Trajectory of a particle in a spinning fluid

In summary, the problem is to show that a particle suspended on a spinning liquid, with constant angular velocity, moves in a spiral motion. This solution does not involve using Lagrangian-Hamiltonian formalism and instead uses weight and buoyant force. The equation for the particle's movement is obtained using a rotating frame and solving for the differential equation along the z-axis. However, there is difficulty in solving for the other two equations, resulting in complicated expressions that do not resemble circular motion over the r-$\theta$ plane. Assistance in solving this problem is requested.
  • #1
rulo1992
15
0

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


[/B]
Weight and Bouyant force

The Attempt at a Solution


[/B]
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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  • #2
rulo1992 said:
suspended on a spinning liquid
Not sure what that means. Is it floating on the surface? Sinking?
rulo1992 said:
spiral .
Any particular kind of spiral?
 

1. What is the trajectory of a particle in a spinning fluid?

The trajectory of a particle in a spinning fluid is the path that the particle follows as it moves through the fluid. This path is influenced by several factors, including the particle's initial position, velocity, and the rotation rate of the fluid.

2. How does the rotation of the fluid affect the trajectory of a particle?

The rotation of the fluid can affect the trajectory of a particle in several ways. Firstly, it can cause the particle to experience centrifugal and Coriolis forces, which can alter its path. Additionally, the rotation of the fluid can create vortices and eddies that can also impact the particle's trajectory.

3. Can the initial velocity of a particle in a spinning fluid affect its trajectory?

Yes, the initial velocity of a particle can have a significant impact on its trajectory in a spinning fluid. A particle with a higher initial velocity will have a larger centrifugal force acting on it, causing it to follow a more curved path. On the other hand, a particle with a lower initial velocity may follow a more linear path.

4. What factors can cause a change in the trajectory of a particle in a spinning fluid?

Apart from the rotation rate and initial velocity of the fluid, other factors such as the size and shape of the particle, the density and viscosity of the fluid, and the presence of other particles or obstacles can all affect the trajectory of a particle in a spinning fluid. These factors can cause the particle to deviate from its original path and follow a different trajectory.

5. Is it possible for a particle to have a circular trajectory in a spinning fluid?

Yes, it is possible for a particle to have a circular trajectory in a spinning fluid. This can occur when the particle's initial velocity and the rotation rate of the fluid are perfectly balanced, resulting in a circular path. However, this is a rare occurrence and is highly dependent on various factors such as the size and shape of the particle and the properties of the fluid.

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