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, . . . (N-1)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 n-gon 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 n-gon.
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
n-gon. Do the perpendicular bisectors of the n-gon's sides all
meet at one point?
Posted by Larry
on 2005-10-18 23:57:32