270
XVI
FLOATING ELASTIC PLATES
smaller still. In this case the pressure inside the plate will
still be distributed over a circular area whose diameter is
approximately equal to the thickness of the plate. We may
roughly represent the case when the weight is as far as
possible concentrated at a point by making R equal hin
the preceding formula; thus we get for the greatest tension
which the weight P can produce at all in the plate
p
3(1-2)P
2πh2
h
log-1.3090
a
00).
For example, in the case of the plates of ice just con-
sidered, we get for a weight of 100 kg. the values
p = 221, 53, 81, 1.9, 0.47 kg/cm³.
The plate 100 mm. thick would certainly bear the weight,
that 50 mm. thick probably not.
3. The force with which the water buoys up the weight
owing to its deformation is
2πS
a4
2 [22pdp = = 20 fps.
2π
V¹z. pdp = -P,
and is therefore equal to the load applied. However great
the load, it will always be supported; the force with which
the plane unloaded plate is buoyed up is immaterial.
If we
place a small circular disc of stiff paper on the surface of
water, we may put at its centre a load of several hundred
grammes, although the force buoying up the paper alone is
but a few grammes. Hence when a man floats on top of a
large sheet of ice, it is in strictness more correct to say that
he floats because by his weight he has hollowed the ice into
a very shallow boat, than to say that he floats because the
ice is light enough to support him in addition to its own
weight. For he would float just as well if the ice were no
lighter than the water; and if instead of the man we placed
weights as large as we pleased upon the ice, they might break
through, but could never sink with the ice. The limit of the
load depends on the strength, not on the density of the ice.
The case is different when men or weights are uniformly