Assignment of Series 6 of Year 33

About the Competition Rules How to Write Solutions Results

Year 33

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Brochure with solutions (cs)

1... gravitational accelerator

3 points

What energy (in electronvolts) will a proton gain during a fall from an infinite distance to the surface of Earth?

We will publish the solution to this problem soon.
~ Kačka saw a vertical accelerator.
Gravitational field

2... under pressure

3 points

The water level in a bathtub is at a height $15.0\,\mathrm{cm}$. The plug is a conical frustum which perfectly fits into the hole at the base of the bathtub. The radii of its bases are $16.0\,\mathrm{mm}$ and $15.0\,\mathrm{mm}$ and the mass of the plug is $11.0\,\mathrm{g}$. What force does the bottom of the bath exert on the plug? Assume that the drain pipe below contains air at atmospheric pressure.

We will publish the solution to this problem soon.
~ Jindra felt under pressure to think up simple problems.
Hydromechanics

3... hung

5 points

What weight can be hung from one end of a coat hanger without turning it over? The hanger consists of a hook made from very light wire, attached to the centre of a straight wooden rod with length $l = 30\,\mathrm{cm}$ and weight $m=200\,\mathrm{g}$. The hook has the shape of a circular arc with a radius $r=2.5\,\mathrm{cm}$ and angular spread $\theta=240^\circ$. The distance between the centres of the arc and the rod is $h=5\,\mathrm{cm}$. Neglect any friction.

We will publish the solution to this problem soon.
~ Dodo is seeking scarce goods.
Mechanics of a point mass Mechanics of a rigid body

4... frightened hair

7 points

Out of joy over the end of an exam period, Danka's hair count began to increase at a constant rate. Later, she noticed that she lost one hair, which scared her. The more hair she lost, the more she felt stressed, which increased her hair loss rate. More precisely, the rate of hair loss is proportional to the amount of already lost hair. The rate of growth of new hair remains the same. Again, we are interested in finding out when all her hair falls out.

We will publish the solution to this problem soon.
~ Jáchym has desired to calculate this for a long time.
Nuclear physics

5... golden nectar

10 points

The magic field of Discworld is so strong that the speed of light no longer has its common meaning. This applies only close to the surface, where the refractive index of the magic field has magnitude $n_0 = 2.00\cdot 10^{6}$. The refractive index decreases with height $h$ as $n(h) = n_0\mathrm{e}^{-kh}$, where $k = 1.00\cdot 10^{-7}\,\mathrm{m^{-1}}$. Calculate the optimal angle (with respect to the vertical direction) under which a light signal should be emitted from one end of the Discworld to reach the opposite end in the shortest possible time. The diameter of the Discworld is $d = 15~000\,\mathrm{km}$ and the speed of light in vacuum is $c = 3.00\cdot 10^{8}\,\mathrm{m\cdot s^{-1}}$.

We will publish the solution to this problem soon.
~ Mirek was waiting for the light from a traffic light to reach him.
Geometrical optics

P... 4D universe

10 points

As you have probably heard, planets and any other bodies in a central gravitational field move along conic sections (in the case of the Solar system, ellipses with small eccentricity). Find out how trajectories would look in a universe where the gravitational force was proportional to the multiplicative inverse of distance raised to the third power (instead of the second power).

We will publish the solution to this problem soon.
~ Matěj likes higher dimensions.
Gravitational field

E... viscosity

12 points

Measure the viscosity (in $Pa.s$) of two different oils using Stokes' method.

Instructions for Experimental Tasks
We will publish the solution to this problem soon.
~ Jáchym stole Jirka's idea to steal this problem from labs.
Hydromechanics

S...

10 points

We will publish the solution to this problem soon.
Other Thermodynamics Gravitational field Astrophysics
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