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## thermodynamics

### (4 points)3. Series 27. Year - 3. cup tubby

Take an empty cylindrical cup. Turn it upside down and push it beneath a calm water surface. How high will the column of air in the cup be depending on the submersion of the cup?

Karel got inspired by the times when he used to play in his bathtub

### (4 points)2. Series 27. Year - 3. torturing the piston

We have a container of a constant cross section, which contains an ideal gas and a piston at a height of $h$. First we compress the air quickly (practically adiabatically) by moving the piston to a height of $h⁄2$, we hold it there until thermal equilibrium with its surroundings is reached, and then we let it go. To what height will the piton rise immediately? What is the height that it will reach after a very long time? Draw a $pV$ diagram.

Karel přemýšlel nad pístem.

### (4 points)1. Series 27. Year - 3. bubble in a pipeline

A horizontal pipeline with a flowing liquid contains a small bubble of gas. How do the dimensions of this bubble change when it reaches a narrower point of the pipeline? Can you find some applications of this phenomena? What problems could it cause? Assume that the flow is laminar.

Karel was thinking about air fresheners.

### (5 points)3. Series 26. Year - 5. Gas crises

A Siberian gas pipeline with a liquefied natural gas needs to be closed. Váňa Vasilijevič decided to do this manually by closing a frictionless valve. What is the work he needed to do so? What is the force he acted on the valve with (choose an appropriate parameter to describe it)? You can imagine the valve as a board that is being inserted into the pipeline (the pipeline is perpendicular to this board). Initially, the pressure inside the pipeline was $p=2MPa$. Its cross-section is square-shaped with a side of length $a=1\;\mathrm{m}$. The board is $d=10\;\mathrm{cm}$ wide, the density of liquefied natural gas is $ρ=480\;\mathrm{kg}⁄m$, and its flow rate is $q=20m3\;\mathrm{s}$.

Ales wanted to know what is it like to live in Russia.

### (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.

### (8 points)4. Series 25. Year - E. boiling water

In this problem you are asked to measure the efficienecy of an electric kettle. You can measure the output power by measuring the temperature increase of water in the kettle per unit time. The input power should be written on the bottom of the kettle. Minimalize the error of measurement and describe the methods you used to achieve this. *Warning* The voltages and currents present are dangerous. Do not use voltmeters and ampermeters without supervision!

### (5 points)4. Series 25. Year - P. energy saving

Some apartment buildings have only one boiler that is used by all the tenants. The way the boiler works is that it keeps the water temperature constant throughout the day. In order to save money the tenants decided to turn off the boiler every night and turn it back on the next morning. Therefore every morning the boiler has to heat up the water that cooled down during the night. Estimate how much energy is saved by this method and suggest a better way to save money without making the living more uncomfortable.

Pikoš platil účet za plyn.

### (4 points)1. Series 25. Year - 3. bicycle pump

What is the temperature of the air leaving a bicycle pump when we want to inflate the tube to a pressure of 3 atm? Assume that the air entering the pump has temperature 20°C.

Lukáš Jáchym

### 6. Series 24. Year - 4. the final solution

How would the power of sunlight hitting the Earth in aphelion change if we were to accelerate the Earth in the direction of motion in such a way to extend a year by a week? Estimate the temperature of the Earth in aphelion and perihelion if its heat capacity is almost zero. For simplicity assume that the original trajectory of the Earth was circular (not the case after the velocity boost).

### 4. Series 24. Year - 4. Home alone

Terka was playing around and spilled five liters of liquid nitrogen in her room. Couple days later she bought five liters of gasoline, brought it to her room and burned it. Could this playing around result in her being sick? To be more concrete describe the change in the temperature, pressure and oxygen concentration in her room (in both cases) if it is perfectly isolated and has dimensions 3$x3x4\;\mathrm{m}$.

Mára