Pendulum, pendulum!

The above figure shows two ball and rod arrangements. In both cases:

  • The balls are to be regarded as identical point masses.
  • The rods are ideal and identical (mass-less and in-extensible).
  • All pivots (one in figure (a) and two in figure (b)) are hinges that can rotate in the vertical plane.
  • For (b) the second pivot is between the upper mass and lower rod.

Let \({ \omega }_{ a }\) and \({ \omega }_{ b }\) denote the critical values of angular velocity of rotation of top pivots for cases \(a\) and \(b\) for which the systems are in stable equilibrium.

Find \(\frac{{ \omega }_{ a }}{{ \omega }_{ b }}\)

Details: Both systems are in the same room on Earth!


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