Finding the maximum acceleration

[haruspex]

I only meant the forces related to the volume within the lens is zero, not that it has to be solved that way. To solve the problem, you can calculate the force as if the spheres didn't overlap, then subtract a force to compensate for the volume within the lens, which is what the OP ended up doing,

Well, it wasn't quite that simple. Treating them as though they don't overlap means taking them to be point masses. The trouble with a portion within the lens is not only that there is no net force between the +ve and -ve elements of the lens, but that there is a reduced force between those elements and the portions outside the lens. Pranav-arora quite correctly ran separate integrals for d-R to R and R to d+R. In the first, only the portion of the other sphere at radius < r contributes to the attraction, so it gets a factor (r/R)².
Yes, the integrals could have dealt with forces instead of potentials. Not sure which is simpler algebraically.

 

[rcgldr]

Well, it wasn't quite that simple. Pranav-arora quite correctly ran separate integrals for d-R to R and R to d+R.

You're correct. This is how I initially thought it could be done, but the OP's method turned out to be bettter.

To maintain the integrity of the spheres, that repulsion must be countered by some unknown structural force.

That was where I was getting it wrong. I kept thinking the volume within the lens would not contribute to any attractive force outside the lens, because I wasn't taking into account that the repulsive force is countered by that unknown structural force. You mentioned this earlier, but I didn't understand this until you mentioned it again.

and this was an intro physics class problem?

 

[Saitama]

and this was an intro physics class problem?

I am not sure how you classify the problems into "intro" and "advanced" but when I looked at the problem, I did not know I will have to do those integrals. Anyways, it was more of a Math problem than Physics. :P

 

[voko]

This problem is definitely not physical in the sense that I cannot think of any real objects keeping a perfectly static shape yet able to interpenetrate.

However, it is very physical in the sense that it examines your knowledge of reference frames, energy conservation, and the relationships between kinetic energy and velocity, potential energy and force, force and acceleration. That was the important part, and I think you struggled with that more than with its mathematical content.

 

[Saitama]

However, it is very physical in the sense that it examines your knowledge of reference frames, energy conservation, and the relationships between kinetic energy and velocity, potential energy and force, force and acceleration. That was the important part, and I think you struggled with that more than with its mathematical content.

Yes, you are right, I have myself noticed that I can do the Math and its the Physics part I have trouble figuring out.

 

[rcgldr]

One part in the solution that I didn't get was the intergral for potential for z < R. The potential for a point in A for z < R makes sense, but I didn't get how the dq (volume) of the spherical cap corresponds to the potential in the x-axis direction, since that volume is spread out over a spherical cap as opposed to being the equivalent of a point charge in that the field related to A with effective radius z.

 

[voko]

I am not sure what x and z are. I do hope that Pranav can address that :)

 

[Saitama]

One part in the solution that I didn't get was the intergral for potential for z < R. The potential for a point in A for z < R makes sense, but I didn't get how the dq (volume) of the spherical cap corresponds to the potential in the x-axis direction, since that volume is spread out over a spherical cap as opposed to being the equivalent of a point charge in that the field related to A with effective radius z.

The potential over a sphere of radius r is constant, so I can take spherical caps and consider them as dq instead of selecting smaller point charges. Is this what you ask?

I found something: http://arxiv.org/pdf/0903.3304.pdf