Chapter 9.5, Problem 19P

For the following problems, use the Lagrangian to find the equations of motion and then refer to Chapter 3, Section 12.

For small vibrations, find the characteristic frequencies and the characteristic modes of vibration of the coupled pendulums shown. All motion takes place in a single vertical plane. Assume the spring unscratched when both pendulums hang vertically, and take the spring constant as $k=mg/l$ to simplify the algebra. Hints: Write the kinetic and potential energies in terms of the rectangular coordinates of the masses relative to their positions hanging at rest. Don’t forget the gravitational potential energies. Then write the rectangular coordinates $x$ and $y$ in terms of $\theta$ and $\varphi ,$ and for small vibrations approximate $\sin \theta =\theta ,\cos \theta =1-{\theta }^2/2,$ and similar equations for $\varphi$.