# 3. Series 28. Year

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### 1. heavy air

What is the weight of Earth's atmosphere? What percentage of weight does it make up? For the purposes of this problem you know only the mass of the Earth $M_{Z}$ and the radius $R_{Z}$ Zeme, gravitational acceleration $a_{g}$ on the surface of the Earth, density of water $ρ$ and you know that near the surface of the Earth at the depth of $h_{1}=10\;\mathrm{m}$ it has the pressure of approximately one atmosphere $p_{a}=10^{5}Pa$.

**Hint:** It is a simple task.
We don't want a perfect solution but a qualified estimation.

Karel saw an interesting misconception according to which a person is lighter on the moon just because the moon is smaller (And what if it were denser?).

### 2. bubbles

Determine the difference in surface energy of a spherical bubble and a bubble n the shape of a regular tetrahedron. Both shapes have the same inner volume $V$.

Karel remembered tetrahedric bubbles from Eureka!

### 3. we're going to bend

Trains, as it is commonly known, don't have a differential and so when they go through a turn the wheels must turn with the same angular velocity. Now assume that the wheels have a cylindrical shape. Thus when going through a turn one wheel has to go a longer path than the other. The axis will have torsion applied on it and in a certain moment the frictional force between the wheel and railway rail won't be large enough and one wheel will slip and the tension on the axis will fall to 0. Determine the distance the individual slippages between depending on the radius of $R_{z}$. The wheel has a raidus of $R$, the axis has a radius of osa $r$, the length of the axis is $L$, the elastic shear modulus of the axis is $G$ (steel), wagon with $N$ wheels has a mass of $M$ and coefficient of static friction between wheels and rails$f$. Finally you can insert realistic values.

**Hint:** For torsion of the$φ$ cylinder of radius $R$, length $l$ and elastic shear modulus $G$, on which we apply the momentum $M$, it is true that

$$\varphi=\frac{2Ml}{G \pi R^4}$$

Vymyslel Lukáš cestou vlakem do Krušných hor.

### 4. fast and beautiful

Teresa was approaching with a relativistic speed $v$ a plane mirror. She was approaching perpendicularly to the mirror's plane. While doing so she is watching herself approach the mirror. What is the actual speed that she is approaching her image with and what is the speed she is observing?

**Bonus:** It isn't a plane mirror but a spherical one

Randomly thought up by Karel when watching Doctor Who (when the colck on the mantelpiece broke).

### 5. spherically symmetrical chickens in a vacuum

Into a container of volume $V=1\;\mathrm{m}$ in which there is a very low pressure (basically a prefect vacuum) we place $V_{0}=1l$ of water at room temperature $t_{0}$. What will be the final state in which the container and the water in it shall find itself in? For the purposes of the calculation assume that the container is prefectly thermally isolated from its surroundings and and has a negligible heat capacity.

Karel was inspired by a problem that one of his classmates in Didactic II. was speculating about.

### P. whistle me something

Explain the principle upon which whistling with your mouth works. Consider first simple models and gradually transfer to more complicated ones. Then Choose the best and on their basis determine the range within which the base frequency can be.(If you know how to whistle you can determine the accuracy of your estimate.)

Mirek wants to inconspicously find out how many others also don't know how to whistle.

### E. sneakers on water

Measure the coefficient of static and dynamic friction between the sneaker (shoe) and a horizontal smooth surface, where the surface is dry and where it is wet. Compare the results and interpret.

Karel slipped on dry land.

### S. numerical

- Look at the equations of the Lorenz model and write a script to simulate them in Octave (maybe even refresh your knowledge of the second part of series). Together with the sketching command your script should have the following form: …

function xidot = f(t,xi)

…

xdot=…;

ydot=…;

zdot= …;

xidot = [xdot;ydot;zdot];

endfunction

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

initialCondition=[0.2,0.3,0.4];

solution=ode45(@f,[0,300],initialCondition,config);

plot3(solution.y(:,1),solution.y(:,2),solution.y(:,3)); </pre> Just instead of three dots fill in the rest of the code (just as in the second part of the series) and use $σ=9,5$, $b=8⁄3.Then$ figure out with a precision of at least units for what positive $r$ the system goes from asymptomatic stopping to chaotic oscillation(it is independent of the initial conditions).

- Here is the full text of the Octave script for simulating and visualising the movement of a particle in a gravitational field of a massive object in the plane $xy$, where all the constants and parameters are equal to one: clear all

pkg load odepkg

function xidot = f(t,xi)

alfa=0.1;

vx=xi(3);

vy=xi(4);

r=sqrt(xi(1)^2+xi(2)^2);

ax=-xi(1)/r^3;

ay=-xi(2)/r^3;

xidot = [vx;vy;ax;ay];

endfunction

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

x0=0;

y0=1;

vx0=…;

vy0=0;

initialCondition=[x0,y0,vx0,vy0];

solution=ode45(@f,[0,100],initialCondition,config)

plot(solution.y(:,1),solution.y(:,2));

pause()</pre>

- Choose initial conditions $x0=0,y0=1,vy0=0$ and and a nonzero initial velocity in the direction $x$ such that the particle will be bound (ie. it won't escape the center.)
- Add to the gravitational force the following force $-α\textbf{r}⁄r^{4}$, where $αis$ a small positive number. Choose gradually increasing $α$ beginning with $α=10^{-3}$ and and show that they cause quasiperiodic movement.