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### 2. Series 22. Year - 2. find the secret of calliper

Explain, how calliper (vernier caliper) works and how it is possible to measure up to 1/10th mm if the main scale is only 1mm!

nad tajemstvími života se zamyslel Marek Scholz

### 1. Series 22. Year - 2. pirate and golden reward

One pirate should get 10 golden coins. However the captain does not want to give money so easily and melts the gold to a cylinder shape. And together prepares similar cylinder, made from brass with identical dimensions. Because the golden cylinder has inside some air, both weight the same. What is the best way to select the golden cylinder?

Úlohu vymyslel kolega Mirka Beláňe.

### 6. Series 21. Year - P. mission impossible

Make a plan how to liberate a Fykos-bid from custody. Do not forget to make plan B and C.

Vyplodil Honza Prachař.

### 5. Series 21. Year - E. milestones of life in Rama

Will have the Rama (brand name for margarine) another physical properties after you will melt it and let it solidify back? We suggest to measure density, viscosity and colour.

Vytlačil Marek Pechal.

### 2. Series 21. Year - E. bubo bubo

Verify experimentally following hypothesis: the rotation of Earth causes water on north hemisphere to swirl to right, on south hemisphere to left. For your conclusion to have relevance, enough number of measurements must be done at different conditions.

### Uranium storage

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.

### Heating wire

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.

### Capacity of a cube

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.

### 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…

### 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.

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