A plane pendulum is made with a plumb bob of mass \(m\) hanging on a Hooke'slaw spring of negligible mass, force constant \(k\), and unstretched length \(\ell_{0}\). The spring can stretch but is not allowed to bend. There is a uniform downward gravitational field \(g\).

(a) Select generalized coordinates for the bob, and find the Lagrangian in terms of them.

(b) Write out the Lagrange equations of motion

(c) Are there any conserved quantities? If so, write down the corresponding conservation law(s).

(d) If the bob is swinging back and forth, find the frequency of small oscillations in the general case where the spring can change its length while the bob is swinging back and forth.