Problem Statement of Series 5, Year 26

Estimate how much energy would be needed to evaporate all oceans (on Earth).

Imagine evaporating a body of liquid with surface area $S$. Assume that during this process all the liquid molecules are separated and tLat each molecule can be considered a small particle with a definite surface area. Obviously, the total surface area of all these molecules is much greater than the surface area of the original body. Given that the latent heat of vaporization of water is $L=2.1\cdot 10^{6}J\cdot \;\mathrm{kg}^{−1}$ and that its surface tension is $α=7.2\cdot 10^{−2}N\cdot \;\mathrm{m}^{−1}$, estimate the size of water molecules.

Two half cylinders are lying on top of each other as shown in the picture. The radius of the smaller one is $r$ and that of the larger one is $R$. Given $r$, what is the condition on $R$ so that the system is stable?

Bonus: If the parameters are chosen so that the system is stable, what will be the period of oscillations of the top half cylinder after it has been slightly shifted from the equilibrium position?

A block of wood or other material placed in front of a wheel is often used to prevent an airplane from moving (see picture). Can such a locking device prevent an airplane from moving even if its engines are on and producing maximal power? Analyze this problem and show how does the answer depend on the material of the wheels, the ground, and the locking device. Will your answer change if the height of the locking device is such that the point of contact with the wheel is at its very top?

On the first day of winter old man Вова wanted to turn on his heater with input power 2 kW but found out that it was not working. Luckily, he realized that there was plenty of heat producing plutonium 237 in the warehouse where he was working. How much plutonium should he bring home in order to replace his old heater? You can assume that the plutonium is almost pure and that Вова has a lot of lead containers that can absorb all the energy radiating from the plutonium.

In 2002 Prague experienced serious floodings. Try to estimate the amount of water that can fit into the Prague subway system. All the important parameters of the subway system like the train sizes, number of stations, length of the tunnels etc. can be found online.

Design an experiment to measure the dependence of the speed of evaporation on the surface area of the evaporating liquid. You should use at least five different containers to do the measurment. What other factors can influence the speed of evaporation? Note that this experiment should run for several days so plan accordingly.

• Go to this site http://fykos.cz/rocnik26/4-compass.dat and download data gathered with a Langmuir probe from the tokamak COMPASS. Draw the volt-ampere characteristic and estimate the value of the floating potential.

• Given the surface area of the probe ($A=6\;\mathrm{mm}^{2})$ and the composition of the plasma (deuterium), analyze the volt-ampere characteristic and determine the value of electron temperature and density.

• Write a short ode describing the invention of the Langmuir probe.