Factfullness

April 7, 2018

Factfullness

Back in December 2010 I wrote about Hans Rosling with a post titled:

Hans Rosling: You’re my hero

Sadly, Hans Rosling died in February 2017.

But aware of his imminent death, he worked with his son and daughter-in-law to write a book which captures some key elements of his world view, which he summarises as…

…Factfulness

The book has two strands that run through all the chapters.

  • The first strand is that we are tremendously ignorant about the world. Repeatedly he expresses his surprise at our levels of ignorance. And given our access to facts, he asks why we are not just randomly ignorant, but have views which are  systematically wrong. He suggests it is a kind of cognitive bias.
  • The second strand is that we can stop being ignorant if we want to. And with this aim, he and his son invented delightful ways to view developmental data.

I won’t try to precis the book, but as someone who experiences intense anxiety in everyday situations, I found the section “Statistics as Therapy” particularly affecting.

As one specific example I went to the download page of Gapminder and downloaded a Powerpoint file with a graph showing how extreme poverty in the world has changed over time.

Extreme Poverty

It is clear that collectively humanity has made truly astonishing progress in reducing the awfulness of crushing poverty. [The Powerpoint file and the web site includes links so you can chase the data sources and check them.]

Why do we not celebrate this fantastic achievement?

This graph tells a good news story compared with which any news story we have experienced in the last 10 years is irrelevant.

This is a story of an epochal and positive change in the world of which most people – including myself – are largely ignorant.

The idea that the world – while acknowledging all its faults and injustices – is dramatically better than it is has ever been, feels like a balm against the ‘news-ification’ of reality that we experience.

Hans Rosling’s Death

I leave you with a video which I find intensely moving.

It shows Hans, his son,and his daughter-in-law explaining why they wrote the book.

Somehow knowing that he is no longer with us feels like a very intense – and despite the fact that I never knew him, personal – loss.

 

Videos

The fact that I feel Hans Rosling’s death personally is probably a cognitive bias caused by his many engaging video appearances.

This page on the Gapminder website contains links to many of his best videos. I cannot recommend them strongly enough.

Air Temperature

April 1, 2018

Recently, two disparate strands of my work produced publications within a week of each other.

Curiously they both concerned one of the commonest measurements made on Earth today – the measurement of air temperature.

  • One of the papers was the result of a humbling discovery I made last year concerning a common source of error in air temperature measurements. (Link to open access paper)
  • On the other  paper I was just one amongst 17 authors calling for the establishment of global reference network to monitor the climate. My guess is that most people imagine such a network already exists – but it doesn’t! (Link to open access paper)

I am writing this article because I was struck by the contrasting styles of these papers: one describing an arcane experimental detail; and the other proposing a global inter-governmental initiative.

And yet the aim of both papers was identical: to improve measurement so that we can more clearly see what is happening in the world.

Paper 1

In the middle of 2018 I was experimenting with a new device for measuring air temperature by measuring the speed of sound in air.

It’s an ingenious device, but it obviously needed to be checked. We had previously carried out tests inside environmental chambers, but the temperature stability and uniformity inside the chambers was not as good as we had hoped for.

So we decided to test the device in one of NPL’s dimensional laboratories. In these laboratories, there is a gentle, uniform flow of air from ceiling to floor, and the temperature is stable to within a hundredth of a degree Celsius (0.01 °C) indefinitely.

However, when I tried to measure the temperature of the air using conventional temperature sensors I got widely differing answers – varying by a quarter of a degree depending on where I placed the thermometer. I felt utterly depressed and humiliated.

Eventually I realised what the problem was. This involved stopping. Thinking carefully. And talking with colleagues. It was a classic case of eliminating the impossible leaving only the improbable.

After believing I understood the effect, I devised a simple experiment to test my understanding – a photograph of the apparatus is shown below.

tubes-in-a-lab-photo.png

The apparatus consisted of a set of stainless steel tubes held in a clamp stand. It was almost certainly the cheapest experiment I have ever conducted.

I placed the tubes in the laboratory, exposed to the downward air flow, and  left them for several hours to equilibrate with air.

Prior to this experience, I would have bet serious amounts of money on the ‘fact’ that all these tubes would be at the same temperature. My insight had led me to question this assumption.

And my insight was correct. Every one of the tubes was at a different temperature and none of them were at the temperature of the air! The temperature of the tubes depended on:

  • the brightness of the lights in the room – which was understandable but a larger effect than I expected, and
  • the diameter of the tubes – which was the truly surprising result.

Results 1

I was shocked. But although the reason for this is not obvious, it is also not complicated to understand.

When air flows air around a cylindrical (or spherical) sensor only a very small amount of air actually makes contact with the sensor.

Air reaching the sensor first is stopped (it ‘stagnates’ to use the jargon). At this point heat exchange is very effective. But this same air is then forced to flow around the sensor in a ‘boundary layer’ which effectively insulates the sensor from the rest of the air.

Air flow

For small sensors, the sensor acquires a temperature close to that of the air. But the air is surprisingly ineffective at changing the temperature of larger sensors.

The effect matters in two quite distinct realms.

Metrology

In metrology – the science of measurement – it transpires that knowledge of the temperature of the air is important for the most accurate length measurements.

This is because we measure the dimensions of objects in terms of the wavelength of light, and this wavelength is slightly affected by the temperature of the air through which the light passes.

In a dimensional laboratory such as the one illustrated below, the thermometer will indicate a temperature which is:

  • different from the temperature of artefacts placed in the room, and
  • different from the temperature of the air.

Laboratory

Unless the effect is accounted for – which it generally isn’t – then length measurements will be slightly incorrect.

Climatology

The effect is also important in climatology. If a sensor is changed in a meteorological station people check that the sensor is calibrated, but they rarely record its diameter.

If a calibrated sensor is replaced by another calibrated sensor with a different diameter, then there will be a systematic effect on the temperatures recorded by the station. Such effects won’t matter for weather forecasting, but they will matter for people using the stations for a climate record.

And that brings me to Paper 2

Paper 2

Hadcrut4 Global Temperature

When we see graphs of ‘global temperatures’ over time, many people assume that the data is derived from satellites or some ‘high-tech’ network of sensors. Not so.

The ‘surface’ temperature of the Earth is generally estimated in two quite distinct parts – sea surface temperature and land surface temperature. But both these terms are slight misnomers.

Considering just the land measurements, the actual temperature measured is the air temperature above the land surface. In the jargon, the measurement is called LSAT – the Land Surface Air Temperature.

LSAT is the temperature which human beings experience and satellites can’t measure it.

LSAT data is extracted from temperature measurements made in thousands of meteorological stations around the world. We have data records from some stations extending back for 150 years.

However, it is well known that data is less than ideal: it is biased and unrepresentative in many ways.

The effect described in Paper 1 is just one of many such biases which have been extensively studied. And scientists have devised many ways to check that the overall trend they have extracted – what we now call global warming – is real.

Nonetheless. It is slightly shocking that a global network of stations designed specifically with the aim of climate monitoring does not exist.

And that is what we were calling for in Paper 2. Such a climate network would consist of less than 200 stations world-wide and cost less than a modest satellite launch. But it would add confidence to the measurements extracted from meteorological stations.

Perhaps the most important reason for creating such a network is that we don’t know how meteorological technology will evolve over the coming century.

Over the last century, the technology has remained reasonably stable. But it is quite possible that the nature of data acquisition for meteorological applications will change  in ways we cannot anticipate.

It seems prudent to me that we establish a global climate reference network as soon as possible.

References

Paper 1

Air temperature sensors: dependence of radiative errors on sensor diameter in precision metrology and meteorology
Michael de Podesta, Stephanie Bell and Robin Underwood

Published 28 February 2018
Metrologia, Volume 55, Number 2 https://doi.org/10.1088/1681-7575/aaaa52

Paper 2

Towards a global land surface climate fiducial reference measurements network
P. W. Thorne, H. J. Diamond, B. Goodison , S. Harrigan , Z. Hausfather , N. B. Ingleby , P. D. Jones ,J. H. Lawrimore , D. H. Lister , A. Merlone , T. Oakley , M. Palecki , T. C. Peterson , M. de Podesta , C. Tassone ,  V. Venema, K. M. Willett

Published: 1 March 2018
Int. J. Climatol 2018;1–15. https://doi.org/10.1002/joc.5458

Obesity Policy

March 6, 2018

BBC Story Extract

Today the BBC are reporting that:

Britain needs to go on a diet, says top health official

The article states that people should allow:

  • 400 kilo-calories for breakfast
  • 600 kilo-calories for lunch
  • 600 kilo-calories for dinner

which adds up to 1600 kilo-calories a day. With this dietary intake, most adults in sedentary occupations will lose weight or maintain a healthy weight.

However, the article then goes on to say:

It is recommended that women should eat no more than 2,000 kilo-calories a day, while men should limit their intake to 2,500 kilo-calories.

No! As I pointed out previously, this is just too many calories for both men and women with sedentary lifestyles.

Any government campaign based on these figures is bound to fail.

Calories versus Age

For someone of my height and weight, the government’s recommended dietary intake is about 30% too high.

Is weight homeostasis possible?

February 28, 2018

I am slightly obsessed with my weight. Forgive me: I am 58 and have spent many decades repeatedly putting on weight slowly, and then losing it rapidly.

For many years I have wondered why can’t I just eat modestly and trust my body to “sort itself out!”

My recent discovery of the Mifflin St Joer equations (link) has allowed me to  simulate my weight over time, and my calculations are allowing me to understanding my own experience.

But my calculations have also raised a profound question:

  • Is homeostasis of weight even possible?

Homeostasis

Homeostasis (or Homoeostasis) is the term given to physiological systems which conspire to keep something constant.

For example, we have systems that maintain our body temperature without any conscious effort. I don’t have to berate myself for being too hot and promise myself that in the future I will try to be cooler.

No. Our bodies sort out their internal temperature. I understand the system consists of temperature sensitive cells and nervous system reflexes that control blood flow, sweat glands, shiver reflexes, and our desire to undertake activity.

Hunger

And I have generally imagined that in a more perfect world, a similar kind of system would underpin my desire to eat.

In this ideal world, I would naturally maintain my weight without any obvious effort on my part – stopping eating when I had eaten ‘enough’.

I had thought such a system actually existed. One part of the system is supposed to arise from the competing actions of hormones such as ghrelin – which makes us experience hunger – and leptin – which makes us feel satiated.

Together, ghrelin and leptin are supposed to act as part of a system of energy homeostasis.

However, having run many simulations of my own weight versus time (see below) and reflected on this, I am sceptical.

“But I know a bloke who…”

We all know people who seem to be able to eat at their ease and not put on weight.

I have no explanation for that, but then I have never experienced that myself.

My experience is that my weight either increases or decreases over time. What I have never observed it to do in all my 58 years on Earth is to stay the same! (I have written about this before: story 1 or story 2.)

What’s the problem?

I programmed the Mifflin St Joer equations into a spreadsheet to see the predicted effect on my weight of various dietary and exercise choices.

You can download the spreadsheet here and perform calculations about yourself in the privacy of your own computer. 

I entered my current age (58.2 years) and weight (74 kg), and I used the MSJ equations to predict what would happen to my weight if I ate 1800 kiloCalories (kCal) a day.

The results are shown below together with the effect of eating 50 kCal/day more or less

Weight versus Age Projection

  • The red line suggests that if I eat 1800 kCal/day then my weight will gradually decline over the next couple of years stabilising at about 71 kg. That would be dandy.
  • However, the dotted green lines show what would happen if I got my calorific intake wrong by ± 50 kCal per day. This is plus or minus half of a small glass of wine, or a half a biscuit either eaten, or not eaten.

These ‘alternate realities’ predict that my weight in three years time might be anywhere between 64 kg and 77 kg – a range of 13 kg!

To be within a kilogram of the predicted weight, my average energy intake would need to match 1800 kCal/day within 10 kCal a day. That is less than a single mouthful of food!

I don’t believe that any autonomic system can achieve that level of control. 

Weight versus Age Projection 2

So what?

Reflecting on these simulations, I don’t believe that the systems within our bodies that mediate ‘energy homoeostasis’ operate well over many years.

At least they don’t operate well in an environment where calories are so easy to obtain.

So I think my experience of slow weight gain over time is not a fault with my autonomic nervous system, or a moral failing on my part. It is just the way things are.

Asking the thinerati

Asking several slim individuals around the coffee machine this morning confirmed my view. They all were either (a) young (b) self-conscious about fitting into clothes or (c) weighed themselves regularly.

Personally I have resolved to keep weighing myself and using this to provide manual feedback.

How is my weight doing? Thank you for asking. It’s been just about stable since Christmas and I intend to keep it that way!

SI at the RI

February 23, 2018

 

MdeP at the RI

On Monday 16th October 2017 I gave a talk at about the International System of Units (the SI) at the Royal Institution (the RI) in London.

It wasn’t a great talk, but it was at the RI. And I stood where Michael Faraday stood!

The RI have now processed the video and produced an edited version: enjoy 🙂

The RI have tended to retain the video of me talking rather than showing the animated PowerPoint slides. If you would like the full multimedia experience, you can download the presentation using the link below.

On the day

I was nervous and arrived ridiculously early with a couple of glass Dewars containing triple point of water cells.

I waited outside the lecture theatre for Martin Davies from the RI to arrive.

When he arrived, he noticed the Dewars and without hesitation he turned to the wall and pointed out the painting above where I was standing, and said:

“What a coincidence: your standing by a picture of Sir James Dewar lecturing in this theatre!”

Henry Dewar at the RI

It’s hard to convey the historical significance of Royal Institution without sounding trite. So I won’t try.

But it is a special place for chemists and physicists alike, and I feel honoured to have even had the chance to stand on that spot.

================

Thanks to Chris Brookes and Martin Davies for a memorable day.

Why is it so hard to lose weight?

February 21, 2018

After several years of looking, I think I have finally found the answer.

So if people follow a calorie-controlled diet based on government guidelines, almost everyone will put on weight.

Let me explain…

Mifflin St Jeor

The minimal basal metabolic requirements (BMR) of a human being have been the subject of scientific study for more than a century.

The best estimate of our requirements are the Mifflin St Jeor (MSJ) Equations which state that the calorific requirements for men and women are given by:

Men:     BMR = 10 × [Weight in kg] + 6.25 × [Height in cm] – 5 × [Age in years] – 5

Women:     BMR = 10 × [Weight in kg] + 6.25 × [Height in cm] – 5 × [Age in years] – 161

This is the amount of food (expressed as kiloCalories (kCal) per day) required to maintain a given weight and do nothing else: no exercise at all.

A sedentary male lifestyle

The MSJ equations are generally multiplied by a factor to reflect the amount of physical activity one undertakes during the day. And there is considerable uncertainty about which factor applies to any particular individual!

The factor 1.2 is commonly chosen to represent a “sedentary lifestyle”. In a moment I’ll come back to whether this factor is justified or not.

But based on this factor, the blue line on the graph below shows how the actual calorific requirements of a man of my weight and height vary with age. The equivalent graph for women is shown in the next section.

Calories versus Age

The most striking thing about this graph is that the actual amount of calories I need to maintain my weight (1860 kCal/day) is 25% less than the government recommend (2500 kCal/day).

The difference is not a rounding error – it amounts to 640 kCal/day which is a reasonably-sized meal!

A man of my age living a sedentary lifestyle and following government guidelines would put on weight at a rate of several kilograms per year.

The second striking feature of the graph is reduction in calorific requirements with age. The slope of the graphs is 50 kCal/day per decade.

This means that if I was maintaining my weight in my forties, then unless I changed either my eating habits or my exercise habits, I would slowly begin to put on weight.

Eating 50 kCal/day too much amounts to putting on weight at around 2 kg per year.

Is the sedentary lifestyle factor 1.2 appropriate?

One way to assess whether the factor 1.2 applied to the MSJ equations is appropriate is to consider the calorific equivalent of some exercise.

For a man of my weight and height, running 1 kilometre uses up about 74 kCal.

So if I were to run 25 km per week, then this would allow me to eat about another 260 kCal/day and still maintain my weight. This is shown as the red line on the graph above.

Most people would consider running 25 km per week to be quite serious exercise. Comparing this amount of exercise to the work done in a sedentary day makes me think that the factor 1.2 is probably about right.

Women

The equivalent graph for women is shown below

Calories versus Age Women

It shows a similar disparity between government recommendations and actual metabolic requirements, but not quite as dramatically wrong as for men.

Government Guidelines

The reason I searched out the MSJ equations was because I know from experience that if I eat anything close to 2500 kCal per day I put on weight.

Calorific intake is notoriously difficult to estimate with an uncertainty better than about 10%,  but the MSJ figure of about 1860 kCal/day for a man of my age weight and height seems about right.

The UK Government guidelines are – frankly – nonsense, and given that the UK has something of a problem with obesity – not least with people of my age – it would seem a sensible first step to just get this simple factual message about right.

One important step would be to emphasise the reducing calorie requirements with age.

Government guides in the US such as this one are closer to reality, but if you want real information I recommend this helpful calculator.

Mugging

February 18, 2018

IMG_6849

After writing about ‘singing glasses’ previously, I was discussing the effect with my friend and colleague Andrew Hanson.

Have you done the mug thing?” he asked. And proceeded to hit the rim of a mug with a spoon.

As he moved the location at which he struck the mug, the pitch of the note changed.

And then he explained. Looking from the top, if the handle of the mug is at 12 O’clock, then:

  • Striking the mug at 3, 6, 9, or 12 produces one note – a lower note.
  • Striking the mug between 1 & 2, between 4 & 5, between 7 & 8 and between 10 & 11 produced a second note – a higher note.

Mug Vibrations 01

Andrew said the explanation was that in a mug, there are two types of flexural oscillations, and the frequency of the oscillations depends on whether the handle moves or not.

I was fascinated. How had I never noticed that before? And why was the note that sounded when the mug was struck ‘on the quarter hours’ lower?

This is a very long article, and I apologise. But the physics of this phenomenon is complex and it took me a long time to get to the bottom of it.

Investigations

I picked 5 mugs from our domestic collection which were as straight-sided as possible, and which had walls which were as thin as possible. I thought these choices would make the vibrational spectrum simple.

First I measured the mugs: their diameter and the wall thickness, and then I picked a wooden striker and started hitting the mugs. (Speadsheet)

The place where one strikes the mug produces an oscillation with the striking location as a local maximum of the oscillation.

For a glass, it wouldn’t matter where one struck – any location on the rim is equivalent to any other. But for mugs, the striking position matters because of the handle.

Mug Vibrations 02

To see if I could understand what was going on I arranged the mugs in size order, from the smallest radius to the largest and struck each one three times at each location.

You can hear the sound here.

I also recorded the ‘spectrogram’ using the wonderful Spectrum View app for the iPhone. A screenshot from the app is shown below together with the mugs which made the noise.Mug Vibrations 03.png

A spectrogram shows:

  • time along the horizontal axis
  • frequency along the vertical axis
  • and the loudness of a sound at a particular frequency and time is shown by the colour: blue is quiet and yellow and red are loud.

On the spectrogram above one can see vertical lines which result from the ‘impulse’ sound of me hitting the mugs. This dies away quickly and one is just left with the ‘ringing’ of the mugs which I have outlined with dotted lines.

One can see that each mug rings at two closely-spaced frequencies. The two notes differ in frequency by between 5% and 15%.

Hitting the rim at either location produces mainly one mode of oscillation, but also a little of the other.

Let’s get numerical!

I used the app to locate the frequency of each note and plotted it on a graph of the frequency versus the mug diameter. I plotted each ringing note as a red dot, and their average as a black dot.

mug-vibrations-04.png

There was a general trend to lower frequency for the larger mugs, but the Toronto mug didn’t fit that trend.

I noticed that the walls of the Toronto mug were much thicker than the other mugs. So I wondered whether I could compensate for this by dividing the frequency by the thickness of the wall.

I did this based on the idea that the speed of the wave would be proportional to the  rigidity of the mug wall against bending. And that rigidity might be roughly proportional to the wall thickness. This seemed to be confirmed because the formerly ‘anomalous’ mug frequency now sat quite sweetly on a smooth trend.

Mug Vibrations 05

Now that the data seemed to fit a trend, I felt I was getting a handle on this problem. Could I understand the dependence of the resonance frequency on diameter?

Frequency

All waves obey the wave formula v = f λ. That is, the speed of the wave, v, is the product of the frequency of the wave, f, and its wavelength  λ.

For the waves on these mugs, the wavelength of the wave which runs around the rim of the cup is just half the perimeter i.e.  λ = π D/ 2 where D is the diameter of the mug.

  • So if all the waves travel with the same speed, then the resonance frequency should vary with diameter as f = 2 v / (π D) i.e. inversely proportional to the diameter.
  • However, for flexural waves, the material supporting the wave becomes floppier at longer wavelengths, and the speed of a flexural wave should fall with increasing wavelength. If this were the case we would expect the resonance frequency to vary inversely as the diameter squared.

Which of the above cases described the data best? I have plotted the two predictions on the graph below.

Mug Vibrations 06

I adjusted the speeds of the waves to match the data for large mugs and then calculated how it should vary for smaller diameter mugs.

Overall I think it is the theory in which the  speed of the flexural wave changes with wavelength that matches the data best.

Test

So now I had a theory that the speed of a flexural wave that runs around the rim of a mug is:

  • Proportional to the wall thickness
  • Inversely proportional to the diameter
    • So the resonance frequencies are inversely proportional to the diameter squared.

After I had finished all these measurements I came across another mug that I could have included in the study – my wife’s ‘Do more of what makes you happy‘ mug.

I decided to see if I could predict its resonant frequency from measurements of its wall thickness and diameter.

I have plotted data on this mug as crosses (×) on the graphs above and I think that overall it fits the trends rather well.

The two notes

If you belong to the minority who have read this far, then well done. It is only you who get to understand the final point about why the ‘quarter hour’ notes are lower.

What is the role of the handle? There are two possibilities.

  • Does it act as an extra mass which slows down the wave?
  • Or does it provide extra stiffness, which would speed up the wave?

Since the quarter hour notes are lower, it seems that it is the extra mass which appears to dominate in the mugs I have examined.

mug-vibrations-07.png

Wonder and Science

January 31, 2018

Recipes for Wonder

My good friend Alom Shaha has a new book out!

And discussing it over dinner the other evening I was struck by an analogy.

Talking and listening and reading and writing 

Children have no problem learning to understand and speak their mother tongue.

All they require is to be exposed to people speaking and they will learn to speak

But this ability does not make them ‘good at languages’.

In contrast with the ease with which children learn to speak, is the great difficulty they have in learning to read and write.

A web search tells me that 15% of the UK population are ‘functionally illiterate’ – a figure which I think has not changed much in recent years.

Reading and writing are hard: they take practice:

  • learning letter shapes.
  • learning the relationship between shapes and sounds.

And it can be a long time before all this becomes automatic and there is a payback on the effort expended.

Nonetheless, widespread literacy is considered essential for a functioning democracy.

And most people who have been taught to read and write are happy with the extra possibilities their new skills enable.

Wonder and Science

Similarly, I think children have an intrinsic sense of wonder.

Or at least they can acquire the sense with ease if they are exposed to adults who express interest in the world around them.

But going beyond the simple pleasure of “Wow!” is hard work.

However, it is that step – from ‘Wow!” to “How?” that is the step from wonder into science.

Why is it hard?

Firstly, imagine how well parents would teach their children to read and write if they were themselves illiterate.

Similarly, scientifically illiterate parents – or more commonly parents lacking confidence in their own abilities – can find teaching science hard.

And secondly, everything is complicated. So it is easy to spread confusion rather than enlightenment.

Consider a ‘simple’ experiment – the kind of activity that people recommend for kids – such as making a wine glass ‘sing’.

Just managing to make this happen is pleasurable – it is intriguing and surprising to hear. It is, literally, wonder-ful.

But when one begins to ‘step beyond’ wonder, it all becomes difficult. I have just spent a happy thirty minutes with my wife investigating. And even with two PhDs, an iPhone equipped with a slow motion camera, and spectrogram software we found it difficult!

For example:

  • Is it the glass or the air in the glass which is vibrating?
  • Why can one see very fine waves running on the surface of water in the glass?

If you search for clues as to what is happening you will find a dearth of answers on the web.

Alom’s book?

As I understand it,  Alom’s aim in writing his ‘recipes for wonder’ is to hold hands with parents and children so that their first steps beyond wonder into science are beguiling and delightful rather than bewildering and demoralising.

Such a book is sorely needed. I hope it does well.

By the way, if you would like to hear Alom talk, he will be appearing at the Royal Institution on March 8th .

P.S. What is happening with the ‘singing’ glass?

I am afraid, the physics is too complicated to explain in full, so here is a summary.

Firstly, the fundamental mode of vibration being excited is a ‘flexural’ oscillation of the glass rim and bowl.

Wineglass

Normally if one calculated the resonant frequency of a sound wave in a glass object of similar dimensions to a wine glass, one might expect a resonance at a very high frequency – perhaps 10 kHz or higher.

This is because the speed of sound in glass is over 4000 metres per second‚ more than 10 times higher than the speed of sound in air.

However, when a material is formed into a ring, it has a ‘soft’ mode of flexing illustrated in the animation above. (The bowl of the glass is not quite a ring, but the upper part of the bowl is ‘almost’ a ring.)

Even if the speed of sound in the material is very high, as the material of the ring becomes thinner, then it becomes easier to flex, and the restoring force pulling the ring back into shape becomes weaker.

This causes the speed a flexural wave in a glass ring to be much lower than the speed of a sound wave in glass. Thus the resonant frequency falls as well.

Once the vibration is established, it vibrates the air around the glass which is what we hear.

But note that this is not a resonance of the air in the glass. If it were, then adding water to the glass would reduce the size of the resonant cavity and cause an increase in the resonant frequency. In fact adding water lowers the resonant frequency.

A spectrogram showing how the frequency of a singing glass is lowered by adding water. Note, the application was paused at 3.8 seconds and then re-started with water in the glass.

A spectrogram showing how the frequency of a singing glass is lowered by adding water. Note, the application was paused at 3.8 seconds and then re-started with water in the glass.

Note also that the gravity capillary waves that can be observed on the surface of the water are also a red herring.

IMG_6825

These waves have a very low speed – about 30 centimetres a second, and so at few hundred hertz, they have a wavelength of much less than 1 millimetre.

Finally, there is also a connection between the noise made by a glass and that made by a xylophone. Xylophone

The vibrations excited by hitting the xylophone keys are not sound waves in the metal but flexural waves.

The speed of flexural waves falls in long thin (floppy) bars – getting less and less for longer bars. So for thin materials, the flexural wave can have a low speed leading to a low resonance frequency.

The fact that a xylophone uses flexural waves explains the relative sizes of the keys.

To make a key for a note one octave lower (i.e. half the frequency) of the top key, one does not have to double the length of the key. In fact one only needs to lengthen the bar by a factor of the square root of two (i.e. make about 41% longer).

Like I said: everything is complicated!

 

 

 

 

Gravity: one more thing

January 28, 2018

I am a great admirer of James Clerk Maxwell.

And amongst his greatest achievements was the prediction that waves in electric and magnetic fields should travel at the speed of light.

He arrived at his prediction by considering the observed strength of static electric magnetic fields.

  • For example, studies had established the strength of the force from a given amount of electric charge at a given distance.
  • This electrical force was characterised by a constant called (for historical reasons) the permittivity of free space. It was given the symbol ε0 – the greek letter ‘epsilon’ with a subscript of zero. It was considered to represent in some way how ‘disturbed’ the space was around an electric charge.
  • Similarly, studies had established the strength of the magnetic force from a given electric current at a given distance.
  • This magnetic force was characterised by a constant called (for historical reasons) the permeability of free space. It was given the symbol μ0 – the greek letter ‘mu’ with a subscript of zero. It was considered to represent in some way how ‘disturbed’ the space was around an electric current.

Maxwell analysed these static experiments and predicted that there should be coupled waves in the electric and magnetic fields and that they would travel with a speed of:

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And when Maxwell calculated this number he arrived at a number very close to the previously measured speed of light.

He observed that this was unlikely to be a coincidence and concluded that light was a wave in the electromagnetic field.

I can still remember how I felt when – aged 19 – I followed Maxwell’s footsteps and ‘discovered’ this connection: I was gob-struck!

Other waves

This type of formula is typical of expressions for the speed of waves. For example, the speed of a wave on a stretched wire or string is given by:

image002

where T is the tension in the string and m is the mass per unit length of the string.  So a wave will travel quickly when the string is taut and low mass.

And in general we expect the speed of waves to reflect how the medium in which the waves travel responds to a disturbance.

Gravity waves

And that is why last years’ announcement (LIGO, Popular Report) that gravity waves travel at the speed of light is so profoundly important.

This discovery implies that there is a connection between:

  • electricity and magnetism – responsible for just about all the phenomena we experience around us – and…
  • gravity – which is associated with space and time and mass.

Alternatively, it could indicate a connection between them both and something else we don’t know about.

But the experimental fact of this connection astounds me as much if not more than the connection that Maxwell made.

And it makes me wonder just what he would have to say about the discovery.

Now I know this connection is not ‘new’: I can remember being told that gravity waves would travel at the speed of light many years ago.

But the discovery of the experimental fact of the speeds of light and gravity being equal seems to me to be more profound than the mere expectation that it should be so.

You can see more about the discovery in the LIGO video below

 

Perspectives on Gravity

January 9, 2018

Gravity is such a familiar force that its utterly mysterious nature can sometimes go unnoticed.

Looking at the picture of Earth and Moon bound together in the solitude of the Universe, it is strange to think that all that holds them together is this apparently weak force.

In this article I will do a couple of calculations using Newton’s law of Universal Gravitation. If you know the maths, please check my calculations, and if you don’t, please trust me.

Not so weak

Many people are familiar with the fact that the average gravitational field strength at the surface of the Earth is approximately 9.8 newtons of force for every kilogram of mass. This is sometimes called one ‘g‘.

(This is sometime expressed as 9.8 metres per second per second, but I don’t think that formulation is as clear in this context.)

But what is the gravitational field strength due to the Earth at the Moon? A simple calculation shows it to be just 0.0027 newtons per kilogram – about 0.02% of g.

And yet this weak field is sufficient to bind the Moon to the Earth with a force of 2 × 1020 newtons.

If gravity disappeared (!) and we applied that force to the Moon with a tensile steel cable, it would need to be 1000 km in diameter and would require about half the mass of the Earth to manufacture!

So weak

Many people are familiar with the fact that the tides on Earth are affected by the Moon.

We can work out the gravitational field strength on the side of the Earth nearest the Moon – where the Moon’s gravity opposes the Earth’s gravity: 9.8134727 newtons per kilogram.

Compare this with the gravitational field strength on the side of the Earth farthest from the Moon – where the Moon’s gravity acts with the Earth’s gravity: 9.8134749 newtons per kilogram.

The gravitational field strengths differ by just 0.2 parts in a million. And yet this difference is sufficient to affect the tides!

So very weak

Many people are familiar with the fact that the Earth is bound to the Sun by gravity. And that the Sun is bound to the Centre of the Milky Way Galaxy by gravity.

We can work out the gravitational field strength at the Earth due to the Sun. It is just 0.0059 newtons per kilogram or about 0.06% of g.

And the gravitational field strength at the Sun due to the Galaxy is a breathtakingly small 0.000000002135 newtons per kilogram or just 0.2 parts per billion of the gravitational field strength at the Earth’s surface.

And the lesson is?

There is no lesson here – it is just surprising to me how weak gravitational fields – billions of times weaker than the fields we are familiar with on Earth – can bind stars into galaxies. That’s all.

Good night.


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