Archive for the Cute Problems Category

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!

 

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A Problem involving Simpson’s Rule

Posted in Cute Problems, mathematics with tags , on March 9, 2018 by telescoper

Since I’m teaching a course on Computational Physics here in Maynooth and have just been doing methods of numerical integration (i.e. quadrature) I thought I’d add this little item to the Cute Problems folder. You might answer it by writing a short bit of code, but it’s easy enough to do with a calculator and a piece of paper if you prefer.

Use the above expression, displayed using my high-tech mathematical visualization software, to obtain an approximate value for π/4 (= 0.78539816339…) by estimating the integral on the left hand side using Simpson’s Rule at ordinates x =0, 0.25, 0.5, 0.75 and 1.

Comment on the accuracy of your result. Solutions and comments through the box please.

HINT 1: Note that the calculation just involves two applications of the usual three-point Simpson’s Rule with weights (1/3, 4/3, 1/3). Alternatively you could do it in one go using weights (1/3, 4/3, 2/3, 4/3, 1/3).

HINT 2: If you’ve written a bit of code to do this, you could try increasing the number of ordinates and see how the result changes…

P.S. Incidentally I learn that, in Germany, Simpson’s Rule is sometimes called called Kepler’s rule, or Keplersche Fassregel after Johannes Kepler, who used something very similar about a century before Simpson…

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!

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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!)

A problem of fluid flowing through a hole

Posted in Cute Problems with tags , , , , on December 19, 2017 by telescoper

I’m sure you’re all already as bored of Christmas as I am so I thought I’d do you all a favour by giving you something interested to do to distract you from the yuletide tedium,
The cute problem of the water tank I posted a while ago seemed to provide a diversion for many – although only about 10% of respondents go it right – so here’s a similar one. It’s not multiple choice so you will have to write your answers to the two parts in the comments box. As a hint, I’ll  say that this is from some notes on dimensional analysis, and it’s one of the harder problems I have in that file!

An incompressible fluid flows through a small hole of diameter d in a thin plane metal sheet. The volume flow rate R depends on d, on the fluid viscosity η and density ρ, and on the pressure difference p between the two sides of the she

(a) Find the most general possible relationship between the quantities  R, d, η,  ρ, and p.

(b) Measurement of the flow rate R1  through this the hole for a pressure difference p1 is made using a particular fluid. What can be predicted for a fluid of twice the density and one-third the viscosity?

 

As usual, answers through the comments box please!

 

 

The Problem of the Water Tank

Posted in Cute Problems on November 26, 2017 by telescoper

Here’s a nice problem I remember hearing in the pub on Friday and figured out this afternoon.

A water tank or sink is open to the air at the top where it can be filled using a tap connected to an infinite reservoir. Water can be drained from the container through an opening at the bottom  by removing a stopper. The effects of viscosity on the outflow from the tank can be neglected.

The time taken for the tank to fill when the tap is fully open and the stopper in place is the  same as the time taken for it to empty from full  when the tap is closed and the stopper is removed.

If the tank is initially empty, the stopper removed and the tap turned full on, how full is the tank when a steady state is reached?

 


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 Cube of Resistance

Posted in Cute Problems with tags , , , on September 14, 2017 by telescoper

It has been brought to my attention that I haven’t posted any cute physics problems recently, so here’s one (which involves applying Kirchoff’s laws) that’s a bit harder than A-level standard which might be of interest to students about to begin a degree in physics this month!


The above image, produced using the advanced computer graphics facilities available at Cardiff University’s Data Innovation Research Institute, represents a cube formed of 12 wires each of which has resistance 1Ω.

What is the electrical resistance between: (i) A and G; (ii) A and H; and (iii) A and D?

As usual, answers through the comments box please!