dimensional analysis | In the Dark

Archive for dimensional analysis

Lecture Summary

Posted in Biographical, Education, Maynooth with tags dimensional analysis on December 16, 2020 by telescoper

Yesterday I gave my last full lectures of the term. The picture shows a summary of my final lecture on Mechanics & Special Relativity for first-year students. It may look a bit thin on content but I’m very tired. How I managed to pad it out to 50 minutes I have no idea.

Today we have presentations by project students via Microsoft Teams which I can watch from home while I await the delivery of a blackboard which I’m going to put in my study at home. (UPDATE: Predictably, the Courier didn’t show up.)

Later today I have my last lecture slot which I will use to pass on some information about the ~~examination~~ online timed assessment in January.

And that will be the end of my teaching for the (calendar) year, which seems to have lasted at least a decade. Let’s hope next year is better.

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A Remnant Problem

Posted in Cute Problems, The Universe and Stuff with tags blast wave, dimensional analysis, Sedov solution, Supernova Remnant 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:

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A Problem of Dimensions

Posted in Cute Problems, Maynooth, The Universe and Stuff with tags astronomy, dimensional analysis, Gravity, orbital period on August 21, 2019 by telescoper

We’ve more-or-less sorted out who will be teaching what next term in the Department of Theoretical Physics at Maynooth University next term (starting a month from now) and I’ll be taking over the Mathematical Physics module MP110, which is basically about Mechanics with a bit of of special relativity thrown in for fun. Being in the first semester of the first year, these is the first module in Theoretical Physics students get to take here at Maynooth so it’s quite a responsibility but I’m very much looking forward to it.

I am planning to start the lectures with some things about units and dimensional analysis. Thinking about this reminded me that I posted a dimensional analysis problem (too hard for first-year students) on here a while ago which seemed to pose a challenge so I thought I would post another here for your amusement.

The period P for an elliptical orbit of semi-major axis a of a moon of mass m around a planet of mass M, depends only on the quantities a, m, M and G (Newton’s Gravitational Constant).

(a). Using dimensional analysis only, determine as completely as possible the relationship between P and these four quantities.

(b). How would the period P compare with the period P′ of a system consisting of a moon of mass 2m orbiting a planet of mass 2M in an ellipse with the same semi-major axis a?

Please submit your efforts through the comments box below.

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Einstein and your Gas Bill

Posted in History, Television, The Universe and Stuff with tags Albert Einstein, calorific value, dimensional analysis, Edward Enfield, Fritz Hasenohrl, Watchdog on October 11, 2011 by telescoper

Taking refuge in my office this lunchtime for a sandwich and a cup of coffee I turned to the latest edition of Physics World and came across an funny little story about a physicist (who is completely new to me) with the splendid name of Fritz Hasenöhrl.

The news story relates to a paper on the arXiv, part of the abstract of which I’ve copied below:

In 1904 Austrian physicist Fritz Hasenohrl (1874-1915) examined blackbody radiation in a reflecting cavity. By calculating the work necessary to keep the cavity moving at a constant velocity against the radiation pressure he concluded that to a moving observer the energy of the radiation would appear to increase by an amount $E=(3/8)mc^2$ , which in early 1905 he corrected to $E=(3/4)mc^2$ …

Since I’ve been doing a bit of dimensional analysis with first-year students, I’m a bit surprised that the authors of this paper read so much into the fact that Hasenöhrl’s formula bears a superficial resemblance to Einstein’s most famous formula $E=mc^2$ , probably the best known and at the same time worst understood equation in physics. In fact any physicist worth his or her salt no matter how incorrect their reasoning would have to get something like $E =\alpha mc^2$ , with $\alpha$ some dimensionless number, simply because the answer has to have the correct dimensions to be an energy.

Expressing energy in terms of the basic dimensions mass $M$ , length $L$ and time $T$ is probability easiest to do when you think of mechanical work (force×distance). Since Newton’s laws give a force equal to mass×acceleration, a force has dimensions $MLT^{-2}$ , so work (a form of energy) has dimensions $ML^{2}T^{-2}$ . Now try to make this out of a combination of a mass ( $M$ ) and a velocity ( $LT^{-1}$ ) and you’ll find that it has to be mass×velocity². You can’t get the dimensionless constant this way, but the combination of $m$ and $c$ must be the way it is in Einstein’s formula.

Anyway, all this suddenly reminded me of a day long ago when I appeared on peak-time television in the consumer affairs programme Watchdog, explaining – or, rather, attempting to explain – the physics behind the way gas bills are calculated. Apparently someone had written in to the programme asking why it was that they weren’t just being charged for the volume of gas that had flowed through their meter, but that the cost involved a complicated calculation involving something called the calorific value of the gas.

The answer is fairly obvious, actually. The idea is that to make competition fairer between different forms of energy (particularly gas and electricity) the bills should be for the amount of energy you have used rather than the amount of gas. Since the source of fuel varies from day to day so does its chemical composition and hence the amount of energy that can be extracted from it when it is burned. Gas companies therefore monitor the calorific value, using it to convert the amount of gas you have used into an amount of energy.

On the programme I was confronted by the curmudgeonly Edward Enfield (father of comedian Harry Enfield) who took the line that it was all unnecessarily complicated and that the bill should just be for the amount of gas used, rather in the same way that petrol is sold. When I tried to explain that the way it was done was really fairer, because it was really the energy that mattered, it quickly became obvious that he didn’t really understand what energy was or how it was defined. He didn’t even get the difference between energy and power. I suspect that goes for many members of the general public.

It was all a bit tongue-in-cheek, but I enjoyed the sparring. Eventually he came out with a question about why energy was given by $E=mc^2$ rather than $mc^3$ or something else. So I launched into an explanation of dimensional analysis and why $mc^3$ couldn’t be an energy because it has the wrong dimensions. His eyes glazed over. The shoot ended. My splendidly erudite and logically rigorous exposition of dimensional analysis never made it into the broadcast programme.