# 4. Series 28. Year

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### 1. square resistance

How does the eletric resistance of a square depend on the length of its side $a?$ All the squares that we are interested in are conductors made of a thin of a thickness $h$ and a resistivity $ρ$. We are interested in the resistance between the opposite sides of a square.

Karel was inspired at the fair of physics teacher's ideas.

### 2. quick beauty reloaded

Terka went on a trip once again. This time she was walking during equinox at twelve o'clock on the Equator. What would her velocity be relative to Ales, if he would want to (foolishly) watch her from the surface of the Sun on the Equator at a point nearest to his object of interest (Terka)? The axial tilt of the sun with respect to the plane of the ecliptic can be considered negligible.

Karel was watching the sun.

### 3. unbreakable bond

Two notebooks of the type A460 we shall insert into each other so that a page of one is always followed by the page of another and we put them on a horizontal table. What is the work we have to do to seperate them if the lists act on each other only with their own weight? Assume that we pull only in the plane of the notebooks by the back of one of them and also assume that in the beginning the pages perfectly cover each other.

Mirek was unsuccesfully dividng analysis and algebra.

### 4. oh the gravity

Determine the acceleration (both due to gravitational and centrifugal forces) on the surface of a neutron star based on what lattitude we are. How large would the tidal forces acting on an object of height $h=1\;\mathrm{m}$ and with a mass $m=1\;\mathrm{kg}$ in the vicinity of it surface be? What would the energy of a marshmallow be if it fell to the surface from a height of $h?$ The neutron star has a radius of $R$ and rotates with a period of $T$. You can consider it spherical even though it is not precisely spherical. Find values that are typical for neutron stars and give general as well as concrete numerical answers.

Karel was dreaming of the devastating power of neutron power and their amazing non-inertiality .

### 5. knife thrower

The throwing knife shall leave the hand in the moment that its center of mass is at the height $h$ and has just a purely horizontal component of velocity $v_{0}$. What must its angular velocity $ω$ be for it to hit and stick in a vertical panel at a distance$dfrom$ the point of escape? To make simplify consider the center of mass to be in the middle of its length$l$ and that the knife shall stay in the vertical if the blade shall hit it before the hilt.

Mirek's experiments with knife throwing were not following his statistical predictions.

### P. unnamed snack bar

Considering biochemical processes in the human body and its mechanics estimate, how much energy is used by a cyclist to rise a thousand vertical meters if the average gradient is 5 %.

Mirek and Tomas were thinking about how many hills can be climbed with a certain snack bar (not naming one due to the lack of funding).

### E. bottled potential

We have a cylindrical container, in which we make a circular hole fromthe side. We shall pour water into. Water shall slowly flow out but at some height above the hole the outpouring of water shall stop. Determine the surface tension of water depending on the height at which it stopped. Repeat the experiment multiple times with three different openings. A plastic bottle would do as a cylinder.

Karel was inspired by what Vojta Zak said he was doing at the Physics seminar.

### S. Ljapunovian

- Assume a pen of length 10 cm with a center of mass precisely in the middle and $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}.Now$ imagine that you put the pen on the table with a null deviation $δx$ with an accuracy of $ndecimal$ places and with a null velocity. How long after making the pen stand can you be sure with just $n-decimal$ places of the nullness of the displacement?

- Consider a model of weather with the biggest Ljapun's exponent $λ=1.16\cdot 10^{-5}s^{-1}$. The weather forecast stops being useful if its error becomes bigger than 20 %. If you had determined the state of the weather with an accuracy of 1 %, how long do you estimate that your forecast would be good for? Give the answer in days and hours.

- Take Lorenz's model of convection from the last part, copy the function $f(xi,t)$ amd simulate and draw the values of the parameters $X(t)$ for two different trajectories using the commands X01=1;

Y01=2;

Z01=5;

X02=…;

Y02=…;

Z02=…;

nastaveni = odeset('InitialStep', 0.01,'MaxStep',0.1);

pocPodminka1=[X01,Y01,Z01];

reseni1=ode45(@f,[0,45],pocPodminka1,nastaveni);

pocPodminka2=[X02,Y02,Z02];

reseni2=ode45(@f,[0,45],pocPodminka2,nastaveni);

plot(reseni1.x,reseni1.y(:,1),reseni2.x,reseni2.y(:,1));

pause()

</pre> Instead of three dots $X02,Y02,Z02you$ have to give the initial conditions for the second trajectory. Run the code for at least five different orders of magnitude that are all still small and note the time, in which the second trajectory shall differ qualitatively from the first(ie will go in the opposite direaction). Don't decrease the deviation under cca 10^{$-8}$, because then the imprecision's of numerical integration start to show. Chart the dependency of the ungluing time on the order of magnitude of the deviation.

**Bonus:** Attempt to use the gained dependency of the ungluing time on the size of the deviation estimate Ljapun's exponent. You will need more than five runs and you can assume that at the moment of ungluing it will always overcome some constant $Δ_{c}$.