5. Series 34. Year

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Post deadline: 29th March 2021
Upload deadline: 30th March 2021 11:59:59 PM (local time in Czech Republic)

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.

2. retarded Jupiter

The sidereal period of Jupiter is approximately $11,9 \mathrm{years}$, the speed of light is $3 \cdot 10^{8} \mathrm{m\cdot s^{-1}}$. Assume the relative distance between the Earth and the Sun to be $150 \cdot 10^{9} \mathrm{m}$. Using these values, calculate how long will the light travel from Jupiter to Earth if Jupiter is located at a point to which it will get from opposition in one quarter of the sidereal period.

Vašek remembered the observations of Ole R\o{}mer.

3. involuntary breatharianism

Lukáš wanted to cook himself a dinner. He put a pot onto a stove, but forgot to fill it with water (or anything else). The teperature of the pot and the air inside stabilized at $100 \mathrm{\C}$ (do not ask, how he managed that without water). Lukáš realized his mistake and removed the pot from the stove. When the pot had cooled down to the room temperature, however, he was unable to remove its lid with the area $S$ and mass $m$. Calculate the force with which the lid resisted being removed if Lukáš put the lit on the pot

  1. just before removing it from the stove and,
  1. before the start of dinner preparation.

Assume the air to be an ideal gas.

Lukáš and his culinary art.

4. period of large oscillations

Assume two half-planes with the angle $2\phi < \pi$ between them. We place them so that the line at their intersection is horizontal and their plane of symmetry is vertical, so they form a kind of valley. Then we take a mass point and throw is with the velocity $v$ from the height $h$ (above the intersection line) in the horizontal direction so that it makes a periodic motion as shown in the picture. What is the magnitude of the velocity that we have to throw it with? Assume the bouncing to be perfectly elastic.

Legolas was bored by the periods of small oscillations.

5. rheonomous catapult

Let us have a thin rectangular panel that rotates around its horizontally oriented edge at a constant angular velocity. At the moment when the panel is in a horizontal position during rotating upwards, we place a small block on it so that its velocity with respect to the panel is zero. How will the block move on the panel if the friction between them is zero? Where do we have to place the block so that it flies away from the panel exactly after a quarter of its turn? Discuss all the necessary conditions that must be met to achieve this. Bonus: What power does the panel transfer on the block and what total work does it do on it?

Vašek was tired of problems with scleronomous bond, so he came up with rheonomous bond.

P. you do not want that

You have probably heard that the shell of an ordinary chicken egg can withstand a significant pressure. Explain how is this possible, knowing that the egg can be cracked quite easily. What is the direction in which the eggshell can withstand the largest pressure? Why and how it cracks, when we overload it? Describe different mechanisms and determine which one is the most likely. Do not forget that we are considering real, not ideal eggs. If possible, try to support your claims with calculations.

Occurred to Jáchym while watching a Czech film classic.

E. do they deceive us?

Measure the capacity of an arbitrary battery (e.g. AA battery) and compare it with the declared value.

S. resonance and damped oscillations

  1. On a tense rope, waves can exist with the deflection $\f{u}{x, t}$ from the equilibrium, that satisfy the wave equation with damping

\[\begin{equation*} \ppder{u}{t} = v^2 \ppder{u}{x} + \Gamma \pder{u}{x}  , \end {equation*}\]

where $v$ is the phase velocity and $\Gamma$ is the coefficient of damping. Do a fourier substitution and find the dispersion relation. Solve it for the wavenumber $k$. What condition, in terms of frequency $\omega$, phase velocity $v$ and the coefficient $\Gamma$, must the waves meet in order to create nodes on the rope (i.e. points in which the rope stays in equilibrium position, but around which the rope is moving)?

  1. Consider a jump rope attached firmly at one end to a fixed wall. At the distance $L$ from the wall, we start moving the rope up and down to create waves. The jump rope has a linear density $\lambda$ and the constant tension $T$ in the direction away from the wall. The deflection then satisfies the equation

\[\begin{equation*} \ppder{u}{t} = \frac{T}{\lambda} \ppder{u}{x}  . \end {equation*}\]

For the deflection of the end of the rope that is moving satisfies $\f{u_0}{t} = A \f{\cos}{\omega_0 t}$. Assume the solution can be written in the form of two planar waves moving in the opposite direction to each other. Find the solution using only the parameters given in this problem statement, that is $T$, $\lambda$, $L$, $A$ and $\omega_0$. For certain frequencies, the solution has a diverging amplitude (i.e. growing beyond any limits). Find their values and the respective wavelenghts.

Štěpán was playing with a jump rope.

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