[ExI] Thermal expansion - Ball and ring experiment

[Christopher Healey]

Emlyn,

I find it handy to come up with a model, even a bad one, and then shoot
as many potential holes in it as possible.

Consider this starting visualization:

1.  Place the ring with the cutout to the right, as in "C"

2.  The ring is circular, and hence left-right symmetrical, except for
the cutout.

3.  Draw two horizontal lines, dividing the ring into 3 regions, with
the middle region the height of the cutout.  This will make the top and
bottom regions mirror images of each other, and the middle region will
contain just the uninterrupted left-hand-side ring segment that mirrors
the "missing" right-hand-side ring segment that was cut out.

4.  Consider the top (or bottom) region so created (and we'll limit
ourselves to vertical expansion).  As this region expands, it will
indeed ingress upon the middle region (pretending the regions are
disconnected for a sec, like in a cad program).  Let's say this ring
region expands vertically by 2mm into the middle region.  The cutout
*would* become 4mm smaller (2mm from each vertical direction), except
for the fact that the left-hand segment *is* still connected, which is
going to push the outside of the whole ring outward 2mm, which will
exactly eliminate the ingress into the middle region.  So no change so
far.

5.   The middle region's expansion should increase the vertical spacing
of the cutout opening by exactly the same amount (since we've canceled
out the expansion in the other regions), but this number is going to be
relatively small, since not much metal will be involved in this part of
the expansion, assuming a relatively small cutout.

COMPLICATIONS- 

1.  The metallurgical process of forming the ring may skew these
results, due to the atomic alignments.  My visualization above is
assuming the ring was carved out of a block.  If you bent a straight rod
into a closed form, then the expansion behavior will potentially be
aggravated along the curved length of the ring, causing the cutout to
get smaller, rather than larger.  Depending on the exact properties of
that particular ring, and the metal involved, it could increase,
decrease, or stay about the same.

2.  Even having been carved out of a block, there will be some bias
toward expanding along the curved length due to differential stresses
that arise during the expansion; so horizontal and vertical expansion
will be coupled together to some extent, and this will increase as the
expansion itself increases.  This goes beyond my ability to factor in,
but maybe others on the list can elaborate on this point.

-Chris