**May. 14, 2024, 23:59, CET**.

# Assignment of Series 6 of Year 37

About the Competition Rules How to Write Solutions Results#### 1... ballons with Martin

#### 3 points

A car is standing on a straight road, with a freely floating helium balloon tied inside. Suddenly, the car starts to accelerate with acceleration $a=5.0\,\mathrm{km\cdot min^{-2}}$. By what angle will the balloon be deflected from the vertical line? What is the direction of the deflection?

*Martin would like to hang on a balloon behind a car.*

#### 2... bombarded organizer

#### 3 points

Estimate how many antineutrinos created in Czech nuclear power plants pass through the body of an average FYKOS organizer in one meeting held for a FYKOS camp. The meeting is 4 hours long and takes place on the tenth floor of the Matfyz building at the Troja campus in Prague.

*Jarda felt under pressure at the meeting.*

#### 3... submarine sickness

#### 5 points

A submarine of volume $V=6\,\mathrm{m^3}$, with solid carbon fiber walls of negligible thickness and internal temperature $t=20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$ was submerged to the depth $d=3\,\mathrm{km}$. Suddenly, the walls ceased to hold and the submarine shrank. What is the temperature inside?

Assume that the submarine did not break, but only shrank (although we know from experience that this is not a realistic assumption), and that the passengers and the cargo put up only negligible resistance to the shrinking (this is a realistic assumption).

*Darwin Awards 2025*

#### 4... infinite pulleys

#### 7 points

*Matěj was looking for the difference between countably and uncountably many pulleys.*

#### 5... oscillating magnets

#### 10 points

Consider two identical dipole magnets, which we fix so that they can rotate in the same plane without friction (their axes of rotation are parallel). If we deflect the magnets slightly out of their equilibrium position, they begin to oscillate. Find the eigenmodes of these oscillations and calculate their frequencies. Discuss what the motion of the magnets will be like for general initial deflection (you don't have to explicitly calculate this case). The magnets have a magnetic moment $m$, a moment of inertia about the axis of rotation $J$ and the mutual distance between their centers is $r$.

*Jirka stole the problem from Výfuk.*

#### P... to boil the ocean

#### 10 points

How long would it take to heat the world's oceans to the boiling point? Consider different energy sources, however, only those that are available on Earth (solar radiation counts).

*Michal is fond of the English quantifier "to boil the ocean".*

#### E... colligative properties of solutions

#### 12 points

Measure the cryoscopic constant, the constant of proportionality between the melting point of a solution and its molality. Measure this constant for several solutions and verify Raoult's 3rd law, which states that the value of the constant does not depend on the solute, but only on the solvent.

Instructions for Experimental Tasks*Tomáš Č. wanted to educate.*

#### S... illuminating units

#### 10 points

- There is an isotropic (its properties depend on the direction) light source perpendicularly above the center of a table. The center of the table is illuminated by $E_1=500\,\mathrm{lx}$. The edge of the table is $R=0.85\,\mathrm{m}$ from the center and is illuminated by $E_2=450\,\mathrm{lx}$. How far from the center of the table is the light source? What is its luminous intensity?
- Measure the luminous intensity of your favorite lamp using one of the visual photometric methods mentioned in the series. Use a tea candle made of white paraffin wax as the unit of luminosity. Remember to describe your experimental setup and attach a photograph or a diagram. How accurate was your result?
- Let's construct the “Earth” system of units using the values of the mean density of the Earth, the standard atmospheric pressure at sea level, the standard gravity of Earth, and the magnetic induction measured at the Earth's south magnetic pole $B_0=67\,\mathrm{\upmu{}T}$. Calculate the values of second, meter, kilogram, and ampere in this system and find the values of the speed of light, Planck's constant, gravitational constant, and vacuum permeability in “Earth” units.

*Dodo's table light at dorms is out.*