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## mechanics of a point mass

### (7 points)1. Series 35. Year - 4. fall to the seabed

A cylindrical capsule (Puddle Jumper) with a diameter $d = 4 \mathrm{m}$, a length $l = 10 \mathrm{m}$ and with a watertight partition in the middle of its length is submerged below the ocean surface and falls to the seabed at a speed of $v = 20 \mathrm{ft\cdot min^{-1}}$. At the depth $h = 1~200 ft$, the glass on the front base breaks and the corresponding half of the capsule is filled with water. At what speed will it fall now? How long will it take for the capsule to sink to the bottom at the depth $H=3~000 ft$? Assume that the walls of the capsule are very thin against its dimensions.

Dodo watches Stargate Atlantis.

### (8 points)1. Series 35. Year - 5. mechanically (un)stable capacitor

Assume a charged parallel-plate capacitor in a horizontal position. One of its plates is fixed and the other levitates directly below it in an equilibrium position. The lower plate is not mechanically fixed in its place. What is the capacitance of the capacitor depending on the voltage applied? Is the capacitor mechanically stable?

Vašek wanted to grill you on a capacitor.

### (3 points)6. Series 34. Year - 2. rotating pendulum

Let us have a mathematical pendulum of length $l$ with a point mass $m$ in a gravitational field with the acceleration $g$. We give the pendulum a constant angular velocity $\omega $ about the vertical axis. Determine the stable positions of the pendulum (expressed as a function of the angle between the pendulum and the vertical).

Jindra wanted to swing on a wrecking ball with a hammer in his hand.

### (9 points)6. Series 34. Year - 5. heavy spring

Let us have a homogeneous spring with stiffness $k$, mass $m$ and its width negligible compared to its length. We grip the spring at one end in a way that it can rotate around and then we spin it with angular velocity $\omega $. By how much does the spring prolong (compared to its initial length) due to the rotation? Neglect the effect of the gravitational field.

Jachym had a very difficult day and wanted to share it with others.

### (7 points)5. Series 34. Year - 4. period of large oscillations

Assume two half-planes with the angle $2\phi < \pi $ between them. We place them so that the line at their intersection is horizontal and their plane of symmetry is vertical, so they form a kind of valley. Then we take a mass point and throw is with the velocity $v$ from the height $h$ (above the intersection line) in the horizontal direction so that it makes a periodic motion as shown in the picture. What is the magnitude of the velocity that we have to throw it with? Assume the bouncing to be perfectly elastic.

Legolas was bored by the periods of small oscillations.

### (10 points)5. Series 34. Year - 5. rheonomous catapult

Let us have a thin rectangular panel that rotates around its horizontally oriented edge at a constant angular velocity. At the moment when the panel is in a horizontal position during rotating upwards, we place a small block on it so that its velocity with respect to the panel is zero. How will the block move on the panel if the friction between them is zero? Where do we have to place the block so that it flies away from the panel exactly after a quarter of its turn? Discuss all the necessary conditions that must be met to achieve this. **Bonus:** What power does the panel transfer on the block and what total work does it do on it?

Vašek was tired of problems with scleronomous bond, so he came up with rheonomous bond.

### (3 points)4. Series 34. Year - 1. two water drops

Suppose two water drops fall in quick succession from a water tap. How will their respective distance change with time? Neglect air resistance. **Bonus:** Do not neglect air resistance, make an estimate of all relevant parameters and find the distance of the water drops after a very long time.

Karel was hypnotized by water.

### (8 points)4. Series 34. Year - 5. Efchári-Goiteía

Efchári and Goiteía are two components of a double planet around recently arisen stellar system. They orbit around a common centre of mass on circular trajectories in the distance $a = 250 \cdot 10^{3} \mathrm{km}$. Efchári has the radius $R_1 = 4~300 km$, density $\rho _1 = 4~100 kg.m^{-3}$ and siderial period $T_1 = 14 \mathrm{h}$. Goiteía is smaller – it has the radius $R_2 = 3~800 km$, but it has a higher density $\rho _2 = 4~500 kg.m^{-3}$ and a shorter period $T_2 = 11 \mathrm{h}$. Rotation axes of both planets and the system are parallel. After several hundred years, the system transfers due to tidal forces into so-called tidal locking. Find the resulting difference in the period of the system, assuming that both bodies are homogeneous and roughly spherical.

Dodo keeps confusing Phobos and Deimos.

### (13 points)4. Series 34. Year - E. breathtaking syringes

Find the magnitude of the friction force between the plunger and the barrel of a syringe.

Dano remembered his trip to Russia.

### (10 points)4. Series 34. Year - P. Fykos bird on vacation

How would aviation work on other planets (with atmosphere)? Consider mainly jet aircraft. Which planetary parameters would influence the aviation positively and which negatively, compared to Earth's?

Karel visited the museum of aviation in Košice.