Archimedes' Principle Simulation

Archimedes’ Principle —

What it says
A body wholly or partly immersed in a fluid experiences an upward force, called the buoyant force F_b, equal to the weight of the fluid it displaces.

Key relations
• Buoyant force: F_b = ρ_fluid g V_displaced
• Weight of object: W = m g
• Apparent weight in a fluid: W_app = W − F_b

Float or sink — the density test
• Float: if ρ_object < ρ_fluid, the object floats. At equilibrium F_b = W and the submerged fraction = ρ_object / ρ_fluid (as a percent, multiply by 100%).
• Sink: if ρ_object > ρ_fluid, the object cannot reach F_b = W before it is fully submerged, so it sinks. When fully submerged, F_b = ρ_fluid g V_object, but F_b < W.

Nuances (common pitfalls students have)

1. “Heavier” is not the criterion — “denser” is. A heavy block of Styrofoam floats; a small iron bead sinks in water.
2. Gravity g cancels in the float/sink decision. Changing g scales both W and F_b together, so the outcome (float or sink) is set by densities. However, the actual force values and the apparent weight do change with g.
3. Shape does not enter F_b directly — only displaced volume matters. Shape and orientation can affect stability and how much volume is submerged at a given tilt, but Archimedes’ relation itself depends on V_displaced.
4. Trapped air changes average density. Ships float because “steel + lots of air” gives ρ_object (average) lower than the fluid.
5. Fluid density matters. Seawater (≈1025 kg/m³) is denser than freshwater (≈1000 kg/m³), so objects float slightly higher in seawater. Mercury is far denser (≈13 600 kg/m³), so even iron can float in it.
6. Surface tension and very small objects: Archimedes’ principle still holds, but for tiny or very light things (paper clips on water), surface tension forces can dominate; we neglect that here.
7. Compressibility/temperature: We treat fluids as incompressible and uniform. In reality, ρ_fluid changes with temperature/salinity; so does ρ_object if it contains air.

How to use the simulation (quick steps)

1. Pick the fluid (Gasoline, Oil, Water, Seawater, Honey, Mercury). The “Fluid Density” slider will match the choice, but you can fine-tune it.
2. Choose an object material (sets ρ_object).
3. Set the object mass m. The simulation keeps density fixed and adjusts volume V_object = m / ρ_object, so the block gets visually bigger for larger m.
4. Set gravity (Moon, Earth, Jupiter or via the slider). This changes the magnitudes of W, F_b, and W_app but not the float/sink decision.
5. Watch the block position, the force arrows, and the readouts update in real time. The water level also rises slightly to represent displaced fluid.

How to read the numbers you see

• “Object Weight” is W = m g.
• “Buoyant Force” is F_b = ρ_fluid g V_displaced.
• “Apparent Weight” is W_app = W − F_b (what a scale would “feel” if it supported the object under the fluid).
• “% Submerged” equals ρ_object / ρ_fluid × 100% when floating; if the object sinks, this reads 100%.

Guided mini-investigations (great for quick checks)
• Ice in water: pick Ice (ρ≈917 kg/m³) and Water (ρ≈1000 kg/m³). Predict ≈91.7% submerged; check the readout.
• Freshwater vs seawater: keep the object the same and switch Water → Seawater; the % submerged should go down (it floats higher).
• Iron in mercury: choose Iron, then Mercury. Even though iron is “heavy,” ρ_iron < ρ_mercury, so it floats.
• Gravity sweep: keep materials fixed and toggle Moon → Earth → Jupiter. Notice “Floats/Sinks” status stays the same, but N-values and apparent weight change with g.
• Sinker sanity check: choose PVC or Brick in Water. It sinks. Note that at full submersion F_b = ρ_fluid g V_object; compare this to W = m g to see why W_app stays positive.