Floating, Sinking & Apparent Weight
- Derive the fraction of a floating object that is submerged from a density ratio
- Compute apparent weight for a submerged object as a force balance
- Predict how a fluid's density changes whether an object floats, sinks or hovers
Buoyancy is a pressure difference, not a special force
Pressure in a fluid increases with depth, so the bottom of a submerged object is pushed up harder than the top is pushed down. That imbalance is the buoyant force, and integrating it gives Archimedes' result: F_b = ρ_fluid V_displaced g — the weight of the fluid pushed aside. Two consequences follow directly. The buoyant force does not depend on the object's own density or what it is made of, only on how much fluid it displaces. And it does not depend on depth, since the pressure difference between top and bottom is the same at any depth.
Float, sink or hover — the density comparison
Fully submerge an object and compare its average density with the fluid's. If ρ_object < ρ_fluid, the buoyant force exceeds the weight and the object rises until part of it is out of the fluid — it floats. If ρ_object > ρ_fluid, weight wins and it sinks. If they are equal, the forces balance at any depth and the object is neutrally buoyant, hovering wherever it is placed. "Average density" is the key word: a steel ship floats because the hull encloses air, and the ship-plus-air average is well below water's.
The floating fraction is a pure density ratio
For a floating object the buoyant force exactly balances the weight: ρ_f V_sub g = ρ_o V_total g. Cancel g and rearrange and everything but the densities disappears: V_sub / V_total = ρ_object / ρ_fluid. So the submerged fraction is nothing but the density ratio. Ice at 917 kg/m³ in water at 1000 kg/m³ floats with 91.7% of its volume below the surface — the origin of "the tip of the iceberg", and a result that holds for any size or shape of iceberg.
A wooden block of density 600 kg/m³ and volume 0.020 m³ floats in water. What fraction is submerged, and what is the submerged volume?
- 1.Floating means buoyant force equals weight: ρ_w V_sub g = ρ_wood V_total g.
- 2.Cancel g and solve for the ratio: V_sub / V_total = 600 ÷ 1000 = 0.60.
- 3.So 60% of the block sits below the waterline.
- 4.Submerged volume: (0.60)(0.020) = 0.012 m³.
Apparent weight is just the force balance
Hang an object from a scale and lower it into fluid. The scale now reads the apparent weight: the tension needed, which is the true weight minus the buoyant force, W_apparent = mg − F_b. This is the standard laboratory method for measuring density — weigh the object in air, weigh it submerged, and the difference is the buoyant force, from which the volume and then the density follow. It is also what makes a heavy stone easy to lift underwater and abruptly heavy the moment it clears the surface.
Buoyant force does not increase with depth. A sunken object 100 m down experiences the same buoyant force as it did at 1 m, because the pressure on both its top and its bottom rose by the same amount. What increases with depth is the absolute pressure, which is a different question.
An object of density 800 kg/m³ floats in water (1000 kg/m³). What fraction of its volume is above the surface?
A 5.0 kg object of volume 0.0020 m³ is fully submerged in water. What does a scale supporting it read? (g = 10 m/s², ρ_water = 1000 kg/m³)
A block floats in water. It is then placed in a denser salt solution. Compared with before, the submerged fraction:
On buoyancy free-responses, start by drawing a free-body diagram with weight down and buoyant force up, plus tension or a normal force if present. Every buoyancy question in this unit is a force balance, and the diagram earns its own rubric point.
Answer the 3 checkpoints as you read.
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