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## gas mechanics

### (10 points)4. Series 37. Year - S. heating and explosions

- Consider a thin-walled glass container of volume $V_1=100 \mathrm{ml}$, the neck of which is a thin and long vertical capillary with internal cross-section $S=0{,}20 \mathrm{cm^2}$, filled with water at temperature $t_1=25 \mathrm{\C }$ up to the bottom of the neck. Now submerge this container in a larger container filled with a volume $V_2=2{,}00 \mathrm{l}$ of olive oil at a temperature $t_2=80 \mathrm{\C }$. How much will the water in the capillary rise?
- In a closed container with a volume of $11{,}0 \mathrm{l}$ there is a weak solution containing sodium hydroxide with $p\mathrm {H}=12{,}5$ and a volume of $1{,}0 \mathrm{l}$. In the region above the surface, we burn $100 \mathrm{mg}$ of powdered carbon. Determine the value of the pressure in the container a few seconds after burning out, after half an hour, and after one day. Before the experiment, the vessel contained air of standard composition at standard conditions; similarly, we maintain a standard temperature around the vessel in the laboratory.
- Describe three different ways in which the temperature of stars can be determined. What are the basic physical principles they are based on, and what do we need to be careful of?

Dodo remembered highschool chemistry.

### (10 points)3. Series 37. Year - 5. air under the water

Assume a cylindrical glass of negligible mass, internal cross-sectional area $S$, and height $h$ that is turned upside down and its open rim aligned with the water level in the reservoir. We start to push slowly downwards. What work will we perform if we move the jar with the air inside so that its base $d>0$ is below the surface? **Bonus::** Let us now consider a more realistic case. How much work must be performed to completely submerge a jar of the same dimensions but mass $m$ to the bottom of a container with area $A$ and initial water level in height $H$? Assume that the jar is completely submerged when it reaches the bottom.

Jarda would not like to visit Titanic…

### (10 points)3. Series 37. Year - S. weighted participants

- According to definitions by International System of Units, convert these into base units
- pressure $1 \mathrm{psi}$,
- energy $1 \mathrm{foot-pound}$,
- force $1 \mathrm{dyn}$.

- In the diffraction experiment, table salt's grating constant (edge length of the elementary cell) was measured as $563 \mathrm{pm}$. We also know its density as $2{,}16 \mathrm{g\cdot cm^{-3}}$, and that it crystallizes in a face-centered cubic lattice. Determine the value of the atomic mass unit.
- A thin rod with a length $l$ and a linear density $\lambda $ lies on a cylinder with a radius $R$ perpendicular to its axis of symmetry. A weight with mass $m$ is placed at each end of the rod so that the rod is horizontal. We carefully increase the mass of one of the weights to $M$. What will be the angle between the rod and the horizontal direction? Assume that the rod does not slide off the cylinder.
- How would you measure the mass of:
- an astronaut on ISS,
- a loaded oil tanker,
- a small asteroid heading towards Earth?

Dodo keeps confucing weight nad mass.

### (3 points)2. Series 37. Year - 2. inflated tyre

It is said that if you want to inflate a car's tires, you should do it when they are cold. Therefore, Jarda drove to a petrol station with a compressor, bought a hot dog, and waited for the tires to cool down. Curious, he measured the tire pressure before and after his snack. It had dropped from $2{,}7 \mathrm{bar}$ to $2{,}5 \mathrm{bar}$. He wondered whether the tire pressure could be determined by the height of the car's body above the road. How much did the body of Jarda's car approach the ground due to the decrease in the tire temperature? The weight of the car is $1{,}3 \mathrm{t}$. The outer radius of the tires is $32 \mathrm{cm}$, the inner radius is $22 \mathrm{cm}$, and their width is $21 \mathrm{cm}$. Assume that the tires deform due to the car's weight only on the underside where they touch the ground.

Jarda would drain his soul (as well as his bicycle's inner tube) for FYKOS.

### (7 points)2. Series 37. Year - 4. perpetuum mobile

Lego wanted to take a break from a problem in his thesis, where a quantum heat machine behaved like a perpetual motion machine. Thus, he came up with a perpetual motion machine in classical physics using the following reasoning. Somewhere in a pit, we use heat to evaporate water. thus he invented „perpetual motion“ in classical physics. Lego's reasoning is as follows: somewhere in a pit (it doesn't even have to be very deep) we evaporate water (to do this we consume some latent heat). The water rises as vapor upwards, where we condense it again (releasing the latent heat). But the water now has a higher gravitational energy! Where did this energy come from? Or should Lego run to the patent office to go down in history as the inventor of the perpetual motion machine? Support your claims with calculations.

Lego was working on his thesis.

### (10 points)1. Series 37. Year - P. rocket

Using current technology, how much fuel would it take to carry an object of mass $m=1 \mathrm{kg}$ into low Earth orbit?

The leprechaun wanted to save on rocket fuel.

### (3 points)4. Series 36. Year - 2. frozen balloon

A balloon of mass $m\_b=2,7 \mathrm{g}$ and volume $V_0=4 \mathrm{l}$ was filled with helium of the same temperature as the surrounding air, i.e., $T_0=20 \mathrm{\C }$. Inside the balloon, the pressure is $\Delta p=2 \mathrm{kPa}$ higher than in the surrounding area. To what temperature do we need to cool the balloon and the gas in it so it stops floating? Assume that there will be atmospheric pressure in the balloon after cooling down.

Vojta trades balloons for inspiration.

### (10 points)4. Series 34. Year - P. Fykos bird on vacation

How would aviation work on other planets (with atmosphere)? Consider mainly jet aircraft. Which planetary parameters would influence the aviation positively and which negatively, compared to Earth's?

Karel visited the museum of aviation in Košice.

### (3 points)1. Series 32. Year - 1. baloons

How many balloons of volume $V=10 \mathrm{\ell }$ filled with helium of density $\rho _{\scriptscriptstyle \rm He} = 0{,}179 \mathrm{kg\cdot m^{-3}}$ are needed to lift Filip, whose mass is $m_{\scriptscriptstyle \rm F} =80 \mathrm{kg}$, and keep him afloat in air of density $\rho _{v} =1{,}205 \mathrm{kg\cdot m^{-3}}$? How many would be necessary to lift Danka, who weighs $m_{\scriptscriptstyle \rm D} =50 \mathrm{kg}$? Neglect the mass of the empty balloons.

Danka gave Filip a promo balloon to lift his mood.

### (10 points)1. Series 31. Year - P. model plane at the ISS

How would a plane behave in microgravity (in other words ignore the effects of gravity)? Describe what effects would the ailerons, the rudder, the elevators and thrust vectoring have of the engines? Which acrobatic maneuvers would be possible? (E.g. a flat spin probably wouldn't be).

Erik read discussions on the internet.