4th Round of 21th year

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1. ... bees and geometry( points)

When you look at honeycomb you can admire its periodic structure. In a cut the walls cell form regular hexagons and fill whole plane.

Why do the bees make cells as hexagons? Why not for example rectangle of pentagon?

2. ... heating of a sphere( points)

In this question we investigate influence of temperature on to moment of inertia of a metal body. Lets have an axe going through the body. How will change the moment of inertia $J$ when the temperature is increased by $ΔT$, if the coefficient of thermal expansion of metal is $α$. If you have problem with a arbitrary shape of body, consider sphere of cylinder.

3. ... roaring volcano( points)

Recently there was a TV document about the eruption of Krakatoa volcano in August 1883. Remarkably the noise of eruption deafened for while people in distance 50 km from volcano. The sound of eruption was heard as distant thunder in town Alice Springs in Central Australia (approx 5 000 km from the volcano).

What was the value of acoustic pressure in dB at the place of eruption?Can we assume, that the decrease of intensity follows inverse square law? Or should we define a different rule in this case?

4. ... save the kidney( points)

Interpol just found that mafia has mobile laser weapons, which are guided from control room located in mountain village Obernieredorf. Control room is in maximum distance of 50 km from the weapons (at longer distance, the signal is unreliable). From control room they observe situation in Carlsbad, where all the weapons are aimed at.

Help the innocent inhabitants of Carlsbad to find a shape and location of continuous mirror surface, which will reflect all laser rays in direction of control room. Solve problem in 2D (in plane), or in 3D, if there is a solution. Obviously, we require a proof of functionality, as we we do not want to invest money without a reason.

P. ... project 5( points)

Suggest a shape of the most fairness cube of 5-sides. We mean to find such 5-sided object, where the probability of stopping on each side is same for all sides.

E. ... rolling resistance( points)

Carefully measure if the rolling resistance of cylinder depends on its radius or not.

S. ... quantum harmonic oscillator( points)

Calculate time dependence of wave function of particle, which is located in potential $V$( $x$ ) = ½ $kx$² and which is at time $τ$ = 0 described by wave function

$ψ$_{$R$} ( $X$, 0 ) = exp ( −(( $X~-~X$_{0} )²) ⁄ 4 ) $,

ψ$_{$I$}~(~$X$, 0 ) = 0 .

It is wave packet with the center not in the origin. We can tell you, that this is so called coherent stat of harmonic oscillator and wave packet should oscillate around origin with angular frequency √( $k$ ⁄ $m~$) same as classical particle.

If you can calculate the previous, then you can try what will be the behaviour of wave packet of different width (e.g. denominator in exponential is different from 4) of how the behaviour will look like with different potential.

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