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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