This represents a circular balance.

                       
         o           /|\           o
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A light circular disk is needle point mounted. It has a graduated scaled track "o" from which identical measuring pans may be suspended.

Suggest the minimum number of pans as well as the minimum weighings needed, and the strategy required to determine which one of 9 coins is in weight variance with the other 8 in a visually identical collection. You should also be able to determine if the variant coin is lighter or heavier than each of the others.

At the end of each weighing, the coins being weighed will be on the balance, and the balance will be in equilibrium.

For additional discussion:
The set of nine has 7 regular coins with one weighing lighter, and the other weighing heavier than the others.
How might the circular balance assist?
Does a disparate difference in weight of the lighter to the normal than the heavier to normal influence the procedure? [This thought was raised at review.]

(In reply to equilibrium? (amendement to solution) by Paul)

Paul,
this took some time to read through.

Having done so I like to firstly comment on your third paragraph.   Because the pans are attached to the balance by a circular track the needle point must always be at a central hub.  In that paragraph you are describing what is analogous to placing the needle point at the centroid of a triangular plate of uniform thickness, but because of variance in weights the effect is to render that "thickness" non-uniform; what you have is a triangle of gravitational forces and you are proposing to locate the centroid of those forces.  Is that a fair assessment?

Paragraph 4 deals with two issues.  The proposal here is what my diagram at a  previous comment (without words) was hinting;  BX is fixed as is the length AX, but the chord length AA is variable.  The position of that chord (but Principle of Moments) determines the equilibrium (I had some difficulty with "lower" and "higher" as I interpreted them as reference to weight rather than elevation above the horizontal).

Paragraph 4 also addresses my discussion question for which I had no answer.  Paul has detailed a scalene triangle of forces which points up how this would be solved.  I will apply Paul's thoughts to detailing the solution to various cases of this scenario.

Well thought out Paul.

Posted by brianjn on 2008-03-27 07:42:57