# Time period of oscillations of a particle revolving in a paraboloid

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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