I found it in Duck Baker’s excellent and thoroughly researched book Complete Fingerstyle Gospel Guitar Solo Collection (Mel Bay, 1997). I was lucky enough to study guitar with Duck for a short while a few years ago, and he told me that his love of gospel music indicates no affiliation with the religion that goes with it. But you can tell from his notes accompanying the score that he was very impressed with the composer of this tune, Shaker church leader Ann Lee. I’m sure he won’t mind if I quote a few lines:

I’ve also included … one of the wordless songs sung by Mother Ann Lee herself. “At the sound of her voice the Pentecostal fire swept the hearts of the congregation, and ‘the spacious apartment would ring with beautiful songs which no man could learn.’” (I am quoting Patterson, who is quoting a contemporary account.) … It is not difficult to credit a state of inspiration for the creation of a melody like “Mother Ann’s Song.” For someone of no musical training or background to compose such an intense, complicated melody beggars belief.

Well, the idea that “no man could learn” Mother Anne’s song may be daunting. Duck’s arrangement is deceptively difficult – it sounds fairly simple but it has taken me a long time to bring it to fluency. It has been no chore though, because its compelling mantra-like quality makes practising it a kind of meditation.  The way I like to play it is to let all the strings ring as much as possible, so that with a little imagination the soundbox of the guitar can evoke the echoey space of a church.

I hope you find it as intriguing as I do.

I was recently reminded that the history of diggers tells a parable about disruptive innovation. Once, there was a highly successful, well-managed, market-leading manufacturer of earth-moving equipment. Its diggers had steel cable pulley mechanisms because this was the best, most mature technology. All its best customers required the superior performance of steel pulleys, so when a few upstart companies started producing diggers operated by feeble hydraulic mechanisms, suitable for nothing but narrow trench digging or heavy gardening, they willingly ceded that end of the market as small fry. Of course the hydraulic technology got steadily better over the years. Every now and then the market leader would look at it and dismiss it; it really was nowhere near as good as their products – perhaps it never would be. The fateful day arrived when cheaper hydraulic diggers were… not “as good as pulley-operated diggers”, but “good enough for most of the market”. Seemingly overnight, the market leader lost the bulk of its market share. Apparently incapable of competing with the new hydraulic technology, the company died.[1]

You have to feel sorry for the old pulley-operated dinosaurs and their makers, outmoded by a scurrying little new breed, incredulous that they faced extinction when their noble claws were still by far the strongest.

Photo taken today, by Jalan Syed Putra, Kuala Lumpur, with my Canon IXUS860IS on a pocket tripod, zoomed right in with the digital zoom from a pedestrian bridge.

I was fascinated by its elegance, and by the strangeness of seeing solid metal, a material you don’t really think of as “bouncy”, rebound almost perfectly.

Sadly, my first one vanished in one of my many removals, so it brought a real wave of nostalgia to find a new Newton’s cradle in a parcel my mum had sent from England to Malaysia for my birthday last year.

As the video shows, if you set it going with one ball, one ball flies off the other end — the epitome of the “equal and opposite reaction” that Newton is famous for. If you drop two balls on one end, two balls fly off the other. The big curiosity for me is this: How does the queue of balls “know” the difference between one ball striking them hard, and two balls striking them softly? Or to put it another way, why don’t all the balls react to the impetus of the striking ball?

Fortunately, I’m lucky enough to have a physicist and mathematician in the family. Despite, or perhaps because of the discouragement offered to him by his early educators (“You won’t benefit from a university education”) my Dad kept at it until he had a PhD in mathematics and a job with IBM at the forefront of computer science. Now in retirement in Whitby, Dad is busier than ever – but never too busy to improve the education of his feckless second son. I started by supposing that it’s something to do with the conservation of momentum, and perhaps the wavelength of the shockwave?

Dad:

Yes, isn’t it an intriguing toy. Newton was good at simple-stupid models which demonstrate vast principles needing little or no specialised knowledge to appreciate. I’m still aghast at Newton’s Bucket, which seems to show the existence of an absolute frame-of-reference in the cosmos, in apparent violation of Mach’s Principle, of which I am a hard-line disciple.

The Cradle (getting back to it) aptly demonstrates the principle, beloved of the Neo-Platonists, that whatever happens in the cosmos happens — and can only happen — in a way which makes the numbers add up. That simple principle expands your mind to comprehend wave equations, quantum mechanics and relativity, whereas “common sense” — intuition induced from day-to-day observation — revolts against such principles. How can those bally balls at the other end “know” that two balls (say) have hit the queue at one end? Why does one not fly off “twice as fast”… whatever that means? I took a glance at Wikipedia, and the treatment rabbits on about shock-waves, totally violating the spirit of conservation of physical invariants.

You are right, it’s to do with Conservation of Momentum. But simultaneously also Conservation of Energy, to the extent that the balls are all the same size and perfectly elastic and non-sticky. If they aren’t, so to that extent the Cradle paradigm breaks down, and the energy of motion of the original ball / two balls / etc., ends up distributed in the system in a chaotic way.

Let m be the combined mass of the balls pulled back by hand. Let v be the velocity with which they strike their brothers. Let M be the combined mass of the balls which fly off the system and V their velocity.

By Conservation of Momentum: mv = MV (1)

Now by lifting a number of balls and letting them go, we deliver kinetic energy to the system:

half m * v-squared

By Conservation of Energy: 0.5 m (v^2) = 0.5 M (V^2)

so (doubling each side) we can write:

mvv = MVV (2)

Now it so happens the *only solution* of (1) and (2) is M=m, V=v. And the only way THAT can physically be achieved is for the same number of balls to fly off the far end of the system, at exactly the same velocity. Nature is squeezed, like toothpaste, into whatever it needs to do to make the numbers add up. The quickest way I’ve been able to dredge from my memory to prove it is:

Substitute mv–>MV by (1) into (2): MVv = MVV
Cancel out MV, leaving v=V,
which substituted back into (1) gives m=M.
QED

The trouble with that simple argument is, it assumes the balls remain together as they fly off. A more comprehensive analysis assigns separate resulting velocities to each of the balls in the system and then uses Diophantine techniques (relying on the solutions being integers, or integrally or rationally related) to show that they must either remain at rest, or fly off with the same velocity. I just haven’t been able to make that one work, though I’ve got close.

me

So if I understand you rightly, if we only needed to worry about conservation of momentum we could have any number of outcomes, such as two balls flying off at half the speed of the one that hit the other side, but we’ve also got to conserve energy, and that has a square power in there, so the system is constrained to a single possible outcome?

By the way, am I right in thinking that momentum is supposed to be conserved even though energy is lost from the system? In that case, where does the momentum “go” as energy is dissipated as sound and heat?

Dad

Yes, that’s the nub of the argument.

However, when I tried to solve the resulting two simultaneous equations, I was dismayed to discover I could not clinch it with more than one moving ball, and began to suspect there may be other solutions. There aren’t if you can assume the balls that fly off stick together (you simply pursue the one-ball argument with “hyper-balls” of mass 2xM, 3xM, etc.)

But why *should* they stick together?

I felt that I could prove that the symmetrical effect was the only one permitted if I knew more about solving Diophantine equations, i.e. which permit only integer solutions, viz no fraction of M. I was then reminded of provocative pictures of droplets falling into water, which are not constrained to integer multiples of M, yet still exhibit “quantized” effects.

I’ve also seen for myself what is effectively the “Galilean Cannon” http://en.wikipedia.org/wiki/Galilean_cannon –in my case a wet rubber ball with a shiny surface (when dry) which when dropped into water from arm-height causes a drop of water to spurt up to a considerable height. It made me wonder about the origin of some earth-grazing comets — you’ll recall my sci-fi novel Interspex offers a suggestion of biological infection of the entire solar system via seawater ejecta. I even see that NASA recently have tried calculating the amount of such ejecta falling on Titan (sorry, don’t have reference to hand).

The Galilean cannon is in effect a Newton’s Cradle with unequal balls: the tiny ball at the end, the bigger ball(s) and the Earth.

Much the same principle applies to theories of formation of the Moon which suppose that a massive body struck the early Earth. Instead of going splat like a bullet hitting a melon, a single lump of roughly equal mass was allegedly blasted off the opposite side, and I’ve heard that simulations have been run which bear this out.

Presumably whatever struck us to form the Moon is still inside us. Is it our iron core? Or maybe it melted to form the water in our oceans? Provocatively the mass of the Moon is 7.4E22 kg, the mass of the oceans 5.2E21 kg — about an order of magnitude less. But you’d expect a lot of the water from the icy missile to boil off subsequently — and there’s no saying how much subterranean water there still is. Water lubrication is conjectured to be at the root of plate tectonics.

Regarding the other half of your question: where does the momentum “go”? — nothing is “lost from the system” if you define your system comprehensively enough. What leaves the vicinity of the balls is energy: radiant heat and sound. Although photons don’t have mass (on current simple theories) they do have momentum, and the same must presumably go for phonons. Also the balls heat up, which although strictly not robbing the system of energy does make it unavailable for propelling mass. Asserting (as physicists do) that Energy & Momentum are physical invariants means that you must add up and balance both quantities independently in all sources/sinks before/after the collision. Heat generated by the balls having a coefficient of restitution less than 1 is the biggest deduction that has to be made from the energy balance-sheet, which will act back (via velocity) on the momentum balance-sheet.

In practice, what I’ve observed in a poor-quality Newton’s Cradle, with lossy balls, is that the balls in the middle don’t remain still. If one ball strikes, they explode away with progressively increasing velocity as you go towards the far end. This is precisely the sort of solution I was hoping to eliminate via my Diophantine argument.

Energy losses due to deformation and molecular friction are therefore presumably the reason why a melon explodes, rather than retaining the bullet and ejecting its mass-equivalent from an “exit wound”. On the other hand, I understand this is precisely the effect designers of armour-plate work hard to avoid, and why modern armour undergoes plastic deformation to absorb and spread the energy of a bullet.

me

So if momentum is conserved in a “lossy” system by the momentum of photons etc., does that mean that the vibration of atoms that constitute heat in matter also embody momentum (even though it’s back-and-forth motion in a solid?)

Dad

Can’t answer that as you’ve phrased it. You’re trying to analyse the problem verbally. A physicist would see it in terms of equations and their solution — and then only try to rationalise the results verbally when attempting to communicate the outcome.

Thus throwing a lossy ball at another one, say a sticky toffee — the ball won’t bounce as much as a perfectly springy one. It may not bounce at all. The relevant simultaneous equations are to me:

…where V u and v are the (vector) velocities, M, N the masses of the balls, and S is the energy mopped up by the “squidge”. I think that’s what you have to solve. I don’t see one needs to consider the “momentum” (if such a thing exists) tied up in the “squidge”. Photons (and phonons) are a different matter: they fly off. So they need their own energy & momentum terms on the RHS of the 1st and 2nd equation respectively.

As I’ve formulated it, there are too many unknown for a unique solution to this pair of equations. Though I think I could solve them if the balls stick together, because then u=v. And if you assert that S is 50% (say) of the energy of impact (=M times the coefficient of restitution), that should give a unique solution.

I’m not going to try to solve it now. (But I’ll report back).

me

Sorry, but there’s still something I’m not quite getting.

If I impart momentum to my Newton’s cradle by setting it in motion, that momentum eventually dies away until it’s at rest again. Where has that momentum gone? Presumably some momentum was imparted to gas molecules, but does that explain all the loss of momentum from the “system”? Where has the rest of it gone?

In terms of energy, we know that the substantial part of the energy was lost in heating up the balls – that is making the atoms vibrate more (faster? with more amplitude? not sure which!) Can we account for the “lost” momentum in the same way?

In algebraic terms, what I’m saying is that: MV = mu +mv + s

where s is the loss of momentum outside the system. Is that not so?

Dad

Your verbal mode of reasoning fails you again, I fear…

Momentum and energy are conserved in “closed” systems only. If you “impart momentum to your Newtons Cradle” (which unavoidably entails feeding in energy) then it is not closed, and you have to expand your system in order to close it. Some physical systems can’t ever be closed: e.g. one which radiates energy into the universe at large.

The word “adiabatic” is used for physical systems in which it’s possible to neglect the effects of heat flow. Otherwise you’re into thermodynamic systems, where the maths is a lot more complicated. I put it to you that the simple conservation laws of energy and momentum we’ve been considering are only applicable to (single) adiabatic collisions. But that little word “adiabatic” is like “Newtonian” or “Euclidean” — it’s an admission that you can whistle for it if you’re hoping for any guarantee your model will actually predict observed velocities.

Physicists always have to simplify a system in order to apply mechanics to it, whether classical or quantum. So the “system” is never more than a mental construct projected onto the real world (whatever that is) — not guaranteed to fit perfectly, or even fit at all. Thus a steel marble is not a (perfect) sphere, a pencil mark on paper is never (perfectly) a straight line or a point. So to treat the asymptotic behaviour of a Newtons Cradle using pure classical dynamics is to push the latter beyond the bounds of validity.

Weasel words, you may think. But if, as a thought-experiment, you want to try to close your system (never guaranteed that you can!) in order to recover some sort of invariant across the event of a collision (total energy or momentum are typical candidate invariants) then you’re better off picking energy than momentum. The reason is that energy is a scalar, and therefore easier to measure, track and account for, whereas momentum is a vector: the vector sum of the momenta of all the components converging on the collision, to be equated to the vector sum of all the bits flying away.

You’ll notice I’m resisting your attempts to force me to account for total momentum before and after a collision and equate the two. That way lies madness.  For one thing, (energy of) linear motion can be converted into (energy of) angular motion — and linear and angular momentum are conserved separately. Any projection of linear momentum is conserved separately: X, Y or Z — or any linear combination of X, Y, Z. Frictional heat is explained in a thermodynamic system as motion of the particles — yes, and they all have momentum. If you want to explain your Cradle in terms of a closed system, then your fingers must be part of the system. The table it’s resting on must be part of the system. If the table had jelly legs, you’d see it wobble as the balls move to and fro… like recoil of a gun. The fact that you don’t is simply due to the fact that the wobble is transferred to the earth… but that doesn’t mean you can ignore it, or imagine it as being “lost from the system” — because then all you’re saying is that the system is not closed, hence momentum cannot be assumed to be conserved.

If you do ever manage to define a closed system to describe your Cradle adequately, you’ll find that at t=0 (when everything is stationary) the (vector sum) momentum is zero. If it isn’t, then you can find a frame of reference in which it is zero (i.e. you can run to keep up with it, or if it’s floating in orbit you can float round with it — in which case it would have to work with springy wires and not gravity.) There’d be no “letting go” with your fingers: you’d need a burning thread, or an electromagnet to serve as a catch. Then as the balls flew together, some sort of recoil of the whole Cradle would take place (so the net vector sum momentum can still be expected to be zero!) And can be expected to remain zero even after the balls collide.

So you see, asking “where momentum goes” is even more hopeless than asking “where the money goes” inside the Ministry of Defence. Yet money, like energy, is scalar, so you can at least picture it to “flow” (though it’s hopeless trying to gauge cash flow by tracking the trajectory of a given pound coin). Momentum on the other hand is not a scalar but a vector: it can’t “flow” — so it can’t “go” anywhere. It simply “is”, before and after a given event — and then only on the plane of existence of your model, not the real world. And to the extent your model is a valid one, then equating momentum before and after a collision will hopefully predict the velocities of the balls before and after a collision.

That’s the best I can do, taking a few moments out to reply before I go out to the shops.

So, what have I learned?

Basically, the system behaves like that because that’s the only outcome that satisfies the physical laws that both energy and momentum are conserved – insofar as we take the toy to conform to an ideal closed adiabatic system.

That’ll do for now!

People frequently ask me why I prefer a vegetarian diet. I don’t want to go into a lot of detail here, but there was a seminal moment a few years ago in Darjeeling. I was visiting a tea plantation and met a youthful Italian couple at the tea tasting house after a tour of the plantation. I forget how the conversation got onto food, but I do recall how these charming young people made a well-reasoned case for avoiding not just meat, but all animal products. Animal farming, they maintained, abuses the animals and the environment, and the vast and diverse plant kingdom contains a substitute for anything you might need.

This argument sat at the back of my mind for a long time before I stopped eating meat, and I continued eating fish and dairy products for a while after that, despite a gnawing doubt that they were, in fact, every bit as bad.

Talking vegan over tea at the plantation in Darjeeling

Then back in March this year I heard Jonathan Saffran Foer on Andrew Marr’s great discussion programme Start the Week, on Radio 4. I liked Saffran Foer’s approach, which was simply to go and look at where his food came from, in order to decide what to feed his young son. I bought his book Eating Animals and read it. Well, there are a lot of very good reasons not to eat animals in that book. Here are a few that I found especially cogent:

• Food crisis? There is plenty of food to feed all seven billion people who live on the planet, but we’re feeding most of it to animals instead. It takes between six and twenty-six calories of feed to produce a calorie of meat. Goats foraging for food on non-arable land are all very well, but that is nothing to do with the food chain that ends on my plate.
• Climate change? You bet. Animal farming emits more equivalent CO2 than all the transport in the world combined. And then some.
• The well-documented, commonplace cruelty in abattoirs is breathtaking, but just as worrying as what happens to the animals is what happens to the people who slaughter them. Their instinct to be humane is frustrated by the industrial imperative to keep the line moving and their harsh working conditions. Where workers cannot be humane, they are dehumanised.
• I knew about cutting down the Amazon rainforest for cattle ranching, and I knew we are fishing the oceans bare, but I didn’t know that our meat supply is creating so much pig/cow/chicken shit that it’s a vast source of environmental pollution, a cost which big food companies easily externalise.
• Big business has taken control of almost all our food production, and the logic of the factory system and the p/l sheet is that sick animals are more profitable than healthy ones. Hence the routine, large-scale addition of antibiotics to animal feed, to keep them alive in their absurdly overcrowded pens and cages. These are the antibiotics we need to fight human disease, and the industry is creating the perfect conditions for antibiotic-resistant diseases to emerge.

Further than this, you’ll have to read the book for yourself. Suffice to say I’m glad I had already stopped eating meat before I read it.

I’ve been vegetarian for two years now, and my friends have been very gracious about accommodating my food choices, but it undoubtedly makes it more work to invite us for dinner. Eating out isn’t as easy as it used to be, though I’m getting adept at ordering off-menu.

My wife is not keen on my vegetarianism. She found it hard to accept, and feels I’m pressuring her to follow suit even though I say I want her to make her own choices. I think she’s uncomfortable with the idea of cooking meat for herself separately from the rest, perhaps because it would violate the social nature of the meal. As a result, she’s also eating largely vegetarian by default, and as I eliminate more and more animal products from my diet, her discomfort has become more acute. She argues that I am rejecting tradition. Perhaps she envisages me visiting her family in Japan and declining even to eat the miso soup because it contains fish stock.

My view on this is that tradition thrives neither when it is followed unthinkingly or when it is neglected entirely, but when it evolves to suit the time. Back in the days when there were only a million people on the planet, and before we started to wage a war of annihilation on the sea with echolocation devices, dragnets and long lines, when fish lived healthy and comparatively unmolested until they were caught by fishermen — if I lived in those days I would fully support eating fish. But times have changed, and so has the food chain, and those traditions are no longer appropriate. Instead, I realise I have to put in some effort to develop something to offer in its place.

Already, I’ve found delightful restaurants and eaten foods that I’d never have gone near without the impetus of the need to find alternatives. There is the vegan canteen at the Chinese temple on Ampang Road (where you get Buddhist chants along with your vegan barbecued pork), our favourite vegan Indian restaurant at the Palace of Fine Arts in Brickfields (where you pay whatever you wish!), and a charming place that does a vegetarian take on the traditional fusion cuisine of Malay and Straits Chinese known as Baba Nyonya. But I can’t rely on these exotic experiences to fill the gap. It’s becoming clear I’ll have to invest some energy in learning to shop and to cook. I believe in principle that home-cooked vegan food can be just as tasty and nutritious as food cooked with animal products. I just need to work out how.

Well, I was at a loose end this weekend, so I took myself down to our local bookshop and found a cookbook called Veganomicon, which despite occasional lapses into folksy Yankee silliness, is bursting with attractive recipes and helpful advice to a novice food bodger like me.
It contains a lot of words I don’t know though. What do agar and tapioca flour look like? What’s the difference between searing, sautéing and braising? And could I tell a mandoline from a crepe pan? I think not.

The only way to learn is to get started. Sitting on a bench outside the bookshop I flicked through the recipes until I found one with reasonably familiar ingredients. I selected Greek-style tomato & zucchini fritters with fresh herbs, and Mediterranean-style cashew-cucumber dip to go with them. It took an hour or so and two supermarkets to find all the ingredients. Thereafter, the book promised that the fritters would take up forty five minutes of my precious time, and the dip could be whipped up in a mere fifteen. After two and a half hours in the kitchen, I finally had the damn things in the oven (having elected to bake rather than fry them).

I never was a big fan of burgers, so I never felt much need for a substitute, but the rissoles turned out quite pleasing nonetheless – very light, and made almost fluffy by the grated courgette in them. The dip had a flavour a bit like Boursin cheese, but more refreshing. Next day, the dip worked beautifully as a base for tomato sandwiches, and a leftover rissole also tasted pretty good with Japanese Okonomiyaki sauce (Worcester-sauce to you and me). Overall, I’d give the results a five out of ten. Not bad, but not really proportionate to the effort.

Nevertheless, I’ve dipped my toe into the chilly water of vegan cookery. I’ve learned what dill and fresh oregano look like, and that breadcrumbs seem to be batter flakes rather than crumbs of bread. I’ve located the kitchen scales and tracked down the obscure supermarket shelf where the greaseproof paper lives. Next time I’m called upon to grate and squeeze a cucumber, I’ll do it with aplomb.

And at the end of the day, nobody got hurt.

Photo courtesy of Tomoko

I’ve just been to a superb performance of Hindustani classical raga by septuagenarian maestro Ustad Usman Khan, who claims dynastic descent from a famed musician of the Mysore Maharaja’s court. You can see it would take the work of several lifetimes to master the instrument and the music so profoundly! He was accompanied on the twin drums called the tabla by a very talented younger chap called Prakash Kandasamy. It was organised by the Temple of Fine Arts (in Brickfields), and held at the Tunku Abdul Rahman Memorial Hall (perhaps worth a future entry of its own) here in KL.

The music was soulful extemporising to complex repeating figures. Khan announced the number of beats for each raga (7, 14 and 16, I recall) – and I found the unfamiliar rhythms were a bit easier to get into when I managed to count along.

As a bit of a guitar player myself I was enthralled by Khan’s repertoire of techniques –  he pulled trick after trick out of the bag as the tunes built steadily to huge crescendos of energy, and it was hard to believe all that breadth of sound was coming from just a few strings and a couple of little drums. The beautiful sobs and coos made by bending the bottom string (up to 7 steps!) put me in mind of blues electric guitar players, the jangling sympathetic strings reminded me of an autoharp, and at times he used doubled octaves Wes Montgomery would’ve been proud of. Most of the melody was played on just the bottom string (called a baajtaar), so there was a lot of to-ing and fro-ing up and down the neck — at times it looked like he was playing a trombone!

The rhythmic precision and interplay with the tabla was astounding too.

These weren't the instruments played, they were just set out on display in the foyer. Photo by Tomoko

The instrument itself is positively Byzantine, with every one of twenty strings. Between sets he retuned the instrument to an entirely different key and voicing, even moving a couple of frets around! Khan said that after he gets all twenty strings perfectly in tune he has to look inside himself and get all his internal strings in tune too before beginning to play. Aw, how sweet. Reminds me of the joke about the genie and the twelve-string guitarist… (oh, never mind).

by Yankra, Wikimedia Commons

The announcer of the evening (an earnest young chap who gushed at length about the artistic merit of the music and the venerable guru) explained that only recently (and in part through Khan’s efforts) has Hindustani music become established in Malaysia, because Malaysia was originally “full of south Indians”.

I’ve heard this before – the majority of Indians in Malaysia seem to be Tamil speakers from the south. I have to say I’ve liked the snippets of Tamil music I’ve heard too. I’m curious to find out how different they sound…