Problem Statement of Series 2, Year 36

Water flows through a water channel of rectangular cross-section, and width $d=10\,\mathrm{cm}$. A leaf falls on its surface and starts moving with a velocity of $60\,\mathrm{cm\cdot s^{-1}}$. The height of the water in the channel is $h=1.3\,\mathrm{cm}$. Estimate how long it will take to fill up a $50\,\mathrm{l}$ bucket. Comment on the assumptions used in comparison with the real situation.

There is a raft in the middle of the river. The mass of the raft is negligible, and it carries a crane on board. The crane moves boxes of building material of mass $m$ from one river bank to another. In one cycle, the crane loads material at one side of the river, rotates to the other river bank, unloads the material there, and rotates back. Calculate the smallest value of angular displacement of the raft from its original position during one cycle. Approximate the crane by a homogenous cylinder of mass $M_{\mathrm{j}}$ and radius $r$, and a rotating jib in the shape of a slim rod of length $kr$. Assume that the velocity of the river and the “friction” between the raft and the water are negligible.

The FYKOS-bird watches in their inertial frame of reference as two point masses move around them on parallel trajectories with constant non-relativistic velocities. They think whether these trajectories could intersect for some other inertial observer. If so, is it possible that the two point masses in question could collide at this intersection given the right initial conditions? Is this consistent with the fact that they are moving in parallel according to the FYKOS-bird?

Consider a thin magnet placed in the middle of a thin hollow rod of length $l$. The material of the rod is capable of shielding the magnetic field. Just beyond the end of the rod, the magnetic field flux is equal to $\Phi$. Calculate the direction and strength of the magnetic field in a plane perpendicular to the rod passing through its center as a function of the distance $r$ from the rod.

What parameters does a planet need to have to keep its atmosphere comparable to the Earth? What conditions are essential for the planet to gain such an atmosphere?

Measure the dependence of sound intensity emitted by your loudspeaker/mobile phone/computer on the distance from the source. Furthermore, determine the dependence of sound intensity on the settings of the output volume. Do not forget to fit the data.

1. Find a beta-carotene molecule and calculate what color should it have or rather what wavelength it absorbs. Use a simple model of an infinite potential well in which $\pi$ electrons from double bonds are “trapped” (i.e., two electrons for each double bond). The absorption then corresponds to such a transition that an electron jumps from the highest occupied level to the first unoccupied level.

   Compare the calculated value with the experimental one. Why doesn't the value obtained by our model come out the way we would expect? (5b)

2. Let's try to improve our model. When studying some substances, especially metals or semiconductors, we introduce the effective mass of the electron. Instead of describing the environment in which the electrons move in a complex way, we pretend that the electrons are lighter or heavier than in reality. What mass would they need to have to give us the correct experimental value? Give the result in multiples of the electron's mass. (2b)

3. If we produce microscopic spheres (nanoparticles) of cadmium selenide /$\ce{CdSe}$) with a size of $2.34\,\mathrm{nm}$, they will glow bright green when irradiated by UV light with a wavelength of $536\,\mathrm{nm}$. When enlarged to a size of $2.52\,\mathrm{nm}$, the wavelength of the emitted light shifts to the yellow region with a wavelength of $570\,\mathrm{nm}$. What would the size of spheres need to be to make them emit orange with a wavelength of $590\,\mathrm{nm}$? (3b)

   Hint $\ce{CdSe}$ is a semiconductor, so it has a fully occupied electron band, then a (narrow!) forbidden band, and finally an empty conduction band. Thus, we must consider that the emitted photon corresponds to a jump from the conduction band (where such states are as in the infinite potential well) to the occupied band. Therefore, all the energies of the emitted photons will be shifted by an unknown constant value corresponding to the width of the forbidden band.

Finally, a bonus for those who would be disappointed if they didn't integrate – the 1s orbital of the hydrogen atom has a spherically symmetric wave function with radial progression $\psi(r) = \frac{e^{-r/a_0}}{\sqrt{\pi}a_0^{3/2}}$, where $a_0=\frac{4\pi \epsilon_0\hbar^2}{me^2}$ is the Bohr radius. Since the orbitals as functions of three spatial variables would be hard to plot, we prefer to show the region where the electron is most likely to occur. What is the radius of the sphere centered on the nucleus in which the electron will occur with a probability of $95\,\mathrm{\%}$? (+2b)