Series 3, Year 27

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(2 points)1. eclipse

A planet is orbiting around a star on a circular orbit and a moon is orbiting around the planet on a circular orbit in the plane of the planet's orbit. We know that, during the eclipse of the sun the angular size of the moon is the same as the angular size of the sun if observed from the surface of the planet (the moon perfectly covers the sun). Furthermore we know that the planet perfectly covers the moon during the moon's eclipse. Determine the ratio of the radius of the planet $R$ and the moon $r$, if the distance of the planet from the star is very large compared to the distance of the moon from the planet $L$ and this is in turn larger by several orders of magnitude than the dimensions $R$, $r$.

Mirek was looking through the archive.

(2 points)2. The Mediterenean sea

How quickly on average does water flow through the Gibralatar Strait if it allows the changing of high and low tide in the Mediteranean Sea? Find the required data on the internet and don't forget to cite them properly!

Lukáš was surprised by the height of the tide

(4 points)3. cup tubby

Take an empty cylindrical cup. Turn it upside down and push it beneath a calm water surface. How high will the column of air in the cup be depending on the submersion of the cup?

Karel got inspired by the times when he used to play in his bathtub

(4 points)4. I have already forgotten more than you ever knew

A hot air balloon with its basket weighs $M$. The basket of the baloon is submerged into a water reservoir and water flows into it. Now we shall raise the temperature a bit and by that we raise the buoyant force acting on the balloon $Mg+F$. The basket has the shape of a rectangular cuboid with a square base which has a side of size $a$ and is submerged into a depth $H$. The openings in the basket make up $p≪1$ of the whole area of the basket about which we assume that it is empty (with the exception of water). Let us neglect the viscosity of water and the volume of the basket itself. How quickly shall it rise depending on the depth of submersion?

Bonus: When shall it emerge?

Tip The expected speed of water flowing from the basket above the water surface is 2/3 of the maximum speed of water flowing out.

Was thought up by Lukáš during watching the movie Vratné lahve.

(5 points)5. mig-mig!

A poor hungry coyote wants to catch the devilish Road Runner and he prepared the following trap to snare him: onto a sturdy rope he fastened a 500 ton anvil and he shall throw it over a branch so that it would hang over the road and he will wait. How many times does he need to make the rope go around the branch so that he can hold it there just using his own weight? Assume that the weight of the rope is negligible compared to the weight of the coyote.

Mirek was always a fan of Wile E. Coyote.

(5 points)P. solar power

Can a plane fly using a solar power?

Dominika with Honza and Michal wanted to fly away to warmer countries.

(8 points)E. viscous

Each liquid has its specific viscosity. Try to make an Ostwald viscometer (capillary viscometer) and measure the relative viscosity of a few (at least three) liquids compared to water. Compare your results with what you find online.

Kiki got frustrated by the fact that everything flows differently during weighing things in a apothecary.

Instructions for the experimental problem

(6 points)S. Aplicational


  • In the text of the seriesy we used the approximative relation √( 1 + $h)$, where $his$ a small value. Determine the precision of such an approximation. How much can $h$ differ from zero so that the approximated value and the precise one shall differ only by 10%? We can make a similar approximation for any „normal“ (read occuring in nature) function using Taylor's series expansion. Try to find the Tylor's series of cos$h$ and sin$h$ on the internet and neglect factors with a higher order than $h$ and find the approximate border value where it differs by approximately 0.1.
  • Considering a wave equation for a classical string from the serial and let the string be fastened on one end in the point [ $x;y]=[0;0]$ a na druhém konci v bodě [ $x;y]=[l;0]$. For what values of $ω,α,aabis$ the expression

$$y(x,t)=\sin ({\alpha} x)\left [a\sin {({\omega} t)} b\cos {({\omega} t)}\right ]$$

a solution of the wave equation? Tip Subsitute into the equation for motion and use the boundary conditions.

  • In the previous part of the series we were comparing different values of action for different trajectories of different particles. Now calculate the value of Nambu-Gota's action for a closed string which from time 0 to time $t$ stands still un the plane ( $x¹,x)$ and has the shape of a circle with radius $R$. Thus we have

$$X({\tau} , {\sigma} )=(c{\tau} , R\cos {{\sigma} }, R\sin {{\sigma} },0)$$

for $τ∈\langle0,2π\rangle$. Furthermore sketch the worldsheet of this string (forget about the last zero component) and how the line of a constant $τ$ and $σ$ look.

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