In the foyer of the Physics Department at the University of Barcelona you will find, as well as a fine refracting telescope, an example of the Atwood Machine. For some years before my current sabbatical I have been teaching Newtonian Mechanics to first-year students in Maynooth and used this as a simple worked example. I have to admit I’ve never seen an actual Atwood machine before, and what I’ve done in lectures is the simplified form on the right rather than the actual machine on the left.
Atwood Machine
The illustration on the right depicts the essential elements, but you can can see that the actual machine has a ruler which, together with a timing device, can be used to determine the acceleration of the suspended mass and how that varies with the other mass. You can work this out quite easily in the simplest case of a frictionless pulley by letting the tension in the string (which is light and inextensible) be T (say) and then eliminating it from the equations of motion for the two masses. I leave the rest as an exercise for the reader. A more interesting problem, for the advanced student, is when you have to take into account the rotational motion of the pulley wheel…
Today I gave a revision lecture/tutorial for my module Advanced Electromagnetism. With the Examination Period starting on Friday, that was the last class I will do for that. One of the topics I’ve been asked to cover in revision was the Method of Images for electrostatics. Preparing for the class I came across this cute problem which I thought I’d share here:
The question concerns a charge +q placed at a distance d as shown above an infinite earthed conducting plane distorted by the presence of a hemispherical bulge with radius R.
Using the method of images, or otherwise, calculate the potential at an arbitrary point above the conducting surface. (HINT: you need three image charges)
Find the magnitude and direction of the electrostatic force on the charge.
If you’re feeling keen you might also find what fraction of the total induced on the conductor is on the hemispherical part.
Answers through the comments box please!
Well, nobody posted an answer so here’s an outline solution.
To solve this problem you need three image charges: one is of charge – q at z=-d to make the plane an equipotential. For an isolated sphere you need a charge of -qR/d at z=-R^2/d (the inverse point of the sphere). But this charge also has an effect on the plane, which you need to correct by placing another image charge of +qR/d at z=-R^2/d. That is, the solution for the potential is due to the original charge plus three image charges. Then the potential is just the sum of four point charges.
You can differentiate the answer to the first bit to get the force, or you could work out the force on the original charge directly by adding the forces in the z-direction from the three image charges, it being obvious by symmetry that there is no other component of the force. For d>R this results in a force which is downward, so the charge is pulled towards the conductor. I’ll leave that as an exercise!
It’s been a while since I posted a question in the Cute Physics Problems folder so I thought I’d offer this one. It’s not particularly hard, but I think it’s quite instructive.
A thin spherical shell of radius r carrying a charge Q spread uniformly with constant surface density is split into two equal halves by a narrow planar cut passing through the centre as shown in the detailed diagram below:
Calculate the force arising from electrostatic repulsion between the two hemispherical shells, expressing your answer in terms of Q and r in SI units.
Answers through the Comments Box please. First correct answer wins a tomato*
I just came across this paradox in an old book of mathematical recreations and thought it was cute so I’d share it here:
Here are two possible solutions to pick from:
Since we are now in the era of precision cosmology, an uncertainty of a factor of 400 is not acceptable so which answer is correct? Or are they both wrong?
The 2022 cycle of Leaving Certificate examinations is under way and the first Mathematics (Ordinary and Higher) were yesterday there’s been the usual discussion about whether they are easier or harder than in the past. I won’t get involved in this except to point you to this interesting discussion based on an archive of mathematics questions, that this year the papers have more choice for students and that, apparently, the first Higher Mathematics paper had very little calculus on it.
Anyway, I was looking through some old Applied Mathematics Leaving Certificate papers, as these cover some similar ground to our first year Mathematical Physics at Maynooth, and my eye was drawn to this question from 2010 about two balls jammed in a cylinder…
I’d add another: does it matter whether or not the cylinder is smooth (as this is not specified in the question)?
Your answers are welcome through the comments box!
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:
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.
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.
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.
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).
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.
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.
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 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.
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?
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:
(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.
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In the Dark 