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I am a math teacher and I have to teach a topic called "Bruchterme" and "Bruchgleichungen" in german (I don't know the english word for it). For example

$$ \frac{x^2 - 3}{(x - 2)x^2} + \frac{4}{x} + 2 $$

is a "Bruchterm" and

$$ \frac{4x}{2x -3} = 4 - \frac{2x}{x-1} $$

is a "Bruchgleichung".

Students have to learn, how to determine for which $x$ the term or equation is defined (i.e. the "singularities") and how to solve such equations.

Now my problem is that most textbooks about this offer little to no interesting applications of this, especially no applications of determining for which $x$ the term is defined.

Now I am looking for interesting examples from classical physics or engineering for this type of problem. Especially examples where singularities occur and are physically "interesting" in some way. Though it is for high school level, I don't want to restrict the question to trivial examples, but to classical physics.

The only physical applications I have in mind are the following:

Two parallel resistors $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$
Same type of equation: $\frac{1}{f} = \frac{1}{b} + \frac{1}{g}$ for the thin lens equation (where $b$ and $g$ are the distances from the image to the lens resp. from the object to the lense)

However here the denominators are rather trivial and I don't see why it would be interesting to determine for which value for example of $b$ (if the other values are fixed) the equation is not defined.

Gravitational force ($\propto \frac{1}{r^2}$) here one might discuss that the force goes to infinity for $r \to 0$. However the point of view of the curriculum is not to discuss such limits but just to look if one is allowed to plug in certain values for the variable or not (in a sense that dividing by 0 is not allowed or not defined without connection to limits).

  1. Are there some interesting examples from physics where students see that it is worth to learn how to solve such equations (from type and perhaps complexity as in my example above)

  2. Are there examples from physics where students see that it is worth to learn to determine for which values the term or equation is defined?

Edit (too long for a comment)

@Danu: I am teaching in the less known part of the german education system, called "Berufliche Schulen" more specifically I am at a school which focus is on technology, engineering and science. The system is a bit complicated with classes on very different levels. To make it short I have the motivation problem described above in a low level class and in a high level class (the level of the second one is equivalent to a German "Gymnasium" or even a bit higher in mathematics and physics) but the students are generally very interested in technology, engineering, physics and computer science (most of them want to take up a profession in this fields. Ranging from electricians or lab technician in the lower level classes up to enegineers or physicists in the higher level classes). Students are from 15 to 18 years old. Even in the lower level classes they are very interested in nontrivial physics even if they cannot grasp it conceptually or mathematically. So for motivation purposes some nontrivial, but for the students interesting examples would be good.

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closed as too broad by ACuriousMind, Kyle Kanos, Qmechanic Aug 12 '15 at 14:03

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Complex analysis has LOTS of stuff with singularities, and also awesome applications. Something tells me though, that you might not be looking for something at that level... Also, the first thing is just a rational function, of $x$. The second, I would call a rational equation (rational is somewhat equivalent to 'Bruch') $\endgroup$ – Danu Dec 1 '13 at 8:56
  • $\begingroup$ Another idea: you could talk about some things like the electric energy of an electron: en.wikipedia.org/wiki/…. Once again, however, this might be too difficult. What level of students are you teaching? $\endgroup$ – Danu Dec 1 '13 at 8:58
  • $\begingroup$ obligatory link to Wikipedia: en.wikipedia.org/wiki/Rational_function#Applications and de.wikipedia.org/wiki/Rationale_Funktion#Anwendungen ; the German one lists some examples, but their singularities aren't particularly interesting as far as I can see $\endgroup$ – Christoph Dec 1 '13 at 10:01
  • $\begingroup$ @Danu: Indeed, the don't know complex numbers so this may be mathematically (not physically) to advanced. $\endgroup$ – Julia Dec 1 '13 at 10:02
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    $\begingroup$ Comment to the question (v3): This looks like a list question with many answers. $\endgroup$ – Qmechanic Dec 1 '13 at 13:37
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I would use the Van der Waals equation. Here you are, in the German wiki: http://de.wikipedia.org/wiki/Van-der-Waals-Gleichung

Also, there's lot of laws expressed as $a = b/c$. They may be too simple for your purposes, but they can be useful to show the application. For example, Ohm's law $I=V/R$, where $I$ is current intensity, $V$ voltage and $R$ is the resistance. Same with resistivity: http://de.wikipedia.org/wiki/Spezifischer_Widerstand Or ideal gas equation: http://de.wikipedia.org/wiki/Thermische_Zustandsgleichung_idealer_Gase

And you can read something about classic field theory theory: http://en.wikipedia.org/wiki/Classical_field_theory With that, you can say that the equations you showed can be, for example, a one-dimension field which determines the velocity of a fluid, and show that is not defined for some values; in that values, you have a vortex or turbulence fluid, and maybe a source/sink of fluid. Also, it can be the expression of an electric field, and in that case, in the points you have positive charges (source) or negatives charges (sink). What it is, you have to calculate using divergences, but I think it's too complicated for your students. http://de.wikipedia.org/wiki/Divergenz_(Mathematik)

I hope this will be useful n.n

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I do not know if my answer can be classified as physics but Julia in her question wrote that

the students are generally very interested in technology, engineering, physics and computer science

and so I offer and answer that maybe can be considered technology or computer science.

The pinhole or projective camera is a purely geometric model that describes the process whereby points in the world are projected into the image. The position in the image depends on the position in the world, and the pinhole camera model captures this relationship.

The pinhole camera model is important in field like Machine Vision (where a camera model is generally needed to perform for example measures on objects under inspection) or Computer Graphics (where a camera model is used to perform the rendering of a scene).

A 3D point $\mathbf{w}=[u,v,w]^T$ (where $u$,$v$,$w$ are expressed, say, in mm) is projected to a 2D point $\mathbf{x}=[x,y]^T$ ($\mathbf{x}$ is a point in the image and so $x$,$y$ are expressed in pixel) by the relations

$x=\frac{\phi_x(\omega_{11}u+\omega_{12}v+\omega_{13}w+\tau_x)+\gamma(\omega_{21}u+\omega_{22}v+\omega_{23}w+\tau_y)}{\omega_{31}u+\omega_{32}v+\omega_{33}w+\tau_z}+\delta_x$

$y=\frac{\phi_y(\omega_{21}u+\omega_{22}v+\omega_{23}w+\tau_y)}{\omega_{31}u+\omega_{32}v+\omega_{33}w+\tau_z}+\delta_y$

All the symbols different from $u$,$v$,$w$,$x$,$y$ are parameters describing the camera (focal length, etc.) and its position (translation and rotation matrix) in the 3D space.

For further details see for example Chapter 14 in the book Computer Vision: Models, Learning, and Inference by Simon J.D. Prince from which I copied the above explanation.

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