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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: 3.8000 (10 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
~ conservation of energy; first thoughts | Comment 4 of 18 |
Not sure yet what the equations will turn out to be, but I'm thinking that minimizing the overall center of gravity of the construct of the 3 balls is one way to go.   The other way is to calculate all the forces.

I struggled with an earlier problem (Bowl and Rod, pid=2667), and found that minimizing the center of mass worked best for me.

The center of the "a" ball lies on a sphere centered where "d" is centered, but with radius "d-a";  similarly for b and c.

Maybe a spherical cooridinate system centered at d's center would be a good way to think about this. 

Like I said, not sure, haven't worked it out.

  Posted by Larry on 2005-09-08 01:38:48
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