5th Round of 28th year

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1. ... stiffness of Mr. Planck(2 points)

Maybe you have heard about the so called Planck's units ie.  units expressed in the form of fundamental physics constants  – speed of light $c~≈~$3.00 · 10^{8} m·s^{ − 1}, gravitational&nbspconstant $G$ = 6.67 · 10^{-11} m³·kg^{ − 1}·s^{ − 2} and  the reduced Planck's constant $h$ = 1.05 · 10&nbsp^{-34} kg·m²·s^{ − 1}. This way Planck's time, Planck's length and Planck's weight are often mentioned. What if we were interested in „Planck's spring constant“? Using dimensional analysis with $c$, $G$ and $h$  the equation of the unit relating to the unit of a spring constant [ $k$ ] = kg·s^{ − 2}. To determine the equation assume that the unknown and from dimensional analysis undeterminable dimensionless constant is equal to 1.

2. ... I hear well, I can't say(2 points)

At a distance $d$ = 5 m from a point-like source of sound we hear a noise of the level of intensity $L$_{1} = 90 dB. At what distance from the source of the sound is the level of intensity of the sound $L$_{2} = 50 dB?

3. ... matfyz tag(4 points)

$N$  people decide to play tag but not the normal variety. At the start they stand in the vertices of a regular $N$-gram of a side $a$. The game then proceeds so that everyone chases (goes to him in a straight line)his neighbour on the right (anti-clockwise). Everyone moves with the same constant velocity $v$. Describe the progress of the game (trajectory on which the players move) and determine how quickly the game will end depending on the parameters $N$, $and$, $v$.

4. ... heavy rain(4 points)

Autumn weather is sometimes as unstable as Spring weather and so it often happens that we can be surprised by an unforeseen torrent of rain. A happy few carried umbrellas. Approximate how large the pressure of heavy rain can be and compare the force of the rain with the gravitational force with which the umbrella is pulled down. Choose the parameters of the umbrella appropriately.

5. ... a lens was floating on water(5 points)

On the surface of water a thin biconcave lens made from a light-weight material is floating. The radii of both surfaces are  $R$ = 20 cm. Determine the distance between the two focal points of the lens, if the index of refraction of the air above the lens is $n$_{a} = 1, index of refraction of the lens is $n$_{l} = 1.5 and index of refraction of water is $n$_{w} = 1.3.

Bonus   Assume that it is a lens of width $T$ = 3 cm, and within it is symmetrically place an air bubble in the shape of a biconcave lens with the radii of curvature $r$ = 50 cm and width $t$ = 1 cm.

P. ... splashed(5 points)

Would it be possible to swim in a pool, if the water in it would behave as a completely ideally incompressible liquid, the visocity of which would approach zero? How would the movement of the swimmer differ from a swimmer that would swim in regular wate? What would happen with the energy of the system if water could flow out of the pool over the edges ? At the beginning the water is level with the edge of the pool.

E. ... Sweetening(8 points)

Determine the dependency of the temperature of the solidification of the aqueous solution of sucrose at atmospheric pressure.

S. ... mapping(6 points)

* Show that for arbitrary values of parameters $K$ and $T$ you can express the Standard map from the series express as

$$x_{n} = x_{n-1} y_{n-1},$$

$$\\ y_n = y_{n-1} K \sin(x),$$

where $x, y$ are somehow scaled  d$φ$ ⁄ d$t~,φ$. Show that the physical parameter $K, x, y$.

  • Look at the model of the kicked rotor from the series and take this time the passed impuls$I$ ( $φ$ ) = $I$_{0}, after the period $T$ then $I$ ( $φ$ ) = $-~I$_{0}, after another one $I$_{0} and this way keep on kicking the rotor on and on.
  • Make a map $φ$_{$n$}$,~$d$φ$ ⁄ d$t~$_{$n$} on the basis of values $φ$_{$n~-~$1}$,~$d$φ$ ⁄ d$t~$_{$n~-~$1} before the doublekick  ± $I$ Why not?
  • Solve $φ$_{$n$}$,~$d$φ$ ⁄ d$t~$_{$n$} on the basis of some initial conditions $φ$_{0}$,~$d$φ$ ⁄ d$t~$_{0} for an arbitrary $n$.
  • $Bonus:$ Try using the ingeredients from this series to design kicking which $will$ result in chaotic dynamics. Take care though because $φ$ is periodic with a period 2π and shouldn't  d$φ$ ⁄ d$t~$ unscrew forever through kicking.
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