Michael.Peshkin@FAS.RI.CMU.EDU (12/03/85)
> Soap bubbles are certainly spherical, even when they sit on circular wire > frames. The force on each piece of the surface is equal to the pressure > (times the area of the piece) and oriented perpendicular to that piece. If > on each piece of a string you put a force (proportional to the length of the > piece) perpendicular to that piece of string, the string would form a > circle. THAT is the correct analogy. (Ken Rimey) Soap bubbles and plastic membranes do not form the same shape. With films that can flow, we have potential energy proportional to area, and therefore force independent of stretch. Inflated on a circular rim, they form a part of the shape that minimizes area: a sphere. Plastic membranes may have an arbitrary relation of force to stretch. By inflating a uniform flat membrane on a circular rim, you will not get part of a sphere. Mirrors made this way approximate spheres (and parabolas) because they are such a small fraction of a whole sphere. The string analogy fails because in 2d stretch of the medium is essential. String (1d) needn't stretch to move, but an initially flat plastic membrane on a circular rim must stretch or it will remain planar. In soap films, force is independent of stretch so stretch is irrelevant. When stretch is relevant, the initial shape of the membrane is relevant. There are toy balloons which inflate to long thin shapes, even though the rubber is uniform in thickness over almost the whole surface. If anyone can calculate the shape of an initially flat membrane inflated on a circular rim, for a simple relation of energy to area (say E=(A-Ao)**2), I would like to hear of it.