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 this has nothing to do with 'goFish'.
I
once proposed the thought of a "Pennyfarthing Bicycle",
Potential/Kinetic Energy and Simple Harmonic Motion as a means to a
solution of this.
I believe that too many of us looked too deeply into this problem and so created a mass of 'red herrings'. I think that we have to construct the three balls as a solid entity and allow them to rest within the bowl.
Consider my specific account and then generalise it.
In
the words of a flamboyant, and somewhat eccentric US physics professor,
Julius Sumner Miller, who made several trips to Australia in the
196070's, "Let us propose an experiment."
Experiment 1.
I have a rod of uniform profile and density which I place on the inner surface of a friction free hemispherical bowl.
I
note that the rod assumes a gravitationally horizontal position with
its Centre of Mass being directly below the centre of the hemisphere.
Experiment 2.
Under
the same conditions I take a triangular plate (for ease of reference it
is a 345 right triangle) and place it so that in its horizontal
position, the three vertices touch the surface.
When I release the plate, the Centre of Mass brings the plate into equilibrium.
The Centre of Mass will again be directly beneath the centre of the hemisphere.
When
this is so the three vertices of my 345 model will be at different
heights from each other above the horizontal plane, the right angle
will be the lowest with the most acute being the highest. It
seems from this circumstance that the vertices may be 'graded' from low
to high according to inverse size of their value. And I would note that
the Centre of Mass lies on a hemisphere which has a radius less than
that of the containing hemisphere.
Experiment 3.
I
am now replacing the plate with three balls whose radii are 1, 2 and 3
units. The Centres of Mass of these three will form my 345 triangle.
I
am anticipating that the Centres of Mass lie in the same orientation to
the System's Centre of Mass (Gravity) as the vertices did as in
Experiment 2. If that is correct, can I now
begin to calculate the location of the vertices in relation to the
System's Centre of Mass, and thereafter the orientation of the place
which intersects the centres of the three balls, thereupon allowing me
to calculate the angle of the plane that Bractals so wishes?
Caution:
I am quite aware that I have made a range of assumptions based upon a specific situation.
I
expect my 'model' to be thoroughly scrutinised for fallacies; while it
looks like the right way to go, I am not completely comfortable with my
proposition. In that context consider that it
is most difficult to 'watch' how these objects behave under such
experimental conditions (I do not have the controls of Prof Julius at
my disposal).
Additionally, Bractals has recent puzzles which appear to be a theme around similar elements. I have not checked on them, but their solutions may add to this.
[Edits here were to remove some unsightly <o.p.> tags]
Edited on February 2, 2006, 12:33 am

Posted by brianjn
on 20051120 08:36:57 