curvature = bending moment :- flexural rigidity
may also be applied to the problem of the flexure in a principal plane
of a very thin rod or wire, for which the curvature need not be small.
When the forces that produce the flexure are applied at the ends only,
the curve into which the central-line is bent is one of a definite
family of curves, to which the name _elastica_ has been given, and there
is a division of the family into two species according as the external
forces are applied directly to the ends or are applied to rigid arms
attached to the ends; the curves of the former species are characterized
by the presence of inflections at all the points at which they cut the
line of action of the applied forces.
[Illustration: FIG. 23.]
We select this case for consideration. The problem of determining the
form of the curve (cf. fig. 23) is mathematically identical with the
problem of determining the motion of a simple circular pendulum
oscillating through a finite angle, as is seen by comparing the
differential equation of the curve
d^2[phi]
EI -------- + W sin [phi] = 0
ds^2
with the equation of motion of the pendulum
d^2[phi]
l -------- + g sin [phi] = 0.
dt^2
The length L of the curve between two inflections corresponds to the
time of oscillation of the pendulum from rest to rest, and we thus
have