Building Software for Damped Pendulum in Electric Field

[s3a]

Firstly, I spoke to a Physics teacher and some strangers on the internet as well as Googled and this is the situation I am in now. I want to build a software simulation for school of a damped-oscillation non-zero charged metallic-sphere pendulum within the uniform electric field of a parallel-plate capacitor. I want electrical and gravitational forces to take effect. I do not want the direction of the electric field to change which means that I expect to have a different damping coefficient per side of the vertical line from which the pendulum oscillates. The vertical line is basically the line the pendulum will align itself with when it is at its lowest height which is also the length of the string below the point it is rotating from.

I am not 100% sure but I think that the damping coefficient is negative when the motion of the pendulum is in the same direction as the electric field lines (for a positive charge) because in that case the electric field helps the pendulum move faster whereas when it goes against the electric field there should be a positive damping coefficient if I'm correct because the electric field is some kind of "resistance" or "friction".

Firstly, I need to know if anything I said so far is wrong because one wrong thing could ruin my entire software and I also need help in finding the damping coefficient whether it's a predefined value on some site (I have Googled but cannot find it) or I have to calculate it myself in which case I have no idea how and would appreciated help on that. Also, what if gravity is not constant or if the user can select which planet their experiment is on? Do I simply change the value of g in the following equation?:

θ = θ₀e^-γt/(2m)sin[√(g/L - γ²/(4m²))*t + ϕ]

where:
θ: the angle the at which the metallic-sphere pendulum bob is at currently.

θ₀: the angle at which the metallic-sphere pendulum bob is released from

ϕ: phase constant

γ: damping coefficient

L: length of the string

As far as the capacitor is concerned, I can ask anything as long as its not too extravagant to the user, since I'm studying the motion of the pendulum.

I need to know:

1) The value of γ for when the charge's motion due to the electric field is in the same direction as that of the natural tendency of the pendulum with gravity.

2) If I can change g such that I can use any planet as well as have gravity not be constant even though it'll change ever so slightly.

3) If my logic is 100% correct in the first place because someone has told me that the fact that both the electric field force and gravity are conservative means that the value of γ is always 0 but I disagree because if it's 0 then that means that it would move as if there was only gravity and no "supporting" or "resisting" force which there is (the force applied by the electric field).

Any help would mean A LOT to me because it's hard to find help for something like this and it would be greatly appreciated!
Thanks in advance!

 

[kuruman]

What makes you think that the electric field will produce a damping force, i.e. a force opposing the motion. You are forgetting about induced charges moving around on the surface of the sphere. Whether the sphere is charged or not, the closer it gets to a plate of some sign, the more induced charge of the opposite sign will collect on the side closer to that plate which will increase the force between the ball and plate. This is the opposite of damping.

See the demo here. Even if the string in your ball is short enough to prevent the ball from touching the plates, you still will not get damped harmonic motion. Instead, the ball will swing and stop at angle closer to one of the plates, probably the one with the opposite sign of the charge that you put on it.