Categories

## A three coffee puzzle

Not a question of how many coffees are acceptable before lunch, but an astronomical conundrum with consequences for your cup.

It starts with gravity. Perhaps you remember that Newton came up with a set of equations describing the laws of gravity. You may even remember the essence of those equations, that the force between two masses is proportional to their product and inversely proportional to the square of the distance between them. If we wanted to phrase it mathematically, the force, F, is given by:

F = GMm/(r x r)

Where G is a constant and r the distance between the masses M and m.

Which is all very well, but suppose we have three masses, or four? M, m and M’, m” for example. If we happened to drop an apple (mass = m) between the moon (mass = M*) and the Earth (mass = M), how exactly, and where exactly, would it fall? How do we add an extra mass into the equation?

It is one of those problems that can seem far removed from your coffee cup, but in fact, the connection is quite close.

But although you may not often drop an apple somewhere between the Earth and the Moon, the question became relevant recently when astronomers observed a dusty disc, the sort of environment that is capable of planet formation, surrounding a three star system. The stars are found in the constellation Orion, which is visible in the evening at this time of year (autumn/winter) from the Northern Hemisphere.

Although these dusty discs are thought to be a host to planetary formation, astronomers have yet to observe any planets actually forming out of the dust. It is thought that in some cases, the gravitational perturbations caused by multiple stars at the heart of the dust clouds could lead to the formation of planets. And so the system in Orion, with three stars in the centre of the dust cloud was perfect to observe the effect of the three stars on the integrity of the disc. Over 11 years, the astronomers recorded the system and then included modelling into understanding how the planetary disc was breaking up. But of course, to do this, they would have needed to understand how the gravitational force is affected by having 3 or more interacting masses.

To solve the problem requires mathematical functions known as a “Bessel functions”. These functions were first described by the astronomer Friedrich Wilhelm Bessel in 1817 who used them for exactly this sort of problem. But they don’t just apply to describing the gravity between three or more objects. They can be used amongst other things to understand heat transfer, to model the microwave fields in a microwave oven and to understand vibrations on your coffee.

Because when you see a series of concentric circles on the surface of your coffee where the table underneath the cup is vibrating, or when you see more complex patterns as you drive a take away cup over a rough table surface, these patterns can be described using exactly the same Bessel functions as would have been used to model the star system in Orion.

And so there is a direct link between the maths describing the planetary formation in a star system visible in our night sky and the patterns of your coffee cup. But if you want to drink your coffee while gazing at Orion, you may want to stick to decaff, or wake before dawn.

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## A Weight-y issue

Ever swung a bucket of coffee round in circles swooping down towards the floor and then over your head? Why would you, you may well ask? Well, the answer may surprise you. It’s all about turbulence.

We have probably all come across turbulence, perhaps by watching how milk is added to a black coffee or seeing the steam interact with the air as it evaporates off a hot mug of tea. But it turns out that there is a lot that we do not yet understand about turbulence and this is where the bucket of coffee comes in.

Waves on the surface of a coffee can be dominated by gravity or capillary effects. Capillary waves are short wavelength (higher frequency) waves that are forced into oscillation by the effects of the surface tension of the liquid pulling the surface of the coffee back into shape once its been distorted. Gravity waves are longer wavelength (lower frequency) waves where the disturbed surface of the coffee is pulled back into shape by gravitational effects rather than surface tension effects.

The frequency at which there is a crossover from gravity dominated waves to capillary dominated waves is dependent on both the density and surface tension of the liquid as well as the strength of the gravitational acceleration experienced by the mug of coffee. (We’re getting to the bucket). On Earth, the gravitational acceleration is 9.8m/s, the ratio of a liquid’s density to surface tension is quite similar for many liquids and so the transition frequency between these two regimes is generally in the region of 10Hz.

What this means is that if you wanted to study the turbulence affecting one type of wave only you could measure at higher frequency (and so measure capillary waves) or measure the turbulence in a liquid in lower gravity eg. on the International Space Station (so that capillary waves dominate at lower frequencies too). But both of these types of measurement don’t give any insight into what’s happening to turbulent waves sustained by gravity, such as Rossby waves which travel the whole circumference of planets with atmospheres and affect the weather in different parts of the globe.

So how could you study turbulence in the gravity dominated surface waves of water? It goes back to the bucket mentioned earlier. By putting a freely moving bucket (the authors called it a ‘gondola’) at the end of the arm of a centrifuge of 8 m diameter, the authors of a recent paper created an effective gravitational force on a liquid of up to 20x the value of the Earth’s gravitational acceleration. It’s sort of like the bucket of coffee being whirled around in a circle apart from a lot bigger and capable of moving at up to 67 rpm! This meant that they could measure the effects of turbulence on gravity driven waves up to about 100Hz allowing them a large frequency range over which to compare their results to theoretical predictions.

And when they did so, they proved one nagging problem for theoreticians studying turbulence: the size of the ‘container’ becomes important, something that models had previously neglected. For the 23cm wide bucket of distilled water used by the authors, this may be something that we can easily visualise but the research has consequences for how we understand the Rossby waves that circle our planet as well as the large wavelength waves in oceans. Slightly more connected with coffee (or at least doughnuts), the results are also important for understanding turbulence in plasma waves in tokamaks.

You may have better things to do over the holidays than swirl a bucket of coffee round and round while watching for the waves on top of it, but if you are stuck for something to do…

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## Telling the time with an Aeropress?

On occasion, it takes a change in our routine for us to re-see our world in a slightly different way. And so it was that when there was an opportunity to borrow an Aeropress together with a hand grinder, I jumped at it. Each morning presented a meditative time for grinding the beans before the ritual of preparing the coffee by a different brew method. Each day became an opportunity to think about something new.

Perhaps it is not as immediately eye catching as the method of a slow pour of water from a swan necked kettle of a V60, and yet making coffee using the Aeropress offers a tremendously rich set of connections that we could ponder and contemplate if we would but notice them. And it starts with the seal. For those who may not be familiar with the Aeropress, a cylindrical ‘plunger’ with a seal tightly fits into a plastic cylinder (brew guide here). The first stage of making a coffee with the Aeropress is to use the cylinder to brew an ‘immersion’ type coffee, exactly as with the French Press (but here, the plunger is on the floor of the coffee maker). Then, after screwing a filter paper and plastic colander to the top of the cylinder and leaving the coffee to brew for a certain amount of time, the whole system is ‘inverted’ onto a mug where some coffee drips through the filter before the rest is forced out using the plunger to push the liquid through the coffee grind.

Immediately perhaps your mind could jump to water clocks where water was allowed to drip out of two holes at the bottom of a device at a rate that allowed people to time certain intervals. It is even suggested that Galileo used such a “clepsydra” to time falling bodies (though I prefer the idea that he sang in order to time his pendulums). With many holes in the bottom of the device and an uneven coffee grind through which the water (coffee) flows, the Aeropress is perhaps not the best clock available to us now. However there is another connection between the Aeropress and the clepsydra that would take us to a whole new area of physics and speculation.

When the medieval thinker Adelard of Bath was considering the issue of whether nature could sustain a vacuum, he thought about the issue of the clepsydra¹. With two holes at the bottom and holes at the top for air, the clepsydra would drip the water through the clock at an even rate. Unless of course the holes at the top were blocked, in which case the water stopped dripping, (a similar thing can be observed when sealing the top of a straw). What kept the water in the jar when the top hole was blocked? What kept it from following its natural path of flowing downwards? (gravity was not understood at that point either). Adelard argued that it was not ‘magic’ that kept the water in when no air could go through, something else was at work.

What could be the explanation? Adelard argued that the universe was full of the four elements (air, water, fire, earth) which are “so closely bound together by natural affection, that just as none of them would exist without the other, so no place is empty of them. Hence it happens, that as soon as one of them leaves its position, another immediately takes its place… When, therefore, the entrance is closed to that which is to come in, it will be all in vain that you open an exit for the water, unless you give an entrance to the air….”²

Now, we would argue that whether the water flows down and out of the Aeropress, or not, depends on the balance of forces pushing the water down and those pushing it up. The forces pushing the water down and out of the clepsydra, or Aeropress, are gravity and the air pressure above the water in the cylinder. Pushing it up, it is only the air pressure from below. Ordinarily, the air pressure above and that below the water in the Aeropress are quite similar, gravity wins the tug of war and the water flows out. In an enclosed system however (if the holes at the top are blocked), were the water to flow out of the bottom, the air pressure above the coffee space would reduce. This makes sense because, if no new air gets in, the same amount of air that we had before now occupies a larger volume as the water has left it, the pressure exerted by that air will have to be less than before. A reduced air pressure means a reduced force on the water pushing it down through the filter and so the force pushing the water down can now be perfectly balanced by the force (from the surrounding air) pushing the water up: the water remains in the Aeropress. The only way we get the coffee out is to change the balance of forces on the water which means pushing down the plunger.

But perhaps it is worth stepping back and imagining what the consequences could be of having the idea that the universe was just full of something that had to be continuous. You may find it quite reasonable for example to consider that vortices would form behind and around the planets as they travelled in their circular orbits through this ‘something’*. Such vortices could explain some of the effects of gravity that we observe and so there would perhaps be no urgency to develop a gravitational theory such as the one we have. There would be other consequences, the world of vacuum physics and consequently of electronics would be significantly set back. In his lecture for the Carl Sagan Prize for Excellence in Public Communication in Planetary Science, The Director of the Vatican Observatory, Br Guy Consolmagno SJ explored previous scientific ideas that were almost right, which “is to say wrong” (You can see his lecture “Discarded Worlds: Astronomical Worlds that were almost correct” here) If it is true that so many scientific theories lasted so long (because they were almost correct) but were in fact wrong, how many of our scientific ideas today are ‘almost correct’ too?

It makes you wonder how our preconceptions of the world affect our ability to investigate it. And for that matter, how our ability to contemplate the world is affected by our practise of doing so. They say that beauty is in the eye of the beholder. For that to be true, the beholder has to open their eyes, look, contemplate and be prepared to be shown wrong in their preconceptions.

What connections do you make to your coffee brew each morning? I’d love to know, here in the comments, on Twitter or over on Facebook.

* Does a connection between this and stirring your freshly brewed Aeropress coffee with a teaspoon trailing vortices stretch the connectivity a bit too far?

¹ “Much Ado about Nothing: Theories of space and vacuum from the Middle Ages to the Scientific Revolution”, Edward Grant, Cambridge University Press, (1981)

Categories

## Light and gravity at Tab x Tab, Westbourne Grove

Earlier this summer, a new café opening on Westbourne Grove attracted a lot of attention. “Tab x Tab” quickly received reviews from Brian’s Coffee Spot (who noted the unusual espresso machine), Bean There at and Doubleskinnymacchiato. Bean There at also suggested that there should be plenty to ponder at Tab x Tab when I finally got the chance to get there. And so, a trip to this café had been on the agenda for a fair while.

Wandering into the café, it seemed exactly as described by the reviews: clean, sharp interiors in a modern building. It was fairly crowded when we arrived just after lunch and so we ordered before taking a seat at the bar (two of the few seats left). I had a long black while my fellow imbiber had a soya hot chocolate. The drinks arrived in those distinctive mugs mentioned by doubleskinnymacchiato (and pictured above). As we had just had lunch, on this occasion we didn’t check the edibles on offer but with plenty of other reviews of the coffee and the cake, I’m sure that you’ll find recommendations there (I understand the avocado on toast with cashew nut is well worth trying).

Sitting down to enjoy our drinks, the first thing to notice was that Bean There at was absolutely right. Despite the slightly minimal and elegant decoration, there were plenty of things dotted around that were slightly quirky. Firstly there was the plant that had been placed on two hexagons of slate that had been ever so slightly displaced from each other, presumably for aesthetic effect. Could this link to graphene and graphite with their strong intra-layer bonding and weak interlayer bonding (so the hexagons of carbon in graphite slide over each other)?

Then there was the selection of items for sale that also provided food for thought. Books and other items from the School of Life, something to think about as you stop with your coffee perhaps. In the other direction, on the counter top, a couple of Venus Fly Traps were waiting for their lunch. There is so much we have yet to learn about the symbiotic relationships between plants and animals and especially between plants and fungi. As we looked further around the café, there was something else a little odd. Just as the name “Tab” was written both the correct way and upside down in the window, so the plants in the hanging baskets were hanging upside down.

This seemed a bit strange in itself. Plants have a tendency to move upwards towards the light. This behaviour of plants (and trees in particular) provides one way to identify which way is south when walking in the country without a compass¹. It is odd to see a plant growing downwards and suggests that the plants in the window are regularly rotated so that they don’t try to reach up. As Simone Weil wrote “Two forces rule the universe: light and gravity”². Which would win in the end? To be fair, Weil was not referring to the light that streamed through the windows in Tab x Tab giving the plants the force they need to move upwards. Nonetheless, whether one is thinking literally or analogously, it is an interesting question what pulls us down, what brings us up?

There is a story that Newton arrived upon his idea of universal gravitation by contemplating a falling apple. Considering that the plants were approximately 2m above the floor level, and using the fact that the acceleration due to gravity, g,  is 10 m/s², if the plants were to fall from their hanging position, they would take:

s = ½gt²

t = 0.6 seconds

to fall and smash to the ground*. While this brings to mind Newton’s experiments dropping pigs bladders filled with liquid mercury from the dome of St Paul’s Cathedral, it is worth instead thinking more about the universal nature of the gravitational force. This is of course what made Newton’s idea of gravity different from the theories that had preceded it. People had known that if an apple fell from a tree (or a plant fell from its hanging basket) it would fall to Earth. What was key to Newton’s idea was that what applied to the apple, applied to all other masses too. The same maths that could be used to calculate how fast a plant dropped, could be applied to the Moon. So, if this was the case, could we calculate the orbital distance of the Moon in the time it took us to enjoy a coffee at Tab x Tab? We know that the Moon’s orbital period is τ = 27.3 days (2.36 x 10^6 seconds) so assuming that the gravitational force acting on the Moon is balanced by the centripetal force, we can equate the two:

Gravity: F = GMm/r²

Centripetal: F = mv²/r

Where, G is the gravitational constant (6.67 x 10^11 Nm²/kg²), M is the mass of the Earth (5.97 x 10^24 Kg), m is the mass of the Moon and r the moon’s orbital distance (which is what we want to calculate). If we assume that the Moon travels in a circular orbit (not quite true but not a bad first approximation), then we know the speed, v, of the moon in terms of the orbit period, it is just:

v = 2πr/τ

A bit of re-arrangement and some plugging in of values leads to a back-of-the-envelope value for the Moon’s orbital distance of 383 000 Km. A value that does not compare badly at all with the average distance of the Moon given by NASA as 384 400 Km.

Perhaps if we’d stayed for an additional flat white we could have refined the calculation somewhat and so obtained a value closer to reality. Nevertheless, the fact that the force that is pulling the plant down at Tab x Tab is the same as is pulling the Moon around the Earth, and that we can quickly check this (and get an approximately correct answer to our calculation), is one of those ‘wow’ moments in physics. Realising the universality, and elegance, of certain mathematical relations. So perhaps it is entirely appropriate that this thought train of mathematical elegance was prompted by the quirky but aesthetic elegance you will find at Tab x Tab.

Tab Tab can be found at 14-16 Westbourne Grove, W2 5RH

¹ The Walker’s Guide to Outdoor Clues & Signs, Tristan Gooley, Hodder & Stoughton, 2014

² Gravity and Grace, Simone Weil, Routledge (1995 vsn)

*Although you could use a more accurate value for g, the error on the estimate of the height of the plants makes such precision potentially misleading. The value 0.6 seconds is absolutely a back-of-the-envelope, calculation.

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## Gravity and Grace at the Wren cafe

There is a lot to like about the Wren cafe. Firstly, there is the space that it occupies (inside St Nicholas Cole Abbey). I went at lunchtime when the way that the light came through the stained glass windows made the cafe a very relaxing and open space. The coffee is from Workshop, complementary water came in 3 flavours (mint, cucumber or lemon) while the food is cooked on site. This is important because it means that they have a great nut policy and could tell me which dishes were likely to contain nuts etc. A further nice feature of the lunch menu at the Wren was that you could select your portion size. Food waste is a major issue for our society and is not helped by the ‘one size’ portions served at many food outlets and cafes. Lunch was offered in two sizes (technically as a side or a main) but the ‘side’ was more than adequate for a mid-week lunch. Sofas in the corner of the room meant that you could relax and take in your surroundings in a comfy environment or, if you were just there for lunch, ordinary chairs and tables were dotted around the room.

Of course, a place such as this will have plenty of things to notice about it. Whether your interest is in architecture or science, there is plenty to observe around you. What I would like to focus on though is a bit of science history that connects the name of this cafe with Isaac Newton, John Theophilus Desaguliers and the dome of St Paul’s Cathedral (which you can see from the front of St. Nicholas Cole Abbey).

Perhaps we all remember the story told to us at school about how Galileo dropped two balls of different mass from the top of the leaning tower of Pisa. According to the story, the balls fell to the earth at the same time, thereby showing that the acceleration due to gravity was independent of the mass of the object and paving the way for Newton’s theory of gravity. Sadly, it seems that Galileo may never have actually performed the experiment (even if it was “re-created” in 2009). However there is evidence that Isaac Newton did perform exactly this experiment in 1710 from the dome of the soon-to-be-completed St Paul’s Cathedral.

“From the top of St Paul’s church in London in June 1710 there were let fall together two glass globes, one full of quick silver [mercury], the other of air”¹. The globes fell 67m before shattering onto the cathedral floor (I’d hate to have written the risk assessment for that experiment). To avoid the possibility of human error, a trap-door mechanism had been designed to ensure that both globes dropped simultaneously. According to the story of Galileo told to us at school, we can calculate how long it would have taken those globes to drop to the floor: 3.7 seconds, independent of mass. So is this what Newton observed? No! The heavy glass globes took 4 seconds to fall, but lighter ones took 8-8.5 seconds! A few years later and Desaguliers repeated the experiment from slightly higher in the dome (but this time with hog’s bladders rather than glass) and obtained the same result.

This surprising result can be explained when we realise that Newton was investigating not gravity, but air resistance. While the gravitational acceleration is independent of mass, the upwards force due to the air resistance depends primarily on the object’s size (and velocity). This means that the deceleration caused by the air resistance will be different for two globes of the same size but different mass (Force = mass x acceleration). Heavy objects will fall faster in air (until the objects reach their terminal velocity).

There is a certain irony in the fact that this result is opposite to what we feel should happen based on what we learned at school of Galileo’s experiments challenging the scientific orthodoxy of the time. However the result of Newton and Desaguliers’ experiments do not contradict the theory of Newton or Galileo, they just add an extra layer to the problem. We do not exist in a vacuum, we need to think about the air around us too.

Both Newton and Desaguliers were regular coffee drinkers albeit at different coffee houses. Desaguliers frequented the Bedford Coffee House in the north east corner of Covent Garden while Newton regularly retired to the Grecian in Devereux Court (just off Fleet Street). Coffee houses were places that the latest science, politics or philosophy were discussed and debated. The Wren describes itself on its website as existing to “serve the ministry of St Nick’s talks“. Sadly I experienced no discussion or debate on my visit (just a very nice, but solitary, lunch and good coffee) but it is interesting to see the tradition of the 17-18th century coffee houses continued in this Wren designed church and cafe.

The Wren cafe can be found inside St Nicholas Cole Abbey, 114 Queen Victoria St. EC4V 7BJ

[1] The Dawn of Fluid Dynamics, Michael Eckert, Wiley-VCH (2006)

Coffee house info: London Coffee Houses by Bryant Lillywhite (pub. 1963)