Where Have All the Clotheslines Gone?

It’s always tempting to use a simplistic view of our surroundings because, in the short term, it seems to conserve energy and appear practical. But in many cases, and certainly in the case of clotheslines, the opposite is true.

Let’s start with an apparently trivial fact about a clothesline. No matter how taut you pull it, the line sags a little under its own weight. The curve is accentuated if the line’s material is heavier or if you start hanging clothes on it. It may superficially resemble the familiar parabola from high school texts, but it’s actually closer to being a catenary described by a hyperbolic cosine function.

Parabolic shapes in both artificial situations and in nature are actually less common than people imagine. To get a parabolic path from, say, a baseball hit upwards at an angle, there would have to be only gravity acting on it. Then the angle in flight would only be the result of the diminishing and then increasing vertical component combined with a constant horizontal component. But in reality air friction and wind change the flight path into a more complicated exponential function.

The tension of a clothesline has a vertical and horizontal component at every point along its curve. Being at equilibrium the chain-tension’s vertical components balances gravity, while its horizontal components are also countered by forces in the opposite direction. The fact that the tension’s two components constantly change with the rope’s varying angle over every little length of the clothesline is what gives rise to the catenary.

Clotheslines have disappeared from many neighborhoods not only because many people do not appreciate the mathematics of pedestrian objects. If they are a rarity it’s partly because of city bylaws inspired by their perceived unsightliness and the way they hinder the view of more pleasant things like trees and sky. What gives people the luxury of giving aesthetics priority is the existence of the clothes dryer. But clothes dryers, as essential as they may be to those with small apartments and living in temperate climates, suck up a great deal of energy, an estimated 6% of all power generated in the province of Ontario, Canada, for example. Compared to the wind-and-sun-option of clothes-drying, the combination of the mechanical dryer’s tumbling action and high heat removes more lint from clothes, wearing them out faster.

While giving convenience priority over environmental matters and household budgets, people also imagine an unnecessary dichotomy between dryers and outdoor clothes lines. But clothes can also be laid out to dry naturally and discreetly in garages and on racks made for decks and balconies. Without intruding on anyone’s views,  CO2 emissions or radioactive wastes will be reduced, assuming that certain homes rely on an electrical grid still dependent on methane combustion, coal or nuclear power. If not, there are still more benefits of natural drying:

  • as we mentioned, the lengthened life cycle of clothes and the associated savings;
  • the energy that a greener grid saves can be distributed to areas with a bigger ecological footprint;
  • a smaller contribution to the heat island effect of urban areas;
  • clothes racks do not need much maintenance;
  • With less use, dryers last a lot longer. We gave a dryer to my parents and since it shares their burden of laundry with outdoor and indoor laundry lines, twenty-eight years later, it is still working.

Keep in mind that real progress lies not in developing technologies in order to be enslaved to them. We progress when we constantly evaluate their interaction with human nature and assess their health/ecological impact and adjust accordingly.

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Origin and Properties of e

This story will not begin with how the irrational side of human nature and money are indeed at the root of much evil. Instead we will look at how a 17th century financially related-insight by Jacob Bernouilli eventually led us to the discovery of the irrational nature and properties of the number e.

Imagine the other extreme of today’s artificially-low interest rates, an annual rate of 100% = 1 ,  compounded twice a year:

A = Ao (1 + 1/2 )2 = 2.25 Ao   

This equation reveals that after a year, the original investment, Ao,  becomes 2.25 times larger than the original. By applying the same interest rate but with twice the frequency, an original investment of $1000  grows to $2250 as opposed to $2000. However, what Bernouilli noticed is that although further increases in compounding keep increasing the factor, the gains become progressively more miniscule. (See table. Needless to say, Bernouilli did not have a computer and did not use a googol in his calculations. 🙂 )

Compounding Type Frequency of Compounding Factor By Which Ao Increases Additional Amount Gained Over Previous Frequency ($1000 invested)
 monthly  12 2.613… =$1000 [ (1+ 1/12)12 – (1+1/2)2= $363.03
 daily  365 2.7145… $101.53
 every second  365(24)(3600)  2.7182… $3.71
a billion times a year  1  000 000 000 2.71828182709… $0.018
a quadrillion times a year 1015  2.718281828459043…  < $0.00000001…
a googol times a year  10100  2.71828182845904523… none, even with all the world’s $

The limiting factor of  2.718281828… is an irrational number like π; it cannot be expressed as a fraction and consequently its decimals are like some staff meetings, going on forever without a pattern.  The number was eventually called e.  When it was used as a base for an exponential function, it became even more interesting as it surfaced not only in financial formulas but in those of chemistry, engineering, biology and physics.

To see why e surfaces in the representation of many natural phenomena we will first express Bernouilli’s insight as a formula—it’s essentially what we have been using all along, but the number of times the interest has been compounded is n, and as n approaches infinity, we get closer to the value of e:e_limit

Next, we will arrive at this same formula by a completely different and far more bumpy route, but an important one which meanders through several key concepts. Among all exponential functions of the form y =ax,  y=ex is special. To understand why, we have to quantify exactly how fast the function grows.

tangent to e
The exponential curve was obtained by plotting only straight lines, a common trick used in both modern sculpture, architecture and in 1960s”hippies-string-art”. Knowing its rate of change and applying y =mx +b, a computer can easily plot various lines corresponding to different values of x1,  using y=ex1*x+(1-x1)*ex1

From the steepness of the tangents at various values of x, we can see that the rate of change for any exponential function (with a base >1) keeps increasing. How do we quantify it? The mathematical details for those interested are shown at the end of this blog entry. It’s a question of deriving an expression for the rate of change of a function y =ax , which in turn is based on the idea that if we zoom in enough on any continuous curve, we can represent it as a sequence of tiny and gradually steeper segments. If we use variables for points that are extremely close to each other, the rate of change- expression will hold for any point on that particular curve. The point’s coordinates will be the only necessary input needed to yield the instantaneous rate at that spot on the curve. For y = 2x, the rate of change is approximately 0.693(2x). For y=3x , it’s about 1.0986(3x). If we try bases bigger than 2 and smaller than 3, we see that it’s possible to have a base that yields an instantaneous rate of change that comes pretty close to exactly 1 times itself.  If we use the base e that  Bernouilli “stumbled upon” along with a small h-value like 10-6, we obtain a value of 1.0000000:

definition ofe

The fact that the instantaneous rate of change of y = ex (1.000…) = ex has many interesting consequences:

(1) For beginners it’s tied in to the coefficients of 0.693….. and 1.0986 for the derivatives of  2 and  3, since those are the exponents required by e  in order to become either 2 or 3, respectively.

(2)When we invert the x and y coordinates for y = ex  and end up with the function y = ln x (which is what we were doing when we obtained 0.693 for 2) we get a reflection of the function, as if the y=x line was acting as a mirror. The rate of change of that new function is simply y’= 1/x.

(3) And if we use limits to get the instantaneous rate of change for the inverse of the general exponential functions, and use the discovered fact that y’= 1/x when y = ln x, we can travel along a different path to reveal again thatCapture(see end of blog for a mental map)

(4) The reason that some form of y = ex is the solution to many differential equations is tied into the fact that many instantaneous changes are proportional to their own instantaneous amounts like a growing, compounded investment; or a multiplying bacteria colony with adequate resources or a decaying radioactive nucleus. In each of those cases when we isolate the variable of time, on the other side of the equation we find the incremental amount of money, bacteria or atoms as dx multiplied by 1/x. Taking the antiderivative of that product, on our way to isolating the variable of time, leads us to ln x and eventually to an expression of its inverse, a function of e.

To an uncritical eye, an outdoor telephone wire or chain sagging from its own weight may seem like a parabolic curve. But it is not. If we balance a chain’s horizontal components of tension and do likewise for its weight and vertical tension- component, upon dividing we get an expression for the tangent-ratio of the angle between a horizontal component and the chain. The former can be expressed as a rate of change between the y and x coordinates. Its derivative of second-order ends up being related to the rate of change of the chain’s arc length. After some sneaky substitutions, the second degree differential equation can be solved and we reveal that the shape of the chain is a function of cosh (δx/H + c), the so-called catenary derived from the Latin word catena for chain. But a cosh x function is simply defined as  0.5(ex + ex ).    (Again see below for details.)

Even as I do the laundry and hang it out to dry (consistent use of the electrical dryer is a waste of energy and removes too much lint from clothes), I cannot escape the beauty of e.

Mathematical Details:

deriv of e

derivof a^x
deriv of ePt2

 

Sources:

Single Variable Calculus. 7E.  James Stewart. Brooks Cole

Differential Equations With Applications. Ritger and Rose. McGraw-Hill

The Number e  School of Mathematics and Statistics
University of St Andrews, Scotland  

The Catenary. David Maslanka

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