Series 6, Year 37

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?

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.

Let us have an infinite system of intangible pulleys as shown in the figure, where the mass of each additional weight is one-third of the weight of the previous one. What is the acceleration of the first weight of mass $m$?

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 $r$.

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).

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.