# Trajectory of a particle in a spinning fluid

## Homework Statement

I need to show that a particle suspended on a spinning liquid (which is spinning with constant angular velocity) describes a spiral .

(I need to solve this without using Lagrangian-Hamiltonian formalism)

## Homework Equations

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Weight and Bouyant force

## The Attempt at a Solution

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I have tried solving this problem using a rotating frame, and subsequently I've obtained the following equation:

$$\vec a^{(eff)}= -g(\frac{ \rho_2 - \rho_1}{\rho_1} )\hat k +\omega^2r\hat r -2\omega\dot r{ \hat{ \theta}}$$
where $\rho_2$ is the liquid's density, $\rho_1$ the particle's density and $\omega$ the constant angular velocity of the liquid.

Hence I solved the differential equation for the particle's movement along the z-axis, but now I'm stuck I cannot solve the other two equations, I just keep getting complicated expressions and nothing resembling circular motion over the r-$\theta$ plane.

Any help will be appreciated.

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