Why does a steel ship float while a steel nail sinks? The answer is buoyancy — the upward push a fluid gives to anything placed in it. Push a beach ball underwater and you feel the fluid shoving it back up; that upward force is buoyancy, and it’s set entirely by how much fluid the object pushes out of the way.
📘 What you need to know
Buoyancy (upthrust) is the upward force a fluid exerts on an object partly or fully immersed in it
It arises from the fluid displaced by the object — the more fluid pushed aside, the greater the upthrust
The buoyancy force is Fb = ρVg, where ρ is the fluid’s density and V the volume displaced
An object floats when the buoyancy force balances its weight: Fb = Fg
Whether it floats or sinks comes down to density — less dense than the fluid floats, more dense sinks
At terminal velocity in a fluid, weight is balanced by buoyancy plus drag
What buoyancy is
Buoyancy is the upward force — also called upthrust — that a fluid exerts on any object partly or fully immersed in it. It exists because the object displaces fluid: when you lower a ball into a bucket, it pushes water aside, and the water pushes back up. That’s what keeps boats afloat and lets a hot-air balloon rise.
The size of the buoyancy force depends only on the fluid displaced, and is found from:
Buoyancy forceFb = ρVg
where Fb is the buoyancy force (N), ρ is the density of the fluid (kg m−3), V is the volume of fluid displaced (m3), and g is the gravitational field strength (m s−2).
The single most-tested trap here: the ρ in Fb = ρVg is the density of the fluid, not the object. A dense steel block still gets an upthrust equal to the weight of water it displaces — that upthrust just isn’t enough to hold its heavier self up. When you substitute, always ask: “whose density is this?” The answer for buoyancy is always the fluid’s.
Floating and sinking
Lower a hollow ball into water and watch what happens. As it goes under, it displaces water, and the buoyancy force pushes up on it. Let go, and if that upward push beats the ball’s weight, the ball accelerates to the surface. It rises until just enough of it pokes out of the water that the buoyancy force exactly equals its weight — then it sits there, floating.
Held under, buoyancy exceeds weight, so the released ball accelerates upward. It rises until enough of it emerges that the buoyancy force drops to exactly equal the weight — and there it floats.
At the floating point, the buoyancy force and the weight are equal and opposite. Since the weight can be written as mg and buoyancy as ρVg, the balance Fb = Fg becomes a comparison of densities: an object floats if its density is less than the fluid’s, and sinks if it’s greater. A ship floats because its overall density (steel plus all the air inside) is less than water’s.
WE 1
An object is fully submerged in water, displacing 2.0 × 10−3 m3 of it. The density of water is 1000 kg m−3. Calculate the buoyancy force on the object. Take g = 9.81 m s−2.
Step 1 — use the buoyancy equation
Fb = ρVg (ρ = water’s density)
Step 2 — substituteFb = 1000 × (2.0×10⁻³) × 9.81Fb = 19.6 NThe density here is the water’s, not the object’s — buoyancy only cares about the fluid displaced.
Drag force at terminal velocity
Buoyancy joins forces with drag when an object falls through a fluid. As a sphere sinks, three forces act on it: its weight pulling down, the buoyancy pushing up, and the viscous drag (from Stokes’ law) also pushing up and growing as it speeds up. When the drag has grown enough that the two upward forces balance the weight, the sphere stops accelerating and falls at a constant terminal velocity.
Force balance at terminal velocityW = Fd + Fb
At terminal velocity the sphere’s weight (down) is balanced by the drag and buoyancy (both up). With no net force, it falls at a constant speed.
Writing each force in full — weight as the sphere’s density times its volume times g, drag as Stokes’ 6πηrv, and buoyancy as the fluid’s density times the same volume times g — and rearranging gives a neat expression for the terminal velocity of a sphere:
Terminal velocity of a spherev = 2r2g(ρs − ρf) ÷ 9η
This shows the terminal velocity is proportional to the square of the radius and inversely proportional to the fluid’s viscosity — bigger spheres fall faster, thicker fluids slow them down.
Floating: the density rule
For a floating object, weight equals buoyancy, and this leads to a tidy result. Since only the submerged part displaces fluid, the fraction submerged equals the ratio of the object’s density to the fluid’s density — a fact that explains why most of an iceberg hides underwater.
An iceberg of density 920 kg m−3 floats in seawater of density 1025 kg m−3. What fraction of the iceberg sits above the water surface?
Step 1 — floating means weight = buoyancy
this gives: fraction submerged = ρice ÷ ρseaStep 2 — find the submerged fraction= 920 ÷ 1025 = 0.898Step 3 — the rest is above water1 − 0.898 = 0.102about 0.10, or 10% above waterRoughly nine-tenths of an iceberg is hidden below the surface — the origin of “tip of the iceberg.”
💡 Top tips
ρ is the fluid’s density in Fb = ρVg — never the object’s.
V is the volume displaced. For a fully submerged object that’s its whole volume; for a floating one, only the submerged part.
Float vs sink is a density comparison — less dense than the fluid floats, more dense sinks.
At terminal velocity, weight = drag + buoyancy. Don’t forget the buoyancy term when working in a liquid.
Quick recap: buoyancy is the upthrust from displaced fluid, Fb = ρVg, using the fluid’s density and the volume displaced. An object floats when buoyancy balances weight, which happens when it’s less dense than the fluid. Falling through a fluid, an object reaches terminal velocity when weight = drag + buoyancy.
⚠ Common mistakes
Using the object’s density instead of the fluid’s in Fb = ρVg
Taking V as the whole volume for a floating object — only the submerged part displaces fluid
Forgetting the buoyancy term in the terminal-velocity balance when the fluid is a liquid
Thinking a heavy object gets no upthrust — it does, just not enough to float
Confusing the fraction submerged with the fraction above — subtract from 1 for the part above water
You’ve now worked with momentum’s cousins — the forces that push, pull, and hold objects in fluids. Next the topic pivots from forces to momentum itself: what it is, why it’s conserved, and how it governs everything from collisions to rocket launches. Up first: Conservation of Linear Momentum.
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