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Kathy is on a swing (a plank suspended on 2 ropes). At high displacements she kneels down, in lowest point she gets up. This movements she periodically repeats. Ration of distance of centre of gravity from the rotation axis at kneeling down and at standing is 2^{1 ⁄ 12} ≈ 1,06. How many times Kathy must swing to double the amplitude of swinging?

• Derive Taylor expansion of exponential and for $x=1$ graphically show sequence of partial sums of series \sum_{$k=1}^{∞}1⁄k!$ with series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$.

Using the same method compare series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$ and series of partial sums of series \sum_{$k=1}^{∞}x^{k}⁄k!$, therefore series {\sum_{$k=1}^{n}x^{k}⁄k!}_{n=1,2,\ldots}$, now for $x=-1$.

• The second task is to find concentration of electrons and positrons on temperature with total charge $Q=0$ in empty and closed cavity (you can choose value of $Q.)$ Further calculate dependence of ration of internal energy $U_{e}$ of electrons and positrons to the total internal energy of the system $U$ (e.g. the sum of energy of electromagnetic radiation and particles) on temperature and find value of temperature related to some prominent temperature and ratios (e.g. 3 ⁄ 4, 1 ⁄ 2, 1 ⁄ 4, …; can this ratio be of all values?).

Put your results into a graph – you can try also in 3-dimensions.

To get the calculation simplified, it could help to take some unit-less entity (e.g. $βE_{0}$ instead of $β$ etc.).

• Solve the system of differential equations for $M(r)$ and $ρ(r)$ in model of white dwarf for several well chosen values of $ρ(0)$ and for every value find the value which it get close $M(r)$ at

$r→∞$. This is probably equal to the mass of the whole star. Try to find the dependence of total weight on $ρ(0)$ and find its upper limit. Compare the result with the upper limit of mass for white dwarf (you will find it in literature or internet). Assume, that the star consists from helium.

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?

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.

Calculate time dependence of wave function of particle, which is located in potential $V(x)=\frac{1}{2}kx$ and which is at time $τ=0$ described by wave function

$ψ_{R}(X,0)=\exp(-((X-X_{0}))⁄4)$,

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.

Integral

Wandering of sailor

Pi-circuit

Epidemic in Prague

• Three cards are randomly selected from 36 cards. Calculate probability of possibility that (i) just one ace was selected, (ii) at least one ace is selected and (iii) no ace is selected.
• $N$ identical particles is in a container. Calculate probability of case, that in the left part of container is $m$ particles more than in the right half. Draw a graph of dependence for $N=10^{10}$. Range of $m$ select so that the probability on the sides will be one tenth of the probability in the middle. How the width depends on $N?$ (Width is difference $m_{2}-m_{1}$, where $m_{2}>0$ a $m_{1}<0$ are values of $m$, for which the probability is half compared to the maximum).
• Estimate size of ln$n!$ (without using Stirling formula).