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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: Some thoughts and a question | Comment 3 of 18 |
(In reply to Some thoughts and a question by Leming)

Leming said:
Is this identical to the balls hanging from strings (length d minus ball diameter) that are anchored at a common point?

Mighty nice way of transforming the problem, great thinking.
It would change the difficulty setting by one or two categories I guess. 
I am not absolutely sure, it is difficult to visualise it , but I believe that it is unfortunately not identical.

As to the problem:
I guess all the necessary information is available to solve the problem.
This said, I don't see myself solving it (certainly not in a reasonable time), but I am very interested in the method used/progress made in solving it.
I would first calculate the function that expresses the coordinates of the center of gravity of the 3 balls in terms of a, b, c and d
Then I would evaluate this function for the center of c going from height c to height say 2c.  The minimum value of this function  would give you an answer where c must be.
a and b coordinates can probably be calculated relatively easy, then the plane, then the angle.

  Posted by Hugo on 2005-09-07 21:12:46
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