= Pendulum with a massless spring-mounted support
{title2=$\omega^2=\frac{(g/\ell)(k/m)}{g/\ell+k/m}$}
For a bob of mass $m$ on a massless rod of length $\ell$, let a massless support slide horizontally against a <spring> of stiffness $k$. With support displacement $s=\ell\eta$ and angle $\theta$ from downward vertical, the quadratic <Lagrangian>, divided by $m\ell^2$, is $(\dot\eta+\dot\theta)^2/2-(k/m)\eta^2/2-(g/\ell)\theta^2/2$. Subtracting its two <Euler-Lagrange equations> gives $\eta=mg\theta/(k\ell)$. There is one finite <normal mode>, with the displayed frequency. A stiff spring fixes the support and recovers the <simple pendulum> frequency. The missing second finite frequency reflects a <singular kinetic matrix in small oscillations>, not an omitted oscillation branch.
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