# Series 1, Year 27

* Upload deadline: -, CET*

### (2 points)1. golden dam

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.

Karel wants to be rich.

### (2 points)2. unstoppable terminator

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?

Karel was watching Futurama

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

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.

Karel was thinking about air fresheners.

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

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 $a$.

Lukáš was staring at a frozen town.

### (5 points)5. a bead

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.

Radomír was rolling in a pipe.

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

What would be the world like if the speed of light was only $c=1000\;\mathrm{km}\cdot h^{-1}$ 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?

Karel came up with an unsolvable problem.

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

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.

Karel spent all of our budget.

### (6 points)S. relativity

- 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 $x¹;$ 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$$ leaves the spacetime interval invariant. * Set $Δx=Δx=0$ 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 ( $Δx^{0},Δx¹;)$ where the spacetime interval ( $Δs)$ is positive? Where negative? What is the curve ( $Δs)=0?$