Two small balls of mass m1 and m2 are floating on thin level soap film.?

Question

Horizontal distance between the balls is R,
surface tension is s=1 and acceleration of gravity
is g=1.

What is the force of attraction between the balls?

Answer 1

Let z = -f(x,y) be the vertical coordinate of the film, and use subscripts (f_x etc.) to denote partial derivatives. The potential energy is (s * area of film):

V = s \integral sqrt(1 + (f_x)^2 + (f_y)^2) dx dy
- g f(x1,y1) m1 - g f(x2,y2) m2

To solve the problem exactly it would be necessary to find the f(x,y) which minimizes this V, then calculate the derivative of V with respect to |x1-x2|.

That's too difficult to do exactly - so this is just an arm waving derivation of the force using the linear approximation. A criterion for its validity is described below. Let's assume |f_x|,|f_y| << 1 so that the square root in the integrand can be approximated:

sqrt(1 + (f_x)^2 + (f_y)^2) --> 1 + (1/2)((f_x)^2 + (f_y)^2)

So V can be written:

V = s \integral dx dy [(f_x)^2/2 + (f_y)^2/2
-g m1 delta(x-X1) -g m2 delta(x-X2)]

where those are two dimensional Dirac delta functions. This leads to the Euler Lagrange equation:

s * Del^2 f = -g m1 delta(x-X1) -g m2 delta(x-X2)

This is the same as two dimensional electrostatics. Each mass contributes linearly to f. Assuming the field produced by each mass is rotationally symmetric we can use Gauss's law to calculate its field:

2 pi r Grad f = -(g m/s)
-->
df/dr = -g m/(2 pi s r)

In other words the slope of the film at a distance r from the mass is gm/(2pi sr) radial toward it. That means the attractive force on either mass due to the other when they're separated by a distance R is:

F = g^2 m1 m2/(2pi s) * (1/R)


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In order for this to be valid |df/dr| << 1 or equivalently the distance between the two masses must be much greater than:

a = gm/s

Actually I can't really be certain that this treatment is valid unless each mass is also required to be a disc of radius >> a, even if their separation distance is >> a. Otherwise there'll be a region around each mass in which the field f is nonlinear.

I can calculate f exactly outside a rotationally symmetric source:

f = -C ln(r + sqrt(r^2-C^2))

where C is a constant. Since this is invalid for r < C it doesn't appear possible to have a true point source.