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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.

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. 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.

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.

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?

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. 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.

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.

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. 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.

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