# 2th Round of 20th year

Series

### 1. ... Cenek's saw mill( points)

Cenek's saw mill is located at the junction of rivers Vydra and Kremelna in the Boheminan forest (mountains). The saw mill is named after wood retailer Cenek Bubenicek, who has built in 19th century. There is water power plant at its former place declared by state as technical monument.

Power plant uses difference of water levels above and under turbine 10 m, electrical output of power station is 96 kW. Water is distributed to the turbine using rectangular open-top canal. Its width is 1 m, and water is  1,5 m deep. The estimated water speed at the middle of surface is  1 m·s^{− 1}. What is the estimated efficiency of water power station?

### 2. ... crushing impact( points)

Find a relation between the speed of meteorite (just before impact) and radius of created crater.

### 3. ... illumination of table( points)

Find such placement of fluorescence tubes at the ceiling of study room, which is  3 m above the top surface of desk, that intensity of illumination will not vary more than 0,1 %.

### 4. ... how far is the Sun?( points)

Let's return back to 18th century, when the value of Newton's gravitation constant was not known yet. And the distance between Sun and the Earth was unknown as well. During this times Edmond Halley (astronomer who discovered that the comet in the sky in 1682 is the same as the one in 1456, 1531 and 1607) suggested to find distance Earth-Sun by measuring time of transit of Venus over the Sun circle. Unfortunately the Venus cross the Sun irregularly, in pair each eight years and then there is century-long break. During Halley life there was no crossing.

However, the idea was not forgotten and with the next crossing was successfully resuscitated and in 1761 a next scientific experiment was prepared. Scientists went to many parts of world, including Siberia, China, South Africa and Indonesia. It was the first international scientific experiment.

After all people have returned, it was concluded that no conclusion can be made. Ironically it was due to too many contradicting observation. Successful was captain James Cook in 1769 on one summit in Tahiti. After his return the astronomers were able to calculate the average distance between the Earth and the Sun as approximately 150 millions kilometers.

It is up to you to calculate distance Earth-Sun using only data known at the time: Earth diameter and orbit time of Venus and Earth.

### P. ... shaking the tea( points)

Explain why the tea in the box after it is shaken is separated such that larger pieces of tea leaves are on the surface. The solution can be enhanced using your observation.

### E. ... waves on the water( points)

Using dimension (unit) analysis find the relation for the speed of waves on the water surface. Verify the theoretical relation and find unknown constants from measuring the speed of waves dependent on wavelength. Do not forget that there are two types of waves – one caused by gravitational field and second by surface tension.

### S. ... particle with 1/2 spin( points)

A particle with spin 1/2 (e.g. electron) can be in two states of projection of spin to the z-axis. Either the spin is pointing up |↑〉 or down |↓〉. These two states create basis for two-dimensional Hilbert space describing particle of spin 1/2.

• Write the operator of identity in this space and language of vectors |↑〉 and |↓〉.
• Find Eigen vector and Eigen number of matrices $S$_{1}, $S$_{2}

and $S$_{3}.

• Lets have operators $S$_{$+$} and $S$_{$-$} in the form

$S$_{$+$} = |↑〉〈↓|, $S$_{$-$} = |↓〉〈↑|. Find its representation in basis of vector |↑〉 and |↓〉 and find how they operate on general vector |$ψ$〉 = $a$|↑〉 + $b$|↓〉. How do look Eigen vectors and what are the Eigen numbers?

• Lets define vectors
 ⊗〉 = ( ↑〉 + ↓〉 ) ⁄ √2 ⊕〉 = ( ↑〉 −

Show that these vectors form basis in our Hilbert space and find relation between coefficients $a$, $b$ in decomposition |$ψ$〉 into original basis and coefficients $c$, $d$ into the new basis |$ψ$〉 = $c$|⊗〉 + $d$|⊕〉.

• Write two spin operators $S$_{1}, $S$_{2} a $S$_{3}

in basis of vectors |⊗〉 a |⊕〉. Find its Eigen vectors and Eigen numbers.

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