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## hydromechanics

### 2. Series 27. Year - 5. Plastic cup on water

A truncated cone that is the upside down (the hole is open downwards) may be held in the air by a stream of water which originates from the ground with a constant mass flowrate and an intial velocity $v_{0}$. At what height above the surface of the Earth will the cone levitate ?

Bonus: Explore the stability of the cone.

### 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.

### 1. Series 27. Year - 4. cube in a pool

Large ice cube placed at the bottom of an empty pool starts to dissolve. Assume that the process is isotropic in the sense that the cube is geometrically similar at all times. What fraction of the cube needs to dissolve before it starts to float in the water? The surface area of the pool floor is $S$, and the length of an edge of the cube before it started disolving was $a$.

Lukáš was staring at a frozen town.

### 6. Series 26. Year - 4. filling a tank

Imagine a large tank containg tea with a little opening at its bottom so that one can pour it into a glass. When open, the speed of the flow of tea from the tank is $v_{0}$. How will this speed change if, while pouring a glass of tea, someone is filling the tank by pouring water into it from its top? Assume that the diameter of the tank is $D$, the diameter of the flow of tea into the tank is $d$, and that of the flow of tea out of the tank is much smaller than $D$. The tea level is height $H$ above the lower opening, and the tank is being filled by pouring a water into it from height $h$ above the tea level. You are free to neglect all friction.

### 5. Series 26. Year - 1. boiling oceans

Estimate how much energy would be needed to evaporate all oceans (on Earth).

Karel says it's too cold for swimming.

### 5. Series 26. Year - E. evaporate!

Design an experiment to measure the dependence of the speed of evaporation on the surface area of the evaporating liquid. You should use at least five different containers to do the measurment. What other factors can influence the speed of evaporation? Note that this experiment should run for several days so plan accordingly.

Kiki was too lazy to go get the rag.

### 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.

### 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.

### 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.

### 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.