Archive for October, 2008

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Going round in circles

Sunday, October 26th, 2008

Circular momentum is one of those things that I whizzed through in studying – but all the really crucial stuff is packed into a page I read too fast (there was a whole book to read too…)

If you can see this Flash movie, then click on it to set the point circulating away. You will be able to see the green velocity vector perpendicular to that of the point, and a tiny red acceleration vector pointing in the opposition direction (it would be bigger if the dot went faster).

If the position of a point is described as the following vector:
\mathbf{r}=R\cos (\alpha t)\mathbf{i} + R\sin (\alpha t)\mathbf{j}

\alpha t = \theta is the angle of the line joining the point to (0,0) with the positive x-axis, with \alpha being a constant, and t being time. When \dot{\theta} is constant, we have uniform circular motion.

The angular speed of the point is \omega=\left|\alpha \right|=\left|\dot{\theta}\right| (in radians).

The vector for the point can be differentiated:

\mathbf{v}=\dot{\mathbf{r}}=-\alpha R \sin(\alpha t)\mathbf{i} + \alpha R \cos(\alpha t)\mathbf{j}

The magnitude of the velocity vector is \omega R

which gives a velocity vector perpendicular to the position vector, and tangential to the circle.

Differentiating the velocity vector gives an acceleration vector which is in the opposite direction to the position vector:
\mathbf{a} = \mathbf{\ddot{r}} =-\alpha ^{2} \mathbf{r} = -\omega ^{2} \mathbf{r}

The magnitude of the acceleration vector is \left|\mathbf{a}\right| = \omega ^{2} R = \left|\mathbf{v}\right| ^{2} / R

While the maths of this is all logical, I have to admit to still finding it tricky to intuitively understand the meaning of the velocity and acceleration vectors. But the first time I read this material, it was the meaning of the angular speed which I missed. The normal understanding of speed is as a linear property, and it’s a subtle leap to think of it in radians.

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Navigating an unfamiliar landscape

Sunday, October 26th, 2008

I’m doing an Open University degree in Computing and Mathematical Sciences. Completion of next year’s module will earn the last 30 points for the qualification.

The problem is that the course has varied its focus each year on either maths or computing. I’ve done little maths for the last two years – last year was Software Engineering, the year before was Developing Internet Applications. Preceding that was Graphs, Networks and Design, which was challenging and lovely, but involved maths of a particular flavour. The upshot: the last time I worked hard at maths learning was 2005, and that was a struggle at the time.

Much of maths involves building on strong foundations, and thinking about things in the appropriately accurate, unambiguous manner. It is possible to march ahead through an accumulation of poorly understood precepts – enough at least to get high marks in assignments and exams. 

The plan is to look over the ground that’s been covered – but this time, to make it familiar territory, rather than viewing it from a window at high speed. This may reveal a direction for next year’s journey, and even provide enough momentum to create a trajectory. It will most likely involve dawdling around some of the simpler areas of the landscape.

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Testing Latex

Saturday, October 25th, 2008

Latex:

\frac12 < \left\lfloor \mathrm{mod} \left( \left\lfloor \frac{y}{17}\right\rfloor 2^{-17\lfloor x\rfloor -\mathrm{mod}(\lfloor y\rfloor , 17)},2\right)\right\rfloor

Code: [tex ]\frac12 < \left\lfloor \mathrm{mod} \left( \left\lfloor \frac{y}{17}\right\rfloor 2^{-17\lfloor x\rfloor -\mathrm{mod}(\lfloor y\rfloor , 17)},2\right)\right\rfloor[ /tex]

WP Math Pub:
(a^2+b^2)=a^2+2ab+b^2

Code: \[pmath \](a^2+b^2)=a^2+2ab+b^2[ /pmath]

WP Math Pub looks nicer, but LaTex seems the standard, offering more portability in the future. MathML is probably the most accessible, but is dependent on browser support and looks cumbersome.

I made two changes to wp-latexrender: 

  1. PHP 5 interprets formfeed characters differently, meaning that \formulabox ends up outputting ormulabox, which ends up on screen. Fixed by amending the wrap_formula function in class.latexrender.php to use single quotes
  2. Changed latex.php so that img tags are xhtml compliant
I didn’t want to use a web service, as this adds load to the remote service, and means messing around with image tags in each post.

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Reality through RjDj

Sunday, October 12th, 2008

rjdj.jpg

I installed RjDj on my iphone yesterday. It takes sound from the environment you’re in, using the samples both as the ingredients of a musical composition, and to influence it.

Some notes:

  • It’s not long before you’re singing along to yourself. This can’t be much fun for anyone listening (I remember when walkmen were fairly new, and my sister singing along in the car without being able to hear herself)
  • Walking past railings is a useful way of generating rhythmic noise
  • Walking around, previously intrusive noises such as car hoots, children wailing and people shouting become welcome material.
  • This causes a new detachment from reality – it becomes much easier to hear it all in a state of detachment, as a rich tapestry
  • I wonder how long it will be before this, or other reality-altering devices, end up with people shaping the reality around them in order to make a more interesting experience. We will end in a narrative that would suit The Dice Man
  • You’re recommended to listen with headphones, presumably to prevent unpleasant feedback loops. But it adds a whole new element to driving if plugged into the stereo. Years ago (early 90’s) I imagined a musical device that would take the motions and actions of driving as the trigger. RjDj seems to do the job.
  • It brings a whole new dimension to watching X-Factor
  • The newfound ability to listen to environmental sounds in an unjudgemental manner continues, at least for a while, into Genuine Unamended Reality

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