A metal ring is heated.

Think of it this way...  All linear distances expand proportionally to
their lengths.  The circumference of the ring is a linear distance, so
it expands proportionally to its length.  This expands its radius.
For that matter, the diameter of the hole is also a linear distance,
and it, too, expands proportionally to its length, making a bigger
hole.

This is verified by the historical use of heat to expand iron hoops to
slip over wooden wheels to make iron-rimmed wooden wagon wheels.  When
the iron cooled down, it shrinks, to bind tightly to the wheel.  I
believe that wire-rim frames to glasses were applied the same way (and
might still).

-- Buddha Buck <bbuck@nyx10.cs.du.edu>

[ Editor: from the October 1994 thread "EXPANSION OR CONTRACTION". ]

<jaiprakash@kuhub.cc.ukans.edu> wrote:
>If you take a steel ring of (say) 10 cm dia, 1 mm x 1 mm rectangular
>cross section and heat it the entire ring will expand - i.e. the O.D.
>and I.D. of the ring will increase.  Now, consider a 10 cm dia and 10
>cm long cylindrical block of steel and drill a 1 cm hole through the
>middle.  Now if you heat this block will the I.D. of the annulus
>increase or decrease?  If it decreases then is there a optimum
>thickness (or cross section) where there is no change in I.D with
>increase in temperature ?

The inside diameter of the annulus increases.  The way to see this is
to note what would have happened to the metal you drilled out if it
were still in place.  Since it would expand, the metal around it also
must expand.

-- Dave Dodson <dodson@convex.com>

This isn't quite right.  The fact is that the bit you drilled out ISN'T
there.  So it doesn't expand.  So it doesn't apply a force onto the
inside of the ring.  The inside of the ring is actually more free to
move towards the centre if it wanted to, in the same way that the
bottom of a quarry will 'heave' upwards because its load has been
lightened.

In fact it does do this (the ring will actually get thicker) but
relative to the circular centre-line of the ring.  The circular
centre-line also expands, however.  (Imagine a circle of people bodily
touching each other - excuse the phrase.  If each suddenly starts
stuffing themselves and grows fat in all directions, you might think
the hole might get smaller.  But in fact each person will be pushing
against each other and so the ring must grow in diameter and
circumference.  I suspect this effect always negates the other one.
Anyone good at geometry out there ?  Of course this is much
over-simplified as there are shear forces, Poisson's ratio effects,
and edge-effects to consider for a steel ring.

-- Devji Vekaria <VekariaD@LILHD.LOGICA.COM>