Archimedes' Principle: Density of a Solid (Floating Sphere)
A mystery sphere floats in water of known density. Read the submerged volume fraction at equilibrium and estimate the sphere’s density from ρ = f·ρ_w, averaged over noisy readings.
Goal
Determine the unknown mass density of a homogeneous solid sphere that floats in fresh water (ρ_w = 1000 kg/m³) using the equilibrium relation ρ/ρ_w = V_submerged / V_total.
Equipment
- Tank of fresh water (ρ_w given)
- Homogeneous mystery sphere
- Scale reading of submerged fraction (simulated)
Theory
For a floating body, weight equals buoyant force: ρ g V = ρ_w g V_sub, hence ρ = ρ_w (V_sub / V). For a uniform sphere, the submerged fraction f = V_sub / V is read from the equilibrium waterline. Random reading error is modelled on f.
Procedure
- The lab uses fresh water with ρ_w = 1000 kg/m³ (shown on the card). The sphere’s density is fixed but hidden from you.
- Inspect the equilibrium: the sphere floats with part of its volume submerged.
- Press “Record reading” to add a row: the simulator estimates the submerged fraction f from the waterline (with small sensor noise).
- Repeat at least 5 times to average out random error.
- The table lists f and ρ = ρ_w·f. Take the sample mean of ρ as your result.
- Compare with the reference density and write the conclusion.
Experiment
Conclusion
The mean density agrees with the reference value within tolerance. Main uncertainties: reading the submerged fraction, assuming pure fresh water, and treating the sphere as perfectly homogeneous.