Archive for Physics

100 Years of Feynman

Posted in Cute Problems, Education with tags , , , , , , on May 11, 2018 by telescoper

Today marks the centenary of the birth of Noble Prize-winning physicist, science communicator and bongo player Richard Feyman. It’s great to see so many articles about him today, so I was wondering how to do my own quick tribute before I head to London for the Royal Astronomical Society Annual General Meeting this afternoon.

With university exams coming up it seemed a good idea to celebrate Richard Feynman’s legacy by combining todays 100th anniversary with some tips (inspired by Feynman) about how to tackle physics problems, not only in terms of how to solve them but also how to present the answer in an appropriate way.

Richard-Feynman-cornellI began with Richard Feynman’s formula (the geezer in the above picture) for solving physics problems:

  1. Write down the problem.
  2. Think very hard.
  3. Write down the answer.

That may seem either arrogant or facetious, or just a bit of a joke, but that’s really just the middle bit. Feynman’s advice on points 1 and 3 is absolutely spot on and worth repeating many times to an audience of physics students.

I’m a throwback to an older style of school education when the approach to solving unseen mathematical or scientific problems was emphasized much more than it is now. Nowadays much more detailed instructions are given in School examinations than in my day, often to the extent that students  are only required to fill in blanks in a solution that has already been mapped out.

I find that many, particularly first-year, students struggle when confronted with a problem with nothing but a blank sheet of paper to write the solution on. The biggest problem we face in physics education, in my view, is not the lack of mathematical skill or background scientific knowledge needed to perform calculations, but a lack of experience of how to set the problem up in the first place and a consequent uncertainty about, or even fear of, how to start. I call this “blank paper syndrome”.

In this context, Feynman’s advice is the key to the first step of solving a problem. When I give tips to students I usually make the first step a bit more general, however. It’s important to read the question too. The key point is to write down the information given in the question and then try to think how it might be connected to the answer. To start with, define appropriate symbols and draw relevant diagrams. Also write down what you’re expected to prove or calculate and what physics might relate that to the information given.

The middle step is more difficult and often relies on flair or the ability to engage in lateral thinking, which some people do more easily than others, but that does not mean it can’t be nurtured.  The key part is to look at what you wrote down in the first step, and then apply your little grey cells to teasing out – with the aid of your physics knowledge – things that can lead you to the answer, perhaps via some intermediate quantities not given directly in the question. This is the part where some students get stuck and what one often finds is an impenetrable jumble of mathematical symbols  swirling around randomly on the page. The process of problem solving is not always linear. Sometimes it helps to work back a little from the answer you are expected to prove before you can return to the beginning and find a way forward.

Everyone gets stuck sometimes, but you can do yourself a big favour by at least putting some words in amongst the algebra to explain what it is you were attempting to do. That way, even if you get it wrong, you can be given some credit for having an idea of what direction you were thinking of travelling.

The last of Feynman’s steps  is also important. I lost count of the coursework attempts I marked this week in which the student got almost to the end, but didn’t finish with a clear statement of the answer to the question posed and just left a formula dangling.  Perhaps it’s because the students might have forgotten what they started out trying to do, but it seems very curious to me to get so far into a solution without making absolutely sure you score the points.  IHaving done all the hard work, you should learn to savour the finale in which you write “Therefore the answer is…” or “This proves the required result”. Scripts that don’t do this are like detective stories missing the last few pages in which the name of the murderer is finally revealed.

So, putting all these together, here are the three tips I gave to my undergraduate students this morning.

  1. Read the question! Some students give solutions to problems other than that which is posed. Make sure you read the question carefully. A good habit to get into is first to translate everything given in the question into mathematical form and define any variables you need right at the outset. Also drawing a diagram helps a lot in visualizing the situation, especially helping to elucidate any relevant symmetries.
  2. Remember to explain your reasoning when doing a mathematical solution. Sometimes it is very difficult to understand what students are trying to do from the maths alone, which makes it difficult to give partial credit if they are trying to the right thing but just make, e.g., a sign error.
  3.  Finish your solution appropriately by stating the answer clearly (and, where relevant, in correct units). Do not let your solution fizzle out – make sure the marker knows you have reached the end and that you have done what was requested. In other words, finish with a flourish!

There are other tips I might add – such as checking answers by doing the numerical parts at least twice on your calculator and thinking about whether the order-of-magnitude of the answer is physically reasonable – but these are minor compared to the overall strategy.

And another thing is not to be discouraged if you find physics problems difficult. Never give up without a fight. It’s only by trying difficult things that you can improve your ability by learning from your mistakes. It’s not the job of a physics lecturer to make physics seem easy but to encourage you to believe that you can do things that are difficult!

 

Advertisements

A Guest Paradox

Posted in Cute Problems, The Universe and Stuff with tags , , , on February 9, 2018 by telescoper

Here’s a short guest post by my old friend Anton. As usual, please feel free to discuss the paradox through the comments box!

–0–

I thought of a physics paradox the other day and Peter has kindly granted me a guest post here about it, as follows. Consider a homogeneous isotropic closed universe as described by general relativity. Let it contain a uniform density of a single species of electrically charged particle, so that this universe has a net charge. The charged particle density is sufficiently low, however, that the perturbation from the regular uncharged metric is negligible. Since this universe is homogeneous and isotropic the electric field in it is everywhere zero. BUT if I consider a conceptual 3-dimensional sphere, small enough for space-time curvature to be neglected, then it contains a finite amount of electric charge, and therefore by Gauss’ theorem a nonzero electric field points out of it at every point on its surface. This contradicts the zero-field conclusion based on the metric.

Here are three responses (one my own) and my further responses to these, in brackets:

  1. In a closed universe it is not clear what is the outside and what is the inside of the sphere, so Gauss’ law is not trustworthy (tell this to a local observer!);
  2. the electric field lines due to the charges inside this (or any) conceptual sphere wrap round the universe an infinite number of times (this doesn’t negate Gauss’ theorem!);
  3. the curved rest of the Universe actually adds a field that cancels out the field in your sphere (neither does this negate Gauss’ theorem!)

The Quickening of the Year

Posted in Education, Maynooth, Music, The Universe and Stuff with tags , , , , on February 1, 2018 by telescoper

It’s 1st February 2018, which means that today is Imbolc, a Gaelic festival marking the point halfway between the winter solstice and vernal equinox. This either happens 1st or 2nd February, and this year it is the former. In this part of the world – I’m in Ireland as I write- this day is sometimes regarded as the first day of spring, as it is roughly the time when the first spring lambs are born. It corresponds to the Welsh Gŵyl Fair y Canhwyllau and is also known as the `Cross Quarter Day’ or (my favourite) `The Quickening of the Year’.

So, talking of quickening, the pace of things is increasing for me now too. This morning at 9am I gave my first ever lecture in Maynooth University in a lecture theatre called Physics Hall, which is in the old (South) part of campus as opposed to the newer North Campus where the Science Building that contains my office is situated.

After that it was back to the Department for some frantic behind-the-scenes activity setting up accounts for the students for the afternoon lab session, which is in a computer room near to my office. Students attend one two-hour lab session in addition to the lecture, on either Thursday or Tuesday. The first lecture being this morning (Thursday) the first lab session was this afternoon, with the same material being covered next Tuesday.

I was far more nervous about this afternoon’s lab session than I was about this morning’s lecture as there seemed to be many things that could go wrong in getting the students up and running on our Linux cluster and getting them started on Python. Quite a few things did go wrong, in fact, but they were fewer in number and less drastic in outcome that I had feared.

So there we are, my first full day teaching in Maynooth. I think it went reasonably well and it was certainly nice to meet my first group of Maynooth students who, being physics students, are definitely la crème de la crème. I’ve got another 6 weeks like this (teaching on Tuesday in Cardiff and on Thursday in Maynooth) before the Easter break so it’s going to be a hectic period. Just for tonight, however, I’ve got time to relax with a glass or several of wine.

Incidentally, I was impressed that Physics Hall (where I did this morning’s lecture) is equipped with an electric piano:

I wonder if anyone can suggest appropriate musical numbers to perform for a class of computational physicists? Suggestions are hereby invited via the Comments Box!

Cosmology: The Professor’s Old Clothes

Posted in Education, The Universe and Stuff with tags , , , , , , , on January 19, 2018 by telescoper

After spending  a big chunk of yesterday afternoon chatting the cosmic microwave background, yesterday evening I remembered a time when I was trying to explain some of the related concepts to an audience of undergraduate students. As a lecturer you find from time to time that various analogies come to mind that you think will help students understand the physical concepts underpinning what’s going on, and that you hope will complement the way they are developed in a more mathematical language. Sometimes these seem to work well during the lecture, but only afterwards do you find out they didn’t really serve their intended purpose. Sadly it also  sometimes turns out that they can also confuse rather than enlighten…

For instance, the two key ideas behind the production of the cosmic microwave background are recombination and the consequent decoupling of matter and radiation. In the early stages of the Big Bang there was a hot plasma consisting mainly of protons and electrons in an intense radiation field. Since it  was extremely hot back then  the plasma was more-or-less  fully ionized, which is to say that the equilibrium for the formation of neutral hydrogen atoms via

p+e^{-} \rightarrow H+ \gamma

lay firmly to the left hand side. The free electrons scatter radiation very efficiently via Compton  scattering

\gamma +e^{-} \rightarrow \gamma + e^{-}

thus establishing thermal equilibrium between the matter and the radiation field. In effect, the plasma is opaque so that the radiation field acquires an accurate black-body spectrum (as observed). As long as the rate of collisions between electrons and photons remains large the radiation temperature adjusts to that of the matter and equilibrium is preserved because matter and radiation are in good thermal contact.

 

Image credit: James N. Imamura of University of Oregon.

Eventually, however, the temperature falls to a point at which electrons begin to bind with protons to form hydrogen atoms. When this happens the efficiency of scattering falls dramatically and as a consequence the matter and radiation temperatures are no longer coupled together, i.e. decoupling occurs; collisions can longer keep everything in thermal equilibrium. The matter in the Universe then becomes transparent, and the radiation field propagates freely as a kind of relic of the time that it was last in thermal equilibrium. We see that radiation now, heavily redshifted, as the cosmic microwave background.

So far, so good, but I’ve always thought that everyday analogies are useful to explain physics like this so I thought of the following.

When people are young and energetic, they interact very extensively with everyone around them and that process allows them to keep in touch with all the latest trends in clothing, music, books, and so on. As you get older you don’t get about so much , and may even get married (which is just like recombination, not only that it involves the joining together of previously independent entities, but also in the sense that it dramatically  reduces their cross-section for interaction with the outside world).  As time goes on changing trends begin to pass you buy and eventually you become a relic, surrounded by records and books you acquired in the past when you were less introverted, and wearing clothes that went out of fashion years ago.

I’ve used this analogy in the past and students generally find it quite amusing even if it has modest explanatory value. I wasn’t best pleased, however, when a few years ago I set an examination question which asked the students to explain the processes of recombination and decoupling. One answer said

Decoupling explains the state of Prof. Coles’s clothes.

Anyhow, I’m sure there’s more than one reader out there who has had a similar experience with an analogy that wasn’t perhaps as instructive as hoped or which came back to bite you. Feel free to share through the comments box…

Hamiltonian Poetry

Posted in Poetry, The Universe and Stuff with tags , , , , , , on January 8, 2018 by telescoper

I posted a couple of items last week inspired by thoughts of the mathematician William Rowan Hamilton. Another thing I thought I might mention about Hamilton is that he also wrote poetry, and since both science and poetry feature quite regularly on this blog I thought I’d share an example.

In fact during the `Romantic Era‘ (in which Hamilton lived) many scientists wrote poetry related either to their work or to nature generally. One of the most accomplished of these scientist-poets was chemist and inventor Humphry Davy who, inspired by his friendship with the poets Wordsworth and Coleridge, wrote poems throughout his life. Others to do likewise were: physician Erasmus Darwin; and astronomer William Herschel (who was also a noted musician and composer),

William Rowan Hamilton interests me because seems to have been a very colourful character as well as a superb mathematician, and because his work relates directly to physics and is still widely used today. Interestingly, he was a very close friend of William Wordsworth, to whom he often sent poems with requests for comments and feedback. In the subsequent correspondence, Wordsworth was usually not very complimentary, even to the extent of telling Hamilton to stick to his day job (or words to that effect). What I didn’t know was that Hamilton regarded himself as a poet first and a mathematician second. That just goes to show you shouldn’t necessarily trust a man’s judgement when he applies it to himself.

Here’s an example of Hamilton’s verse – a poem written to honour Joseph Fourier, another scientist whose work is still widely used today:

Hamilton-for Fourier

If that’s one of his better poems, then I think Wordsworth may have had a point!

The serious thing that strikes me is not the quality of the verse, but how many scientists of the 19th Century, Hamilton included, saw their scientific interrogation of Nature as a manifestation of the human condition just as the romantic poets saw their artistic contemplation. It is often argued that romanticism is responsible for the rise of antiscience. I’m not really qualified to comment on that but I don’t see any conflict at all between science and romanticism. I certainly don’t see Wordsworth’s poetry as anti-scientific. I just find it inspirational:

I HAVE seen
A curious child, who dwelt upon a tract
Of inland ground, applying to his ear
The convolutions of a smooth-lipped shell;
To which, in silence hushed, his very soul
Listened intensely; and his countenance soon
Brightened with joy; for from within were heard
Murmurings, whereby the monitor expressed
Mysterious union with its native sea.
Even such a shell the universe itself
Is to the ear of Faith; and there are times,
I doubt not, when to you it doth impart
Authentic tidings of invisible things;
Of ebb and flow, and ever-during power;
And central peace, subsisting at the heart
Of endless agitation.

The Problem of the Spinning Tube

Posted in Cute Problems with tags , , on November 22, 2017 by telescoper

It’s been a while since I posted a problem in the folder for cute physics problems so here’s a nice little one for you to have a go at:

A vertical cylindrical tube of height 12cm and radius 6cm, sealed at the bottom and open at the top,  is two-thirds filled with a liquid and set rotating with a constant angular velocity ω about a vertical axis.  Neglecting the surface tension of the liquid, estimate the greatest angular velocity for which the liquid does not spill over the edge of the tube.

Answers through the comments box please!

 

A Problem with Spitfires

Posted in Cute Problems, History, The Universe and Stuff with tags , , , , , on July 25, 2017 by telescoper

This problem stems from an interesting exchange on Twitter last night, prompted by a tweet from the Reverend Richard Coles:

I think his clerical vocation may be responsible for the spelling mistake. The answer to his question doesn’t require any physics beyond GCSE but it does require data that I didn’t have access to last night.

Here’s a version for you to try at home with all the necessary numbers (though not necessarily in the right units):

A model of a Mark VI Spitfire showing its two 20mm cannons.

A Supermarine  Mark VI (Type 350) Spitfire fighter aircraft weighing 6740 lb is initially travelling at its top speed of 354 mph. The aircraft is armed with two Hispano-Suiza HS.404 20mm cannons, one on each wing, each of which is fed by a drum magazine containing 60 rounds. Each projectile  fired from  the cannon weighs 130 grams, the rate fire of each cannon is 700 rounds per minute and the muzzle velocity of each shell is 860 m/s.

(a) Calculate the reduction in the aircraft’s speed if the pilot fires both cannon simultaneously until the magazines are empty, if the pilot does nothing to compensate for the recoil. Express your answer in kilometres per hour.

(b) Calculate the average deceleration of the aircraft while the cannons are being fired, and express your result as a fraction of g, the acceleration due to gravity at the Earth’s surface which you can take to be 9.8 ms-2.

(c) A Mark 24 Spitfire – which is somewhat heavier than the Mark VI, at 9,900 lb (4,490 kg) – is armed with 4×20mm cannons, two on each wing. The inboard cannon on each wing has a magazine containing 175 rounds; the outboard one has 150 rounds to fire. Repeat the above  analysis for these new parameters and comment on your  answer.

Answers through the comments box please!