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## mechanics of a point mass

### (3 points)6. Series 37. Year - 1. ballons with Martin

A car is standing on a straight road, with a freely floating helium balloon tied inside. Suddenly, the car starts to accelerate with acceleration $a=5{,}0 \mathrm{km\cdot min^{-2}}$. By what angle will the balloon be deflected from the vertical line? What is the direction of the deflection?

Martin would like to hang on a balloon behind a car.

### (3 points)6. Series 37. Year - 2. bombarded organizer

Estimate how many antineutrinos created in Czech nuclear power plants pass through the body of an average FYKOS organizer in one meeting held for a FYKOS camp. The meeting is 4 hours long and takes place on the tenth floor of the Matfyz building at the Troja campus in Prague.

### (7 points)6. Series 37. Year - 4. infite pulleys

Let us have an infinite system of intangible pulleys as shown in the figure, where the mass of each additional weight is one-third of the weight of the previous one. What is the acceleration of the first weight of mass $m$?

### (3 points)5. Series 37. Year - 2. basic problem of acoustics

Adam can take meaningful notes at the speed $v_1$. Unfortunately, his calculus professor speaks at the speed of $v_2$. There is an airflow in the lecture hall, moving from Adam towards the professor, with the air flowing at a velocity of $v_3$. At what velocity and in which direction along a straight line intersecting Adam and the lecturer should Adam move to transcribe everything the lecturer says into his notebook?

Adam likes the word \uv {meaningful}.

### (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.

### (3 points)4. Series 37. Year - 1. the flight over the moon

One day, the FYKOS-bird was watching the sky during a full moon. An airplane just passed over the moon in $0{.}35 \mathrm{s}$, and the perpendicular distance of its flight path from the center of the moon was $1/3$ of the full moon's radius. This plane flies typically with a speed of $800 \mathrm{km\cdot h^{-1}}$. The FYKOS-bird wondered what altitude the plane was at so he could fly with it next time. Like him, determine this altitude.

Jarda was sunbathing in the garden.

### (3 points)4. Series 37. Year - 2. they got off in Hněvice

Tomáš got into the train wagon in the shape of a rectangular cuboid and decided to take a nap. When he woke up, he found that he was alone in the wagon, which was suspended at its geometric center on a cargo crane and rotating around the hinge axis at an angular velocity of $\omega $. Tomáš didn't notice it at first since he was sitting in the wagon's centre with a width of $d$. When he realized it, he was pleased because he thought of using one of his kilogram standards, which he carries around for situations like this, to measure the length of the carriage. After a few attempts, he managed to throw the standard at an initial velocity of $\vec {v}$ so that after two revolutions of the wagon, the standard hit the far corner of the wagon and broke the window. Neglecting air resistance, what length $L$ of the wagon did he determine?

Tomáš fell asleep on the train and was thrown off by the conductor.

### (6 points)4. Series 37. Year - 3. step here, step there

Consider a homogeneous magnetic field of induction $B_1$, which spans a half-space bounded by the plane of interface $y=0$, beyond which is an equally oriented, also homogeneous magnetic field of induction $B_2$. An electron flies out of the plane perpendicularly to it and the field lines (as in the figure) with velocity $v$. Determine the size and the direction of its average velocity parallel to the plane of the interface.

*Bonus:* Consider now that the magnitude of the field changes linearly as $B = B_0 \(1+\alpha y\)$ and its direction is in the positive direction of the $z$-axis. Again, determine the magnitude and direction of the average velocity of the electron parallel to the interface plane. The electron is initially emitted as in the previous case.

Jarda is going one step forward and two steps back

### (5 points)3. Series 37. Year - 3. randomly you get further

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 $D \approx 10^{-9} cm^2.s^{-1}$ and the rate of active transport speed $u\approx 10^{-6} m.s^{-1}$. 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.

### (10 points)3. Series 37. Year - S. weighted participants

- 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 confucing weight nad mass.