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Little Jagr and his friends would like to go out to play ice hockey. However, it has only started freezing recently, so they don't know if the ice on the pond is already thick enough. Calculate how long it takes for a deep pond to freeze sufficiently; if you know that the water temperature is $0 \mathrm{\C }$ at the beginning, the air is kept at a constant $-10 \mathrm{\C }$ and the minimum ice thickness for safe skating is $10 \mathrm{cm}$. Neither the density of the water nor the ice formed changes with depth. The heat transfer between air and ice and water and ice is much faster than heat conduction in ice. You will need to look up the necessary thermal properties of ice.

1. Consider a thin-walled glass container of volume $V_1=100 \mathrm{ml}$, the neck of which is a thin and long vertical capillary with internal cross-section $S=0{,}20 \mathrm{cm^2}$, filled with water at temperature $t_1=25 \mathrm{\C }$ up to the bottom of the neck. Now submerge this container in a larger container filled with a volume $V_2=2{,}00 \mathrm{l}$ of olive oil at a temperature $t_2=80 \mathrm{\C }$. How much will the water in the capillary rise?
2. In a closed container with a volume of $11{,}0 \mathrm{l}$ there is a weak solution containing sodium hydroxide with $p\mathrm {H}=12{,}5$ and a volume of $1{,}0 \mathrm{l}$. In the region above the surface, we burn $100 \mathrm{mg}$ of powdered carbon. Determine the value of the pressure in the container a few seconds after burning out, after half an hour, and after one day. Before the experiment, the vessel contained air of standard composition at standard conditions; similarly, we maintain a standard temperature around the vessel in the laboratory.
3. Describe three different ways in which the temperature of stars can be determined. What are the basic physical principles they are based on, and what do we need to be careful of?

Danka has a humidifier in her dorm room, which evaporates water from its boiling point to create warm steam. The device can hold a maximum of $V = 3,8 \mathrm{l}$ of water, which it uses up in $t = 24 \mathrm{h}$. What is its efficiency, i.e., what fraction of the energy drawn from the electrical grid it uses to convert the water to steam? The input power of the humidifier is $P = 260 \mathrm{W}$, and Danka put water at $T_0 = 20 \mathrm{\C }$ inside. All the necessary properties of water can be looked up.

In the microworld of cells, there are two types of transport: transport by free diffusion, also known as Brownian motion where the motion uses the energy of the environment directly. The second type, so-called active transport, requires, among other things, a motor protein moving at a constant speed along the cytoskeletal filament. Consider the typical value of the diffusion constant $D \approx 10^{-9} cm^2.s^{-1}$ and the rate of active transport speed $u\approx 10^{-6} m.s^{-1}$. For which distances is the Brownian motion more time efficient than the active transport? Assume that the transport is happening in only one direction.

Attach a string at two points at a fixed distance $L$ and ensure it is always taut during measurement. Determine the dependence of the fundamental frequency of its oscillations on temperature.

In the winter, Matěj found a $0{,}5 \mathrm{m^3}$ bale of polystyrene and decided to use it. He made a cube-shaped box out of it. Then he cut the ice from a frozen pond, which he stored in the polystyrene cube in the cellar, where the temperature is constant $9{} \mathrm{\C }$. How big should Matěj make the cube so that he has the largest amount of ice left in it after half a year? And how many kilograms of ice will he have left? Suppose that the ice from the pond has a temperature of exactly $0{} \mathrm{\C }$. Ignore the volume of polystyrene used for the edges of the cube.

Hint:: The thermal conductivity coefficient is the easiest parameter of polystyrene to find.

Using current technology, how much fuel would it take to carry an object of mass $m=1 \mathrm{kg}$ into low Earth orbit?

What is the smallest amount of gadolinium $148$ needed to put together to cause local melting from the heat generated by its nuclear decay? Assume that only $\alpha $ decays take place and the material is at room temperature in the air.

Estimate the upper limit of work that can be done on Earth over the long term. The planet must remain habitable and, if possible, with the same climate for future generations.

The FYKOS-bird arrived at his cottage in the middle of winter with only $T_1 = 12 \mathrm{\C }$ indoors. So he lit a fire in the fireplace by using wood of heating value $Q_0 = 14{,}23 \mathrm{MJ\cdot kg^{-1}}$. How much wood does he need to burn to heat the air inside to $T_2 = 20 \mathrm{\C }$? The cottage is in the shape of a rectangular cuboid with dimensions $a = 6 \mathrm{m}$, $b = 8 \mathrm{m}$ and $c = 3 \mathrm{m}$. A roof is in the shape of an irregular recumbent triangular prism with a height of $v = 1{,}5 \mathrm{m}$, the upper edge of which is the axis of the cottage layout. The air occupies $87 \mathrm{\%}$ of the volume of the cottage and its specific heat capacity is $c_v = 1 \mathrm{007 J\cdot kg^{-1}\cdot K^{-1}}$. Does the result match the expectation? Discuss the simplicity of the model used.