Autumn Leaves

 Autumn Leaves (or Hojas de Otoño)

     The streets in our Colonia in Mexico City (Colonia San Miguel Chapultepec) are typically lined with trees. They nestle in amongst the houses. Their roots thrust up the concrete slabs of the pavement and gradually mould themselves to the spaces they have thus created. Their branches tangle with the myriad phone lines that still drape themselves from pole to pole and cluster round the high eaves of the apartment blocks. Where a protective iron railing once circled a young tree, now the tree has engulfed the iron; the smooth bark gently creeping round till it meets bark and fuses. 

     Trees here are valued for their shade, respected and preserved by common consent. If a trunk leans over the pavement the pedestrians step aside, or duck. If you want to roof-over the yard to make a new room, the tree remains in the centre of the new room, finding its way up through the roof, the terra cotta tejas pushed back each year and the floor slabs chipped to free the trunk. I have seen a tree that grows through an iron fence, and another the grows in line with a boundary wall, which obligingly curves out and round the tree. 

     Popular species include a thin-leaved ash (Fraxinus uhdei) that towers upward and throws a light and dappled shade. And a small-leaved weeping fig (Ficus benjamina), which forms a dense canopy and throws a deep shade, the roots of which, when given space, come welling up and over the pavement, writhing in slow motion. Also popular is the Liquidambar (Liquidambar styraciflua) with its three-pointed leaves, fine autumn colour and prickly fruit. You will find the occasional Purple-Orchid tree (Bauhinia blakeana) and the yellow-flowered Tulipán mexicano (Hibiscus rosa-sinensis), obviously planted for their flowers.  Occasional here, but magnificent elsewhere in the city, you can see the astonishing, jaw-dropping Jacaranda (Jacaranda mimosifolia) whose lavender-coloured flowers in early spring completely cover the canopy. 

     Just now, in the last weeks of October, the streets are littered with fallen leaves, dry and yellow. For the Fraxinus and the Ficus, though evergreen, are also deciduous; the new leaves emerge even as the old leaves are falling. Housewives and caretaker are, for the next four weeks, repeatedly sweeping the pavement outside their property. I wrote before about the "Street sounds in our Colonia" but omitted to mention the gentle swishing sound that, night-after-night, I head from outside on the street; though puzzling at at 3 a.m., it turned out to be a night porter whiling away an hour or two of vigilance by sweeping his bit of pavement. 

Pocket-calculator Science (1)

 Pocket-calculator Science (1)    

    I enjoy viewing the earth as a sphere, for then I can calculate its volume, surface area, etc using the tricks I learned in middle school. 

    OK, it is not quite spherical, for the circumference through the poles is 39,942.209 km, while the circumference at the equator is 40,074.156 km. (I shall sometimes simplify by assuming it is a sphere with an average diameter of 40,008 km).

    The force of gravity (Fg) was found by Newton to follow the equation 

Fg =G.m1.m2/r^2

where G= the universal gravitational constant, m1 and m2 the mass of the man and the earth respectively, and r is the distance between the two centres of mass.  Imagine a man of mass 72 kg. Clearly, when standing at the north pole he is closer to the centre of the earth than when standing at the equator.  

    Taking the gravitational constant (G) as 6.674 x (10)^-11 ( N. m^2/Kg^2)

and the mass of the earth (m2) as 5.972 x (10)^24, and Newton's relation: F =G.m1.m2/r^2, from these I calculate the gravitational force at the north pole (Fnp) to be 9.8628 Newtons;  while the gravitational force at the equator (Feq) to be 9.798 N. (An average value is often quoted as 9.81 Newtons.)

    Applying this geometrical effect alone (ignoring any centrifugal effect) I would expect the man (of mass 72kg) would weight at north pole 72 x 9.8628 = 710 N; and at the equator 72 x 9.798 = 705 N. 

    However, the earth rotates on its axis once a day, relative to the sun. (Relative to the fixed stars it rotates a bit more because of its rotation round the sun in one day; approximately 361º in a 24 hour day, or 15.04º in one hour.)

    At the equator (circumference 40,074,156 m`), we are rotating eastwards with an angular velocity (𝜔) of 2𝛑 radians per day, but a quasi-linear velocity of 40,074.156 ÷ 24 = 1669.57 km/hr. The inertial centrifugal force (Ff), regarded as acting radially, is usually calculated  as F=m.r .𝜔^2 (where m signifies mass and r radius). A man of mass 72 kg (weight = 710 Newtons at the north pole) would find his weight reduced at the equator not only because of his greater distance from the centre of the earthe, but by a small extra amount due to the centrifugal force acting (as far as he is concerned) vertically upwards.

    If T is the rotational period (1 day, or 86,400 s), we can write the rotational velocity (𝜔) as 2𝛑 ÷ 86,400 radians per second.

    We can then write the centrifugal force (Ff) as:

    Ff = m.r.𝜔^2 = m.r.(2𝛑 /T)^2

Taking r as 40,074.156/2𝛑 km = 6,378 km; or rather 6,378,000 m:

    Ff = 72 x 6,378,000 x (2𝛑 ÷ 86,400)^2 Newtons = 2.4286 Newtons. (Note that at the north pole the man rotates at the same speed, but r= 0, so he experience no centrifugal force.)

    So, combining the geometrical effect of the greater radius at the equator and the spin which is negligible at the north pole, a man that weighed 710 Newtons at the north pole would be expected to weigh only 705 - 2.43 = 702.53 Newtons at the equator. 

(Next post Coriolis, if I succeed in understanding it!)

(Comments are welcome to cawstein@gmail.com)

The Brick Wall

 The Brick wall

    I suppose it is the same for most retired folk, but it seems to me that the world in going to the dogs, going off the rails. If we continue in this direction we are heading for an impasse. To make the image more visible I will call it a brick wall.

    Take for example our UK National Health Service. It seems to be running out of money and giving less and less satisfaction. I wrote about this in 2013, a period of austerity following the banking fiasco of 2008 and the 2010 election that produced a ConLib coalition. I was then arguing against austerity cuts and calling for an increase in taxation 'specifically for the NHS'.  A call subsequently made by others.

    In a similarly outrageous companion piece I expressed my alarm at 'mission creep' of the NHS. New expensive tools and therapies are constantly being developed. Since the NHS was conceived, we have developed CT (1971) and MRI (1977) scanners, kidney (1960), liver (1968), and heart (1968) transplants, in vitro fertilization (1977), and now gene therapy (2023). There is no foreseeable limit to the development of more and more costly therapies. (Thus, there are 19,000 other protein-coding genes, all mutating at the standard rate of 1/100,000,000 to 1/1,000,000,000 per generation.) A billionaire could have his genome repaired every year, but that service cannot be available to all. 

    Deciding where to stop repairing a failing human being and to let nature take its course opens up a new area of ethics, into which we have barely taken the first tentative step. Or rather, deciding the point at which it is not the responsibility of the state to fund the repair work (which can perfectly well proceed with private funds, if those are available). I suggested, in 2013, that the state's responsibility might be deemed to cease when the patient reached the age of 70. (I now wonder about suggesting the age of 80 years.)

    In January 2025, when I reached the age of 83, I found I wanted the surgical repair of bilateral inguinal hernias, and found myself queuing at the local NHS hospital. Very straight-forward, my sister-in-law reassured me; absolutely routine. Yet the NHS could not find a bed for me. I was scandalised that the (free) NHS, to which I thought of myself as being loyal, was failing so lamentably. 

    After waiting 4 months I remembered (with some embarrassment) my earlier conclusion that it may not be sufficiently in the state's best interest to spend public money repairing a 70 year old citizen, let alone an 83-year-old. Due to my persistent good fortune, I had plenty of money in my "re-roofing fund"; indeed, it was totally untouched. So I went private, and got both sides done in the same operation for a cost of some £4500.

    My updated suggestion, therefore, is that the present system may not be so bad. Perhaps GPs could give a stronger hint to well-off, elderly, patients that they could quite honourably jump the queue, and have a private bed if they were prepared to pay. They could think of it as a voluntary contribution to the coffers of the NHS. 

    It is marvellous to find that the majority of citizens in Britain are willing to spend public money to help less-well-off citizens. But public money is largely other people's money. It would be even more to the nation's credit if there were some citizens willing to go over and above the average donation, and voluntarily to forego their fair share of the public benefits. 

    (Comments are welcome to: cawstein@gmail.com)

     

Age

Age & Aging

    Until  the last fortnight, I think I treated life as everlasting, and myself as immortal. Not explicitly and to the letter, of course, for I know that there is an allotted span to the life of man; that, barring accidents, "the days of our years are threescore years and ten" ***. I know also that "80 is the new 70", or some such figure, for we are definitely living longer than our grandparents (on average). But in many other ways I completely ignored the fact that I was hurtling towards my end. As a youth I did not contemplate death, but life rather; nor did I notice forgetting, but rather I thought of myself as learning. And, until quite recently, I thought of myself as learning still. Also, that I was playing my fiddle better each year; by guile, perhaps, if not by dexterity. I did not write a will until I was 60, and even then I wrote it only because I was setting off on an adventure. (A minor adventure, perhaps, but still cut off from the security of 'home'. And every billion plane embarkations yields 73 fatalities.)

     I do not know what, in the last fortnight, triggered this change of viewpoint. Perhaps I suffered a mild stroke? But I think not. Or was there a step-change in my biometric data? Again I think not.  I packed my suitcase for 10 weeks in Mexico as I have for the 10 years past, using my list of things-taken-last-year. And still I forgot to pack my UK ➙ USA adaptors; no doubt because I was concentrating so hard on switching off the gas, while clutching my passport. On my last evening with my son and daughter-in-law in Brookline we tuned up two fiddles and a mandolin only for me to discover with horror that I could no longer sight-read; even in the first-position. What then is there left to live for?

    Imagine a stream, which I could normally take in a stride. I have slowed of course, but still can leap the stream. Then comes the day when such a leap seems out of the question, and I am cut off. It is an all-or-nothing thing. 

    For a week now I have been living with this new undercurrent to my thoughts. That, if I only have another decade, and if I am to write anything worthy, I shall have to plan my time rather rigorously. If, for example, I am to solve (for myself) the Coriolis force and the pKA of water, if I am to tighten, lay-out and transmit my (perhaps important) views on morality, let alone politics; I must focus. I must forego those happy but time-wasteing computer-sessions with Sudoku, or Duolingo, and those TV drama series set on Caribbean islands. I must delegate (if I can) the time-consuming tasks I do for the Historical Society and the local Quaker meeting. 

    But this morning, as my accustomed time for morning coffee found me roaming the cereal section of a supermarket, I realised I was feeling a whiff of faintness. Two minutes later, as I watched the second-hand bookseller raise the shutters of his higgledy-piggledy shop, a second deeper wave forced me to push rudely past him muttering something about "Silla" and "Necesito sentarme".  And I thought to myself, what if it is not even a decade but three years! What if it is only one year!!

    I went my wiggly way up to 'La Milla' and ordered an americano and an almond croissant which was brought to me up on the little roof garden that looks out over Calle General Molinos Del Campo. There, no longer experiencing a trace of faintness, nor indeed of morbidity,  I sketched these few paragraphs. And plotted an Essay on "The Brick Wall", and another on "Pocket-calculator Science", and another on "The Search for New Music". 

*** Psalm 90, v. 10