A homogenous thin rod of mass $M$ and length $l$ is constrained to rotate in a horizontal plane about its centre $O$. A bead of mass $m$ is set to slide along the rod without friction. The bead is attracted to $O$ by a force resulting in a potential $k x^{2} / 2$, where $x$ is the distance from $O$.

(a) Identify suitable generalized coordinates and write down the Lagrangian of the system.

(b) Identify all conserved quantities.

(c) Derive the equations of motion and show that one of them can be written as

$$m \ddot{x}=-\frac{\partial V_{\mathrm{eff}}(x)}{\partial x}$$

where the form of the effective potential $V_{\text {eff }}(x)$ should be found explicitly.

(d) Sketch the effective potential. Find and characterize all points of equilibrium.

(e) Find the frequencies of small oscillations around the stable equilibria.