Problem Statement of Series 6, Year 39

Martin attached a weight to a swing with a suspension length $l = 2\,\mathrm{m}$ and released it from the horizontal position. What is the mass $m$ of the attached weight if the massless suspension, with a maximum tension $T_{\mathrm{max}} = 1\,\mathrm{kN}$, broke at the moment when it formed an angle $\varphi=20 ^\circ$ with the vertical? Assume that the only object with mass is the weight itself, and that the suspension remains taut at all times.

Both plates of a capacitor are perforated with a small hole at the same height. The plates are then brought very close to each other and charged to a potential difference $U$, thereby forming a charge bilayer. An electron approaches the hole in the plates with velocity $v$, at an angle $\alpha$ with respect to the normal, and is closer to the positive plate. At what angle does it emerge from the hole on the other side? What would this angle be if, initially, it were directed toward the negative plate?

Jarda decided to use centrifugal force to make a pancake. Onto a smooth, symmetric circular pan of radius $R$, rotating with angular velocity $\omega$, he quickly poured batter of mass $m$ onto the center. Initially, it formed a cylinder of height $h_0$ and radius $r_0$ ($h_0 \ll r_0 < R$). The surface tension between the batter and the surrounding atmosphere is $\sigma_1$, between the batter and the pan is $\sigma_2$, and between the pan and the atmosphere is negligible. Energy is supplied to maintain the rotation of the pan with constant power $P$. Assume that immediately after being poured onto the pan, the layer of batter rotates together with it, and gravity ensures only that the layer always has the shape of a cylinder; otherwise, the gravitational potential energy may be neglected. After how long from the start of rotation does the batter reach the edge of the rotating circle?

The day came when Greta achieved her greatest triumph. She convinced the entire population of 8 billion people that it is necessary to take action regarding the current state of the climate. The plan was radical: to organize a massive march (run) in the direction opposite to the Earth's rotation, halt the alternation of day and night, and thereby cast one hemisphere into perpetual shadow for maximum cooling.

Thanks to advances in genetic engineering, Greta replaced the entire population with clones of Usain Bolt—each has a mass of $100\,\mathrm{kg}$ and can reach a speed of $45\,\mathrm{km\!\cdot\! h^{-1}}$. Can humanity in this configuration stop the rotation of the planet? How many such „super-runners“ would actually be required? Assume a uniform distribution of people over the planet.

Jarda's jealous neighbor had always envied his beautiful and carefully maintained garden. In his own overgrown garden, he collected slugs into a bucket, and when night came, he dumped them right in the middle of Jarda's nicest flower bed. Let us define the linear (i.e., defined on a line) quantities $z(x,t)$ and $s(x,t)$ describing Jarda's flower bed. The quantity $z$ represents the greenery at a given location, and initially (at the moment the slugs are dumped) it is homogeneous, with value $z(x,0)=z_0$. The quantity $s$ is the slug concentration per unit length; immediately after being dumped from the bucket, it has value $s_0$ on an interval of length $l$, and it is zero elsewhere (Jarda has already picked all his own slugs).

The slugs immediately got to work—the rate of greenery destruction is equal to their concentration at the given point multiplied by the voracity coefficient $\zeta$. The slug flux is equal to the product of their mobility $\mu$ and the gradient of the greenery concentration. Finally, the continuity equation holds for the slugs, and we consider their concentration to be a continuous quantity. What will the flower bed look like after four days, when Jarda returns from problem selection? What conditions must the given constants ($z_0$, $s_0$, $\zeta$, $\mu$) satisfy for the problem to make good sense?

Bonus: What would happen if the neighbor dumped the slugs onto a circular patch in the middle of the garden, so that the problem were planar?

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

What determines the current limits on the maximum strength of magnets? We are interested in both permanent magnets and electromagnets. Support your solution with calculations.

Determine the mass of the spring $m$ using a dynamic method based on the period of its oscillations $T$. Perform the measurement for various values of the known mass of the weight $M$.

Hint: The mass of the spring contributes to its period by one third, thus $T=2\pi\sqrt{( M+ m/3)/k}$, where $k$ is the spring constant.

While sorting through old optical elements, we obtained an echelle grating with a blaze angle $\alpha=70^\circ$ and 80 grooves per millimeter. The grating has a width $s=5\,\mathrm{cm}$ along the grooves and a length $l=10\,\mathrm{cm}$ perpendicular to them. The goal is to use it for a construction of an echelle spectrograph with a wavelength range from $375\,\mathrm{nm}$ to $690\,\mathrm{nm}$ as shown in the attached figure. Light is introduced at point $A$ using an optical fiber with diameter $100\,\mathrm{\upmu{}m}$. The output beam with an $f\!/8$ focal ratio reflects off the left side of the collimating parabolic mirror $B$ and strikes the grating $C$ as a collimated beam. After reflecting from the grating and again from the collimator $B$, the beam is directed by the plane mirror $D$ back to the right side of the collimator. The parallel beam is then refracted by the prism $E$ made of BK7 optical glass, with its dispersion direction perpendicular to that of the grating. The resulting beam is imaged by the objective lens $F$ onto a camera chip $G$, a $2048 \times 2048$ array of square pixels with pixel size $a=13.5\,\mathrm{\upmu{}m}$. Your task is to calculate

• the focal length $F$ and diameter of the collimator $B$,

• the prism base length $b$ and its apex angle $\beta$,

• the focal length of the objective $f$ and its diameter $d$.

Try to obtain the greatest possible resolving power while fully covering the required wavelength range while fully utilizing input light beam with the highest possible luminous efficiency. What resolving power will such an instrument have?

Bonus: Construct the resulting echellogram, i.e. the image of the source containing all wavelengths on the chip, with the wavelengths at the center and at the edges of the orders marked.