Archive for the Cute Problems Category

A Question of the Past

Posted in Biographical, Cute Problems, Education, Maynooth, The Universe and Stuff with tags , , , , , on April 20, 2022 by telescoper

I was tidying up some old files earlier today and came across some old examination papers, including those I took for my final examinations in Part II of the Natural Sciences Tripos in 1985. There were six of these, in the space of three consecutive days…

I picked one of the questions to share here because it covers similar ground to my current (!) Advanced Electromagnetism module for final-year students in Maynooth. Sorry it’s a bit grubby!

It’s been a long time since I took my finals and I’d largely forgotten what the format was. The question above was taken from Paper II which consisted of nine questions altogether in three Sections, A (Solid State Physics), B (Statistical Physics) and C (Electromagnetism, from which Q9 above was taken; I think the course was actually called Electrodynamics & Relativity). The examination was 3 hours in duration and students were asked to answer four questions, including one from each Section. That means each question would be expected to take about 45 minutes.

Looking at the paper in general and the above question in particular, a number of things sprang to mind about differences between then in Cambridge and now in Maynooth:

  1. Our theoretical physics papers in Maynooth are 2 hours in duration in which time students are to answer four questions, so that the questions are a bit shorter – 30 minutes each rather than 45.
  2. Our papers are also on a single subject rather than a composite of several; we typically don’t offer the students choice; my Advanced Electromagnetism paper has four questions and students have to answer all four for full marks.
  3. The questions on the old Tripos papers are less structured. There is no indication of the marks allocated to each part of the question in the question above.
  4. As far as I can recall there was no formula booklet back in 1985, though there was a sheet of physical constants. My Advanced Electromagnetism examination this year comes with a couple of pages of useful formulae from vector calculus and key equations in EM theory. One might argue that the old Cambridge papers relied rather more on memory (especially when you take into account that everything was in the space of three days).
  5. Back to Question 9, it is true that this along with the other Electromagnetism questions is at a similar level to what I have been teaching this Semester. If I recall correctly the relevant course in Cambridge was of 24 lectures, the same length as the course I’m teaching this year.
  6. Students taking my course should know how to do both parts of Question 9 without too much difficulty.

On the final point, the easiest way to tackle this sort of problem is to do what the question says: determine the electric and magnetic potentials, derive the electric and magnetic fields from them, then work out the Poynting vector quantifying the energy flux. The part of this that survives in the far-field limit gives you the radiated power then – Bob’s your Uncle – the answer is basically the Larmor Formula which is ubiquitous in problems of this type. The case of an oscillating dipole is a standard derivation but this method works for any time-varying source, as long as you remember to include the retarded potentials if it’s not periodic.

Had I been writing this question for a modern exam I think I would at very least have ended the first part with “Show that the radiated power is…” and then given the formula, so that it could be used for the second part even if a student could not derive it.

Elliptical Discussions

Posted in Cute Problems, Education, Maynooth, The Universe and Stuff with tags , , on April 7, 2022 by telescoper

It’s the ninth week of Semester 2 and I’m coming to the end of lectures and laboratory sessions for my Computational Physics module; for the remaining three weeks (plus the Easter vacation) the students in this class will be mainly concentrating on the mini-projects that form part of the assessment.

This afternoon, though we had a session on how to transform higher-order differential equations into sets of coupled first-order ODEs suitable for vectorization and consequent solution using standard techniques. The problem we focussed on today was the simple problem of orbital motion of a test particle under the gravitational force in plane-polar coordinates, which can be prepared for physical solution thusly:

This sort of thing reminds me of my undergraduate theory project at Cambridge, where I did a similar thing to solve the equations governing the action of a four-level laser, though that was in Fortran rather than Python. In my own solution I used Python’s off-the-shelf solver odeint.

I like the orbital motion problem a lot because it’s a bit more than a coding exercise: students have to think about how to choose initial data, how to test the their code and interpret the results. Even before that there’s the issue of what units to use; SI units are a bit daft for astronomical problems. For solar system calculations it makes sense to use Astronomical Units for distances and years for time; in such a system it’s easy to work out that GM is just 4π2, which avoids having to deal with ridiculously large or ridiculously small numbers.

Anyway, the fun thing about this lab was that once everyone had got their code working they could try setting initial data to get a circular orbit as a special case, explore how the shape of elliptical orbits depends on the input data, how to make an unbound orbit, and so on. It’s important to understand the output of a numerical calculation in terms of basic physical principles. All that led to a discussion in class of solar system exploration, transfer orbits, what would happen if the mass of the Sun suddenly changed, or if G was a function of time, and lots of other things.

I find sessions like this that encourage students to explore problems themselves very rewarding and I think they add a valuable extra dimension to standard teaching formats. I hope the projects that they’ll be doing from now on – involving topics in areas ranging from atomic physics, cosmology, particle physics and climate science, and done in groups – will provoke even more discussion of this type.

The Hardest Problem

Posted in Cute Problems, Education, mathematics with tags , , , on November 19, 2021 by telescoper

The following Question, 16(b), is deemed to have been the hardest problem on the Maths Extension 2 paper of this year’s HSC (Higher School Certificate), which I think is the Australian Equivalent of the Leaving Certificate. You might find a question like this in the Applied Mathematics paper in the Leaving Certificate actually. Since it covers topics I’ve been teaching here in Maynooth for first-year students I thought I’d share it here.

I don’t think it’s all that hard really, probably because it’s really a physics problem (which I am supposed to know how to solve), but it does cover topics that tend to be treated separately in school maths (vectors and mechanics) which may be the reason it was found to be difficult.

Anyway, answers through the comments box please. Your time starts now.

A Question of Balance

Posted in Cute Problems, The Universe and Stuff with tags , , on November 3, 2021 by telescoper

Here’s an interesting physics problem for you, based on the idea that the mass of a set of bodies changes if the energy of their mutual interactions changes according to Einstein’s famous formula “E=mc2“.

Four identical masses are placed at rest in pairs either side of an extremely sensitive balance in a symmetrical way such that the distance between the members of a pair is identical for each pair and the centre of mass of each pair is equally spaced from the fulcrum of the balance. In this configuration the system is in equilibrium and the balance is level.

As illustrated schematically in the graphic, one pair of weights is adjusted by displacing each weight slightly away from the centre of mass of the pair by an equal and opposite distance, thus keeping the position of the centre of mass of the pair constant. The other pair of weights is not adjusted.

Assuming that the balance is sufficiently sensitive to detect the slight change in mass associated with the gravitational interactions between the masses in each pair, does the balance move?

If it does move which pair moves up: the displaced pair or the undisturbed pair?

Oh Larmor! Energy in Electromagnetic Waves

Posted in Cute Problems, The Universe and Stuff with tags , , , on April 16, 2021 by telescoper

This week I started the bit of my Advanced Electromagnetism module that deals with electromagnetic radiation, including deriving the famous Larmor Formula. It reminded me of this little physics riddle, which I thought I’d share again here.

As you all know, electromagnetic radiation consists of oscillating electric and magnetic fields rather like this:

Figure10.1

(Graphic stolen from here.) The polarization state of the wave is defined by the direction of the Electric field, in this case vertically upwards.

Now the energy carried by an electromagnetic wave of a given wavelength is proportional to the square of its amplitude, denoted in the Figure by A, so the energy is of the form kA2 in this case with k constant. Two separate electromagnetic waves with the same amplitude and wavelength would thus carry an energy = 2kA2.

But now consider what happens if you superpose two waves in phase, each having the same wavelength, polarization and amplitude to generate a single wave with amplitude 2A. The energy carried now is k(2A)2 = 4kA2, which is twice the value obtained for two separate waves.

Where does the extra energy come from?

Answers through the Comments Box please!

The Mechanics of Nursery Rhymes

Posted in Cute Problems, The Universe and Stuff with tags , , , , on December 30, 2020 by telescoper

I’ve always been fascinated by Nursery Rhymes. Some people think these are little more than nonsense but in fact they are full of interesting historical insights and offer important advice for the time in which they were written. One such story, for example, delivers a stern warning against the consequences of placing sleeping babies in the upper branches of trees during windy weather.

Another important role for nursery rhymes arises in physics education. Here are some examples that students of elementary mechanics may find useful in preparation for their forthcoming examinations.

1. The Grand Old Duke of York marched 10,000 men up to the top of a hill and marched them down again. The average mass of his men is 65 kg and the height of the hill is 500m.

(a) Estimate the total work done in marching the Duke of York’s men up to the top of the hill.

(b) If, instead of marching down again, the men take turns sliding down a frictionless slide back to where they started, estimate the average speed of a man when he reaches the bottom of the hill.

(You may assume without proof that when they were up they were up, and when they were down they were down and, moreover, when they were only half way up they were neither up nor down.)

2. By calculating the combined rest-mass energy of half a pound of tuppenny rice and half a pound of treacle, and assuming a conversion efficiency of 10%, estimate the energy released when the weasel goes pop. (Give your answer in SI units.)

3. The Moon’s orbit around the Earth can be assumed to be a circle of radius r. A cow of mass m is standing on the Earth (which has mass M, and radius R). Derive a formula in terms of r, R, M, m and Newton’s Gravitational Constant G for the energy the cow needs in order to jump over the Moon.

(The Earth, Moon and cow may be assumed spherical. You may neglect air resistance and udder frictional effects. )

Feel free to contribute similar problems through the Comments Box.

A Problem of Resistance

Posted in Cute Problems with tags , on November 29, 2020 by telescoper

Bizarrely, last night I dreamt of this physics problem. This mean that I’ve seen it before somewhere, but if that’s the case then I’ve forgotten where. In the dream the problem of electrical resistance was muddled up with the problem of how to calculate the Euler Characteristic of a structure defined on a grid*, which is something I have used in the past. Anyway, with apologies for the poor quality of the drawing, here is the set up.

Twelve identical resistors R are arranged in four squares with common edges thus:

Yes, they’re meant to be identical squares!

What would be the effective resistance of this circuit measured between A and B?

Please post your answers through the comments box, with appropriate explanations. Bonus marks for elegant (i.e. short) solutions.

(In my dream this problem came up in contrast with the case where the four internal resistors and their connecting wires were absent, so the circuit was just a ring.  The Euler Characteristic of the original connected set of squares is 1 while that of the ring is 0, not that it’s relevant to the problem in hand!)

 

A Remnant Problem

Posted in Cute Problems, The Universe and Stuff with tags , , , on November 16, 2020 by telescoper

 

I haven’t posted any physics problems for a while so here’s  a quickie involving dimensional analysis. You have to assume that the supernova remnant mentioned in the question is roughly spherical, like the one shown above (SNR 0500-67.5):

As usual, answers and comments through the box below please!

Click on the `continue reading’ thing if you would like to see my worked solution:

Continue reading

Marginalia

Posted in Cute Problems, Education, The Universe and Stuff with tags , , , , , on August 12, 2020 by telescoper

While this morning’s repeat exams were going on I was leafing through an old second-hand text book, one of many I have acquired over the years looking for nice problems and worked examples. The good thing about old books is that solutions to the problems are usually not available on the internet, unlike modern ones. The book concerned this morning is a classic: Statics by Horace Lamb, which you can still get via Cambridge University Press. I have the first edition, published in 1912.

Looking through I was somewhat alarmed to see what had been pencilled in some of the margins:

Of course anyone who has been to India knows that the swastika isn’t necessarily a Nazi symbol: you find it all over the place in the Indian sub-continent, where it is used as a symbol for good luck. I remember being given a very nice conference bag in Pune many years ago with a swastika on it. I didn’t use it back home, of course.

The first owner of the copy of Statics that I have was acquired in 1913 by a J.H.C (or G) Lindesay of Sidney Sussex College, Cambridge. I know because he/she inscribed their name in the front. That doesn’t look to me like an Indian name, but I think it’s a fair bet that the book passed through many hands before reaching me and that one of the past owners was Indian. I haven’t tried any of the problems marked with the swastika, but perhaps they are difficult – hence the `good luck’ symbol? I notice though that the symbol at the bottom of the page has a chirality different from the others. Is this significant, I wonder?

All of which irrelevance reminded me of an discussion I’ve had with a number of people about whether they like to scribble in the margins of their books, or whether they believe this practice to be a form of sacrilege.

I’ll put my cards on the table  straightaway. I like to annotate my books – especially the technical ones – and some of them have extensive commentaries written in them. I also like to mark up poems that I read; that helps me greatly to understand the structure. I don’t have a problem with scribbling in margins because I think that’s what margins are for. Why else would they be there?

This is a famous example – a page from Newton’s Principia, annotated by Leibniz:

dsc00469

Some of my friends and fellow academics, however, regard such actions as scandalous and seem to think books should be venerated in their pristine state.  Others probably find little use for printed books given the plethora of digitial resources now available online or via Kindles etc so this is not an issue..

I’m interested to see what the divergence of opinions is in with regard to the practice of writing in books, so here’s a poll for you to express your opinion:

The Geostationary Orbit

Posted in Cute Problems, The Universe and Stuff with tags , , , on May 16, 2020 by telescoper

I’m was mucking out this blog’s blocked comments folder and unsurprisingly found a few from Mr Hine, a regular if sadly deranged correspondent.

One of his blocked comments begins

In the forlorn hope that Mr Hine might some day learn something scientifically correct I thought I’d repost this problem, which is very easy if you have a high school education in physics or applied mathematics but no doubt very difficult if you’re Mr Hine.

Verify that the radius of a circular geostationary orbit around the Earth is about 42,000 km, i.e. find the radius of a circular orbit around the Earth which has a period of 24 hours so that its orbital period matches the Earth’s rotation period, thus ensuring that an object travelling in such an orbit in the same direction as the Earth’s rotation is always above the same point on the Earth’s surface.

(You will need to look up the mass of the Earth.)