V
159
CONTACT OF ELASTIC SOLIDS
about that value.
As a second example, consider a number
of steel spheres pressed by their own weight against a rigid
horizontal plane. We find that the radius of the circle of
pressure in millimetres is very approximately a =
Hence for spheres of radii
1000R¹.
1 mm.,
a becomes about
1 m.,
1 km.,
1000 km.,
1000 mm.,
10 mm.
100 m.,
1000 km.,
or
a =
1000,
100,
t
of the
of the radius. For spheres whose radius exceeds 1 km.
the radius of the circle of pressure is more than
radius of the sphere. Our calculations do not apply to such
ratios, for we presupposed the ratio to be a small fraction.
But the very fact that for such large spheres equilibrium is
no longer possible with small deformations shows that equili-
brium is altogether impossible. Consider further two steel
spheres of equal radius touching one another and pressed
together only by their mutual gravitational attraction. In
millimetres we find the radius of the circle of pressure to be
p=0·000000378R. If the radius of the two spheres is 4.3
kilometres, then p=R; if it is 136 kilometres, then
p=R. That value of R, for which the elastic forces cease
to be able to equilibrate gravitational attraction, will lie
between the above values and nearer to the greater. If steel
spheres of greater radius be placed touching each other, they
will break up into pieces whose dimensions are of the order of
the values of R just mentioned.
1
Finally, we shall apply the formulæ we have obtained to
the impact of elastic bodies. It follows, both from existing
observations and from the results of the following considera-
tions, that the time of impact, i.e. the time during which the
impinging bodies remain in contact, is very small in absolute
value; yet it is very large compared with the time taken by
waves of elastic deformation in the bodies in question to
traverse distances of the order of magnitude of that part of
their surfaces which is common to the two bodies when in
1 In these calculations ´´
kg/mm², its density 7.7
v of steel is taken to be 20,000
2
earth 6.