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 Three Balls in a Bowl (Posted on 2005-09-07)
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)

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 re: Red Herrings | Comment 16 of 18 |
(In reply to Red Herrings by brianjn)

Yes.  You can see where the experiments are going.  Centre of Gravity directly under centre of D.

Now instead of using a triangular inset, use a circular inset.  Now the centre of gravity(G) and the centre of the plane (O) containing the three centres of the ball can be represented as two points on a line (chord) in a circle.

and tan (alpha) = OG/h where h is the (perpendicular) distance of the plane from the centre of D as I illustrated in the two ball case.

Note that the plane used is NOT the plane containing the three points of contact with the outside hemisphere D but rather that containing the three centres.  The physical contraint that the balls "fit" in D is dealt with by the conditions that (d-a)^2 = OA^2+h^2 etc.

As I presented for two balls, a general algebraic solution is possible in a, b c and d but is not very elegant.  The three ball case greatly increases the length of the expression needed to identify G on the chord and does not increase its beauty.

 Posted by goFish on 2005-11-20 09:47:17

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