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Three Balls in a Bowl (Posted on 2005-09-07) Difficulty: 5 of 5
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 ?

No Solution Yet Submitted by Bractals    
Rating: 3.6667 (9 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: How about Two Balls in a Bowl | Comment 13 of 18 |
(In reply to How about Two Balls in a Bowl by Leming)

There is an algebraic solution for 3 balls but if you consider the formula for finding the centre of a circle from 3 given points, this is itself awkward and irreductible.

You then need to extend this circle to touch the walls of the outside hemisphere D and you get another more complicated and irreductible expression.

When you have found this circle, you can find an equally long expression for the centre of gravity on this circle.

Next you need to align the CG so it lies directly below the geometric centre (of the hemisphere) . 

Finally you need to relate the geometric centre of the circle,  the CG on the circle and the geometric centre of the hemisphere to calculate the angle.

There are surds everywhere.  I will be most impressed if anyone, even Bractals has an elegant solution.

The steps do not significantly change for two balls, it is not the third ball that is the problem but the awkwardness of the equation referred to in the first para of this comment. 

But I could be wrong!  What if I start with ....

Edited on November 11, 2005, 4:26 am

Edited on November 11, 2005, 4:27 am
  Posted by goFish on 2005-11-11 04:17:20

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