Solving for Electric Force Balance in a Suspended Sphere Setup"

[Brothas]

First of all, hey everyone.
I was wondering if any of you could help me solve this. I've tried but i keep getting insane equations so I might be doing something wrong.

Two equal spheres with a mass of m have equal charges q. They're suspended by two ropes with length L in points distanced d apart. Calculate the distance between the (centers) of the two spheres when this setup is in balance.
Picture: https://dl.dropbox.com/u/29642931/phys.png
What I've tried so far is:
The forces acting on the ball are the electric forces from the charges, the gravity and the tension frmo the rope. I figured the sum of these forces should be 0 for this setup to be balanced, which led me to: F_g + F_el+F_rope with |F_rope| = |F_g|/cos(θ) + |F_el|/sin(θ), but this is insanely hard to solve..

 

[Doc Al]

The forces acting on the ball are the electric forces from the charges, the gravity and the tension frmo the rope. I figured the sum of these forces should be 0 for this setup to be balanced,

Good.

which led me to: F_g + F_el+F_rope with |F_rope| = |F_g|/cos(θ) + |F_el|/sin(θ),

I don't understand that last equation. What's the direction of the electric force?

 

[Brothas]

The charges are equal, so the electrical forces act 'outwards', right?
I got that last equation by fiddling a bit, I'm not too sure how to find the rope tension force..

 

[kushan]

use the diagram to find the forces at equillibrium

 

[Doc Al]

The charges are equal, so the electrical forces act 'outwards', right?

Right, which is horizontal.

I got that last equation by fiddling a bit, I'm not too sure how to find the rope tension force.

Redo it by considering horizontal and vertical components separately.

 

[Brothas]

use the diagram to find the forces at equillibrium

Using this diagram, I get the conditions:
|F_el|= q²/(4π*ε*r²) = |T|sin(θ)
|F_g| = mg = |T|cos(θ)

And I'm stuck here: how do I find r (or θ?) from this?

Edit : Dividing top by bottom I get : tanθ = q²/(4π*ε*r²*m*g)
I can write r = d + 2Lsin(θ) -> r² = d²+4dLsinθ + 4L²sin²(θ)
If I denote q²/(4π*ε*m*g) = C (for my ease of writing mainly), I get:
C*cotθ = d²+4dLsinθ + 4L²sin²(θ), which I can then solve for θ (somehow)?

 

[kushan]

you can try to eliminate T , so that you are left with only one variable (θ)

 

[Doc Al]

I assume you are given θ and L, and must solve for d in terms of θ and L.

 

[Brothas]

I assume you are given θ and L, and must solve for d in terms of θ and L.

What I assume is L,d,q and m are given. (d being the distance between the fixpoints of the two ropes). I then have to solve for the distance r between the centers of the spheres (which can be found by finding θ.)

 

[Doc Al]

What I assume is L,d,q and m are given. (d being the distance between the fixpoints of the two ropes). I then have to solve for the distance r between the centers of the spheres (which can be found by finding θ.)

Ah, OK. Do you need to solve it analytically, or are you given values for those givens? (Which you can then plug into a smart calculator, which can solve for θ.)

 

[Brothas]

Ah, OK. Do you need to solve it analytically, or are you given values for those givens? (Which you can then plug into a smart calculator, which can solve for θ.)

No values are given, so analytically I suppose. But I get the feeling this spits out a very ugly equation if I do it exactly..
So maybe the answer to this question just is "For a θ which satisfies this equation, the distance is given by r = d + 2Lsinθ"?

Or, maybe, if I assume θ is small, I can get a neat approximate solution?
Maybe something like: C*cotθ = (d+2Lsinθ)² ~ d² for small θ, so θ ~ arctan(C/d²) = arctan(q²/(4π*ε*m*g*d²))?