What is the Ratio of Angles and Distance Between Two Charged Spheres?

[jackiefrost]

Homework Statement

Two small charged spheres (S1 and S2) are suspended from a common point P by cords of equal length l, as shown in the figure below, and make small angles, a1 and a2, with the vertical through P. The charge on S1 is Q and the charge on S2 is 2Q (same polarity). The mass of S1 is m1 and the mass of S2 is m2. For this particular problem, the mass of S2 is given as twice that of S1.

http://home.comcast.net/~perion_666/Files/phys1.jpg

The problem:
Part A - We are asked to find the ratio of the two angles a2/a1.
Part B - Estimate the distance d between the two spheres.

Homework Equations

http://home.comcast.net/~perion_666/Files/eq1.jpg

The Attempt at a Solution

I reasoned that each sphere's equilibrium position would be the result of where along the swing arc the tangential components of the graviatational and repulsive electrical force acting on it (Fg_ and Fe_) would cancel each other out. These tangential components are shown above as Fgt1 & Fet1 for S1 and Fgt2 & Fet2 for S2. Note that the magnitude of electrical force acting to repel each sphere is given by Coulomb's Law using Q and 2Q as the charge values and d as the distance between the sphere's centers.

The magnitude of the tangential components of the electrical and gravitational forces acting on S1 and S2 are:
http://home.comcast.net/~perion_666/Files/eq1_2.jpg

Since the spheres are at rest, Fet1 = -Fgt1 and Fet2 = -Fgt2:
http://home.comcast.net/~perion_666/Files/eq3_4.jpg

Dividing eq. 3 by eq. 4:
http://home.comcast.net/~perion_666/Files/eq5.jpg

Rearranging:
http://home.comcast.net/~perion_666/Files/eq6.jpg

Before I show where I'm stuck for Part B, does eq. 6 answer Part A or have I messed up somewhere?

Thanks

 

[Mindscrape]

Wow, that's a very involved figure! Um, which angles are you solving for again? I see ten angles on the picture, and I am not really sure which ones you are solving for. I'm not sure if this is the figure you were given, or one you drew, but I (personally) would try and break it up before your head explodes from so much going on.

I am not quite sure about your answer. Your reasoning is definitely correct though, you need to find when all the forces sum to zero. For example, on charge 1 there is an electric field vector pointing NW, a gravity vector pointing down, and (don't forget!) a tension vector pointing NE. These all sum to zero.

 

[jackiefrost]

Sorry if that figure is too complex looking. It's really not all that complicated. I labeled the angles because the first part of the problem is looking for the ratio of a1 to a2. [I think I mis-stated the ratio as a2 to a1 in the first post but but that's not a big deal]. They are the angles that each sphere swings from vertical (the dashed red line) due to the electrical repulsion (Fe1 and Fe2) working against their weight (Fg1 and Fg2). I labeled the other triangle's angles because I tried (unsuccessfully) using the Law of Sines for the second part of this problem, which wants me to estimate the distance between the spheres, d.

I came up with the ratio of the angles as tan a1/tan a2 (for the given mass ratio (m2/m1 = 2) and charge ratio (Q2/Q1=2). Actually, in my analysis, the charge ratio wasn't a factor in obtaining the angle ratio - see my first post.

Also, I only summed the components of the forces that act in the direction tangent to a sphere's circular swing at the rest position of the sphere. In this direction (the green dashed lines), the string tension is zero. I did this because, since the electrical force is repulsive, the components of electrical repulsion plus weight along the direction of the string will equal the string tension and won't contribute to the swing of the sphere. The only force components that affect the swing angle are the tangential compnents of weight and electrical repulsion. At least, I think that's the way it would work - never totally sure of anything :)

I'm going to post one more time for the second part of this problem where I don't have any equations to mess with since I conclude that the textbook is asking for something that can't be found with what's given.

 

[jackiefrost]

Part B - Two Suspended Charged Spheres

Part B

http://home.comcast.net/~perion_666/Files/phys1.jpg

For this part of the problem, I'm asked to "estimate" the distance d (see the diagram above) between the centers of the two charged spheres when the charge on S2 is twice that of S1 and the mass of S2 is twice that of S1 suspended on cords of length l. After a lot of fruitless algebraic and trigonometric juggling I've come to the conclusion that d can't possibly be calculated (or even estimated) when only given the mass ratio and charge ratio (and cord length). For any particular mass ratio there are an infinite number of actual mass values, m1 and m2, that could produce that ratio. The same goes for the charge ratio. But those different mass and charge values, though satifying the required ratios, would determine how much each sphere must swing until the electrical and gravitational tangential components might cancel and thus determine the final separation distance d. That's probably not what the textbook author had in mind - am I missing something here?

So, to answer Part B, I could "estimate" the distance d as: 0 < d < 2l [yuk, yuk]

Even if I'm right in all this, the fact remains that I failed! Even if not require for the textbooks problems, I still should be able to obtain an equation for calculating d as a function of m1, m2, Q1, Q2, and cord length l. I've tried several approaches but I always end up with some incredibly complicated mess when I try to get rid of the angles. If you're bored and want to give me a shove in the right direction I'd appreciate it.

jf

 

[denverdoc]

It does say to estimate the distances. First estimate would be just to add the chords from the original position to final in which case d=L(a1+a2)

More accurate estimate might be to consider d as equal to d1+d2, where di is the horizontal distance to the center line from the lefthand ball in which case a1*L/d1=cos a1 and a2*L/d2=cos a2 and d in this case would be d1+d2.

I suppose this could be further refined by considering the difference in height the two balls might have or even possibly even considering tilting the triangle under consideration, which would be that LDL using your segments and looking at potential energy sums relative to the ball on the right.

Have you done the more trivial case where both are of the same charge and mass?