Drop something into water and the water pushes back up on it — that upward push is buoyancy (or upthrust). It’s what keeps boats afloat and lets balloons rise, and its size depends only on the fluid and the volume displaced.
📘 What you need to know
Buoyancy is the upward force on a body that is partly or fully immersed in a fluid, caused by the fluid it displaces.
Equation: Fb = ρVg.
ρ = density of the fluid (kg m⁻³), V = volume of fluid displaced (m³), g = 9.81 m s⁻².
An object floats when its weight is balanced by the buoyancy: Fb = Fg.
At terminal velocity in a fluid, weight is balanced by drag + buoyancy: W = Fd + Fb.
The buoyancy force
When a body is immersed in a fluid, it pushes some of the fluid out of the way. The fluid pushes back, producing an upward force called buoyancy or upthrust:
Buoyancy forceFb = ρVg
The crucial detail: ρ is the density of the fluid, and V is the volume of fluid displaced — not the density of the object. A fully submerged object displaces its own volume of fluid; a floating one displaces only as much as the part beneath the surface.
Floating: weight balanced by buoyancy
Push a hollow ball under water and you feel it resist — the buoyancy force pushes it back up. Let go and it accelerates to the surface, then settles and floats. At that point it is in equilibrium: the upward buoyancy exactly balances the downward weight.
Floating conditionFb = Fg
A floating object has its weight exactly balanced by the buoyancy of the fluid it displaces.
Drag force at terminal speed
Buoyancy is the key to terminal velocity. As a sphere falls through a fluid it speeds up, the drag grows, and eventually the forces balance so it stops accelerating. At that point:
At terminal velocityW = Fd + Fb
At terminal velocity the downward weight is balanced by the drag and buoyancy acting upward.
Writing each force in full — weight W = (4/3)πr³ρsg, drag Fd = 6πηrv (Stokes’ law), and buoyancy Fb = (4/3)πr³ρfg — and solving for the terminal velocity gives:
Terminal velocity of a spherev = 2r²g(ρs − ρf) ⁄ 9η
So the terminal velocity is proportional to the square of the radius and inversely proportional to the viscosity — bigger spheres fall faster, and thicker fluids slow them down.
Quick reference: buoyancy Fb = ρVg (ρ = fluid density, V = volume displaced) • floats when Fb = Fg • terminal velocity when W = Fd + Fb.
Worked examples
WE 1
How much of an iceberg shows?
An iceberg of volume Vi floats in seawater. Ice has a density of 917 kg m⁻³ and seawater 1020 kg m⁻³. What fraction of the iceberg is above the water?
ρ is the fluid’s density, not the object’s — this is the single most common slip.
V is the volume displaced. Fully submerged → the object’s volume; floating → only the submerged part.
For floating problems, set buoyancy equal to weight and the fraction submerged becomes ρobject / ρfluid.
For terminal velocity in a liquid, remember buoyancy as well as drag — both act upward against the weight.
In air, buoyancy is often negligible compared with weight, so it’s sometimes ignored — but not in a dense liquid.
⚠ Common mistakes
Using the object’s density in Fb = ρVg. It must be the fluid’s density.
Using the whole volume for a floating object. Only the submerged volume is displaced.
Forgetting buoyancy in terminal-velocity problems in liquids, leaving the force balance wrong.
Confusing mass with weight when equating forces — keep everything as forces (N).
Notice WE 1 never needs the actual volume — the fraction submerged is just the ratio of densities, which is why almost exactly 90% of every iceberg sits hidden underwater. Up next we move from forces to momentum, starting with the Conservation of Linear Momentum.
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