§ 7. Archimedes' Principle & Flotation Conditions (Grade 7)
Buoyant force F_A = ρgV and equilibrium at the waterline. A body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the submerged portion of the body.
Theory Summary & Scientific Mechanism
Hydrostatic pressure in a stationary fluid of density ρ_fluid increases linearly with depth: p(h) = p₀ + ρ_fluid g h. Because the upward pressure on the bottom surface of a submerged body exceeds the downward pressure on its top, integrating pressure forces over the body’s surface yields an upward Archimedean buoyant force F_A.
By Archimedes’ principle, the buoyant force equals the weight of the displaced fluid: F_A = ρ_fluid g Vₛub. Vertical motion of a body of total volume V and density ρ_body is governed by the interplay between weight mg = ρ_body g V, buoyancy F_A, and fluid drag.
Three regimes emerge from the density ratio: 1) if ρ_body ρ_fluid, it sinks to the bottom with apparent weight Wₐpp = (ρ_body - ρ_fluid)gV.
Everyday Analogy
Why does a massive steel ship or cork float while a small pebble sinks? Water pushes harder on the bottom of an object than on the top, buoying it upward with a force equal to the weight of the displaced water!
Interactive Question & Derivation
A solid cube of density ρ_body = 600 kg/m³ is placed into fresh water (ρ_water = 1000 kg/m³). What fraction of the cube's volume will be submerged in equilibrium?
- A: Exactly half of its volume (50%), regardless of the body's density
- B: The submerged volume fraction equals the density ratio: ρ_body/ρ_water = 60%
- C: The cube will sink completely to the bottom of the vessel (100%)
Study Hint: Compare the body density ρ_body with the fluid density ρ_fluid: when does Archimedes’ buoyant force F_A balance gravity mg?
Proof & Explanation: In static equilibrium of a floating body, gravity is balanced by the Archimedes buoyant force: ρ_body g V₀ = ρ_fluid g Vₛub. Hence the submerged fraction is Vₛub/V₀ = ρ_body/ρ_fluid = 600/1000 = 0.6 (60%). If ρ_body > ρ_fluid, the body sinks.
Key Equations & 3D Simulation Parameters
ρ_body g V₀ = ρ_fluid g Vₛub ⇒ (hₛub)/(H) = (ρ_body)/(ρ_fluid)
- Body density ρ_body: 200 … 1800 kg/m³ (default: 600 kg/m³)
- Fluid density ρ_fluid: 700 … 2000 kg/m³ (default: 1000 kg/m³)
- Cube side length a: 0.8 … 1.6 m (default: 1 m)
- Fluid viscosity η: 0.1 … 1.2 Pa·s (default: 0.35 Pa·s)