**Apr. 9, 2024, 23:59, CET**.

# Assignment of Series 5 of Year 37

About the Competition Rules How to Write Solutions Results#### 1... annexation of Kaliningrad

#### 3 points

The commander of the operation to take over the Russian enclave is chilling in his recreational boat, which has the shape of a block with a base area $S$ and height $H$. Suddenly, directly below him at the bottom of the Vistula Lagoon, a group of saboteurs punches a hole in the alcohol pipeline – a pipeline bringing a high-quality, scarce Czech commodity with a density $\rho_{\mathrm{B}}$ from Budějovice to Královec. Determine the conditions under which the boat sinks, assuming that it was submerged to a depth $h$ before the accident and that the layer of beer on the surface after the accident is $\Delta h$.

*Adam has a vivid imagination but doesn't want to circumvent physics with it.*

#### 2... basic problem of acoustics

#### 3 points

Adam can take meaningful notes at the speed $v_1$. Unfortunately, his calculus professor speaks at the speed of $v_2$. There is an airflow in the lecture hall, moving from Adam towards the professor, with the air flowing at a velocity of $v_3$. At what velocity and in which direction along a straight line intersecting Adam and the lecturer should Adam move to transcribe everything the lecturer says into his notebook?

*Adam likes the word “meaningful”.*

#### 3... bowling

#### 6 points

Jirka was bowling with his friends. He was throwing the ball so that when it hit the lane it had a horizontal velocity $v_0$ and glided on the lane without spinning. However, there was a coefficient of friction $f$ between the lane and the ball, hence after time $t^\ast$ the ball started to roll without slipping. Determine the final velocity $v^\ast$ at this equilibrium, the time $t^\ast$ and the distance $s^\ast$ the ball travels before reaching the equilibrium. The ball is solid, with radius $r$ and mass $m$.

*Jirka didn't trust the lecturer, so he made up his own problem.*

#### 4... centrifuge

#### 7 points

Consider a centrifuge of length $L = 30\,\mathrm{cm}$ filled with a solution in which there are homogeneously distributed small spherical particles of radius $r = 50\,\mathrm{\upmu{}m}$ and mass $m = 5.5\cdot 10^{-10}\,\mathrm{kg}$. The density of the solution is $\rho_{\mathrm{r}} = 1~050\,\mathrm{kg\cdot m^{-3}}$ and its viscosity is $\eta = 4.8\,\mathrm{mPa\cdot s}$. The container with the solution is in a horizontal position and suddenly begins to rotate at an angular velocity of $\omega = 0.5\,\mathrm{rad\cdot s^{-1}}$. Determine how long it will take for $90\,\mathrm{\%}$ of all the particles to reach the end of the centrifuge. Do not consider interparticle collisions and movement of the particles due to diffusion. The container rotates around a vertical axis located at one of its ends.

*Jarda loves to make enriched uranium.*

#### 5... tuning a circuit

#### 9 points

*Jarda wanted to have as many different sources in the circuit as possible.*

#### P... CERN on Mercury?

#### 10 points

On the surface of Mercury, the atmosphere is approximately as dense as the vacuum tubes at CERN, in which scientists conduct experiments to investigate particle physics. Would it be a good idea to move the experiments to Mercury and perform them on its surface? Mention as many arguments as you can and elaborate on them.

*Bonus:* Suggest the best place to build an accelerator.

*Karel was looking at the table of pressures.*

#### E... gooey

#### 12 points

Measure the dependence of a cooking oil's dynamic viscosity $\eta$ on temperature $T$. Fit the measured data to function \begin{equation*} \eta = \eta_0 \exp\!\left(\frac{T_0}{T}\right) \,, \end{equation*} and calculate the values of the parameters $\eta_0$ and $T_0$.

*Hint:* When fitting the results, plot the horizontal axis as $1/T$.
Then, it is possible to fit the data with the required curve even in less
advanced software, such as *Excel*.

*Petr was preparing for laboratory practice.*

#### S... we are spending electricity

#### 10 points

- The aluminum smelter annually produces $160~000\,\mathrm{t}$ of aluminum, which is produced by electrolysis of alumina using a DC voltage of $U=4.3\,\mathrm{V}$. Determine how many units of nuclear power plant with a net electrical output of $W_0=500\,\mathrm{MW}$ are equivalent to the energy consumed by the aluminum smelter.
- A DC current of magnitude $I$ is applied to a tangent galvanometer with $n$ turnings of radius $R$. The compass needle is deflected by an angle $\alpha$ from the equilibrium position. Determine the relationship needed to calculate the flowing current.
- Measuring the temperature $T$ using a thermistor to determine its resistance $r(T)$ utilizes a Wheatstone bridge with three resistors of known values $R_1$, $R_2$, $R_3$. What voltage $U(T)$ do we measure on the voltmeter in the middle of the bridge?
- In the second half of the last century, conventional electrical units were based on the values of the frequency of the cesium hyperfine transition $\nu_{\mathrm{Cs}}=9~192~631~770\,\mathrm{Hz}$, the von Klitzing constant $R_{\mathrm{K}}=25~812.807\,\mathrm{\Omega}$ and the Josephson constant $K_J=483~597.9\cdot 10^{9}\,\mathrm{Wb^{-1}}$. Determine the value of the coulomb $1\,\mathrm{C}$ using these constants.

*Dodo has dead batteries.*