Solving Dynamics Problem: Coriolis & Gravitational Forces on Rotating System

[lovelyrita24]

I have a problem involving the coriolis and gravitational forces on a rotating coordinate system:

A free particle of mass m is release from a state of rest on a rotating, sloping, rigid plane. The angular rotation rate about a vertical axis is omega and the angle formed by the plane with the horizontal is theta. Friction and centrifugal forces are negligible. What is the maximum speed acquired by the particle, and what is its maximum downhill displacement?

- I am pretty sure the answers are derived symbolically by determining when the coriolis force comes into balance with the component of the gravitational force down the slope.

Any insight would be much appreciated...

 

[Nate810]

The maximum speed acquired by the particle can be determined by setting the net force acting on it equal to zero. This would give us the equation:$m \omega^2 r \sin(\theta) - mg \cos(\theta) = 0$where $r$ is the distance of the particle from the axis of rotation. Solving for the maximum speed, we get:$v_{max} = \frac{mg \cos(\theta)}{m \omega \sin(\theta)}$The maximum downhill displacement can be determined by noting that the total energy of the system is conserved. The initial energy is zero, so the final energy must also be zero. We can calculate the final energy as:$E_{total} = \frac{1}{2} m v^2 - m g y \cos(\theta)$where $y$ is the displacement of the particle down the slope. Setting this equal to zero and solving for $y$, we get:$y_{max} = \frac{v^2}{2 g \cos(\theta)}$Substituting in our expression for $v_{max}$, we get:$y_{max} = \frac{\left (\frac{mg \cos(\theta)}{m \omega \sin(\theta)} \right )^2}{2 g \cos(\theta)} = \frac{g}{\omega^2 \sin^2(\theta)}$

 

[quicknote]

I would approach this problem by first considering the forces acting on the free particle. In this case, we have the gravitational force pulling the particle down the slope, and the coriolis force acting perpendicular to the particle's velocity due to the rotation of the coordinate system.

To determine the maximum speed acquired by the particle, we can use the principle of conservation of energy. At the top of the slope, the particle has only potential energy, and as it moves down the slope, this potential energy is converted into kinetic energy. At the bottom of the slope, all of the potential energy has been converted into kinetic energy, and the particle has reached its maximum speed.

To find this maximum speed, we can equate the change in potential energy to the change in kinetic energy, and solve for the velocity. However, we must also take into account the coriolis force, which will affect the particle's velocity. This can be done by setting the coriolis force equal to the component of the gravitational force down the slope, and solving for the velocity.

To find the maximum downhill displacement, we can use the equations of motion and solve for the distance traveled by the particle. Again, we must take into account the coriolis force, which will affect the particle's motion.

It is important to note that these solutions will be derived symbolically, as you mentioned. The exact values will depend on the specific parameters given in the problem, such as the mass of the particle, the rotation rate, and the angle of the slope. Therefore, it is important to carefully consider these values and plug them into the equations to obtain accurate results.

I hope this provides some insight into approaching this problem. It is important to carefully consider all of the forces at play and use the appropriate equations to solve for the maximum speed and displacement of the particle. Further analysis and calculations may also be necessary to fully understand the behavior of the particle in this rotating system.