Buoyant Force
When you place a rubber duck in a bathtub, it floats — and when you try to push a beach ball underwater, you feel a strong upward resistance. Both experiences are examples of buoyancy, the upward force that a fluid exerts on any object submerged (fully or partially) within it. This force doesn't come from magic; it arises directly from the way fluid pressure increases with depth, and it acts on every face of the submerged object simultaneously. Buoyancy is not limited to boats and rubber ducks. It governs why hot-air balloons rise, why icebergs float with only about one-ninth of their volume above the waterline, and why your own body feels lighter when you stand in a swimming pool. Understanding buoyancy requires connecting two ideas you already know: fluid pressure and Newton's laws of force balance. Once you see the pressure argument, the famous result — Archimedes' Principle — follows almost automatically. In this article we will derive the origin of the buoyant force from first principles by examining pressure differences, state Archimedes' Principle precisely, apply it to both floating and sinking objects, and work through a quantitative example. We will also look at the condition for an object to float, sink, or remain neutrally buoyant, expressed as a simple comparison of densities.
- Pressure Increases with Depth
- Pressure on the Bottom Face Exceeds Pressure on the Top Face
- Archimedes' Principle
- Floating, Sinking, and Neutral Buoyancy
- The Floating Fraction Rule
Pressure Increases with Depth
In a static fluid, the gauge pressure at depth h below the surface is P = ρ_fluid · g · h, where ρ_fluid is the fluid's density and g is gravitational acceleration. This means the deeper you go, the higher the pressure pushing inward on any surface. Every face of a submerged object experiences this pressure acting perpendicular to and into that face.
Pressure on the Bottom Face Exceeds Pressure on the Top Face
Consider a rectangular block submerged with its top face at depth h₁ and its bottom face at depth h₂ = h₁ + Δh. The downward pressure force on the top face is F_top = ρ_fluid · g · h₁ · A, and the upward pressure force on the bottom face is F_bottom = ρ_fluid · g · h₂ · A. Because h₂ > h₁, we have F_bottom > F_top. The net upward force from this pressure difference is F_b = ρ_fluid · g · (h₂ − h₁) · A = ρ_fluid · g · Δh · A. But Δh · A is just the volume V of the block, so F_b = ρ_fluid · g · V.
Archimedes' Principle
The result F_b = ρ_fluid · g · V is Archimedes' Principle: the buoyant force on an object equals the weight of the fluid displaced by that object. For a fully submerged object, V is the entire object's volume. For a floating object, V is only the submerged portion. Notice that the buoyant force depends entirely on the fluid and the displaced volume — not on the object's material, weight, or internal structure.
Floating, Sinking, and Neutral Buoyancy
The object's weight is W = ρ_object · g · V_object. Comparing this with F_b tells us the outcome. (a) If ρ_object < ρ_fluid, then W < F_b when fully submerged, so the object accelerates upward and floats, settling at an equilibrium where only enough volume is submerged to make F_b equal W. (b) If ρ_object > ρ_fluid, then W > F_b at all submerged volumes, so the object sinks. (c) If ρ_object = ρ_fluid, the object is neutrally buoyant and hovers at any depth. Fish use a swim bladder to fine-tune their average density to achieve neutral buoyancy.
The Floating Fraction Rule
For a freely floating object, setting the buoyant force equal to the object's weight gives ρ_fluid · g · V_submerged = ρ_object · g · V_total. Dividing both sides: V_submerged / V_total = ρ_object / ρ_fluid. Ice has a density of about 917 kg/m³ and seawater about 1025 kg/m³, so the submerged fraction is 917/1025 ≈ 0.895, meaning roughly 89.5% of an iceberg lies below the surface — exactly why icebergs are so dangerous to ships.
The Key Equation
- F_b — Buoyant Force (N (newtons)): The net upward force exerted by the fluid on the submerged or partially submerged object, resulting from the pressure difference between the bottom and top of the object.
- \rho_{\text{fluid}} — Fluid Density (kg/m³): The mass per unit volume of the surrounding fluid (e.g., ~1000 kg/m³ for fresh water, ~1025 kg/m³ for seawater, ~1.2 kg/m³ for air at sea level).
- g — Gravitational Acceleration (m/s²): The acceleration due to gravity, approximately 9.81 m/s² near Earth's surface. It appears because fluid pressure is generated by the weight of the fluid above.
- V_{\text{displaced}} — Displaced Volume (m³): The volume of fluid pushed aside by the object. For a fully submerged object this equals the object's total volume; for a floating object it equals only the submerged portion.
Worked Example
Problem. A solid aluminium sphere of radius 0.10 m is completely submerged in fresh water (density 1000 kg/m³). The density of aluminium is 2700 kg/m³. (a) Calculate the buoyant force on the sphere. (b) Calculate the sphere's weight. (c) Find the net force on the sphere and state whether it sinks or would need to be held down.
Answer. Buoyant force ≈ 41.1 N; Weight ≈ 110.9 N; Net downward force ≈ 69.8 N — the sphere sinks.
Origins · 1 minArchimedes of Syracuse and how this was discovered
Archimedes rigorously proved that the upward force on a body submerged in a fluid equals the weight of fluid displaced — a result he stated in his treatise 'On Floating Bodies' around 250 BCE. He derived this from basic assumptions about fluid pressure without any of the calculus or algebraic notation we use today, relying instead on geometric arguments. His principle remains one of the oldest quantitative laws in physics still in everyday use. According to the Roman writer Vitruvius, Archimedes discovered his principle while stepping into a bath and noticing the water level rise — reportedly running through the streets of Syracuse shouting 'Eureka!' ('I have found it!'). Whether or not the story is literally true, it captures how displacement and volume are at the heart of buoyancy.
Related Concepts
- Fluid Pressure and Depth — Buoyancy arises directly from the increase of fluid pressure with depth; understanding P = ρgh is the essential first step to deriving the buoyant force.
- Density and Specific Gravity — Whether an object floats or sinks is determined entirely by comparing the object's density to the fluid's density, making density a central concept in any buoyancy problem.
- Newton's Second Law — The floating or sinking condition is a force-balance problem; applying ΣF = ma (with a = 0 for equilibrium) connects buoyancy to Newton's laws.
- Fluid Dynamics and Continuity — After mastering static fluid behaviour (buoyancy), the natural next step is studying fluids in motion, where continuity and Bernoulli's equation govern pressure in flowing fluids.