Challenging Problem Equations of Motion for Spherical Magnetics Pendulum

[tymbark]

Homework Statement

I struggle to write equations of motion for spherical magnetic pendulum.
The forces acting on the pendulum are: gravity, tension in the rope, and 4 repelling forces
Here is the picture for the problem: first attachment
(I didn’t add forces to not mess it up)
There are no data given. I have to assume some realistic values.​

Homework Equations

second attachment have used those equation of motion presented on this site to do the project in 2D but I didn’t really helped me in the 3D case.
Then I have found this pdf
-the problem is almost done using Lagrange mechanics but I don’t know how to add repealing forces to the equations.​

The Attempt at a Solution

I solved it in 2D case(attached picture). I don’t know how to add equations of motion to the equations given in the pdf from the attachments. However of course it does not need to be done using Lagrange mechanics.​

please help

 

[KataKoniK]

me

Thank you for reaching out for help with writing equations of motion for a spherical magnetic pendulum. This is a complex problem that requires careful consideration of all the forces acting on the pendulum, as well as any assumptions made about the system.

First, let's discuss the forces acting on the pendulum. As you mentioned, there are three main forces: gravity, tension in the rope, and repelling forces. The gravity force can be calculated using the equation Fg = mg, where m is the mass of the pendulum and g is the acceleration due to gravity. The tension in the rope can be calculated using the equation Ft = Tsinθ, where T is the tension in the rope and θ is the angle between the rope and the vertical.

Now, let's talk about the repelling forces. Without knowing the specific details of the system, it is difficult to provide specific equations for these forces. However, we can make some assumptions to help us write the equations of motion. For example, we can assume that the repelling forces are proportional to the distance between the pendulum and the magnetic field, and are directed away from the magnetic field. We can also assume that the strength of the repelling forces is dependent on the strength of the magnetic field and the charge of the pendulum.

With these assumptions in mind, we can write the equations of motion for the system. Using Newton's second law, we can write the equations for the x, y, and z components of the forces acting on the pendulum. The equations will look something like this:

∑F = max = m(d^2x/dt^2)
∑F = may = m(d^2y/dt^2)
∑F = maz = m(d^2z/dt^2)

where ∑F is the sum of all the forces acting on the pendulum, m is the mass of the pendulum, and x, y, and z are the position coordinates of the pendulum.

To add the repelling forces to these equations, we can simply add them to the ∑F term. For example, if we have a repelling force Fmag acting in the x direction, we can add it to the equation for ax, like this:

∑F = Fg + Ft + Fmag = max = m(d^2x/dt^2)

Similarly, we can add the repelling forces