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## gravitational field

### (5 points)4. Series 27. Year - P. the true gravitational acceleration

Faleš wanted to determine the *gravitational* acceleration from an experiment in Prague(V Holešovickách 2 in the first floor/ground floor). In the experiment he was dropping a round ball from a height of a couple of meters above the Earth. Think about what kind of corrections he had to apply when analysing the data. Then think up your own experiment to determine g and discuss its accuracy.

Karel was thinking about the difference between gravitational acceleration and gravitational force

### (2 points)2. Series 27. Year - 2. Flying wood

We have a wooden sphere at a height of $h=1\;\mathrm{m}$ above the surface of the Earth which has a perimeter of $R_{Z}=6378\;\mathrm{km}$ and a weight of $M_{Z}=5.97\cdot 10^{24}\;\mathrm{kg}$. The sphere has a perimeter of $r=1\;\mathrm{cm}$ and is made of a wood which has the density of $ρ=550\;\mathrm{kg}\cdot \mathrm{m}^{-3}$. Assume that the Earth has an electric charge of $Q=5C$. What is the charge $q$ that the sphere has to have float above the surface of the Earth? How does this result depend on the height $h?$

Karel přemýšlel, co zadat jednoduchého.

### (2 points)6. Series 26. Year - 2. stupid wire

What is the minimal length of a wire so that if you hang it from a ceiling, it will break due to its own mass? The wire's density is $ρ=7900\;\mathrm{kg}\cdot \mathrm{m}^{-3}$, it has a diameter $D=1\;\mathrm{mm}$, and it breaks at $σ_{max}=400MPa$. Assume that everything takes place in a homogeneous gravitational field $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}$.

**Bonus:** If the wire's length is maximal possible so that it does not break, how much will it stretch (in percents)? Young's modulus of the wire's material is $E=200GPa$.

Karel stuck a wire into his eye

### (2 points)5. Series 26. Year - 1. boiling oceans

Estimate how much energy would be needed to evaporate all oceans (on Earth).

Karel says it's too cold for swimming.

### (4 points)4. Series 26. Year - 5. How to build a bridge

Imagine a cross section of a bridge as depicted in the picture. It consists of massless rods attached at the points $\bodA$, $\bodB$, $\bodC$, $\bodD$ and $\bodE$. Determine which rods would exhibit pressure forces and which pulling forces if a car of mass $m$ is placed on the rod $\bodBC$. You should use the picture to estimate the lengths of the rods.

**Bonus:** Assume that the linear density of the rods is $λ$ instead of zero.

Karel spying at a construction site.

### (5 points)4. Series 26. Year - P. Mrazík

In the fairy tale Mrazik, Ivan fought several bandits, stole their clubs, and threw them so high up into the sky that they did not fall back until half a year later. What is the altitude the clubs had to reach in order to stay in the air for so long? Make a first guess and then go on and improve it. Carefully analyze all the approximations you made and explain why are these estimates most likely wrong. Furthermore, explain why it makes no sense for the clubs to fall back at the same spot where Ivan threw them.

Lukáš was watching fairy tales.

### (5 points)2. Series 26. Year - P. messing with gravity

What if the gravitational constant suddenly doubled (without affecting the value of other physical constants)? What if it increased a hundred times? Discuss the impact the change would have on the life on the Earth and on the trajectories of bodies in the universe.

### (2 points)6. Series 25. Year - 2. space station

Estimate the minimal energy needed to put a space station on an orbit around the Earth. You can work with the values valid for the International Space Station which orbits the Earth at height approximately $h=350\;\mathrm{km}$ and has mass $m=450000\;\mathrm{kg}$. Explain why is this estimate minimal and why, in reality, much more energy is needed.

Astrokarel.

### (4 points)4. Series 25. Year - 5. gas leakage

What is the mass percentage of Earth's atmosphere that escapes to the outer space each year? Assume the atmosphere reaches 10 km above the ground, the pressure is everywhere the same (equal to the pressure at sea level) and it consists of ideal gas at tepmperature 300K whose molecular speeds obey the Maxwell-Boltzmann distribution. Also assume that the gravitational field is homogeneous.

Aleše napadlo při úniku.

### (5 points)1. Series 25. Year - P. Cubeworld

Imagine that the Earth is not a sphere but a cube. Would it be able sustain this shape? If so for how long and on what parameters would this time depend on? What about the life on such a planet? What gravitational force would you feel while walking around this planet?

Mysterious…