 Elastic pendulum

A point mass attached to a massless straight spring can change its radial distance $r$ from a fixed support while swinging in a vertical plane. Its lower equilibrium length is $r_0=l+mg/k$. Linearizing the Euler-Lagrange equations there gives independent radial and angular normal modes with squared angular frequencies $k/m$ and $g/r_0$. The formal upper stationary state has $r=l-mg/k$ and exists in positive-length coordinates only when $l>mg/k$; its angular mode is unstable.

An elastic pendulum is a mechanical system that combines the principles of a traditional pendulum with elastic properties, typically involving a mass (or bob) suspended from a spring or elastic material. The elastic pendulum demonstrates interesting dynamics because the motion is governed by both gravitational forces and spring (or elastic) forces.