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 ?
Three balls in a bowl.
Without loss of generality, we may place the (centres of the ) balls A, B and C on the xy 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 = (a1b1)^2 + b2^2
(5) (b+c)^2 = (b1c1)^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 xy 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 20051120 11:23:58 