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## hydromechanics

### (4 points)5. Series 26. Year - P. Prague is flooded!

In 2002 Prague experienced serious floodings. Try to estimate the amount of water that can fit into the Prague subway system. All the important parameters of the subway system like the train sizes, number of stations, length of the tunnels etc. can be found online.

Karel was drowning.

### (5 points)2. Series 26. Year - 5. the U tube

Imagine a U-tube filled with mercury, and a bubble of height $h_{0}$ that floats inside (see the attached picture). Describe what would happen if we changed the surrounding atmosphere in the following ways. Assume that the density of mercury is independent of temperature. The same is valid for the glass the tube is made of. Also assume that the surrounding air behaves as an ideal gas. The initial state of the atmosphere is described by temperature $T_{0}=300K$, and pressure $p_{a}=10\cdot 10^{5}Pa$. Furthermore, assume that the system is in a thermodynamic equilibrium at all times, and that the bubble has a cylindrical shape.

- Both ends of the tube are open, and the temperature doubles.
- Both ends of the tube are closed, and the temperature doubles.
- Only one of the ends of the tube is closed, and the temperature doubles. For each of these cases, determine the new size of the bubble, and the height difference between the mercury columns in the two branches.

**Bonus:** Repeat the calculation assuming that the volume of mercury grows linearly with temperature.

### (4 points)6. Series 25. Year - 4. intelligent polar bears

A sphere of ice is floating in the Arctic Ocean. Calculate the fraction of its surface that is above the sea level. The density of ice is 917 kg ⁄ m and the density of salty water is 1025 kg ⁄ m.

Dominika chatted in a zoo.

### (2 points)5. Series 25. Year - 1. the flu pill

Some pills against flu dissolve in water making it to fizz. At first the pill is on the bottom of the glass but after a while it rises to the surface. Why?

Lukáš wanted to avoid the flu.

### (2 points)3. Series 25. Year - 2. river ride

Lock is placed in a dam on a river in order for ships to be able to pass through. Assume the river has a flow rate of $Q=200m⁄s$ and that the lock connects places with height difference $H=4\;\mathrm{m}$ and has dimensions $s=100\;\mathrm{m}$, $d=20\;\mathrm{m}$. How many ships per day can this lock transport from the lower place to the higher place if the maximum flow rate in and out of the lock is $Q_{Z}=250m⁄s?$

Napadlo Michala při čtení tajného časopisu.

### 5. Series 24. Year - 1. warm up

*blood sedimentation*

Given a test tube full of human blood, how long does it take for the red cells to settle at the bottom (the so called blood sedimentation)? The usual method for such a measurement is to let the blood cells settle for an hour and than to measure the height of the cells that are already at the bottom (usually about 10 mm). We are interested in an approximate calculation. You may need to use the Stoke's relation $F=6$ π η r v$ and the value of dynamical viscosity of blood plasma $η$ = 2 Ns/m²$$.

*different eyes*

Aleš was sitting in a tram and the Sun was about 60° to the left. Because he was staring at a hot blond in front of him one of his eyes was in the shadow of his nose. When the blond noticed he was staring at her he turned his eyes to the right and he found out that he saw different color shades in each of his eyes. Describe the difference between the shades he observed with his left eye compared to his right eye. Why did this happen?

archive, Aleš

### 4. Series 24. Year - 3. Old clock

Design a shape for a sandglass so that the dependance of the height of sand on time is linear. If you do not neglect friction etc. you can gain some extra points.

from Nečada couple

### 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 - 1. Warm-Up

*Dr. Nec.*

There are two ways to measure the amount of wood in a pile of trees. Either as the volume of pure wood in the pile or as the volume of wood together with the empty spaces in the pile. Find the conversion factor between these two units assuming the trees are cylinders of radius $r$ that are layed one on top of the others.

*Bubbles.*

A spherical cap of radius $r$ is made by blowing air into a circular surface of soap water. Estimate the velocity of air molecules hitting the surface?

Jakub

### 3. Series 24. Year - P. Water, water and water

There are many interesting properties of water that other liquids do not possess. Some of them are listed at http://www.btinternet.com/ martin.chaplin/anmlies.html. Think of some consequences of these anomalies for the life on Earth, humans and technology.

Mára was listening to Meteor.