Time period of oscillations of a particle revolving in a paraboloid

Classical Mechanics Level 5

Consider a vertical paraboloid \(y = r^2\). A small particle kept on its smooth inner surface at \(y_{0} = 1 m\) is given an angular velocity of \(\omega = \omega_{0}(1+ \eta)\), where \(\eta <<1\), and \(\omega_{0} = \sqrt{19.62} \text{rad}/s^2\). It has no vertical velocity initially.

Find the time period of its vertical oscillations in \(\text{ seconds }\). Assume \(g = 9.81 m/s^2\).

Hint : Use conservation of angular momentum and conservation of energy. Afterwards, use approximations in your integral (\(\eta<<1\))

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