Assignment of Series 3 of Year 20

The food on a transatlantic ship is prepared by a chef Thomas. He has very userful spring-based device to hold plates. The holder holds the top plate always in the same height regardless the number of plates in holder. The distance between plates is 1 cm. The weather is stormy and the plates are oscillating. What is the frequency of the oscillations?

The space probe Huygens (named after the discoverer of Titan) has landed on Titan on Friday 14th of January 2005. The mother ship Cassini was travelling 7 years to reach Saturn. It is the most distant landing of man-made probe in history.

The landing module of nett weight (without fuel) $m$, equipped with reactive engine, levitated at fixed position above the surface of the moon (gravitational acceleration is $g)$. The module had available fuel of mass $M$ and energy $E_{0}$ which is used to accelerate the fuel (the speed and amount of fuel which is ejected from the engine can be regulated without any limitations). What is the maximum time the module can stay in constant height? Suggest how the speed and amount of fuel should be programmed to reach this maximum time.

We have calculated spectral class of two stars forming binary star from reduced star spectra (from present spectral lines which does not change its position in time). From spectral class we have estimated its masses as 2 and 3 times the mass of the Sun. From the observation with telescope of focal length 3 m we know that the stars really orbit in constant angular distance of 5 angular minutes. One orbit takes 50 years.

Are you able to calculate distance of this binary star from the Sun? If yes, state which information you need and the result round accordingly. Comment on the precision of the result and how the uncertainty of input information (mainly the masses) impacts on accuracy.

Albert Einstein in his retirement (in contrary to his peers working in the back gardens) still enjoyed solving difficult puzzles. In winter he noticed that water heated directly in fire gets war very slowly and the efficiency is small.

He decided to try other method: Take ideal heat machine and use boiler and air as hot and cold reservoir. Then use the gain work to another ideal heat engine which will get heat from air and transfer it to water. If the temperature of boiler, water and air is $T_{1}$, $T_{2}$ and $T_{3}$, what is efficiency of water heating? Does it conflict with second thermodynamic law?

Olympic discipline freestyle skiing (or acrobatic skiing) seems to contradict basic law of physics. Skier first speeds up and then comes on to jump ramp. Between take-off and landing he manages to make several flips and turns. Explain how and what he must make in order to spin in the way he wants and spectators expect. How would you falsify following opinion: "according to the momentum conservation law the skier must remain at rotation along the same axis at constant speed".

Suggest and make adequate theoretical justification for methods suitable to measure Planck constant which can be realized at home or in school laboratory. Realize at least one of them. All physical quantities measure with highest accuracy (consider using statistical averages etc.) and estimate value of this fundamental constant including relevant experimental error.

Hint: LED diode with resistor costs approximately 5 Kč ( 0.10Eur).