  We perform exact Statistical Mechanics calculations for a system of elongated
objects (hard needles) that are restricted to translate along a line and rotate
within a plane, and that interact via both excluded-volume steric repulsion and
harmonic elastic forces between neighbors. This system represents a
one-dimensional model of a liquid crystal elastomer, and has a zero-tension
critical point that we describe using the transfer-matrix method. In the
absence of elastic interactions, we build on previous results by Kantor and
Kardar, and find that the nematic order parameter $Q$ decays linearly with
tension $\sigma$. In the presence of elastic interactions, the system exhibits
a standard universal scaling form, with $Q / |\sigma|$ being a function of the
rescaled elastic energy constant $k / |\sigma|^\Delta$, where $\Delta$ is a
critical exponent equal to $2$ for this model. At zero tension, simple scaling
arguments lead to the asymptotic behavior $Q \sim k^{1/\Delta}$, which does not
depend on the equilibrium distance of the springs in this model.
