The two pendulums shown are coupled by means of a weak spring $k$, which is unstrained when
the two pendulum rods are in the vertical position.
(a) Determine the normal modes of vibrations.
(b) For the initial conditions $\theta_1(0) = A$, $\theta_2(0) = 0$, $\dot{\theta}_1 = \dot{\theta}_2 = 0$ and a very soft spring, show
that beating phenomenon takes place between the two pendulums.