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## hydromechanics

### (3 points)5. Series 37. Year - 1. annexation of Kaliningrad

The commander of the operation to take over the Russian enclave is chilling in his recreational boat, which has the shape of a block with a base area $S$ and height $H$. Suddenly, directly below him at the bottom of the Vistula Lagoon, a group of saboteurs punches a hole in the alcohol pipeline – a pipeline bringing a high-quality, scarce Czech commodity with a density $\rho \_B$ from Budějovice to Královec. Determine the conditions under which the boat sinks, assuming that it was submerged to a depth $h$ before the accident and that the layer of beer on the surface after the accident is $\Delta h$.

### (7 points)5. Series 37. Year - 4. centrifuge

Consider a centrifuge of length $L = 30 \mathrm{cm}$ filled with a solution in which there are homogeneously distributed small spherical particles of radius $r = 50 \mathrm{\micro m}$ and mass $m = 5,5 \cdot 10^{-10} \mathrm{kg}$. The density of the solution is $\rho \_r = 1~050 kg.m^{-3}$ and its viscosity is $\eta = 4,8 \mathrm{mPa\cdot s}$. The container with the solution is in a horizontal position and suddenly begins to rotate at an angular velocity of $\omega = 0,5 \mathrm{rad\cdot s^{-1}}$. Determine how long it will take for $90 \mathrm{\%}$ of all the particles to reach the end of the centrifuge. Do not consider interparticle collisions and movement of the particles due to diffusion.

Jarda loves to make enriched uranium.

### (12 points)5. Series 37. Year - E. gooey

Measure the dependence of a cooking oil's dynamic viscosity $\eta $ on temperature $T$. Fit the measured data to function \[\begin{equation*} \eta = \eta _0 \f {\exp }{\frac {T_0}{T}} , \end {equation*}\] and calculate the values of the parameters $\eta _0$ and $T_0$.

**Hint:** When fitting the results, plot the horizontal axis as $1/T$. Then, it is possible to fit the data with the required curve even in less advanced software, such as *Excel*.

### (10 points)3. Series 37. Year - 5. air under the water

Assume a cylindrical glass of negligible mass, internal cross-sectional area $S$, and height $h$ that is turned upside down and its open rim aligned with the water level in the reservoir. We start to push slowly downwards. What work will we perform if we move the jar with the air inside so that its base $d>0$ is below the surface? **Bonus::** Let us now consider a more realistic case. How much work must be performed to completely submerge a jar of the same dimensions but mass $m$ to the bottom of a container with area $A$ and initial water level in height $H$? Assume that the jar is completely submerged when it reaches the bottom.

Jarda would not like to visit Titanic…

### (10 points)2. Series 37. Year - 5. ferry

Imagine a ferry in the shape of a rectangular cuboid with a weight $M$, length $L$, width $W$, and height $H \ll L$ from the keel to the deck. After docking at the pier, passengers gradually exit through the back of the deck so that the empty front part of the deck becomes larger and the area density of people on the filled part does not change in a different way. Find the maximum weight of passengers the ferry can carry so that no part of the deck is below the surface when people disembark. Consider that the ship is stable in the transverse direction and that people get off slowly.

After quite some time, Dodo was at sea again.

### (13 points)6. Series 36. Year - E. ripples

Build an apparatus that can measure the smallest possible ripples on the surface of the liquid. You can choose the container yourself – it can be a cup, a bottle, or something else. Thoroughly describe and take a picture of the whole apparatus. Determine the minimum amplitude you are able to measure.

Karel was staring into space… he was writing his dissertation thesis.

### (10 points)4. Series 36. Year - P. the boat is sailing

Discuss what physical phenomena affect the cruising speed of a ship and submarine. What resistive forces act on them? What is the highest cruising speed that a ship or submarine can sail?

Jindra went punting on the river Cam.

### (3 points)3. Series 36. Year - 1. creative problem-solving

Danka attached a garden hose with an inner diameter of $1{,}5 \mathrm{cm}$ to a tap in her dorm room and placed the other end on the edge of a window on the eighth floor, $23 \mathrm{m}$ above the ground. What is the necessary volumetric flow rate of the water tap so that Danka can spray a stream of water on the people disturbing the night's silence? They are standing below the window at a horizontal distance $9 \mathrm{m}$ from the building. Is Danka able to achieve this if water is being sprayed horizontally from the hose and there is no wind?

**Bonus:** Where is the farthest these people can stand so Danka can still spray them if the volumetric flow rate of the tap is $0{,}4 \mathrm{l\cdot s^{-1}}$? Danka can now set the end of the hose so that water sprays at an arbitrary angle to the horizontal plane.

Danka is annoyed by the noise below the windows at night.

### (3 points)2. Series 36. Year - 1. water channel

Water flows through a water channel of rectangular cross-section, and width $d=10 \mathrm{cm}$. A leaf falls on its surface and starts moving with a velocity of $60 \mathrm{cm\cdot s^{-1}}$. The height of the water in the channel is $h=1{,}3 \mathrm{cm}$. Estimate how long it will take to fill up a $50 \mathrm{l}$ bucket. Comment on the assumptions used in comparison with the real situation.

Dodo was cooling his horsefly bite.

### (3 points)1. Series 36. Year - 2. weighing an unknown object

Let us have an ideal scale which we calibrate using a state standard (etalon) with a mass $m\_e = 1,000~000~165 kg$ and a density $\rho \_e = 21~535,40 kg.m^{-3}$. By calibration, we mean that after placing the standard on the scale, we assign to the measured value the mass $m\_e$. The unknown object is weighed under the same conditions in which its volume is $V_0 = 3,242~27 dl$. What mass did we measure if we measured the weight $G = 1,420~12 N$? What is the actual mass of the object? The experiment is conducted at a place with standard gravitational acceleration $g = 9,806~65 m.s^{-2}$ and air density $\rho \_v=1,292~23 kg.m^{-3}$. Take into account that the calibration is linear, and the unloaded scale shows zero.

Karel wanted to use a standard.