Problem Statement of Series 3, Year 36

About the Competition Rules How to Write Solutions Results

Year 36

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Text of Serial Number 3 Brochure with solutions (cs)

1... creative problem-solving

3 points

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 $0.4\,\mathrm{l\cdot s^{-1}}$? Danka can now set the end of the hose so that water sprays at an arbitrary angle to the horizontal plane.

We will publish the solution to this problem soon.
~ Danka is annoyed by the noise below the windows at night.
Mechanics of a point mass Hydromechanics

2... heating in the cottage

3 points

Danka arrived at her cottage in the middle of winter with only $T_1 = 12\,\mathrm{^\circ\mskip-2mu\mathup{C}}$ indoors. So she lit a fire in the fireplace by using wood of heating value $Q_0 = 14.23\,\mathrm{MJ\cdot kg^{-1}}$. How much wood does she need to burn to heat the air inside to $T_2 = 20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$? The cottage is in the shape of a rectangular cuboid with dimensions $a = 6\,\mathrm{m}$, $b = 8\,\mathrm{m}$ and $c = 3\,\mathrm{m}$. A roof is in the shape of an irregular recumbent triangular prism with a height of $v = 1.5\,\mathrm{m}$, the upper edge of which is the axis of the cottage layout. The air occupies $87\,\mathrm{\%}$ of the volume of the cottage and its specific heat capacity is $c_{\mathrm{a}} = 1~007\,\mathrm{J\cdot kg^{-1}\cdot K^{-1}}$. Does the result match the expectation? Discuss the simplicity of the model used.

We will publish the solution to this problem soon.
~ Danka gets cold at the cottage.
Thermodynamics

3... bobsled

5 points

Matěj and David are sliding on bobsleds down the hill. The hill with a slope of $\alpha=29\,\mathrm{^\circ}$ turns into the horizontal ground at the bottom of it. Both of them started from rest from the same height. Matěj's bobsled always travels the same distance $l$ on an inclined plane as well as in a horizontal part. Since the bobsled digs deeper into the snow at higher loads, assume the coefficient of friction to be proportional to the normal force as $f(F)=kF$, where $k$ is a positive constant. Determine how many times Matěj will travel farther from the bottom of the hill than David if David's mass (including the bobsled) is $12\,\mathrm{\%}$ greater than Matěj's. Also, assume that bobsledders don't lose any energy at the bottom of the hill.

We will publish the solution to this problem soon.
~ Matej likes to talk about bobsled.
Mechanics of a point mass

4... escape to Tau Ceti

7 points

Since our Sun will explode one day, it will be necessary to organize the construction of an evacuation spacecraft, in which at least $0.000~001\,\mathrm{\%}$ 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 $v = 0.75 c$ 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.

We will publish the solution to this problem soon.
~ Karel wanted to escape in time, but it didn't work.
Relativity

5... guitar

10 points

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 $10\,\mathrm{^\circ\mskip-2mu\mathup{C}}$? Will the guitar still sound in tune? The distance between the string attachment points is $d = 65\,\mathrm{cm}$. The strings have a density $\rho = 8~900\,\mathrm{kg\cdot m^{-3}}$, a Young's modulus of elasticity $E = 210\,\mathrm{GPa}$ and a thermal expansion coefficient $\alpha = 17\cdot 10^{-6}\,\mathrm{K^{-1}}$.

We will publish the solution to this problem soon.
~ Honza's guitar is out of tune again.
Mechanics of a rigid body Oscillations Other

P... absurd pendulum

9 points

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.

We will publish the solution to this problem soon.
~ Kačka was wondering what she could write in the discussion.
Gravitational field Other

E... game of discharges

13 points

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: \begin{equation*} Q = Q_0 \mathrm{e}^{-\frac{\sigma}{\epsilon}t}\,, \end{equation*} where $Q_0$ is the initial charge, $\epsilon$ 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.

Instructions for Experimental Tasks
We will publish the solution to this problem soon.
~ Jarda tried to measure the charge for so long that he changed the entire problem to measure the conductivity.
Electric field

S... quantum of orbital

10 points

  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. $(4~b)$

    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? $(2~b)$

  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 $250\,\mathrm{nm}$.

     $(4~b)$

  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? $(2~b)$

Text of Serial Number 3
We will publish the solution to this problem soon.
~ Mikuláš gives presents again, this time at the right time of the year, almost.
Molecular physics Wave optics
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