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## thermodynamics

### (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 $T\_{s} = 25 \mathrm{\C }$.

### (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?

### (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.

Karel was thinking about the Fermi paradox.

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

- 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

- $0,2 \mathrm{\micro m}$,
- $5 \mathrm{\micro m}$.

- How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?
- 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.

### (10 points)3. Series 35. Year - S. igniting

- Determine the reach of helium nuclei in central hot spot (using the figure ).
- What energy must be released in the fusion reactions in order for the fusion to spread to the closest layer of the pellet? How thick is the layer?
- Estimate the most probable amount of energy transferred from helium nucleus to deuterium. How many collisions on average does the helium nucleus undergo in the central hot spot before it stops?

### (10 points)2. Series 35. Year - S. compressing

What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? **Note:** Assume that the initial plasma is a monatomic ideal gas.

### (14 points)1. Series 35. Year - E. Is the pasta ready?

Measure the dependency of the time it takes for water to start boiling on its volume. Repeat the measurement several times for at least five different volumes. Pay attention to the consistency of the external conditions, especially the criterion you use for assessing when the water starts boiling and the initial temperature of the water, vessel and stove. Try to explain the resulting relation.

Dodo's fight with the stove at the dormitory.

### (10 points)1. Series 35. Year - S. commencing fusion

- Determine the energy gain of the following reactions and the kinetic energy of their products

\[\begin{align*} {}^{2}\mathrm {D} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {T} + \mathrm {p} ,\\ {}^{2}\mathrm {D} + {}^{2}\mathrm {D} &\rightarrow {}^{3}\mathrm {He} + \mathrm {n} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {T} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {n} ,\\ {}^{3}\mathrm {He} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + 2\mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {n} + \mathrm {p} ,\\ {}^{3}\mathrm {T} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + {}^{2}\mathrm {D} ,\\ \mathrm {p} + {}^{11}\mathrm {B} &\rightarrow 3\;{}^{4}\mathrm {He} ,\\ {}^{2}\mathrm {D} + {}^{3}\mathrm {He} &\rightarrow {}^{4}\mathrm {He} + \mathrm {p} . \end {align*}\]

- By using the graph of fusion reaction rate (sometimes called volume rate) as a function of temperature in the Serial study text, derive the Lawson criterion for the inertial-confinement-fusion time for a temperature of your choosing, while considering the following reactions:

- deuterium - deuterium,
- proton - boron,
- deuterium - helium-3.

Determine the product of the size of a fuel pellet, and the density of a compressed fuel for each case. Are there any advantages of these reactions compared to the traditional DT fusion?

- What form would the Lawson criterion take for the non-Maxwellian velocity distribution, considering the case with the following kinetic energy of a particle

- $E\_k = k\_B T^\alpha $,
- $E\_k = a T^3 + b T^2 + c T$.

Could such a fusion be even possible? If so, what (the fuel) should drive the fusion reaction, what is the ideal size of the fuel pellet and what density should it be compressed to?