Rod pendulum A uniform rod of length 2a is suspended from a fixed point O by a light inextensible string of length b attached to one of its ends. The system moves in a vertical plane through O. Take as coordinates the angles θ, Φ between the string and the rod respectively and the downward vertical. Show that the equations governing small oscillations of the system about θ = Φ = 0 are
bθ·· + aΦ·· = -gθ,
bθ·· + (4/3)aΦ·· = -gΦ.
For the special case in which b = 4a/5, find the normal frequencies and the forms of the normal modes. Is the general motion periodic?