This may be a second copy. If so
apologies.
Hello everyone,
I have what I think should be a simple question but I can't
find a direct answer in any of the atm books that I have. Why does the
mirror become concave when it is ground on top of the tool, and the tool
convex? The books note that the center of the mirror gets more grinding
than the edges, because it is always on the tool. However, the same is
true of the center of the tool. It is always on the mirror.
So, both centers should grind out faster, leading to TWO concave
surfaces. But of course then neither would be in
contact, and grinding would move to the edges until the centers are in contact
again, etc. In equilibrium, the result would seem to result in continued flat
surfaces for both the tool and the mirror.
So there must be another factor.
Two possibilities that come to mind are rotation and gravity. Clearly
rotation of the tool and mirror are crucial to producing a sphere, but since
both mirror and tool are rotated after every few manual strokes, the effect
appears to be symmetric. So, it seems unlikely to me that the rotation
itself is what makes the mirror concave and the tool convex. On the
other hand, gravitational effects are asymmetric. When the tool is
fixed, it is essentially part of the mass of the earth. The
mirror on top, on the other hand, has tiny mass in comparison. The
result I think is that the grinding force is largest on the portion of the
mirror at the edge of the tool when the mirror is maximally extended over the
tool. The effect is a grinding away of the outer areas of
the tool and scooping out of a region of the mirror between the center and the
edge. Note that, when the mirror is fully extended, the force on the
tool is least at the other edge of the mirror (the on the
tool). So, the gravitational effect is not symmetric for the
mirror and tool. In other words, the action of the earth's greater
gravity is to make the mirror concave and the tool convex, I
think.
If there were no rotation on either disk
or of the line of the stroke motion, it seems the result would be a
mirror in the shape of a concave cylinder, and a tool in the shape of a convex
cylinder. It rotation is taking place, the mirror becomes a concave
sphere.
This explanation predicts, for example, that if a robot is
standing fixed to the earth, rubbing two identical disks together vertically
by moving both of its hands back and forth along a line (i.e., controlling for
differential mass/gravity effects), they would remain
flat indefinitely...
I would really like to understand this but the books that
I've looked at don't really speak to it. If I am wrong about this, could
someone please explain why, not in terms of a metaphor please, but in
terms of the physics principles that actually do lead to a concave mirror and
a convex tool.
Thank much!
Tim Rickard