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## thermodynamics

### (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

### 3. Series 24. Year - 3. Aleš, the Drug Addict

Aleš stores toluen in a cylindrical bottle. When he left the room 90% of the bottle was occupied by toluen. When he returned after the weekend he noticed that the level of toluen was lower. He obviously accused his roommate from stealing. Then he realized that over the weekend the temperature in the room had risen by 20$\celsius$.
**Help:** Aleš solve this mystery. Was his roommate really stealing or could the temperature rise be responsible? You can use data from http://en.wikipedia.org/wiki/Toluene (data page).

Mára was filling a bottle with toluene.

### 2. Series 24. Year - 3. Percolator

While enjoying his daily coffee, Lukáš decided to tune up his percolator a bit. He placed a bent tube with a short wire wrapped around it to the bottom of the main vessel (see the picture). The wire was placed height $d$ above the bottom and the vessel's water level was in a height $h$. Parameters of the tube were chosen in such a way that the water vapor created by boiling water close to the wire pushed the water above it. What is the power we need to provide to the wire in order to see water coming from the tube in a height $l?$

eee

### 1. Series 24. Year - 4. rocking horse

Massless rod of length $L$ is attached to an arch of top angle 2φ and radius $R$. A mass $m$ is glued to the rod's end (see picture). Assume that the body's motion is two-dimensional.

- Find the condition necessary for the system to be stable.
- What is the period of oscillations of the body about its equilibrium point?