Archimedes' Principle | Physics Library

Archimedes’ Principle states that

When a floating body of mass M is in equilibrium with a fluid of constant density, then it displaces a mass of fluid M_d equal to its own mass; M_d = M.

Archimedes’ principle can be justified via arguments using some elementary classical mechanics. We use a Cartesian coordinate system oriented such that the z-axis is normal to the surface of the fluid.

Let g be The Gravitational Field (taken to be a constant) and let Ω denote the submerged region of the body. To obtain the net force of buoyancy F_B acting on the object, we integrate the pressure p over the boundary of this region

∫ FB = − pn dS ∂Ω

Where n is the outward pointing normal to the boundary of Ω. The negative sign is there because pressure points in the direction of the inward normal. It is a consequence of Stokes’ theorem that for a differentiable scalar field f and for any Ω ⊂ ℝ³ a compact three-manifold with boundary, we have

∫ ∫ ∂Ωf n dS = Ω ∇f dV

therefore we can write

∫ FB = − ∇p dV Ω

Now, it turns out that ∇p = ρ_fg where ρ_f is the volume density of the fluid. Here is why. Imagine a cubical element of fluid whose height is Δz, whose top and bottom surface area is ΔA (in the x − y plane), and whose mass is Δm. Let us consider the forces acting on the bottom surface of this fluid element. Let the z-coordinate of its bottom surface be z. Then, there is an upward force equal to p(z)ΔAe_z on its bottom surface and a downward force of −p(z + Δz)ΔAe_z + Δmg. These forces must balance so that we have

p(z)ΔA = p(z + Δz )ΔA − Δm |g |

a simple manipulation of this equation along with dividing by Δz gives

p(z + Δz ) − p(z) Δm ρ ΔA Δz -----------------= -------|g| = --f------|g| = ρf|g| Δz ΔA Δz ΔA Δz

taking the limit Δz → 0 gives

∂p- = ρ |g| ∂z f

Similar arguments for the x and y directions yield

∂p- = ∂p- = 0 ∂x ∂y

putting this all together we obtain ∇p = ρ_fg as desired. Substituting this into the integral expression for the buoyant force obtained above using Stokes’ theorem, we have

∫ ∫ F = − ρ g dV = − ρ g dV = − ρ gVol (Ω) B Ω f f Ω f

where we can pull ρ_f and g outside of the integral since they are assumed to be constant. But notice that ρ_fVol(Ω) is equal to M_d, the mass of the displaced fluid so that

F = − M g B d

But by Newton’s second law, the buoyant force must balance the weight of the object which is given by Mg. It follows from the above expression for the buoyant force that

Md = M

which is precisely the statement of Archimedes’ Principle.

"Archimedes' Principle" is owned by joshsamani.
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