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## thermodynamics

### (9 points)6. Series 36. Year - 5. gadolinium sphere

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.

Karel was thinking about elements, but Matěj Rz. changed that.

### (10 points)6. Series 36. Year - P. Earth at full throttle

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.

### (3 points)3. Series 36. Year - 2. heating in the cottage

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.

Danka gets cold at the cottage.

### (10 points)2. Series 36. Year - P. planetary atmosphere

What parameters does a planet need to have to keep its atmosphere comparable to the Earth? What conditions are essential for the planet to gain such an atmosphere?

### (6 points)1. Series 36. Year - 3. canning jam

A cylindrical jar made of glass has a height $h = 7,0 \mathrm{cm}$ and an inner radius $r = 2,5 \mathrm{cm}$. We pour hot apricot jam at temperature $T_0 = 80 \mathrm{\C }$ into the jar, we close the lid and let it cool down. Note that we didn't fill the jar to the top, but left some air between the jam and the lid. If a force of at least $F = 4 \mathrm{N}$ is applied, a sound is heard as the lid suddenly incurves. We heard this sound $t\_i = 30 \mathrm{min}$ after the jar had been closed. If jam hardens at temperature $T\_h = 60 \mathrm{\C }$, was it to be already hard when the lid incurved?

Bonus: How long after closing the jar will the jam harden? Assume that the temperature is evenly distributed throughout the jar and that the cooling rate only depends on the difference in temperatures of the jar and its surroundings $T\_{s} = 25 \mathrm{\C }$.

Jarda's apricot trees froze this year and he dreams about last year yield.

### (8 points)1. Series 36. Year - 5. U-tube again

We have a U-tube with length $l$ and cross-sectional area $S$. We pour volume $V$ of water into the tube. The volume $V$ is large enough that the whole U-turn is filled with water but $Sl > V$. When water levels in both arms of the tube are at rest, we seal one of the arms. What is the period of small oscillations of water in the tube?

Karel went crazy again.

### (6 points)5. Series 35. Year - 3. under the lid

A lid has a shape of a hollow cylinder of radius $6,00 \mathrm{cm}$. The lid under which is an air of atmospheric pressure $1~013 \mathrm{hPa}$ is placed in a horizontal washbasin. While doing the dishes, we start filling the washbasin with water at room temperature. The water also gets under the lid and compresses the air trapped inside. At a certain moment, the lid starts floating. At what height is the water level at that moment? The lid weighs $200 \mathrm{g}$, its height is $2,00 \mathrm{cm}$ and negligible volume.

Danka was doing the dishes.

### (10 points)5. Series 35. Year - P. hot asteroid

Come up with as many physics reasons as possible on why an asteroid might have a higher temperature than its surroundings.

### (10 points)5. Series 35. Year - S. stabilizing

1. What intensity must a laser with a wavelength of $351 \mathrm{nm}$ have in order to stabilize a Rayleigh-Taylor (RT) instability using the surface ablation of a fuel pellet? Suppose the boundary between the ablator and DT ice is corrugated with a wavelength of
1. $0,2 \mathrm{\micro m}$,
2. $5 \mathrm{\micro m}$.
1. How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?
2. What else can help us minimize the RT instability?

### (10 points)3. Series 35. Year - 5. blacksmith's

Gnomes decided to forge another magic sword. They make it from a thin metal rod with radius $R=1 \mathrm{cm}$, one end of which they maintain at the temperature $T_1 = 400 \mathrm{\C }$. The rod is surrounded by a huge amount of air with the temperature $T_0 = 20 \mathrm{\C }$. The heat transfer coefficient of that mythical metal is $\alpha = 12 \mathrm{W\cdot m^{-2}\cdot K^{-1}}$ and the thermal conductivity coefficient is $\lambda = 50 \mathrm{W\cdot m^{-1}\cdot K^{-1}}$. The metal rod is very long. Where closest to the heated end can gnomes grab the rod with their bare hands if the temperature on the spot they touch is not to exceed $T_2 = 40 \mathrm{\C }$? Neglect the flow of air and heat radiation.

Matěj Rzehulka burnt his fingers on metal. 