Problem Statement of Series 6, Year 26

Many years ago you drank 2 dcl of water. Imagine that since then all the water on the Earth has mixed. If you drink 2 dcl of water today, how many molecules from the original water you drank does it contain?

What is the minimal length of a wire so that if you hang it from a ceiling, it will break due to its own mass? The wire's density is $\rho =7900\mathrm{kg}\cdot {\mathrm{m}}^{-3}$, it has a diameter $D=1\mathrm{mm}$, and it breaks at ${\sigma }_{max}=400MPa$. Assume that everything takes place in a homogeneous gravitational field $g=9.81\mathrm{m}\cdot {\mathrm{s}}^{-2}$.

Bonus: If the wire's length is maximal possible so that it does not break, how much will it stretch (in percents)? Young's modulus of the wire's material is $E=200GPa$.

If an object is placed a distance $p$ from a thin glass lens (index of refraction $n_s)$, we can see its image on a screen that is placed a distance $d$ from the lens. Without altering any distances, we immerse this system into a liquid (index of refraction $n)$. Under what conditions can we still observe the object's image on the screen, and how far from lens would this image be?

Imagine a large tank containg tea with a little opening at its bottom so that one can pour it into a glass. When open, the speed of the flow of tea from the tank is $v_0$. How will this speed change if, while pouring a glass of tea, someone is filling the tank by pouring water into it from its top? Assume that the diameter of the tank is $D$, the diameter of the flow of tea into the tank is $d$, and that of the flow of tea out of the tank is much smaller than $D$. The tea level is height $H$ above the lower opening, and the tank is being filled by pouring a water into it from height $h$ above the tea level. You are free to neglect all friction.

Let us consider the following model of a baseball player hitting a ball. Baseball bat is a thin homogeneous rod of length $L$ and mass $m$. The bat can only rotate around an axis perpendicular to the axis of the bat that is located at its end. The bat is rotating with an angular velocity $\omega$. How far from the end of the bat should the player hit the ball in order to minimize the force with which the bat acts on the player's hands?

How many people per second can be killed by a nuclear reactor without any protective walls?

A loaded falling balloon will eventually reach certain constant terminal velocity. Measure how does this velocity depend on the balloon size, and on the mass of its load.

• Spočtěte dobu udržení energie v tokamaku COMPASS, kde je energie plazmatu 5 kJ a ohmický ohřev 300 kW.

• Spočtěte, jaký alfa ohřev by byl v tokamaku COMPASS, pokud by v něm hořela DT směs. Typická teplota plazmatu je 1 keV, hustota ${10}^{20}\text{jd}{m}^{-3}$, objem plazmatu cca 1 m. Při uvážení ohmického ohřevu z předešlého příkladu spočtěte $Q$.

• S využitím obrázku v textu seriálu a znalosti DD reakce

$$^{2}_{1}D + ^{2}_{1}D → ^{3}_{2}He + n + 3,27 MeV (50 \%),$$

$$^{2}_{1}D + ^{2}_{1}D → ^{3}_{1}T + p + 4,03 MeV (50 \%),$$

kde opět $\frac{3}{4}$ energie v první reakci odnáší neutron, spočtěte celkový ohřev plazmatu, který se vyvine během jedné DD reakce (uvažujte, že následně proběhne i DT fúze s produktem druhé reakce) a odhadněte nároky na dobu udržení při hustotě ${10}^{20}\text{jd}{m}^{-3}$ a teplotě 10 keV.