Here is a problem I have been developing. Maybe somebody can tell me if it can be solved or if more information is needed.

Three solid balls of radii a, b, and c are placed in a bowl whose inner surface is a hemisphere of radius d. The following information is known:

1) a < b < c < d,

2) d is large enough so that each ball touches a point on the inner surface of the bowl,

3) a is large enough so that each ball 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 bowl are directed along the lines determined by their centers.

After the balls come to rest, what is the angle between the plane determined by the centers of the balls and the horizontal in terms of a, b, c, and d ?

(In reply to re: Hugo's Challenge from CB by Hugo)

In developing my thoughts, I simplified the concept to a 'pennyfarthing' bike which had solid disks as wheels. I reasoned that for this system to achieve equilibrium, the PE of each wheel would need to be the same.  This would mean that the line join joining the 'axles' of the wheels would therefore not be horizontal. 

I figured that if this situation was true, then it would be relatively simple (for one better equipped than me) to extrapolate the ideas behind it and apply them to the 3 sphere system.

I had considered the 'problem' which Cee-Anne raised re having the centres of the three spheresin the same vertical plane.  There would need to be some condition which prevented this.  A plane being parallel to the maximum circumference of the bowl (radius d) is eliminated because the three spheres have different radii.

 

Posted by brianjn on 2005-09-20 00:59:22