1th Round of 27th year

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Series

1. ... golden dam(2 points)

How many bricks of 24-karat gold can you fit into the Orlík dam? What would be the pressure acting on a brick placed at the deepest point? The dimensions of a brick are 10 cm, 3 cm a 1 cm.

2. ... unstoppable terminator(2 points)

How fast does the boundary between regions with and without sunlight move on the surface of the Moon? Is it possible to run away from dark when you are at the equator?

3. ... bubble in a pipeline(4 points)

A horizontal pipeline with a flowing liquid contains a small bubble of gas. How do the dimensions of this bubble change when it reaches a narrower point of the pipeline? Can you find some applications of this phenomena? What problems could it cause? Assume that the flow is laminar.

4. ... cube in a pool(4 points)

figure

Large ice cube placed at the bottom of an empty pool starts to dissolve. Assume that the process is isotropic in the sense that the cube is geometrically similar at all times. What fraction of the cube needs to dissolve before it starts to float in the water? The surface area of the pool floor is $S$, and the length of an edge of the cube before it started disolving was $a$.

5. ... a bead(5 points)

figure

A small bead of mass $m$ and charge $q$ is free to move in a horizontal tube. The tube is placed in between two spheres with charges $Q$ = $-~q$. The spheres are separated by a distance 2$a$. What is the frequency of small oscillations around the equilibrium point of the bead? You can neglect any friction in the tube. Hint   When the bead is only slightly displaced, the force acting on it changes negligibly.

P. ... speed of light(5 points)

What would be the world like if the speed of light was only $c$ = 1 000 km·h^{ − 1} while all the other fundamental constants stayed unchanged? What would be the impact on life on Earth? Would it even be possible for people to exist in such a world?

E. ... bend it but don't bend it!(8 points)

Your task is to measure the spacing of a diffraction grating using the light from three different LED-diodes. In case your interested, send us an email at experiment@fykos.cz and we will send you the LED diodes, resistor, wires, and, of course, the diffraction grating. The only thing you will need to buy is a 9 V battery.

S. ... relativity(6 points)

* Any theory of quantum gravity is useful only when we deal with very small distances where the effects of gravitation are comparable to quantum effects. Gravitation is characterized by the gravitational constant, quantum mechanics by the Planck constant, and special relativity by the speed of light. Look up numerical values of these constants, and, using standard algebraic operations, combine them to obtain a quantity with the dimensions of length. This is the length scale where both quantum mechanics and gravitation are important.

  • Prove that the special Lorentz transform (i.e. a change of the reference frame to one that is moving with speed $v$ in the $x$¹; direction)

$$x^0_\mathrm{nov}=\frac{x^0-\frac{v}{c}x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^1_\mathrm{nov}=\frac{-\frac{v}{c}x^0 x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^2_\mathrm{nov}= x^2\,,\quad x^3_\mathrm{nov}= x^3$$ ~~clear~~ leaves the spacetime interval invariant. * Set~$Δx$³~=~$Δx$² = 0 in the definition of a spacetime interval. You should get

$$(\Delta s)^2 = -\(\Delta x^0\)^2 \(\Delta x^1\)^2\,.$$

What is the region of the plane ( $Δx$^{0}$,~Δx$¹; )  where the spacetime interval ( $Δs$ ) ² is positive? Where negative? What is the curve ( $Δs$ ) ² = 0?

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