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## gravitational field

### 3. Series 35. Year - 1. Where my center of gravity is?

We can find an unofficial interpretation that the red, blue and white colors on the Czech flag symbolize blood, sky (i.e. air) and purity. Find the position of the center of mass of the flag interpreted in this way, assuming that purity is massless. The aspect ratio is $3:2$ and the point where all three parts meet is located exactly in the middle. Look up the blood and air densities.

**Bonus:** Try to calculate the position of the center of mass of the Slovak flag as accurately as possible. You can use different approximations.

Matěj likes to have fun with flags.

### 3. Series 35. Year - P. artificial gravitation

How could artificial gravity be implemented on a spaceship? What would be the advantages and disadvantages depending on the different characteristics of the spacecraft? Is it realistic to have gravity in different directions on different floors of the spaceship or for it to change rapidly, as we can sometimes see in sci-fi movies when „artificial gravity fails“?

Karel was day-dreaming while watching sci-fi.

### 1. Series 35. Year - 5. mechanically (un)stable capacitor

Assume a charged parallel-plate capacitor in a horizontal position. One of its plates is fixed and the other levitates directly below it in an equilibrium position. The lower plate is not mechanically fixed in its place. What is the capacitance of the capacitor depending on the voltage applied? Is the capacitor mechanically stable?

Vašek wanted to grill you on a capacitor.

### 5. Series 34. Year - 1. the charge of the Earth

Find the total electric charge, that the Earth would need to let all electrons close to its surface fly away. How would this charge differ if it had to deflect protons?

Karel likes planetary problems.

### 1. Series 34. Year - S. oscillating

Let us begin this year's serial with analysis of several mechanical oscillators. We will focus on the frequency of their simple harmonic motion. We will also revise what does an oscillator look like in the phase space.

- Assume that we have a hollow cone of negligible mass with a stone of mass $M$ located in its vertex. We will plunge it into water (of density $\rho $) so that the vertex points downwards and the cone will float on the water surface. Find the waterline depth $h$, measured from the vertex to the water surface, if the total height of the cone is $H$ and its radius is $R$. Find the angular frequency of small vertical oscillation of the cone.
- Let us imagine a weight of mass $m$ attached to a spring of negligible mass, spring constant $k$ and free length $L$. If we attach the spring by its second end, we will get an oscillator. Find the angular frequency of its simple harmonic motion, assuming that the length of the spring does not change during the motion. Subsequently, find a small difference in angular frequency $\Delta \omega $ between this oscillator and the one in which the spring is substituted by a stiff rod of the same length. Assume $k L \gg m g$.
- A sugar cube with mass $m$ is located in a landscape consisting of periodically repeating parabolas of height $H$ and width $L$. Describe its potential energy as a function of horizontal coordinate and outline possible trajectories of its motion in phase space, depending on the velocity $v_0$ of the cube on the top of the parabola. Mark all important distances. Use horizontal coordinate as displacement and appropriate units of horizontal momentum. Neglect kinetic energy of cube motion in the vertical direction and assume it remains in contact with the terrain.

Štěpán found a few basic oscillators.

### 6. Series 33. Year - 1. gravitational accelerator

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

Kačka saw a vertical accelerator.

### 6. Series 33. Year - P. 4D universe

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

Matěj likes higher dimensions.

### 6. Series 33. Year - S.

*We are sorry. This type of task is not translated to English.*

### 3. Series 33. Year - 5. probability density of water

Imagine a container from which continually and horizontally flows out water stream with constant cross-section area. Velocity of the stream randomly fluctuate with uniform distribution from $v_1$ to $v_2$. Water from the container continually freely falls onto a horizontal floor below. Figure out arbitrary area of the floor to which falls exactly $90 \mathrm{\%}$ of water.

Another from a list of problems, which crossed Jachym's mind while being on a toilet.

### 5. Series 32. Year - S. heavenly-mechanic

- Consider a cosmic body with the mass of five Suns surrounded by a spherically symmetrical homogenous gas cloud with the mass of two Suns and radius $1 \mathrm{ly}$. The cloud starts to collapse into the central cosmic body. Neglect the mutual interactions of particles in the cloud (excluding gravity). Find how long it will take for the whole cloud to collapse into the central body. Do not solve this problem numerically.
- Show that the Binet equation solves following the differential equation, which describes the motion of a mass point of mass $m$ in a spherically symmetrical central-force field. \[\begin{equation*} \dot {r}^2 = \frac {2}{m} \(E - V(r) - \frac {l^2}{2mr^2}\) \end {equation*}\] Where $r$ is the length of the radius vector, $E$ is the total energy, $l$ is angular momentum, and $V(r)$ is the potential energy of the mass point.
- Set up the Lagrangian for the Sun-Earth-Moon system. Assume the Sun to be motionless. The Earth and the Moon move under the influence of both the Sun and each other. While setting up the Lagrangian, think about whether you are using an appropriate number of generalized coordinates.