Assignment of Series 5 of Year 36

Vojta plays the cello. He lightly places his finger on a string, tuned to the frequency $f$, at a distance of $1/n$ of its length from the head of the instrument and sounds it, hearing a tone of fundamental frequency $f_1$. He then presses the string fully against the fingerboard at the same point and sounds it again. This time the instrument produces a tone of fundamental frequency $f_2$. Determine the ratio of the frequencies $f_1/f_2$ as a function of the natural number $n$.

Every second, a material of mass $\mu$ falls vertically onto a moving horizontal conveyor belt and falls away at its end. A resistive force $F_{\mathrm{odp}}=kv$, which is directly proportional to the belt speed $v$ through the constant $k$, acts on the belt. At what speed does thevbelt move if

• a constant driving force $F$ acts on it?
• it is driven by a motor of constant output power $P$?

Karel uses an elevator in a building with a ground floor and $12$ floors above it, while the height between floors is $h=3.0\,\mathrm{m}$. Consider that the elevator accelerates half the time and decelerates half the time at a constant acceleration of $a=1.0\,\mathrm{m\cdot s^{-2}}$ and that there is a $50\%$ probability that the elevator is stationary on the ground floor. The rest of the probability is evenly distributed among the other floors. What is the expected waiting time for the elevator on each floor of the building? Neglect the time needed for opening doors.

Bonus Let us have $2$ elevators in a twelve-story building. One elevator is always recalled to the ground floor. To which floor should we send the second one to minimize the average waiting time? Similarly, assume that half of the rides will start on the ground floor and the other half, with equally distributed probability, will start on any other floor.

FYKOS plans to send its own satellite into space. It will be powered by solar cells; hence, it cannot stay in the Earth's shadow for too long. What is the height above the Earth's surface for which the time of the satellite passing through the Earth's shadow is the shortest? In your calculations, assume (same as the organizers did) that the Earth is a perfect sphere, that sunrays close to Earth's surface are parallel, and that the Sun, the Earth and the satellite's trajectory are in the same plane.

Bonus While solving the problem, you will encounter an analytically unsolvable equation. Do not use online solvers, but try to create your own solution.

A once positively ionized xenon atom flew out from the center of a large cylindrical coil with velocity $v=7\,\mathrm{m\cdot s^{-1}}$ and began to move through a homogeneous magnetic field, which is in a plane perpendicular to the magnetic lines of force. At a certain point the coil is disconnect from the source, thus its induction begins to decrease exponentially according to the following equation $B\!\left(t\right)=B_0\mathrm{e}^{-\Omega t}$, in which $B_0=1.1\cdot 10^{-4}\,\mathrm{T}$ and $\Omega =600\,\mathrm{s^{-1}}$. What is the deviation from the initial direction after the atom is stabilized?

Describe as many natural influences as possible that cause uprooting/severe damage to a lone tree in a meadow. Try to analyze one of them qualitatively as best as you can. What is the difference between a broadleaved tree and a conifer?

Bonus Discuss some of the influences quantitatively.

Use diffraction on the grating to determine the density of data written to the CD. Try to compare the results with the DVD.

The binding energy of a fluorine molecule is approximately $37\,\mathrm{kcal/mol}$. Assuming the range of binding interactions to be approximately $3\,\mathrm{\AA}$ from the optimum distance, what (average) force do we have to exert to break the molecule? Calculate the “stiffness” of the fluorine molecule if such an average force was applied in the middle of this range. What would be the vibrational frequency of this molecule? Compare this with the experimental value of $916.6\,\mathrm{cm^{-1}}$. ($4\,\mathrm{pts}$)

Using Psi4, calculate the dissociation curve $\mathrm{F_2}$ and fit a parabola around the minimum. What value will you get for the energy of the vibrational transitions this time? ($3\,\mathrm{pts}$)

You are given two bottles of alcohol that you found suspicious, to say the least. After taking them to the lab, you obtain the following Raman spectra from them. Using the Psi4 program, calculate the frequencies at which the vibrational transitions of both the methanol and ethanol molecules occur. Use this to determine which bottle contains methanol and which one contains ethanol. You can use the approximate geometries of ethanol and methanol, which are included in the problem statement on the web. ($3\,\mathrm{pts}$)

\begin{figure}[htbp] \hfill \begin{minipage}{0.45\textwidth} \includegraphics[width=0.95\linewidth]{problem5-8_raman1} Raman spectra of bottle A \end{minipage} \hfill \begin{minipage}{0.45\textwidth} \includegraphics[width=0.95\linewidth]{problem5-8_raman2} Raman spectra of bottle B \end{minipage} \hfill \end{figure}