Problem Statement of Series 5, Year 39

Marek bought enchanted beans from a strange trader at the train station; once they grow, they are supposed to lead to a castle of magical giants somewhere high in the sky. How high can the giants be so that Marek can reach them, if all the carbon for the bean plant comes from atmospheric carbon dioxide? Assume that the stem is pure cellulose with density $\rho =1.56\,\mathrm{g\cdot cm^{-3}}$ in the shape of a cylinder with base $R = 1.0\,\mathrm{km}$. Estimate the amount of carbon dioxide in the atmosphere and compare it with more accurate data.

Jindra needs to move an iron beam of mass $m = 50\,\mathrm{kg}$. You may consider the beam to be a massive line segment lying on the ground. The coefficient of static friction between the beam and the ground is $f = 0.80$. Jindra can exert a maximum pulling or pushing force $F_{\mathrm{Jin}} = 340\,\mathrm{N}$ in any direction. Can Jindra move the beam by his own strength without using tools or asking other people for help? Support your answer with a calculation showing whether Jindra can or cannot move the beam.

Tadeáš is sitting at his desk in his room, positioned against a wall, directly adjacent to the outdoors, and he feels a draft. Calculate the heat flux density $q$ between the interior of Tadeáš's room and the outdoors through the wall next to which the desk stands.

Suppose that the temperature in Tadeáš's room is $T_1$ and the outdoor temperature is $T_2$. For simplicity, assume that the wall consists of a two-layer thermal insulation made of materials with known thermal conductivities $\lambda_1$, $\lambda_2$ and thicknesses $d_1$, $d_2$. The heat transfer coefficient between the indoor air and the first insulation layer, and the heat transfer coefficient between the second insulation layer and the outdoor air, are $\alpha_1$ and $\alpha_2$, respectively. Neglect any wall curvature and assume that the system is in thermodynamic equilibrium.

Hint: Use thermal resistances.

A permanent bar magnet with dipole moment $\mu$, mass $m$, radius $r$, and length $l$ is attached horizontally to a refrigerator. What is the heaviest weight that can be hung from its end if the coefficient of friction between the magnet and the refrigerator is $f$? For simplicity, assume that the refrigerator forms a half-space of perfectly magnetizable metal and that the magnetic field of the magnet is dipolar and symmetric with respect to its body.

Hint: Use a point dipole.

This problem has an open solution, so be sure to cite all sources used.

Let a Tsar Bomba be detonated near Halley's comet. What happens to the comet depending on the distance at which the explosion occurs? What is the maximum possible change in its orbital velocity if the explosion occurs at the comet's perihelion?

Measure the tensile strength limit of an inflatable balloon (while inflating). Choose appropriate approximations. Pay attention to safety while experimenting.

1. 1. Derive the differential equation for the oscillation of a cantilever in a general medium with a viscous damping coefficient $b$. Assume that the cantilever with a tip is driven by a force $F = A\cos(\omega t)$. – 1 point
   2. Solve the derived differential equation. To solve the differential equation, the materials cited in the serial text will be useful. Introduce the following notation into the derived equation and use the approximations below \begin{align*} \omega_0 &= \sqrt{\frac{k}{m}}\,,\\ Q &= \frac{m \omega_0}{b}\,,\\ Ae^{i\omega t} &= A(\cos(\omega t) + i\sin(\omega t))\,. \end{align*} Further, assume a solution of the form $z = \overline{B}e^{i\omega t}\,$ and the approximation $\omega \approx \dot{\omega}_1$. $\overline{B}$ denotes the complex conjugate, so treat it accordingly. Also, attempt to explain the meaning of the $Q$ factor and $\omega_0$. When is the approximation $\omega \approx \dot{\omega}_1$ valid? You should obtain a defining relation for the amplitude $B$ and the phase. – 3 points

   3. Plot the dependence of the amplitude $A$ on the angular frequency for various $Q$ and values $\omega_0 = 1.2; B = 1$. You should obtain a graph similar to the one in Fig.. Briefly comment on the dependence on the choice of constants. – 1 point

      

   4. How does the shift of the maximum depend on the magnitude of $Q$ (see Fig.)? Would any shift be observed when using AFM in castor oil with a viscosity of $600\,\mathrm{mPa\cdot s}$? What about for UHV? Search online for cantilevers and state in your answer the material (and ideally also the information source) that you selected for the calculation. If the cantilever is coated, you may neglect this coating. – 2 points

2. Using the SEM image (BSE) and the model EDX spectrum, determine which elements are present in the material and which regions of the samples correspond to them. What else do you observe in the image? How should a real EDX spectrum differ from the model one? Justify all your statements in detail. The necessary images can be found at the end of the serial text.