  We study the vibrations of a hanging thin flexible rod, in which the dominant
restoring force in most of the domain is tension due to the weight of the rod,
while bending elasticity plays a small but non-negligible role. We consider a
linearized description, which we may reduce to an eigenvalue problem. We solve
the resulting singularly perturbed problem asymptotically up to the first
modification of the eigenvalue. On the way, we illustrate several important
problem-solving techniques: modeling, nondimensionalization, scaling, and
especially use of asymptotic series.
