Perpetual Motion - A Second Case Study


Being of a naturally inquisitive mind, hearing the random expostulations of those around me tends to inspire curiousity, and has been known to make me sit down to over-analyze some ridiculously oversimplified idea.

That being said... I'm sure we've all heard the cliché. A cat, dropped from any height, will always land on its feet. Similarly, a piece of buttered toast, dropped from any height, will always land buttered-side down. These are naturally accepted laws of the universe; while perhaps not governed by any of Newton's abstracts, they are nonetheless empirically sound. Accepting these, one could logically derive that some force overcomes mere probability, and some force acts upon these objects to ensure that a cat will never land on its head, and buttered toast will never be preserved upon falling. Therefore, by strapping a piece of buttered toast to the back of a cat, and dropping the whole mechanism, we would be left with a device which cannot logistically contact the ground. Ever. So long as the toast remains firmly attached to the cat, they will simply float, with all downward kinetic motion directed into a spinning movement. Technically, since this defeats the laws of gravity, one or more of these apparatus could support an infinite amount of mass, leading to the conjecture of cat-toast-monorails and the like.

Now, this is a fine little ideal, for the uneducated. I'm left with something of a sour taste in my mouth, however, which requires a bit of rectification.

For one, how does one define the point at which the forces of gravity cease to affect this cat-toast composite? Logically, it stands to reason that the "ground" defined is any large, horizontal plane upon which one stands. (Double points if carpeted, since the attraction of toast to white carpeting is another unexplained phenomenon.) Therefore, the "ground" is simply an arbitrarily defined plane in a three-dimensional space, given in reference to the vertical location of the toast-cat contraption. Since gravity always pulls downward, there must be a gradual progression whereby the unknown toast-cat forces begin to overpower these gravatic forces, eventually reaching equilibrium. These could exist in one of two relationships. The representation of the two forces in tandem could either be linear (F(gravity) = (Distance from ground)(-x)), or, far more likely, could exist in an exponential function (F(gravity) = (x)^(Distance from ground)).

Now, as any first year economics major knows, the number which seems to dominate most exponential/logarithmic equations is e, or 2.718. We'll factor that into our equation, to arrive at gravity's percent net effect on the mechanism to be F(unknown force) = e^(-distance from ground), and, correspondingly, F(gravity) = 1 - (F(unknown force)), where 0 <= F(unknown force) <= 1. This fails to take into account the effects of acceleration in relation to this unknown force (heretofore referred to as "u"), but since we have little knowledge of u's characteristic behaviors, and whether or not it is governed by the standard 9.8 m/s/s standard we all know and love, we'll temporarily ignore it. Also, this standard lacks a standard unit of measurement; without empirical data, we can only derive a formula, not the distance constants involved therein. For the sake of simplicity, we'll assume the units of distance to be meters, the age-old favorite of the metric system.

Now, we have an object which, as it approaches the "ground", rapidly decelerates, in a smooth curve. As it approaches this ground, it slows more and more-- gravity's effect at 5 m from the ground is still a strong 99.994%, but by 3 m has dropped to 95.1%, and at 1 m is only 63.3%. As the apparatus nears the ground, it slows further and further, but never reaches zero. The important thing to note here is that exponential equations never truly reach zero. At .0000001 meters, gravity is 99.99999% eclipsed, but the apparatus can neither ever truly contact the ground, nor stop falling.

This leaves an interesting conundrum. Firstly, if the unit of distance derived from experimental evidence (which I sadly lack the capital to finance) proves to be less than a meter, and winds up being, say, a centimeter, then we have trouble. This apparatus would appear to hover the tiniest fraction of a centimeter over the ground--but would this be the axis of rotation, the point closest to the ground, or the point furthest from the ground? God help the poor beast if it's not the axis of rotation...

And, as long as we're at it, think about the rotation going on here. Clearly, we have two distinct forces at work. One, the cat ("C"), has an innate inclination to rotate along a single axis until the feet face the ground. The other, the toast ("T"), can rotate in any direction with no ergonomic loss of efficiency. Now, assume that both forces begin acting at the same moment to stop all downward momentum. Clearly, both acting on each other will negate all downward movement, but what then? Depending on the angle at which the apparatus began falling, C's centripetal force may move at a vector identical to T, or exactly opposite to it. This means that the apparatus will either completely fail to spin at all (since opposite vectors cancel each other), or will *not* slowly spin, but rather spin with the combined force of gravity acting upon both objects, directed upward. That could make for a great deal more motion than we had banked on, and will undoubtedly make for a very sick and disoriented cat.

But, these are mere speculations that I can't hope to extrapolate without data. Given the difficulty of strapping anything to a cat, much less something as elusive as buttered toast, I'd say it's a safe bet that this will never come to pass under my observation. However, another aspect of the whole mess draws my attention.

In the buttered toast case, it's the butter that causes it to land buttered side down - it doesn't have to be toast, the theory should work equally well with Ritz crackers. So to save money you just miss out the toast - and butter the cats.

Also, should there be an imbalance between the effects of cat and butter, there are other substances that have a stronger affinity for carpet. Probability of carpet impact is determined by the following simple formula: p = s * t(t1)/t2 where p is the probability of carpet impact, s is the "stain" value of the toast-covering substance - an indicator of the effectiveness of the toast topping in permanently staining the carpet. Chicken Tikka Masala, for example, has a very high s value, while the s value of water is zero. t2 and t(t1) indicate the tone of the carpet and topping - the value of p being strongly related to the relationship between the color of the carpet and topping, as even chicken tikka masala won't cause a permanent and obvious stain if the carpet is the same color. So it is obvious that the probability of carpet impact is maximized if you use chicken tikka masala and a white carpet - in fact this combination gives a p value of one, which is the same as the probability of a cat landing on its feet. Therefore, a cat with chicken tikka masala on its back will be certain to hover in mid air, while there could be problems with buttered toast as the toast may fall off the cat, causing a terrible monorail crash resulting in nauseating images of members of the royal family visiting accident victims in hospital, and politicians saying it wouldn't have happened if their party was in power as there would have been more investment in cat-toast glue research. Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet.

--Erik, bringing you the very best in scientific abstracts since 1984, 5/6/01