Problem Statement of Series 6, Year 28

The Turtle A'Tuin, on the shell of which the four elephants that carry on their backs Discworld stand, isn't tiny. Let us assume that we would be bored with the sphericity of our Earth and we would want to exchange it for a circular disc with the same mass and density and with the width $h=1\;\mathrm{km}$ carried by the same turtle-elephant band. In case that the turtle would have hit with the tip of its tale into a planetoid, how long would it take for it to notice the impulse of pain, given that her tail and her brain are connected by a very long neuron? (This neuron is approximately as long as the diameter of the disc) How much earlier/later would A'tuin realise this pain (the length of the neuron is equivalent to its length 18 000 km)? For a numerical estimate assume that the speed of the spread of the signal in the large animals is the same as with normal land animals who experience a speed of $v≈120\;\mathrm{m}\cdot \mathrm{s}^{−1}$.

Mage Greyhald celebrated his 100th birthday a long time ago and has begun to fear that Death will pay him a long delayed visit. He decided thus that he will ecase himself into a magic chest, where Death can't reach him. Unfortuntely he forgot to tell the craftsmen to add breathing holes. Air in the chest takes up a volume;$V_{0}=400l$, the percentage of the volume that is oxygen is $φ_{0}=0,21$. With every breath he uses up only $k=20%$ of the volume's oxygen $V_{d}=0,5l$. The frequency of breaths of the mage after the closing of the chest rises according to the relation

$$\\f(t)=f_0 \cdot \frac{\varphi_0}{\varphi (t)}\,,$ wheref_{0}=15breaths\cdot min^{−1}is$ the initial brath frequency is $φ(t)$ and the percetage of the volume that is oxygen at time case $t$. Determine how long until Death will come for Greyhald if the minimum volume of oxygen in the air required for survival is $φ_{s}=0,06$.

One of the offices of lord Vetinari has a circular layout with a perimeter $R$ and is placed on bearings, thanks to which it can turn around its axis. To ensure maximum rotation it uses an engine that can apply any moment of force. During the turning the room has a torque of friction acting on it $M_{0}$, independent of speed, which is equivalent to the static friction torque. The chair for visitors is positioned so that a person will feel the rotation only if it will exceed $ε_{0}$. Determine what is the smallest time it will take to turn by 180°, so thatthe visitor won't notice and what is the work required to achieve it. The mass of the room, that can be approximated to a homogeneous disc, is $m$.

Bonus: Assume that the visitor will feel the roation only if the change in angular acceleration is greater than $j_{0}$.

Before the edge of Discworld was reached and overcome and scientific expeditions were made to confirm the existence of the four elephants, turtle and determine its gender some primitve tribespeople thought that the force that kept them on the surface was due to a disc made from a superdense Wasneverwasium. It was truly a very primitive idea because as we know today the expedition, that confirmed the turtle's existence, infamously ended when its boat tore apart and fell or rather did not fall;... Nevertheless we would be interested what kind of surface density would such a disc have to if an object in its middle should experience, while neglecting magical forces, attracted with the same force as the gravitational force on Earth. Assume that, as the legends told, the disc is very thin and is placed ;$H=8^{4}m=4096\;\mathrm{m}$ under the surface of Discworld. The Disc should be hmomogeneous and the masses of other bodies negligible. Neglect the movement of the turtle and the elephants. If you haven't read the works of a genius for whom Death came recently then just replace Discworld with Earth. Discworld has a diameter of precisely $d=10000\;\mathrm{km}$ for our purposes.

During his visit to Ankh-Morpork Two flower also visited a pub. It wouldn't have been a good pub if a general brawl didn't occur. A brawl during which chairs, bottles and other things fly fromone side of the pub to the other. Twoflower obviously documented everything with his camera. Now he is currently taking a picture of a ball of radius $Rthat$ is flying with a velocity $v$ (which is close to the speed of light $c)$. Even in such establishments the theory of relativity is valid and from it stems that Twoflower could have measured the Fitzgerald contraction of the ball in his rest frame in the direction of movement by a factor of

$$
\sqrt{1- \frac{v^2}{c^2}}$$

What radius of the ball was documented by the camera with a negligibly small exposition?The position of the camera is general.

We all know very well how well the water supply is arranged in Discworld. And none of us need to know how. What if somethig serious would happen and magic would stop working? How long would it take until Discworld would be without water? For simplicity you can assume a pesimistic situation where nobody would hold the water in any way. You know very well that Discworld has a diameter of $d=10000\;\mathrm{km}$, a homogeneous gravitational acceleration $g≈10\;\mathrm{m}\cdot \mathrm{s}^{−2}acts$ everywhere and is perfectly circular. The true complete volume and the distribution of water on Discworld is unknown to everybody so we can consider water being homogeneously distributed over Discworld, that can be considered flat and water has a height of $H=5\;\mathrm{m}$ (that is very pesimistic because everybody would have to be standing on stakes to stay out of the water or being completely submerged in it). The aim of the task is to find an approximate model that would give a good estimation as to the time it would take to lose all the water..

On Discworld it is not unusual to be an alchemist. So we have decided that you should try it. Imagine that you are sitting an exam to enter the guild of alchemists. Together with the brochure of the series you got three wrapped pieces of metal. They are thin plates of metal so be careful with them so that yu won't destroy them and ideally don't touch them. It is your task to find out which (precious?) metals we sent you. We don't require you return the metals and so you can use whatever procedures to determine that, even destructive processes but we shall acknowledge only the sufficiently scientific ones. Your solution will be the description of the procedure required and to determine as precisely as possible the cmposition of the individual specimens and you should menntion the label that was on their packaging. Don't forget that it is even good to determine what metals they aren't.

Note If someone wouldlike to become a new participant in this seminar and they would like to solve this task then they should write an email to alchymie@fykos.cz and they will recieve the package from a week later up to 10 days.

Copy the function $iterace_stanMap$ from the series and using the following commands choose ten very close initial conditions for some $K$.

K=...;

X01=...;

Y01=...;

Iter1 = iterace_stanMap(X01,Y01,1000,K);

...

X10=...;

Y10=...;

Iter10 = iterace_stanMap(X10,Y10,1000,K);

</pre> Between $Iter1$ and $Iter10$ there are hidden a thousand iterations of given initial conditions using the Standard map. As to see how the ten points look after the $nth$ iteration, you have to write

n=...;

plot(Iterace1(n,1),Iterace1(n,2),"o",...,Iterace10(n,1),Iterace10(n,2),"o")

xlabel ("x");

ylabel ("y");

axis([0,2*pi,-pi,pi],"square");

refresh;

</pre> we write $"o"$ into $plot$ so that the points will draw themselves as circles. The rest of the commands is then included so that the graph will include the whole square and that it would have the correct labels.

• Set some strong kicks, $Kat$ least approx. -0,6, and place the 10 initial conditions very close to each other somewhere in the middle of the chaotic region (ie for example „on the tip of a pen“). How do the ten iteration's distances with respect to each other change? Document on graphs. How do the ten initially very close initial conditions change after 1 000 iterations? What can we learn from this about the „willingness to mix“of the given area?
• Take again a large kick and set your ten initial conditions along the horizontal equilibrium of the rotor ie $x=0$, $y=0$. How will these ten initial conditions change in time with respect to each other? What can we say about their distance after a large amount of kickso?

**Bonus:** Try to code and plot the behaviour of some other map. (For inspiration you can look at the sample solution of the last series.)