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NB: This is not a homework question. I am not searching for any solution of a math problem.

I found something incorrect to do always in the nature of two charged pith balls hanging from a light string. After both the pithballs are charged should not they be allowed to stay as far as they can be from each other. The angle ABC should be near 180° not 180° because if that happens there will be a denial of the pithballs' weight. So I guess the Angle ABC have to always be 179.99° . Still there will be balanced state. So theta will be 89.99° each. If the tension of the string is T then Tcos89. 99=W Tsin89.99°= columb force between the pithballs enter image description here

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    $\begingroup$ What's the actual question? $\endgroup$
    – Kyle Kanos
    Dec 11 '18 at 11:07
  • $\begingroup$ Why do most of the time we write theta 30, 40,50° etc. It always should be 89.99° $\endgroup$
    – ffahim
    Dec 11 '18 at 11:11
  • $\begingroup$ Why "always ~90 degrees"? Doesn't it depend on the charges? $\endgroup$
    – Kyle Kanos
    Dec 11 '18 at 11:11
  • $\begingroup$ Why should it "always" be 89.99°? It will be whatever is needed for the tension, weight and electrostatic force to balance each other. $\endgroup$
    – Aetol
    Dec 11 '18 at 11:13
  • $\begingroup$ No. Why should it depend on charges. Like charges always try to stay as far as they can. So in case of balance they can't have theta to be 90° because there is a vertical component of tension which is its own weight.(having theta 90 is impossible because Tcos90 will be 0). So the possible way is 89.99° .it should always be. Dose not mater +5c, +5c or 100c +100c charge are present. $\endgroup$
    – ffahim
    Dec 11 '18 at 11:15
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You are correct that they will always try to be as far apart as possible. However, there are constraints on what is possible. For example, why do they not move infinitely far away from each other? That seems obvious: it's because of the strings. But more generally, the strings are creating a force that works to prevent them from moving away. If you had some incredibly large charges on each ball, the strings would snap and the balls would begin to move away from each other.

The reason that they don't move 180˚ apart is that gravity is also acting on the balls in a downward direction. Usually the charge is not large enough to push the balls that far apart. To do so would require that the repulsive force between the balls be much more important (that is to say larger) than the effect of gravity. The force between the balls relies directly on the charges on the balls and the distance between them and typically is not large enough to push them that far apart.

However, if we look at things at the very small scale (molecules), we can see that they do something similar to what you're asking. CO$_2$ is a molecule with one carbon atom in the middle, an oxygen atom on either side and no unpaired valence electrons. If you look up a diagram of CO$_2$, you should find that it is drawn as a linear molecule. The oxygen atoms push against each other to try to move 180˚ apart. However, they are still tethered to the main carbon atom so they can't move further apart. In this case, you would find that the force of gravity on each oxygen atom is extremely small compared to the repulsive force they exert on each other due to being very close together. The fact that they have very little mass, and thus are much less affected by gravity is an important distinction between the two cases.

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Try the following at home.
Construct a simple pendulum with an (heavyish) object hanging from a piece of string.

Push the object a few degrees to one side with a finger.
Now push the object many more degrees to one side with a finger.
You should find that as the angle increases you need a greater force to push the object to one side.

Going back to your example.
Given your two pith balls you need a greater force of repulsion between them to increase the angle between the two strings.
To get that greater force of repulsion you need a greater charge on each of the pith balls.
So the final angle between the strings, depends amongst other things, on the charge on the pith balls.

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