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## hydromechanics

### (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.

### (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

### (10 points)2. Series 32. Year - P.

Create an accurate weather forecast for address V Holešovičkách 2, Prague 8, for Wednesday 14th of November from 12:00 to 15:00. How will the weather change throughout the whole day? You are allowed to use previous data about the weather in this area (remember you are only permitted to use data until 10th of November). It is necessary to justify your weather prediction, write down references and ideally to use as many data and resources as possible.

Karl listened to radio on a motorway

### (9 points)5. Series 31. Year - P. floating mercury

Try to invent as much „physics tricks“ as possible thanks to which mercury would float on the liquid water for at least a limited time. The more permanent solution you find, the better.

### (3 points)5. Series 30. Year - 2. spheres in viscous fluids

When solving problems involving drag in air or in general a fluid, we use Newton's resistance equation

$$F=\frac{1}{2}C\rho Sv^2\,,$$

where $Cis$ the drag coefficient of the object in the direction of motion, $ρis$ the density of the fluid, $Sis$ the cross-section area and $vis$ the velocity of the object. This is usually quite accurate for turbulent flow. We are interested in a sphere, for which $C=0.50$. In the case of laminar flow, we usually use Stokes' law

$$F = 6 \pi \eta r v\,,$$

where $ηis$ the dynamic viscosity of the fluid and $ris$ the radius of the sphere. Is there a velocity for which the two resistance forces are equal for the same sphere?. How will this velocity depend on the radius of the sphere?

Karel heard at a conference that people struggle with equations.

### (8 points)1. Series 30. Year - P. The sky is falling

Did you ever think about, why the clouds simply don't fall down, when they consist of water, which is much denser than air? The raindrops fall to the ground in minutes, so why not clouds? Try to physically explain this. Support all of your claims with calculations.

Mirek se zadíval na nebe a dostal strach.

### (4 points)4. Series 29. Year - 4. Bubbles reunited!

What is the smallest number of equally sized soap bubbles with the diameter $r$, that would make a single bubble with the diameter at least 3$r?$ Expect the air in the bubbles to have a constant temperature.

### (5 points)4. Series 29. Year - 5. Slide

There are two identical blocks with the mass $m$ and one of the sides of lenght $lon$ a horizontal plane. The distance between the closest two faces is 2$x_{0}$. Suddenly we start pouring water between them with the volume flow $Q$. At two sides of the blocks there is a barrier keeping the water in the place between the two blocks. The coefficient of static friction between the block and the plane is $f_{0}$ and the of the kinetic friction is $f$. There is no friction between the barriers and the blocks. What is the condition for $f_{0}$ that would keep the blocks in place? In the case of sufficiently small $f_{0}$, determine the acceleration of blocks as a function of position and also the distance, where the blocks eventually stop moving. Consider all the movement of the water reasonably slow, for any eddies to appear, for any heating of the water solely from its movement to take place or for any significant kinetic energy possesion. For the same reason of very slow $Q$, we can approximate there is no contribution of adding any water past the point where the blocks started moving.
**Bonus:** Find a condition for turning the block over.

### (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.