XVI
271
FLOATING ELASTIC PLATES
distributed over the surface, or when the radius of the plate
is not very large, that is, not many times a.
4. If we consider the latter case, that of a finite plate,
more closely, we get the above-mentioned paradoxical result.
For the free edge of the circular plate these conditions¹ must
hold
-
1 მა
μ
a
(a) ²z-
=
др
0, (b)
²% = 0.
др
P
The
At the centre we have the same condition as before.
solution is to be compounded of the three integrals of the
equation *z+a+z = 0, which are finite at the origin; and
is completely determined by the given conditions. According
as at the edge of the plate z is negative or positive, that is,
according as the edge is above or below the surface of the
water, the plate will or will not float without further aids
(without buoyancy of its own). The case (c) when 2 = 0 at
the edge, is a limiting case. When we enquire under what
conditions the equations (a) (b) (c) are simultaneously possible,
we are led to the vanishing of a determinant involving the
radius of the plate as the unknown. This determinant in
fact vanishes for certain values of R, and with a little patience
we find that the least value of R for which this occurs lies
between 2.5a and 2·8a. It is equal to 2.5a when μ = 0,
and to 2.8a when μ If we suppose the plate to be of
=
με = 1/
the same density as water, then, so long as the radius is less
than the above value, for every load the edge must be below
the surface of the water in the position of equilibrium, and
the plate will be unable to support even the slightest load.
When the radius just has the critical value, then for every
load the edge is at the surface of the water, and thus the
plate suddenly becomes able to support every weight which
does not exceed the elastic limit. When the radius is still
larger, z is negative at the edge and we can now distribute a
certain additional weight uniformly over the plate without
lowering the edge below the surface of the water; i.e. we
may suppose the plate somewhat heavier than water to begin
with. If we suppose such a plate, which by itself could not
float, to be loaded at the centre with a sufficient weight and
1 Clebsch, Theorie der Elasticität, § 73.