160
CONTACT OF ELASTIC SOLIDS
closest contact, and which we shall call the surface of impact.
It follows that the elastic state of the two bodies near the
point of impact during the whole duration of impact is very
nearly the same as the state of equilibrium which would be
produced by the total pressure subsisting at any instant
between the two bodies, supposing it to act for a long time.
If then we determine the pressure between the two bodies by
means of the relation which we previously found to hold
between this pressure and the distance of approach along the
common normal of two bodies at rest, and also throughout the
volume of each body make use of the equations of motion of
elastic solids, we can trace the progress of the phenomenon
very exactly. We cannot in this way expect to obtain general
laws; but we may obtain a number of such if we make the
further assumption that the time of impact is also large com-
pared with the time taken by elastic waves to traverse the
impinging bodies from end to end. When this condition is
fulfilled, all parts of the impinging bodies, except those infinitely
close to the point of impact, will move as parts of rigid bodies;
we shall show from our results that the condition in question
may be realised in the case of actual bodies.
We retain our system of axes of xyz. Let a be the
resolved part parallel to the axis of z of the distance of two
points one in each body, which are chosen so that their
distance from the surface of impact is small compared with
the dimensions of the bodies as a whole, but large compared
with the dimensions of the surface of impact; and let a' denote
the differential coefficient of a with regard to the time. If
dJ is the momentum lost in time dt by one body and gained
by the other, then it follows from the theory of impact of
rigid bodies that da'k,dJ, where k, is a quantity depending
only upon the masses of the impinging bodies, their principal
moments of inertia, and the situation of their principal axes of
inertia relatively to the normal at the point of impact.¹
On
1 See Poisson, Traité de mécanique, II. chap. vii. In the notation there
employed we have for the constant ki
(b cosy - ccos B) (c cos a - a cos y)2 (a cos ẞ-b cos a)²
1
k₁ =
M
+
A
1
(b'cos y'-c'
+
+
M'
+
+
C
(a' cos B' - b' cos a')2
+
C'