Assignment of Series 1 of Year 23

Information system in Prague underground has following feature: during scrolling text on the display to the left, the letters are tilted. Is there a simple way how to do this function on "hardware" level? What effect will have this at text scrolling in vertical direction? Assume, that information display consists from LEDs spaced in orthogonal grid.

A rope is placed on two wedges (see figure) in such way, that in center is not touching the base. The situation is symmetric in left-right direction. Calculate maximal part of rope can be free standing in static equilibrium. Bonus: for which angle is the ratio biggest?

The movement of point mass is restricted on straight line by 2 end points. At the beginning the point mass $m$ is moving at speed $v$. The end point starts to move away at speed $v_{1}<<v$. How will the energy of point mass change?

Lets have a tall cylindrical vessel filled with water (radius $r$, height of water $h)$ and spin it around its axis at angular speed $ω$. At the centre of bottom is a small orifice of surface area $S$, while the vessel is still rotating. How much water will escape the vessel?

The capillary of medical thermometer is at its bottom thinner, to stop mercury to return back to the reservoir at the base, which allows us to read the maximum reached temperature. From June 2009 it is forbidden to sell such thermometers. At this historical occasion, take the opportunity and explain us, why the capillary narrowing works only in one direction – only when heating. Because while cooling down the mercury cannot go through the narrow point back.

We are going to the north pole. Having snowmobile helping to bring them north, we need into the cooler some antifreeze. What is the ideal concentration of alcohol in water, more precisely what is dependence of melting point on alcohol concentration? If you have good freezer, then measure, what concentration will freeze at some specific temperature.

Do not forgot, that this task is experimental.

• What will you see when standing between two vertical mirrors connected at right angle?

• Lets have plane mirror inclined at angle 45°, moving to the left at the speed $v$. From the right there is a light ray of speed $c$ (e.g. angle of incidence is 45°) and is reflected upwards. Using Huygens principle calculate angle between incoming and reflected ray, e.g. correct the law of reflection for mowing mirrors.