernoulli equation and Reynolds number calculators when the next topic is moving-fluid behaviour.
• • Auditable for lab reports: Every output is a closed-form evaluation of Archimedes' principle plus the spherical-cap volume, so the calculation can be repeated by hand on the same lab sheet.

The tool is intentionally narrow: it solves one specific two-liquid floating problem well, with default values that match a published reference so students can compare their hand calculation to a known answer. It does not try to model surface tension, viscosity, or temperature-driven convection, which would obscure the simple force balance the experiment is designed to demonstrate.

When the buoyancy experiment graduates into a moving-fluid discussion, the density values you just measured feed straight into the reynolds number calculator to characterise the flow regime of the same liquids.

What changes the answer this calculator returns, and what it cannot capture because the model is intentionally simple.

• • It assumes a single perfectly spherical object with uniform density. Real objects with internal voids or non-uniform composition can sit at the interface with a different effective cap height than the formula predicts.
• • Surface tension at the interface can pull the meniscus up the side of the ball and add a small extra force that the Archimedes-only balance does not capture, so the top density may read slightly off in narrow containers.

Use the tool as the backbone of a lab calculation; the formula-only answer is usually within 1 to 2 percent of the measured top density, well inside the noise of a hand-measured cap height.

As published by Wikipedia (Buoyancy), the buoyancy force on a submerged object equals the weight of the displaced fluid.

According to Wikipedia (Spherical cap), the volume of a spherical cap of height h sliced from a sphere of radius r is V = pi * h^2 * (3r - h) / 3.

When the buoyancy experiment is reframed as a sum-of-forces problem, the weight, the bottom-liquid buoyant force, and the top-liquid buoyant force all become inputs to the forces and newton's laws calculator for a second look at the same equilibrium.