Problem Statement of Series 1, Year 38

What would be the gravitational acceleration on the surface of Jupiter's moon Europa if it had the same radius but consisted entirely of liquid water (for simplicity, consider water under normal conditions)? How would this change if it were made entirely of ice? How do the results differ from the actual value?

Jarda is bringing Victor a new philodendron in the trunk of his car. It is planted in a flowerpot with a circular base with a radius of $6\,\mathrm{cm}$ and the center of gravity of the philodendron and the flowerpot is at a height of $7\,\mathrm{cm}$. Jarda is driving at a speed of $90\,\mathrm{km\cdot h^{-1}}$, but he is approaching a turn with a radius of curvature of $10\,\mathrm{m}$. To ensure that the philodendron prospers, it cannot even tilt along the journey. What is the shortest distance from the turn that Jarda has to start braking? He wants to drive through the turn at a constant speed.

Jarda is strolling through the square at a speed of $1.5\,\mathrm{m\cdot s^{-1}}$ toward his favorite cake shop. The door to the cake shop is only $50\,\mathrm{m}$ ahead of him when he notices a teacher leading a line of preschool children across his path. The teacher is positioned midway between Jarda and the door. The children are following her at a speed of $0.5\,\mathrm{m\cdot s^{-1}}$ perpendicular to Jarda's path. Jarda wants to avoid passing through a ten-meter-long line of children to prevent them from joining him by mistake and to avoid inviting them for cake. When is the earliest Jarda can reach the cake shop if he does not change his speed?

We are holding an object in one hand at a distance $D$ from the ceiling. Where on the vertical axis passing through the object do we have to place a contact lens with a focal length $f$, so that a focused image is created on the ceiling?

Now, we will place the lens at this distance. The object accidentally falls out of our hand, i.e., it is in free fall. How do we need to move the lens so that the image remains focused? Will the position of the lens approach some specific value after a very long time? Assume that the height of the room is much greater than both $D$ and $f$.

Let us consider a plate air capacitor whose plates are connected to the rest of the circuit with springs of stiffness $k$. In the resting position, the plates are at a distance $d$ and have an area $A$ ($A \gg d^2$). We start charging the capacitor so that the plates begin to attract each other and get closer. Determine the work necessary to charge the capacitor to a charge $Q$. What is the maximal voltage that we can create between the plates?

Determine the most efficient drive for a car. More exactly, determine such a drive, which consumes the least amount of energy per $1\,\mathrm{J}$ of work done by the engine, considering the entire process from fuel production to engine efficiency. You can compare options such as gasoline, diesel, electricity, hydrogen, or feed for a harnessed horse.

Measure the density of grains of raw rice. Do not forget to thoroughly describe your solution and estimate its uncertainty.

1. To get well acquainted with concepts such as oxidation or cathode, we should solve a few chemical equations on our own. For the following chemical equations, determine the oxidation numbers of the individual atoms, determine, what oxidizes and what reduces, write both of the half-reactions, balance them out, and write the total equation of the reaction:
   1. $\ce{Cu^{2+}(aq) + Cr(s) -> Cu(s) + Cr^{3+}(aq)} $,
   2. $\ce{Fe(s) + O_2(g) -> Fe^{2+}(aq) + H_2 O(l)} $ in an acidic solution,
   3. $\ce{Pb(s) + PbO_2(s) + H_2 S O_4(aq) -> PbSO_4(s)}$,
   4. $\ce{\left(MnO_4\right)^-(aq) + Cr\left(OH\right)_3(s) -> MnO_2(s) + \left(CrO_4\right)^{2-}(aq)}$ in a basic solution.

   5. Bonus: Determine the same for the reaction $\ce{CH_3OH(l) + O_2(g) -> H_2 O + CO_2(g)}$.

2. Let us consider the production of gas chlorine from $26\,\mathrm{wt\%}$ concentrated kitchen salt solution. Through the circuit, a current of $6\,\mathrm{kA}$ is flowing under a voltage of $3.4\,\mathrm{V}$.
   1. Determine, what will be the weight of the chlorine produced in the device after one day.
   2. Considering that the reaction also produces $\ce{H_2}$ and $\ce{NaOH}$, write the total reaction of this process and determine by what amount does the weight of the water drop after one day.
   3. How long would it take to fill a $50\,\mathrm{l}$ bottle with the produced chlorine, if it was stored inside under normal conditions?
   4. The amount of chlorine in the bottle felt low, so before filling the bottle, we have isothermically pressed it to a pressure of $8\,\mathrm{bar}$. What is the amount of work necessary (for the electrolysis and the pressing) to fill the mentioned $50\,\mathrm{l}$ bottle?
   5. Another option for chlorine production is the electrolysis of melted salt, which produces liquid sodium as well. Why is this type of chlorine production less common?