Wednesday, September 17, 2014

The Ugly Index: Part IV

Every year Professors take first year engineering and physics students to the woodshed and disabuse them of notions like "the simplest way to do something is always the right way to do it."

The case of mechanical springs and electrical resistances  provides an interesting contrast.

Picture two plates.  One plate is solidly fixed.  The other rests on two springs that are positioned between the two plates.  Suppose both springs have a spring rate of 10 pounds/inch.  What is the combined spring rate seen by the upper plate?

Well, it depends.  If the springs are in parallel then the two spring rates are simple added together to yield an effective spring rate of 20 pounds/inch.

If the springs are stacked one atop the other then they are in series and one must invert the spring rate to produce a compliance rate (0.1 inches/pound).  Then one adds the compliances together (0.1 + 0.1 = 0.2 inches/pound).  To get the final answer one must invert the results to change the compliance back to a spring rate on one learns that the final spring rate is 5 pounds/inch.  Contrary to intuition, adding two stiffness elements in series makes the system softer.

Sidebar:  Amateur gunsmiths are perpetually surprised to learn that  cutting coil springs results in shorter springs but higher spring rates.

Out of simple perversity, the calculations for resistors are exactly the opposite.  In this case consider two resisters with a resistance of 10 Ohms (Volts/Amp).

In the case of parallel resistors on must invert the resistance to calculate the equivalent conductivity.  0.1 Amp/Volt +0.1 Amp/Volt = 0.2 Amps/Volt.  Invert the result to convert it back to resistance and you get 5 Ohms.

In the case of resistors in series (connected end-to-end) the resistances simple add.  10 Ohms + 10 Ohms = 20 Ohms.

One cannot simply add parameters willy-nilly.  It does not always work.  One must either know what they are doing or draw some pictures to gain an understanding of the underlying physics of the problem before proceeding.

So it begs the question, in the majority of "real life" selection problems should one add "ugly" or should one add "beauty", the inverse of ugly?

Let me offer a simple word exercise rather than attempt a rigorous proof.

Imagine a large...50 gallons easily, crystal punch bowl nestled in ice shavings flown in from the most pristine of Greenland's glaciers.  The finest ingredients are added to the bowl.  The finest rum.  Vernor's aged ginger ale.  Squirt.  Freshly squeezed juice from passion fruit picked by nubile young women.  Nectars and ambrosias of fruits from the Hindu Kush and Pacific tropical islands are added to the punch to enhance the body and add heavenly scents.  Blossoms of orchids and viola and scoops of the richest ice cream float on the surface of the punch.  Every ingredient is the finest, rarest, most aesthetically titillating and the most epicureanly divine.  Beauty in liquid form.

Assume that next to this punch bowl is a water tap that is connected to the municipal water supply.

A very, very small bird flies overhead and defecates.  The turd deposits itself in the punch bowl.  The deposit adds nothing to the beauty.

Given that information, most people will choose to drink tap water rather than the punch.  That suggests that most people guide their decisions not by the simple addition of beauty, but rather by the cumulative summation of the ugly.

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