**Jan. 2, 2024, 23:59, CET**.

# Assignment of Series 3 of Year 37

About the Competition Rules How to Write Solutions Results#### 1... it's too dry in here

#### 3 points

Danka has a humidifier in her dorm room, which evaporates water from its boiling point to create warm steam. The device can hold a maximum of $V = 3.8\,\mathrm{l}$ of water, which it uses up in $t = 24\,\mathrm{h}$. What is its efficiency, i.e., what fraction of the energy drawn from the electrical grid it uses to convert the water to steam? The input power of the humidifier is $P = 260\,\mathrm{W}$, and Danka put water at $T_0 = 20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$ inside. All the necessary properties of water can be looked up.

*Danka has to use a humidifier in her dorm room during winter.*

#### 2... stable sheep

#### 3 points

*Dodo watched sheep on a hillside.*

#### 3... randomly you get further

#### 5 points

In the microworld of cells, there are two types of transport: transport
by *free diffusion*, also known as *Brownian motion* where the motion
uses the energy of the environment directly. The second type,
so-called *active transport*, requires, among other things, a motor
protein moving at a constant speed along the cytoskeletal filament. Consider
the typical value of the diffusion constant $D \approx 10^{-9}\,\mathrm{cm^2\cdot s^{-1}}$
and the rate of active transport speed $u\approx10^{-6}\,\mathrm{m\cdot s^{-1}}$. For which
distances is the Brownian motion more time efficient than the active transport?
Assume that the transport is happening in only one direction.

*Marek J. read Sekimoto.*

#### 4... size matters

#### 8 points

A sphere with a radius $r$ rolls on a horizontal surface with a speed $v$. However, its path is blocked by a perpendicular step with the height $h$. Find the conditions under which the ball rolls onto the step and starts rolling along it without losing contact with the step. Under these conditions, determine its speed after it has crossed the step. Assume that all collisions are perfectly inelastic and the friction between the ball and the step is high. The step is angular and is oriented perpendicular to the direction of the sphere's motion.

*Dodo had small wheels.*

#### 5... air under the water

#### 10 points

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...*

#### P... by a flash

#### 9 points

What determines the width of a lightning channel in a thunderstorm? Create a quantitative model.

*Karel stumbled upon a claim about the Sky Tower lightning rod.*

#### E... acoustic thermometer

#### 12 points

Attach a string at two points at a fixed distance $L$ and ensure it is always taut during measurement. Determine the dependence of the fundamental frequency of its oscillations on temperature.

Instructions for Experimental Tasks*Honza Benda is nuts.*

#### S... weighted participants

#### 10 points

- According to definitions by International System of Units, convert these into base units
- pressure $1\,\mathrm{psi}$,
- energy $1\,\mathrm{foot-pound}$,
- force $1\,\mathrm{dyn}$.

- In the diffraction experiment, table salt's grating constant (edge length of the elementary cell) was measured as $563\,\mathrm{pm}$. We also know its density as $2.16\,\mathrm{g\cdot cm^{-3}}$, and that it crystallizes in a face-centered cubic lattice. Determine the value of the atomic mass unit.
- A thin rod with a length $l$ and a linear density $\lambda$ lies on a cylinder with a radius $R$ perpendicular to its axis of symmetry. A weight with mass $m$ is placed at each end of the rod so that the rod is horizontal. We carefully increase the mass of one of the weights to $M$. What will be the angle between the rod and the horizontal direction? Assume that the rod does not slide off the cylinder.
- How would you measure the mass of:
- an astronaut on ISS,
- a loaded oil tanker,
- a small asteroid heading towards Earth?

*Dodo keeps confusing weight nad mass.*