Problem Statement of Series 1, Year 36

Jarda decided to bake a cake but he found out that the battery in his kitchen scale was dead, so he can't weigh $300\,\mathrm{g}$ of flour. However, he had the idea that he could use a block of butter instead. The packaging said its weight is $m = 250\,\mathrm{g}$. Fortunately, he found a suitable spring and a stopwatch. He put a heap of flour in a very light bowl, attached it to the spring, perturbed it and measured the period of oscillations $T_1=2.8\,\mathrm{s}$. He repeated the same process with the cube of butter and measured $T_2 = 2.3\,\mathrm{s}$. How much flour does Jarda need to add or remove?

Let us have an ideal scale which we calibrate using a state standard (etalon) with a mass $m_{\mathrm{e}} = 1.000~000~165\,\mathrm{kg}$ and a density $\rho_{\mathrm{e}} = 21~535.40\,\mathrm{kg\cdot m^{-3}}$. By calibration, we mean that after placing the standard on the scale, we assign to the measured value the mass $m_{\mathrm{e}}$. The unknown object is weighed under the same conditions in which its volume is $V_0 = 3.242~27\,\mathrm{dl}$. What mass did we measure if we measured the weight $G = 1.420~12\,\mathrm{N}$? What is the actual mass of the object? The experiment is conducted at a place with standard gravitational acceleration $g = 9.806~65\,\mathrm{m\cdot s^{-2}}$ and air density $\rho_{\mathrm{v}}=1.292~23\,\mathrm{kg\cdot m^{-3}}$. Take into account that the calibration is linear, and the unloaded scale shows zero.

A cylindrical jar made of glass has a height $h = 7.0\,\mathrm{cm}$ and an inner radius $r = 2.5\,\mathrm{cm}$. We pour hot apricot jam at temperature $T_0 = 80\,\mathrm{^\circ\mskip-2mu\mathup{C}}$ into the jar, we close the lid and let it cool down. Note that we didn't fill the jar to the top, but left some air between the jam and the lid. If a force of at least $F = 4\,\mathrm{N}$ is applied, a sound is heard as the lid suddenly incurves. We heard this sound $t_{\mathrm{i}} = 30\,\mathrm{min}$ after the jar had been closed. If jam hardens at temperature $T_{\mathrm{h}} = 60\,\mathrm{^\circ\mskip-2mu\mathup{C}}$, was it to be already hard when the lid incurved?

Bonus How long after closing the jar will the jam harden? Assume that the temperature is evenly distributed throughout the jar and that the cooling rate only depends on the difference in temperatures of the jar and its surroundings $T_{\mathrm{s}} = 25\,\mathrm{^\circ\mskip-2mu\mathup{C}}$.

There is a town on the slope of a hill whose shape is a cone with apex angle $\alpha = 90^\circ$. On the other side of the cone, right opposite to the town in the same altitude, lies a train station. Mayor of the town decided to build a road to the station. They can either drill a tunnel or build a road on the surface of the hill. What is the maximum ratio of per-kilometer prices for the tunnel and for the surface road, so that building a tunnel is cheaper? The road can be built anywhere on the hill.

We have a U-tube with length $l$ and cross-sectional area $S$. We pour volume $V$ of water into the tube. The volume $V$ is large enough that the whole U-turn is filled with water but $Sl > V$. When water levels in both arms of the tube are at rest, we seal one of the arms. What is the period of small oscillations of water in the tube?

Estimate the consumption of electrical energy for one trip of the IC Opavan train. The train set consists of seven passenger cars, a 151-series locomotive and is capable of reaching a speed of $v_{\mathrm{max}} = 160\,\mathrm{km\cdot h^{-1}}$. For simplicity, consider that all passengers are going from Prague to Opava.

Measure the density of ice.

Find the Rydberg constant's value and determine which hydrogen spectral lines belong to the visible spectrum. These lines are the only ones Rydberg could use to discover his formula, as neither UV nor IR spectra could yet be measured. What color are they, and which transitions in the Bohr model do they correspond to? (3pts)

Calculate de Broglie wavelength of your body. How does this value compare to the size of an atom or atomic nucleus? (3pts) Assume you have a cuvette with $10\,\mathrm{ml}$ of fluorescein water solution. Then, you point an argon laser at the cuvette. The laser is characterized by a wavelength of $488\,\mathrm{nm}$ and a power of $10\,\mathrm{W}$. At the same time, the fluorescein molecule fluoresces at a wavelength of $521\,\mathrm{nm}$ with a quantum yield (proportion of absorbed photons that are emitted back) of $95\,\mathrm{\%}$. If the initial temperature of the cuvette is $20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$, how long will it take for its contents to start boiling? Assume that the cuvette is perfectly thermally insulated, the laser beam is fully absorbed in it, and the amount of fluorescein is negligible in terms of heat capacity. (4pts)