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## gravitational field

### 1. Series 21. Year - S. gravitation

Download program Planeta.pas, ThreeStars-TriHvezdy.pas. As an excerscice try to understand the programs, modify them and change them.

• First task is to add into at least two of programs something new. For example change origin conditions, add another planet or star. Also try to change gravitational law – calculate with force $F=A⁄R+B⁄R$, where $A$ and $B$ are constants etc.
• Assume two same mass stars, which orbit around each other on circle. In axis of this circle the third star is approaching, at the beginning with the same speed as the two stars are orbiting. It has also the same mass. Create simulation what will happen.

As solution of both problems please send us pictures with extensive commentary. State at least short explanation what is at the images. Also mention how have you proceed during calculations and which computation system have you used.

### 3. Series 20. Year - 2. landing on Titan

The space probe Huygens (named after the discoverer of Titan) has landed on Titan on Friday 14th of January 2005. The mother ship Cassini was travelling 7 years to reach Saturn. It is the most distant landing of man-made probe in history.

The landing module of nett weight (without fuel) $m$, equipped with reactive engine, levitated at fixed position above the surface of the moon (gravitational acceleration is $g)$. The module had available fuel of mass $M$ and energy $E_{0}$ which is used to accelerate the fuel (the speed and amount of fuel which is ejected from the engine can be regulated without any limitations). What is the maximum time the module can stay in constant height? Suggest how the speed and amount of fuel should be programmed to reach this maximum time.

Úlohu vymyslel Marek Pechal.

### 1. Series 20. Year - 1. mysterious matter

<h3>Deep space, year 2224</h3>

Federation got a message: in sector 0056 is happening some collection of mysterious matter with fascinating properties. Matter is black, as it does not interact with electromagnetic radiation, its particles interacts only with gravity. Federation had sent a small two-crew space ship to explore. During a trip the space hit asteroid and was deviated out of its original direction.

The crew managed to land on small unknown planet of class „M“. During the free fall the space-ship shield was heated up due to the contact with the planet's atmosphere. From the speed of free fall the gravitation acceleration on the planet was established as 0.5 $g$.

Planet is covered by very strange trees. Observation showed that they are very similar to foliage-trees known from the Earth and they reach maximum possible height. However the branches are covered by leafs, also strobiles grows on them.

During one of exploring trip on the surface of this new planet humanoid creatures were discovered. They lived in heights of trees, which can be used to send SOS signal. The humanoids used some sort of color stones to send color-coded signals. The stones are transparent and allow to pass some colors. If the light from different stones is combined they give different (other) colors.

After sending SOS-signal from the top of trees the space ship USS Odyssey came to collect stranded crew. They then sent all the results from this adventure trip back to headquarters on the Earth.

A cloud of mysterious intergalactic matter is isotropic, homogeneous and spherical shape formed by gas. At the beginning the whole cloud is completely in rest, without any movement of matter, with total mass of $M$. Find out how the local speed of collapse will change during gravitational collapse. Comment on the speed of collapse at the moment just before collapsing into a single point.

Úlohu vymyslel Pavel Brom.

### 1. Series 19. Year - 1. can Saturn float?

Do you believe that average density of Saturn is lower than density of water?

You can look at Saturn in a telescope. Except the rings you will see several moon (if they are not behind Saturn in the moment – in such case you have to wait maximum 6 hours, which is the time of transit over the planet circle). You will find, that Titan orbits around Saturn once in 16 days. Can you calculate average density of Saturn from observation of Titan. If not, justify why and if yes, calculate it. You will notice one curiosity!

Při pozorování Saturnu vymyslel Pavel Brom.

### 1. Series 19. Year - P. tide on Bali

After the international physics olympiad on Bali all olympionics went relax to the beach on the south side of the island in Indonesia. They saw coral reef disappearing in the tidal wave and realized that, after full moon night and summer day, high tide happened only once (during 24-hour period). Aboriginals confirmed this observation, but could not explain it. Can you do it?

Honza na MFO na Bali.

### 6. Series 18. Year - 2. how to make a black hole

When the star is compressed (or any other body) into a sphere of radius $r_{g}$, the black hole is created. So called Schwarzschild's radius $r_{g}$ can be interpreted in classical physics as a radius of a body of mass $M$, where the exit velocity is equal to the speed of light $c)$.

Knowing the mass of the star $M$ calculate Schwarzschild radius $r_{g}$ and critical density of the star $ρ$, at which it collapses into black hole. Solve for arbitrary values and then for the masses of Earth, Sun and the galaxy nucleus 100 billions heavier than Sun.

Jarda

### 5. Series 18. Year - 4. misbehaving gravity

During the long term observations of Jupiter's moon Io it was noticed that the the orbit time of the Io around Jupiter (e.g. time between two consecutive 'disappearances' of the moon behind Jupiter) is oscillating regularly between 42 h 28 min 21 s and 42 h 28 min 51 s (with the measurement error of 2 s).

Try to explain qualitatively and quantitatively observed changes. By quantity we mean to determine the magnitude of the source of this oscillations from the measurements. And of course including the estimation of the experimental error.

Pavel Brom

### 2. Series 18. Year - E. a mass is not a mass

Check experimentally the equivalence of inertial mass (the mass in second Newton's law) and gravitational mass (the one in Newton's gravitational law).

Vymyslel Jarda Trnka na přednášce z relativity.