Question

Write the equation needed to calculate the distance between the centers of the balls, R, as a function of the locations of the center of each ball where X is the left ball

Answer 1

You do not give any detail in connection with the process of your balls undergo: circular movement, electrostatic interraction, etc. Are they placed symmetrically from the vertical (passing through the suspension point) or theta is different for each of them?

Anyway, it seems to be a problem of Geometry rather than Physics.

If you set the system of coordination with the 0x-horizontal; 0y-passing through the suspension point, the left side ball will have negativ x coordinate and right side ball will have positive x coordinate.

If they are at the same heigh, y=0 for both of them (choose the 0y axis to fulfill this condition, for simplicity's sake).

In this case R=x2-(-x)=x2+x.

OBS: if you have x, x2 and you have to calculate R, I really don't understand why all the other parameters are mentioned (theta, L).

OBS No 2:

I just observed your comments from the top. Sorry, I did not see them when I answered (I do not expect such a text for a question!).

An idea: you have to make a calibration of the system. Smth like knowing the distance between the charges when they have a certain value, or find the time to discharge them when they have a certain value. If you make these graphs, you may then find any requested information when the visitors are 'playing' with the 'toy'.

I don't understand what should the student have to do exactly: 'invent' methods to fulfill the tasks or he has some instruments already offered to solve the problem?

An idea I see now:

1. To find the charge of the balls you use the geometrical information (L, theta).

Measure the angle and find q.

2. use a condenser for finding the charges of the balls: after being 'electrified' with the piece of fur, place one ball between the 2 plates of a condenser. Adjust the voltage applied to the plates in such a way to make the particle stationary (friction force=0); thus, you may find the charge by using the formula:

(d=the distance between the plates of the condenser)

Thus, q=mgd/U

The geometrical information (L, theta) will be used to find the time neceessary for discharging the balls (suppose a uniform movement): how long it takes to touch each other.

I would prefer the first method.