A circular centrifuge has 30 slots spaced evenly around its circumference. Thirty samples need to be spun in the centrifuge, their masses being 1g, 2g, 3g, . . . 29g, 30g. How can all the samples be placed in the centrifuge at once while keeping it balanced properly?
For what other values of N is it possible to balance an N slot centrifuge with samples weighing 1g, 2g, 3g, . . . (N1)g, Ng?
My first thought was a sum of X and Y components.
But maybe one can imagine a graphical vector solution, where each
vector force has magnitude or length equal to 1, 2, 3, ... centimeters
; with a change in angle from the last vector of 2 pi / n. In
other words, construct an ngon where sides are of lengths 1, 2, ...,
n; all angles are equal. The problem then becomes finding an
ordered set of values for lengths that lead to a closed ngon.
The math will probably be the same, but maybe there's a geometry trick
that makes it easier. Like considering the n triangles
formed by drawing spokes from some central point to the vertices of the
ngon. Do the perpendicular bisectors of the ngon's sides all
meet at one point?

Posted by Larry
on 20051018 23:57:32 