Problem Statement of Series 3, Year 35

We can find an unofficial interpretation that the red, blue and white colors on the Czech flag symbolize blood, sky (i.e. air) and purity. Find the position of the center of mass of the flag interpreted in this way, assuming that purity is massless. The aspect ratio is $3:2$ and the point where all three parts meet is located exactly in the middle. Look up the blood and air densities.

Bonus Try to calculate the position of the center of mass of the Slovak flag as accurately as possible. You can use different approximations.

Vašek likes to plays with keys by swinging them around on a keychain and then letting them wrap around his hand. We will simplify this situation by a model, in which we have a point mass $m$ in weightlessness attached to an end of massless keychain of length $l_0$. The other end of the keychain is attached to a solid cylinder of radius $r$. The keychain is taut so that it is perpendicular to the surface of the cylinder at the attachment point, and the point mass is then brought to velocity $\vect{v_0}$ in the direction perpendicular both to the axis of the cylinder and to the direction of the keychain. The keychain then starts to wrap around the cylinder. What is the dependence of the velocity of the point mass on the length of the free (not wrapped around) keychain $l$?

Hint Find a variable that remains constant during the wrapping process.

Bonus How long does it take for the whole keychain to be wrapped around the cylinder?

Consider two coils wound around a common paper roll. First coil has a winding density of $10\,\mathrm{cm^{-1}}$ and the second coil has a winding density of $20\,\mathrm{cm^{-1}}$. The paper roll is $40\,\mathrm{cm}$ long and has $1\,\mathrm{cm}$ in diameter. Both coils are wound along the whole length of the roll, with the second coil wound around the first one. Considering the dimensions of the roll, we can neglect the boundary effects and assume that the coils behave as perfect solenoids. Now consider connecting the coils in series. This configuration can be substituted by a circuit with a single coil. What is the inductance of the substituting coil?

Close to the shore, the speed of sea waves is influenced by the presence of the sea bed. Assume that the speed of waves $v$ is a function of the gravity of Earth $g$ and the water depth $h$. We have $v = C g^\alpha h^\beta$. Using dimensional analysis, determine the speed of the waves as a function of the depth. Constant $C$ is dimensionless, and cannot be determined using this method.

Besides the speed of the waves, swimming Jindra is also interested in the direction of incidence of the waves. Let's define a system of coordinates, where the water surface lies in the $xy$ plane. The shoreline follows the equation $y = 0$, the ocean lies in the $y > 0$ half-plane. The water depth $h$ is given as a function of distance from the shore $h = \gamma y$, where $\gamma = \const$. On the open ocean, where the speed of the waves is constant $c$ (not influenced by the depth), plane waves are propagating at incidence angle $\theta_0$ to the $x$ axis. Find a differential equation \begin{equation*} \der{y}{x} = f\!\left(y\right) \end{equation*} describing the shape of the wavefront close to the shore, but do not attempt to solve it, it is far from trivial. Calculate the incidence angle of the wavefront at the shoreline.

Bonus Solve the differential equation and find the shape of the wavefront close to the shore.

Gnomes decided to forge another magic sword. They make it from a thin metal rod with radius $R=1\,\mathrm{cm}$, one end of which they maintain at the temperature $T_1 = 400\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. The rod is surrounded by a huge amount of air with the temperature $T_0 = 20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$. The heat transfer coefficient of that mythical metal is $\alpha = 12\,\mathrm{W\cdot m^{-2}\cdot K^{-1}}$ and the thermal conductivity coefficient is $\lambda = 50\,\mathrm{W\cdot m^{-1}\cdot K^{-1}}$. The metal rod is very long. Where closest to the heated end can gnomes grab the rod with their bare hands if the temperature on the spot they touch is not to exceed $T_2 = 40\,\mathrm{^\circ\mskip-2mu\mathup{C}}$? Neglect the flow of air and heat radiation.

How could artificial gravity be implemented on a spaceship? What would be the advantages and disadvantages depending on the different characteristics of the spacecraft? Is it realistic to have gravity in different directions on different floors of the spaceship or for it to change rapidly, as we can sometimes see in sci-fi movies when “artificial gravity fails”?

Measure the rotation of the plane of polarization as a function of the sugar concentration in the solution.

1. Determine the reach of helium nuclei in central hot spot (using the figure \ref{R35S3S-A}).
2. What energy must be released in the fusion reactions in order for the fusion to spread to the closest layer of the pellet? How thick is the layer?
3. Estimate the most probable amount of energy transferred from helium nucleus to deuterium. How many collisions on average does the helium nucleus undergo in the central hot spot before it stops?