Problem Statement of Series 5, Year 38

Jindra owns a sheet of paper with a square hole cut out in the middle. The hole has a side length $a = 3\,\mathrm{mm}$. It is a beautiful sunny day, so Jindra takes this sheet of paper outside to project his square onto the sidewalk. First, he placed the paper about $2\,\mathrm{mm}$ above the sunlit surface. What shape does the light spot on the wall caused by the square hole in the paper have? Jindra then moved his favorite piece of paper to a distance of about $1.5\,\mathrm{m}$ above the sidewalk. What shape does the light spot on the sidewalk have now? Explain why the shadow behaves this way.

We fill a hose with water—one end is submerged in a water reservoir, while the other end is sealed airtight. We then start pulling the hose vertically upward. This water has a temperature of $20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. How much would the height change if we used water at $90\,\mathrm{^\circ\mskip-2mu\mathup{C}}$ instead, i.e., what is the difference between these two heights?

Hint: Consider how strong the vacuum will be at the sealed end.

Tomáš boarded a train, sat in one of the carriages, and decided to take a nap. When he woke up, he found out he was alone in the carriage, and the carriage, along with him, was orbiting the Earth at an altitude of $h = 400\,\mathrm{km}$ above the planet's surface in a state of weightlessness. The carriage was oriented perpendicular to both the orbit of its center of mass and the radial direction.

Tomáš was excited because he realized he could take advantage of the inhomogeneity of the Earth's gravitational field to measure the length of the carriage. He took two $1\,\mathrm{kg}$ calibration weights he always carried around for such occasions out of his bag and placed them on the opposite ends of the carriage. He placed a laser telemeter between them and measured their relative distance. He then started a stopwatch. After $t = 60\,\mathrm{s}$, he measured their distance again–-it has since changed by $\Delta l = 4\,\mathrm{cm}$. What is the length of the carriage $L$ he measured? Assume the force acting on the weights was constant. Earth's mass is equal to $M = 5.97\cdot 10^{24}\,\mathrm{kg}$ and the mean radius of the Earth is equal to $R = 6~371\,\mathrm{km}$.

Maybe you have gone minigolfing at a course where one of the holes featured a see-saw obstacle. We can model the see-saw as a board of mass $M$ and length $l$. This board can rotate around a horizontal axis passing through the center of the board; the axis is located at height $h$ above the ground. A golf ball of mass $m$ and radius $r$ rolls toward the see-saw from a direction perpendicular to the axis of rotation, moves onto the see-saw, and gradually tips the board, allowing the ball to to descend on the other side. The ball rolls without slipping throughout the process. Formulate the equations of motion for this system – you do not have to solve them.

Marek has two identical metal cubes with constant heat capacity $C$, one at temperature $T_1$ and the other one at $T_2$. What is the highest and lowest temperature at which they can both stabilize if he brings them into contact and only uses them to power a heat engine?

Hint: If you get stuck, remember that entropy will never decrease.

Which isotope is the most dangerous in terms of nuclear power plant accidents? Consider the amount that could be released during an accident, the likelihood of a leakage, the spreading of specific isotopes and their impact on the human body.

Try to measure your latitude as accurately as possible without using GPS or any other method that relies on information about your location. You can assume your longitude to be a known quantity.

1. In the Levič equation, physical quantities on the right-hand side appear with non-integer exponents. Verify that both sides of the equation have the same units. – 1 point

2. In a beaker intended for a rotating disc electrode, we dissolved $0.63\,\mathrm{g}$ of seventy percent perchloric acid in $750\,\mathrm{ml}$ of pure water and mixed everything thouroughly. Then, on a platinum working electrode with a circular shape and a diameter of $5.0\,\mathrm{mm}$, we varied the voltage for the hydrogen formation reaction until we reached the limiting current of $0.29\,\mathrm{mA}$. After measuring it, we started spinning the electrode at a frequency of $3~600\,\mathrm{rpm}$, where the limiting current was $11.5\,\mathrm{mA}$. Determine the diffusion coefficient and the thickness of the diffusion layer before spinning. The kinematic viscosity of water is $\nu = 0.9\,\mathrm{mm^2\cdot s^{-1}}$. – 3 points

3. Find the highest power that a galvanic cell with the following parameters can provide and determine the corresponding load. For simplicity, consider the Tafel regime with a Tafel slope of $100\,\mathrm{mV/dec}$ and a parameter $I_0 = 2\cdot 10^{-8}\,\mathrm{A}$. The ohmic resistance is $R_\Omega = 55\,\mathrm{m\Omega}$. The open-circuit voltage is $1.18\,\mathrm{V}$. Neglect the diffusion regime. – 3 points

4. Derive the Koutecký-Levič equation as presented in the series text. Start from the derivative of the Levič equation in the case where $c(z=0) = c^s \neq 0$. – 3 points