A dancing marble

A massive hemispherical bowl of radius \(R\) is placed on the table as shown. A small marble is move steadily in the bowl create a circular path in xy plane with radius \(r\). If the marble is given a tiny kick so the marble has small z velocity component (parallel to the axis of the bowl). The angular frequency of small oscillation can be written as: \(\omega = F(r) \times \sqrt{\frac{g}{R}}\). Find \(F(r)\) when \(r = 0.5R\). Round your answer to 2 decimal places!

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