Series 5, Year 37

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 $\rho \_B$ 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 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?

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 $m$.

Consider a centrifuge of length $L = 30 \mathrm{cm}$ filled with a solution in which there are homogeneously distributed small spherical particles of radius $r = 50 \mathrm{\micro m}$ and mass $m = 5,5 \cdot 10^{-10} \mathrm{kg}$. The density of the solution is $\rho \_r = 1~050 kg.m^{-3}$ 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.

Consider a series circuit with a resistor of resistance $R$, a coil, and a capacitor with the capacitance $C$. AC voltage sources with identical amplitudes $U$ are connected in series with these components. These sources vary in frequency by being multiples of $\omega _0$, where $n$ represents an integer. What frequency, denoted by $\omega _0$, would allow us to find a coil possessing an inductance $L$, such that voltages with frequencies different from $N \omega _0$ are suppressed by at least $90 \mathrm{\%}$ on the resistor? $N$ is a positive natural number known in advance (i.e., the value of $\omega _0$ may depend on it), and we do not want to suppress the voltage with frequency $N\omega _0$ by more than $90 \mathrm{\%}$.

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.

Measure the dependence of a cooking oil's dynamic viscosity $\eta $ on temperature $T$. Fit the measured data to function \[\begin{equation*} \eta = \eta _0 \f {\exp }{\frac {T_0}{T}}  , \end {equation*}\] 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.

1. The aluminum smelter annually produces $160~000 t$ 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.
2. 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.
3. 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?
4. In the second half of the last century, conventional electrical units were based on the values of the frequency of the cesium hyperfine transition $\nu \_{Cs}=9~192~631~770 Hz$, the von Klitzing constant $R\_K=25~812.807 \ohm $ and the Josephson constant $K_J=483~597.9e9 Wb^{-1}$. Determine the value of the coulomb $1 \mathrm{C}$ using these constants. {enumerate}