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### 2. Series 21. Year - S. cutting of wild plains

<h3>Uranium storage</h3>

Very important question is storing of radioactive waste. Usually it is stored in cylindrical containers immersed in water, which keeps the surface at constant temperature 20 °C. Your task is to find the temperature distribution inside containers of square base of edge length 20 cm. Container is relatively long, therefore just temperature distribution in horizontal cross section is of interest. Uranium will be in block of square base of edge 5 cm. From the experience with cylindrical capsules we know, that it will have constant temperature of about 200 °C.

<h3>Heating wire</h3>

Lets have a long wire of circular cross section and radius $r$ from a material of heat conductivity $λ$ and specific conductivity $σ$. Then a electric field is applied. Lets the electric field inside the wire is constant and parallel with the axis of the wire and the strength is $E$. Then the current through wire will be $j=σE$ and will create Joule's heat with volume wattage $p=σE$.

Because the material of the wire has non-zero temperature conductivity, some equilibrium gradient of temperature will form. The gradient fulfills Poisson's equation $λΔT=-p$. Assume, that the end of wire is kept at temperature $T_{0}$. This gives a border condition needed to solve the equation. Due to symmetry we can take into account only two dimensions: on cross section of wire (temperature will be independent of shift along the axis of symmetry). Now it is easy to solve the problem with methods described in text.

However, we will make our situation little bit more complex and will assume, that specific electrical conductivity $σ$ is function of temperature. So we will have a equation of type Δ$T=f(T)$.

Try to solve this equation numerically and solve it for some dependency of conductivity on temperature (find it on internet, in literature of just pick some nice function) and found temperature profile in wire profile. Try to change intensity of electric field $E$ and plot volt-amper characteristics, you can try more than one temperature dependency. $σ(T)$ (e.g. semiconductor which conductivity increase with temperature, or metal, where conductivity is decreasing) etc.

Do not limit your borders, we would be glad for any good idea.

<h3>Capacity of a cube</h3>

Calculate capacity of ideally conductive cube of edge length 2$a$ (2Ax2Ax2A). If you think, it is simple, try to calculate for cuboid (AxBxC) or other geometrical shapes.

**Hint:**
Capacity is a ration of the charge on the cube to the potential on the surface of cube (assuming that the potential in infinity is zero). Problem can be solved by selecting arbitrary potential of cube and solving Laplace equation Δ$φ=0$ outside of the cube and calculating total charge in cube using Gauss law. E.g. calculating intensity of electrical field and derivating potential and calculation of flow through nicely selected surface around the cube.

Final solution is finding a physical model, its numerical solution and realization on computer. More points you will get for deeper physical analysis and detailed commentar. For algorithm you can also get extra points.

Zadal spoluautor seriálu Lukáš Stříteský.

### 1. Series 21. Year - E. catch your snail

Measure the smallest movement which can be registered by human eye. To be precise, measure minimal angular velocity of an object relatively to the background, which your constantly open eye can detect within 5 seconds.

Several tips for slow motion: snail movement, movement of Sun against horizon during sunset and sunrise, movement of hands on your watch, growth of flowers, growth of animals, movement of stars…

Napadlo Honzu Prachaře.

### 5. Series 20. Year - 3. resistor sequence

Lets assume you are a director of company which started as first to produce resistors for mass market. From some marketing research it was discovered that demand for resistors is uniformly distributed between 1 Ω –10 MΩ. For technical reasons you can manufacture only finite number of types of resistors, lets say 169.

If customer demands resistor of value $R_{p}$ and you offer him resistor of value $R_{n}$, his dissatisfaction will be given by ( 1$-R_{p}⁄R_{n})$. The question is what are optimal 169 values of resistors to make maximum customer satisfaction. To make it more simple assume that first and last resistors from you portfolio must have values 1 Ω and 10 MΩ.

Návrh Pavla Augustinského.

### 4. Series 20. Year - P. greasy paper

When the drop of oil falls onto a sheet paper it makes it transparent. Explain what happens. Find in your life another examples of the same effect, but in different situation.

Na problém narazil Peter Zalom při čtení o sněhových vločkách, když mu kapka oleje dopadla na papír.

### 1. Series 20. Year - 4. captains diary

Contribute by some interesting record to the diary of the expedition (image, artistic creature, adventure story of length of daily observation, physical observation, …).

Napadlo Honzu Prachaře.

### 1. Series 20. Year - E. collection of strobiles

The number of spirals made from scutes of strobiles (pine-cones) coming from the center is not arbitrary, but is either 1, 2, 3, 5, 8, 13, 21, … These are numbers of Fibonacci sequence, where the next term is generated by adding two previous terms and first two terms of sequence are 1 and 1. As every rule also this one has its exceptions. Sometimes the number of spirals is equal to 1, 3, 4, 7, 11, …, which is Lucas series. Lucas series is derive in similar order as Fibonacci series, the only difference is starting numbers 1 and 3.

Your task is to find how often and at which conditions this exception occurs on the Earth. More detailed information can be found at http://artax.karlin.mff.cuni.cz/ zdebl9am/phyllot.pdf.) Explore most possible number of parameters (e.g. if the tree grows in middle of woods, is stand alone etc.).

Úlohy vymyslela Lenka Zdeborová.

### 6. Series 19. Year - E. discover your body

As the last experimental question in this year (volume) we have following simple experiment. Chose two of you body fluids (saliva, blood, urine, sweat, tears, gastric juice) and measure at least one of its physical properties (density, viscosity, electrical conductivity, refractive index, boiling point). Follow the saying „The more is better“.

Tuto hovadskou úlohu vymysleli Jarda s Honzou po ICQ těsně před tiskem.

### 5. Series 19. Year - P. about a lost well

An old couple is occupying theirs old house with their own well for drinking water. One day the water from the well stopped running, probably because of some problem in the well itself. Unfortunately, no one knows where the well is located.

From the pump, which is inside the house, 1.5 inch pipe goes approximately one meter underground and turns at right angle and continues in direction 'outside of the house' in horizontal direction. The well is filled with some material, and it is not clear, if is in the garden or under the house.

Suggest several possibilities of finding the well which are easy to realise.

V reálném životě na problém narazil Marek Scholz.

### 4. Series 19. Year - E. how the eyes are misleading

Balloons often assume that the constellation of stars close to the horizon look bigger then when located high at the sky.

Perform a measurement on the Earth and decide if it is really misleading observation. Measure angular distance $α(t_{1})$ of two selected stars which are approximately above each other and angular distance $β(t_{1})$ of other two stars at approximately the same height above horizon (i.e. independent check in two separate directions) at the moment, when the stars are as low above horizon as possible. Later on, find the same stars as high as possible and repeat the measurement recording $α(t_{2})$ and $β(t_{2})$ parameters. Try to measure with the highest precision.

We will specially recognise if you are able from known catalogued coordinates of stars calculate its theoretical angular distance. Do not forget to describe used equipment and discuss pros and cons. Plot schematic sketch of the sky around the stars and record time, direction etc. of the measurement. Finally estimate errors of measurement and discuss and compare the results.

Zformuloval Pavel Brom inspirován dotazem na hvězdárně.

### 3. Series 19. Year - E. vine analysis

Measure the alcohol content in cheap table vine and compare the content to the value declared on the packaging.

Našel Jarda Trnka na internetu a vzpomněl si při tom na jednoho z organizátorů.