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## hydromechanics

### (2 points)3. Series 29. Year - 1. a crazy fish

In an aquarium of a spherical shape with the radius of $r=10\;\mathrm{cm}$ which is completely filled with water, swim two identical fish in opposite directions. The fish has a cross-sectional area of $S=5\;\mathrm{cm}$, Newton's drag coefficient $C=0.2$ and it swims with a speed of $v=5\;\mathrm{km}\cdot h^{-1}$ relative to the water. How long have the fish to swim in the aquarium to increase the temperature of the water by 1 centigrade?

### (8 points)3. Series 29. Year - E. hydrogel

Examine the dependence of a weight of a hydrogel ball on a time of submersion in a water and on a concentration of salt dissolved in water.
*Note* We do not send the experimental material abroad, therefore the hydrogel you buy must be described in detail.

### (2 points)2. Series 29. Year - 2. numismatic

Once in a while, a situation may occur, that the nominal value of coins is lower that their manufacturing costs. Assume we have two coins, made of a gold-silver alloy. The first one has diameter $d_{1}=1\;\mathrm{cm}$, second one $d_{2}=2\;\mathrm{cm}$, both have thickness $h=2\;\mathrm{mm}$. If we submerge them in mercury, the smaller one sinks to the bottom, whilst the larger one rises to the surface. If we submerge both coins, smaller one on top of the larger, they neither rise nor sink. Assuming the smaller coin is made of pure gold, determine the fraction of silver in the larger coin (in percent of mass).

**Bonus:** How would the result change if the smaller one could contain silver as well?

Mirek má radši mince než bankovky.

### (3 points)1. Series 29. Year - 3. golden sphere

A golden sphere has a mass of $m_{1}=96,25g$ while in the air. When sunken into water it is balanced out by a weight with the mass $m_{2}=90,25g$. Determine whether the item is hollow. If it is then determine the volume of the hollow part.The density of gold is $ρ_{Au}=19,25g\cdot \;\mathrm{cm}^{-3}$, density of water $ρ_{H_{2}O}=1,000g\cdot \;\mathrm{cm}^{-3}$. The graviational acceleration is $g=9,81\;\mathrm{m}\cdot \mathrm{s}^{-2}$.

Fales was reading about Archimedes.

### (5 points)6. Series 28. Year - P. waters of Discworld

We all know very well how well the water supply is arranged in Discworld. And none of us need to know how. What if somethig serious would happen and magic would stop working? How long would it take until Discworld would be without water? For simplicity you can assume a pesimistic situation where nobody would hold the water in any way. You know very well that Discworld has a diameter of $d=10000\;\mathrm{km}$, a homogeneous gravitational acceleration $g≈10\;\mathrm{m}\cdot \mathrm{s}^{-2}acts$ everywhere and is perfectly circular. The true complete volume and the distribution of water on Discworld is unknown to everybody so we can consider water being homogeneously distributed over Discworld, that can be considered flat and water has a height of $H=5\;\mathrm{m}$ (that is very pesimistic because everybody would have to be standing on stakes to stay out of the water or being completely submerged in it). The aim of the task is to find an approximate model that would give a good estimation as to the time it would take to lose all the water..

Karel was curious about how water flows from Discworld

### (5 points)5. Series 28. Year - P. splashed

Would it be possible to swim in a pool, if the water in it would behave as a completely ideally incompressible liquid, the visocity of which would approach zero? How would the movement of the swimmer differ from a swimmer that would swim in regular wate? What would happen with the energy of the system if water could flow out of the pool over the edges ? At the beginning the water is level with the edge of the pool.

Chemical physicist floats.

### (2 points)3. Series 28. Year - 1. heavy air

What is the weight of Earth's atmosphere? What percentage of weight does it make up? For the purposes of this problem you know only the mass of the Earth $M_{Z}$ and the radius $R_{Z}$ Zeme, gravitational acceleration $a_{g}$ on the surface of the Earth, density of water $ρ$ and you know that near the surface of the Earth at the depth of $h_{1}=10\;\mathrm{m}$ it has the pressure of approximately one atmosphere $p_{a}=10^{5}Pa$.

**Hint:** It is a simple task.
We don't want a perfect solution but a qualified estimation.

Karel saw an interesting misconception according to which a person is lighter on the moon just because the moon is smaller (And what if it were denser?).

### (2 points)1. Series 28. Year - 2. streaimng streamlines

Draw streamlines into the picture. Into both openings with an arrow the same amount of water flows, All the water then flows out through one opening, the third one. The flow is stable and is slow enough that we can consider it not to be turbulent. When drawing follow the rules that dictate the shapes of the streamlines and write these rules down as comments to the picture. We don't expect that this problem will be solved quantitatively.
*Comment* Draw into the bigger picture available from the website.

kolar

### (3 points)1. Series 28. Year - 3. accelerating

Explain why and how the following situations occur:

- In a cistern of a rectangular cuboid shape that is filled with water a ball is floating on the surface of the water. Describe the movement of the ball if the cistern starts moving with a constant acceleration small enough that the water shall not flow over the edge.
- In a cistern of a rectangular cuboid shape that is filled with water a ball filled with water is floating. Describe the movement of the ball if the cistern starts moving with a constant acceleration small enough that the water shall not flow over the edge.
- In a closed bus a ballon is floating near the ceiling. Describe its movement if the bus starts accelerating constantly

Dominika and Pikoš during a physics exam

### (4 points)1. Series 28. Year - 4. doom of the Titanic

Náry always wanted a boat and so one beautiful day he bought himself one in the shape of a cuboid without a top side (like a bath) with outer sides of $a$, $b$, $c$ and with a width of wall of $d$, which was created from scented wood of a density $ρ$ (bigger density than water). The second day he took his boat outfor a ride on the water and he found out that it has a small hole on the bottom through which water flows with a flow rate of $Q_{1}$. That was unfortunate but since he was a man of action he started calculating how long until water starts entering the boat from the top. The same question is asked by this task.Conider also the situation where Náry of a mass $m$ would have sat in the boat and while calculating would spill water out of the boat with a flow rate of $Q_{2}$. The boat is horizontal the whole time.

Kiki heard about the problem that nearly all tasks are thought up by Karel.