I am dreaming of three balls on a hoop.

Three solid balls of radii a, b, and c are placed on a hoop of radius d. The hoop is a thin, rigid wire. The balls have been drilled so they fit perfectly on the wire hoop and slide without friction. The hoop passes through the center of gravity of each ball. The hoop is oriented vertically so the balls slide to the bottom.
The following information is known:

1) a, b, and c are all < d,


2) d is large enough so that each ball is entirely below the center of the hoop,

3) the central ball is large enough so that it touches the other two balls,

4) the balls are made of the same material so that their weights are proportional to their volumes,

5) the forces that the balls exert on each other and the hoop are directed along the lines determined by their centers.

After the balls come to rest, what is the angle between the vertical and a line from the center of the hoop to the center of each ball?

Inspired by Three Balls in a Bowl.

Here's my first cut at a solution.

Let A, B, and C be the centers of the balls
with radii a, b, and c respectively. Their
weights are a constant k times their radii
cubed. Let H be the center of the hoop and
L the lowest point on the hoop. Define

    t_X = <LHX  for X = A, B, and C

Assume ball with center B lies between the
other two balls. Thus

    t_A = t_B - d_A

        and

    t_C = t_B + d_C

        where

    (a+b)^2 = 2d^2[1 - cos(d_A)]

        and

    (b+c)^2 = 2d^2[1 - cos(d_C)]
   
We want to maximize the following,

    k*a^3*d*cos(t_A) + k*b^3*d*cos(t_B) + k*c^3*d*cos(t_C)

        or

    k*d*[a^3*cos(t_B-d_A) + b^3*cos(t_b) + c^3*cos(t_B+d_C)]

This occurs when,

    a^3*sin(t_B-d_A) + b^3*sin(t_B) + c^3*sin(t_B+d_C) = 0

        or

                   a^3*sin(d_A) - c^3*sin(d_C)
    tan(t_B) = -----------------------------------
                a^3*cos(d_A) + b^3 + c^3*cos(d_C)

Posted by Bractals on 2006-11-10 15:27:52