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?

(In reply to vector sum ideas by Larry)

I think Larry is right.  The vector sum must come back to zero.

If the centrifuge is centred at (0,0), then the centre of gravity resulting from placing the weights must be (0,0) for it to remain balanced.

In particular the x-components (a_i Cos( 2PI/30 i)) must sum to zero and the y-components (a_i Sin(2PI/30)) must sum to zero where a_i are the weights 1...30.

The geometrical approach he suggests, looks good in that I do not think it is possible to construct a closed polygon of vectors for the distinct integer values.

 

Posted by goFish on 2005-10-30 04:45:33