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## gravitational field

### (6 points)6. Series 29. Year - P. iApple

Think up and describe a device that can deduce its orientation relative to gravitational acceleration and convert this information to an electrical signal. Come up with as many designs as you can. (An accelerometer-like device that is in most smart phones.)

### (3 points)2. Series 29. Year - 3. fatal fall

From a spaceship on a circular orbit with height $h=2000\;\mathrm{km}$ above the surface of Earth a screwdriver is thrown with speed $v=5\;\mathrm{km}\cdot h^{-1}$ relative to the rocket towards the center of the Earth. Determine when will the screwdriver hit the surface?

Karel nemá rád šroubováky.

### (8 points)1. Series 29. Year - E. small g

Measure the local gravitational acceleration with at least two different methods. Then compare these two methods in detail.

Viktor heard the complaint of the participants that they don't want to constantly be knee deep in water.

### (5 points)6. Series 28. Year - 4. Unbearable weight

Before the edge of Discworld was reached and overcome and scientific expeditions were made to confirm the existence of the four elephants, turtle and determine its gender some primitve tribespeople thought that the force that kept them on the surface was due to a disc made from a superdense Wasneverwasium. It was truly a very primitive idea because as we know today the expedition, that confirmed the turtle's existence, infamously ended when its boat tore apart and fell or rather did not fall;… Nevertheless we would be interested what kind of surface density would such a disc have to if an object in its middle should experience, while neglecting magical forces, attracted with the same force as the gravitational force on Earth. Assume that, as the legends told, the disc is very thin and is placed ;$H=8^{4}m=4096\;\mathrm{m}$ under the surface of Discworld. The Disc should be hmomogeneous and the masses of other bodies negligible. Neglect the movement of the turtle and the elephants. If you haven't read the works of a genius for whom Death came recently then just replace Discworld with Earth. Discworld has a diameter of precisely $d=10000\;\mathrm{km}$ for our purposes.

Karel likes gravity

### (2 points)6. Series 27. Year - 1. anticore

There are two homogenous non-rotating planets the shape of perfect sphere's with outer radii $R_{Z}$. The first of which is a perfect sphere with a density of $ρ$ a on its surface the gravitational acceleration is $a_{g}$. The second is hollow to half its radius and then its full.

• If both planets would be out of the same homogenous material, on the surface of which planet shall the gravitational acceleration be greater and what shall be the ratio of the two gravitational accelerations on the two planets?
• If the gravitational acceleration on the surface of the second planet will be $a_{g}$, what does the density of the second planet have to be?

Karel created something astrological again with a hollow earth.

### (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 $h?$

### (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.