Archimedes' Principle
Archimedes' Principle states that
When a floating body of mass
is in equilibrium
with a
fluid of constant density, then it displaces a mass of fluid
equal to its own mass;
.
Archimedes' principle can be justified via arguments using some
elementary classical mechanics. We use a Cartesian coordinate system
oriented such that the
-axis is normal to the surface of the
fluid.
Let
be The Gravitational Field
(taken to be a constant)
and let
denote the submerged region of the body. To obtain
the net force of buoyancy
acting on the object, we
integrate the pressure
over the boundary
of this region
Where
is the outward pointing normal to the boundary of
. 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
and for
any
a compact three-manifold with
boundary, we have
therefore we can write
Now, it turns out that
where
is the volume
density of the fluid. Here is why. Imagine a cubical
element of fluid whose height is
, whose top and bottom
surface area is
(in the
plane), and whose mass is
. Let us consider the forces acting on the bottom surface
of this fluid element. Let the z-coordinate of its bottom surface
be
. Then, there is an upward force equal to
on its bottom surface and a downward force of
. These forces
must balance so that we have
a simple manipulation of this equation along with dividing by
gives
taking the limit
gives
Similar arguments for the
and
directions yield
putting this all together we obtain
as
desired. Substituting this into the integral expression for the
buoyant force obtained above using Stokes' theorem, we have

Vol
where we can pull
and
outside of the integral
since they are assumed to be constant. But notice that
Vol
is equal to
, the mass of the
displaced fluid so that
But by Newton's second law, the buoyant force must balance the
weight of the object which is given by
. It follows
from the above expression for the buoyant force that
which is precisely the statement of Archimedes' Principle.
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As of this snapshot date, this entry was owned by joshsamani.