# Trajectory of a particle in a spinning fluid

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1. Feb 24, 2017

### rulo1992

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

Weight and Bouyant force

3. The attempt at a solution

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.

2. Feb 24, 2017

### haruspex

Not sure what that means. Is it floating on the surface? Sinking?
Any particular kind of spiral?