A particle \(P\) is attached to a fixed point \(O\), which is \(0.8 \, \text{m}\) above a smooth horizontal plane, by a light inextensible string of length \(1.0 \, \text{m}\).

The particle moves in a circle with angular speed \( \omega \, \text{rad s}^{-1}\). Given that \(P\) stays in contact with the plane, find the maximum value of \(\omega ^2\).

**Note**: Let \(g = 9.8 \text{m s}^{-2}\).

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