How Are Theta_1 and Theta_2 Related in Suspended Charged Spheres?

[redbeard]

Homework Statement

2 small spheres of charge "q" suspended from strings of length "l" are connected at a common point. one sphere has mass m, the other 2m. Assume angles theta_1 and theta_2 that strings make with vertical are small. (a) how are theta_1 and theta_2 related? (b) show that the distance r btw spheres is (3k*q^2*l/(2mg))^(1/3)

Homework Equations

Refer to question

The Attempt at a Solution

My attempt involved taking the cos and sin of both angles and getting particular geometric lengths. Ultimately, from these I got the angle from the horizontal that the higher ball is at with respect to the lower ball. I won't list this here as this is largely a dead end.

It would be far too hard to separate theta 1 and theta 2 in the component eqns. The math should be far easier than this considering all the other questions were much simpler (and this is only question 3 of 7). My guess is I can use the approximation somewhere. Assuming that the 2 balls are roughly on the same horizontal plane makes for an easy solution (angle1 = 2*angle 2), but I was hesitant to employ this particular approximation.



I realize this question is more complicated, so I hope there are any takers out there.

 

[Delphi51]

Quite an interesting problem!
I get the answer when I assume that

strings of length "l" are connected at a common point.

means that the two strings are hanging from a common point, not that the spheres are touching.

It is one of those problems where you work with the forces on the charged spheres. The electric force pushes them apart, the gravitational forces pull them down and the tension forces in the strings pull them at an angle. You must make a force diagram for each sphere showing all those forces. Write that, in each case, the horizontal forces total zero and the vertical forces total zero (because they are not accelerating and F = ma). For each sphere, solve one formula for the tension and sub that into the other to get a formula for the electric force. The electric force on one sphere is the same as on the other, so you quickly get a relationship between the two angles to answer part (a).

(b) involves some tricky work with the same equations. It says the angles are small, so you can assume that tan θ is approximately equal to sin θ, which can be expressed as the distance of the sphere from the center line divided by the length of the string. That's how you get the L in the formulas. That 3/2 comes from adding the two distances from the center line to get r, the total distance between the spheres.

Can't give more details until I see your try!