Series 4, Year 37

One day, the FYKOS-bird was watching the sky during a full moon. An airplane just passed over the moon in $0{.}35 \mathrm{s}$, and the perpendicular distance of its flight path from the center of the moon was $1/3$ of the full moon's radius. This plane flies typically with a speed of $800 \mathrm{km\cdot h^{-1}}$. The FYKOS-bird wondered what altitude the plane was at so he could fly with it next time. Like him, determine this altitude.

Tomáš got into the train wagon in the shape of a rectangular cuboid and decided to take a nap. When he woke up, he found that he was alone in the wagon, which was suspended at its geometric center on a cargo crane and rotating around the hinge axis at an angular velocity of $\omega $. Tomáš didn't notice it at first since he was sitting in the wagon's centre with a width of $d$. When he realized it, he was pleased because he thought of using one of his kilogram standards, which he carries around for situations like this, to measure the length of the carriage. After a few attempts, he managed to throw the standard at an initial velocity of $\vec {v}$ so that after two revolutions of the wagon, the standard hit the far corner of the wagon and broke the window. Neglecting air resistance, what length $L$ of the wagon did he determine?

Consider a homogeneous magnetic field of induction $B_1$, which spans a half-space bounded by the plane of interface $y=0$, beyond which is an equally oriented, also homogeneous magnetic field of induction $B_2$. An electron flies out of the plane perpendicularly to it and the field lines (as in the figure) with velocity $v$. Determine the size and the direction of its average velocity parallel to the plane of the interface.

Bonus: Consider now that the magnitude of the field changes linearly as $B = B_0 \(1+\alpha y\)$ and its direction is in the positive direction of the $z$-axis. Again, determine the magnitude and direction of the average velocity of the electron parallel to the interface plane. The electron is initially emitted as in the previous case.

A polarized beam of light coming from a material with refractive index $n_1$ is incident on a planar interface of a material with refractive index $n_2$ such that it does not lose intensity after passing through. It then reaches the parallel interface with refractive index $n_3$, again passing through without any loss, and so on. Find a sequence $n_i$ that satisfies this.

Little Jagr and his friends would like to go out to play ice hockey. However, it has only started freezing recently, so they don't know if the ice on the pond is already thick enough. Calculate how long it takes for a deep pond to freeze sufficiently; if you know that the water temperature is $0 \mathrm{\C }$ at the beginning, the air is kept at a constant $-10 \mathrm{\C }$ and the minimum ice thickness for safe skating is $10 \mathrm{cm}$. Neither the density of the water nor the ice formed changes with depth. The heat transfer between air and ice and water and ice is much faster than heat conduction in ice. You will need to look up the necessary thermal properties of ice.

Describe the basic physical principles of the various methods of producing artificial lighting. Calculate the efficiency for at least three of them, i.e. how much energy supplied is actually converted into visible light. Compare with actual data.

Measure the period of the torsion pendulum oscillations as a function of the length of the thread. Use at least two types of thread materials. Determine as accurately as possible all the relevant parameters on which the period depends.

1. Consider a thin-walled glass container of volume $V_1=100 \mathrm{ml}$, the neck of which is a thin and long vertical capillary with internal cross-section $S=0{,}20 \mathrm{cm^2}$, filled with water at temperature $t_1=25 \mathrm{\C }$ up to the bottom of the neck. Now submerge this container in a larger container filled with a volume $V_2=2{,}00 \mathrm{l}$ of olive oil at a temperature $t_2=80 \mathrm{\C }$. How much will the water in the capillary rise?
2. In a closed container with a volume of $11{,}0 \mathrm{l}$ there is a weak solution containing sodium hydroxide with $p\mathrm {H}=12{,}5$ and a volume of $1{,}0 \mathrm{l}$. In the region above the surface, we burn $100 \mathrm{mg}$ of powdered carbon. Determine the value of the pressure in the container a few seconds after burning out, after half an hour, and after one day. Before the experiment, the vessel contained air of standard composition at standard conditions; similarly, we maintain a standard temperature around the vessel in the laboratory.
3. Describe three different ways in which the temperature of stars can be determined. What are the basic physical principles they are based on, and what do we need to be careful of?