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## hydromechanics

### (5 points)4. Series 33. Year - 3. uuu-pipe

What period of small oscillations will water in a glass container (shown on the picture) have? The dimensions of the container and the equilibrium position of water are shown. Assume that there is room temperature and standard pressure and that water is perfectly incompressible.

### (3 points)3. Series 33. Year - 1. fountain with nozzle

Let us have a fountain with $N$ nozzles of the same cross section. These nozzles are fed by one pump with constant volumetric flow rate, which leads to water streaming to the height $h$. Find this height in case of all nozzles with the exception of one being blocked.

Lukáš experimented in the town square.

### (5 points)3. Series 33. Year - 3. umbrella

As you probably have noticed, water flow creates a mushroom-like shape when a teaspoon is placed against it (e.g. while washing the dishes). Assume (for simplification) planar round-shaped teaspoon of small radius. When placed perpendicularly to the flow, falling from rest from the height $h$, a wonderful rotational paraboloid would form. Find the optimal height to put the teaspoon in to maximalise the distance from the original flow axis to the place, where the falling water touches the surface (e.g. of the sink). Assume water to be an ideal liquid (uncompressible, zero viscosity, no inner friction).

**Bonus:** Find optimal height to maximalise the volume of the paraboloid.

Matěj washed the dishes.

### (9 points)3. Series 33. Year - 5. probability density of water

Imagine a container from which continually and horizontally flows out water stream with constant cross-section area. Velocity of the stream randomly fluctuate with uniform distribution from $v_1$ to $v_2$. Water from the container continually freely falls onto a horizontal floor below. Figure out arbitrary area of the floor to which falls exactly $90 \mathrm{\%}$ of water.

Another from a list of problems, which crossed Jachym's mind while being on a toilet.

### (10 points)1. Series 33. Year - S. slow start-up

*We are sorry. This type of task is not translated to English.*

Karel wants to have the longest problem assignment.

### (6 points)6. Series 32. Year - 3. range

A container is filled with sulfuric acid to the height $h$. We drill a very small hole perpendicularly to the side of the container. What is the maximal distance (from the container) that the acid can reach from all possible positions of the hole? Assume the container placed horizontally on the ground.

Do not leave drills where Jáchym may take them!

### (6 points)5. Series 32. Year - 3. border

Imagine an aquarium in the shape of a cube with edge length $a = 1 \mathrm{m}$, which is separated into two parts via a vertical partition perpendicular to sides of the aquarium. Let us assume that the partition can move in the direction perpendicular to the plane of the partition, but it is fixed in other directions. Also, it can't rotate. We pour $V_1 = 200 \mathrm{\ell }$ of water (density $\rho \_v = 1 000 \mathrm{kg\cdot m^{-3}}$) into the first part and $V_2 = 230 \mathrm{\ell }$ of oil (density $\rho \_o = 900 \mathrm{kg\cdot m^{-3}}$) into the second part. In which position is the partition in mechanical equilibrium? In what height will be the surfaces of the liquids?

**Bonus:** Find frequency of small oscillations of the partition. Assume, that mass of the partition is $m = 10 \mathrm{oz}$ and the liquids move without friction or viscosity.

Michal cleaned an aquarium.

### (8 points)5. Series 32. Year - 4. splash

Consider a free water droplet with radius of $R$. We start to charge the drop slowly. Find the magnitude of the charge $Q$ the drop needs to splash.

### (3 points)3. Series 32. Year - 1. discounted bananas

Mikulas put bananas into a carry bag in a grocery store and before he had weighted them, he got an idea. If he fills the bag with helium instead of air, the bananas will weigh less. Mikulas bought the helium in a sale - one CZK for a litre at standard pressure. Calculate the prize of the bananas so that this „bluff“ pays off.

**Bonus:** Find a gas for which it would pay off when the price of bananas is 30 CZK per kilogram. Do not forget to cite references.

What do you think about while weighting bananas?

### (12 points)2. Series 32. Year - E.

Measure an average vertical velocity of falling leaves. Use leaves from several different trees and discuss what impact the shape of a leaf has on the velocity. How should an ideal leaf look like when we want it to fall as slow as it is possible?

Jachym got this idea, when he asked his friend, whether he knew any interesting experiment