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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)

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Solution! | Comment 17 of 18 |

Three balls in a bowl.

Without loss of generality, we may place the (centres of the ) balls A, B and C on the x-y plane such that

A = {a1, 0, 0}

B= {b1, b2, 0}

C= {c1, c2, 0}

D= { 0, 0, h}

We next add constraints so that the balls "fit" in D:

(1) (d - a)^2 = a1^2 +h^2

(2) (d - b)^2 = b1^2+b2^2+h^2

(3) (d - c)^2 = c1^2+c2^2+h^2

and so that they touch each other

(4) (a+b)^2 = (a1-b1)^2 + b2^2

(5) (b+c)^2 = (b1-c1)^2 + (b2 - c2)^2

(6) (a+c)^2 = (a1 -c1)^2+c2^2

The six equations would allow us to eliminate five variables {a1, b1, b2, c1, c2} and express h in terms of a,b,c,d. We note that h is fixed for a given {a,b,c,d}. However they are useful for computing the centre of gravity,

We denote the centre of gravity by G and since it is proportional to the volume of the spheres we have

G (a^3+b^3+c^3) = A a^3 + B b^3 + C c^3. We note that G lies on the x-y plane.

Then g= |OG| = Sqrt[((a^3*a1 + b^3*b1 + c^3*c1)^2 + (b^3*b2 + c^3*c2)^2)/(a^3 + b^3 + c^3)^2].

Now it is a simple matter to calculate t = tan(alpha) = g/h. I will post separately an expression for h but note that there seem to be possibly 4 different intervals where other expressions apply.

Edited on November 20, 2005, 1:04 pm

Edited on November 20, 2005, 1:07 pm
  Posted by goFish on 2005-11-20 11:23:58

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