Problem Statement of Series 3, Year 36

Danka attached a garden hose with an inner diameter of $1.5\mathrm{cm}$ to a tap in her dorm room and placed the other end on the edge of a window on the eighth floor, $23\mathrm{m}$ above the ground. What is the necessary volumetric flow rate of the water tap so that Danka can spray a stream of water on the people disturbing the night's silence? They are standing below the window at a horizontal distance $9\mathrm{m}$ from the building. Is Danka able to achieve this if water is being sprayed horizontally from the hose and there is no wind?

Bonus Where is the farthest these people can stand so Danka can still spray them if the volumetric flow rate of the tap is $0.4\mathrm{l}\cdot {\mathrm{s}}^{-1}$? Danka can now set the end of the hose so that water sprays at an arbitrary angle to the horizontal plane.

Since our Sun will explode one day, it will be necessary to organize the construction of an evacuation spacecraft, in which at least $0.000001\mathrm{\%}$ of humanity can escape. To escape, people choose the star Tau Ceti which is $12\mathrm{ly}$ away at that time. They build engines that accelerate the spacecraft to a cruising speed of $v=0.75c$ in very little time. Unfortunately, at half a distance to their destination, they observe both the Sun's and Tau Ceti's explosions. How long before the observation the explosions happened in an inertial coordinate frame of the spacecraft? And when in the coordinate system fixed with the stars? Presume that the Sun and Tau Ceti are static relative to each other.

Assume you have a guitar that is perfectly tuned at room temperature. By how many semitones (in tempered tuning) will the individual strings be out of tune if we move to a campfire, where it is cooler by $10{}^{\circ }\mathrm{C}$? Will the guitar still sound in tune? The distance between the string attachment points is $d=65\mathrm{cm}$. The strings have a density $\rho =8900\mathrm{kg}\cdot {\mathrm{m}}^{-3}$, a Young's modulus of elasticity $E=210\mathrm{GPa}$ and a thermal expansion coefficient $\alpha =17\cdot {10}^{-6}{\mathrm{K}}^{-1}$.

What phenomena can affect the measurement of gravitational acceleration using a pendulum? Estimate how many valid digits your result would have to contain to measure them. Consider also the phenomena that you usually neglect.

Charge the object by rubbing it and then measure the dependence of its self-discharge on time. Determine the electrical conductivity of air. Consider that the magnitude of the charge varies according to: $$Q=Q_0{\mathrm{e}}^{-\frac{\sigma }{ϵ}t},$$ where $Q_0$ is the initial charge, $ϵ$ is the permeability of the air, and $\sigma$ is the conductivity we are looking for.

Hint: Hang a small metallic object (e.g. a nut) on a thin, long filament. Then take a straw, rub it to charge it, and transfer some of the charges to the object. It should begin to repel away from the straw. Afterwards, you can determine the product of the charges and the conductivity from their relative distances.

1. Similarly to the series, use the Hückel method to create the Hamiltonian matrix for the cyclobutadiene molecule and verify that its eigenvalues are $\alpha +2\beta$, $\alpha$, $\alpha$, $\alpha -2\beta$. Sketch the diagram of the final energies in the resulting orbitals. And show how the electrons will occupy them. $(4b)$

   Bonus What is the main difference in the characterics of these orbitals and their occupancy compared to a benzene molecule we showed in the series? What are the consequences for the cyclobutadiene molecule? $(2b)$

2. Try going back to the beta-carotene molecule and calculate again at what wavelength it should absorb using the Hückel method. What should the value of the parameter $\beta$ be equal to in order to be consistent with the experimental result?

   Alternative If you encounter a problem with the diagonalisation of the hamiltonian, solve the problem statement with the hexa-1,3,5-triene molecule. The experimentally determined absorption value in this case is at a wavelength of $250\mathrm{nm}$.

    $(4b)$

3. What happens to a molecule (a molecule with only simple bonds is sufficient) if we use UV light to excite an electron from the $\sigma$ to the ${\sigma }^{\ast }$ orbital? $(2b)$