Problem Statement of Series 3, Year 22

Fykos organisers were left with a container with compressed hydrogen. And they plan to transfer all hydrogen into a light balloon (so it will have atmospheric pressure). Is such balloon able to lift the container, if the temperature is constant?

Honza travels home by train at speed $v_{0}$. From his baggage on top shelf is hanging down a mathematical pendulum. The train starts suddenly to break (at acceleration $a$ for duration $t)$. Can the pendulum go around of 180 °? Assume, that pendulum is fully fixed above.

They have a new attraction in "Dolni Dvur": Helium filled soap bubble, which are just levitating in the air. What is heavier? Helium in bubble or its wall?

Jakub took chain and held it at two places by fingers and started to spin it at the angular speed $ω$, same as at the picture. Marek saw it and asked, what is the shape of the chain 2 dimensions. What is the correct answer, if neglecting gravitation field?

Titan (a satellite of Saturn) is very cold (surface temperature is approximately 94 K) and has bulky nitrogen atmosphere, icy surface with lakes made from hydrocarbons. Radar on Cassini probe orbiting Titan found, that objects on the surface rotate faster then the moon itself (approximately by 0$,36°\;\mathrm{year}^{−1})$. Scientific explanation is, that the icy layer on the surface of oceans is affected by a wind. The assumption about moon rotation is, that it is synchronised with orbiting around Saturn

Another hint for existence of under-surface ocean was given by Huygens probe, which landed on the moon surface. During the descent it measured strong electromagnetic waves at frequency approximately 36 Hz. Such waves are amplified at the interface between e.g. water and ice under the surface.

Suggest some methods, to confirm or disprove the existence of hidden ocean on Titan.

Try following experiment: Fill the same glasses one with water, second with wine up to the top. Put a sheet of paper on the glass with water. Then turn the glass with water upside down and face it with the glass with wine (final state is on the figure). Now try to remove the paper and you will see very strange effect: the content of glasses will change, without mixing (it takes a time, be patient). Find out, why it is as it is. How is the time for change dependent on surface area, alcohol concentration and other parameters? Will this happen also for other liquids? E.g. coloured water, milk, oil...

* Imagine a strong laser at wavelength 400 nm, and shine it at the Moon. On its surface the light will reflect and come back. Assuming circular orifice of diameter of 1 cm through which the beam is going, what will be the diameter of the reflection on the Earth? Hint: It will be much more, than 1 cm.

• In this task assume, that the aether really exists and predict, what will happen, if Mr. Michelson would make its measurement by other means: one arm would be 5 meters long and other 10 meters long. Such apparatus would create some interference pattern. Then he would rotate whole experiment by 90$°$, so both arms changed its positions. During rotating the experiment, we would see changes in interference patterns (assume rotating doubleslit). How would the interference patterns move at above rotation? How long would have to be the longer arm to inverse the interference fringes (e.g. minima would become maxima)?

• In the following task again assume existence of aether and that a body moving in aether is pulling it completely with to body, so the relative speed of aether to the body is zero. What would be then phase shift between two beams in the system in above figure? The light is splitted at semitransparent mirror into two beams and continues at perfectly rectangular path back to the semitransparent mirror, where it reaches screen at which interference fringes are observed. On the way are both beams three times reflected by a mirror and are going through the cylinder of length $L$, filled with water. Whole system is moving relatively to aether at speed $v$ to the right (do not forgot, that the cylinder is not moving relatively to aether!).