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## mechanics of a point mass

### 3. Series 35. Year - 2. playing with keys

Vašek likes to plays with keys by swinging them around on a keychain and then letting them wrap around his hand. We will simplify this situation by a model, in which we have a point mass $m$ in weightlessness attached to an end of massless keychain of length $l_0$. The other end of the keychain is attached to a solid cylinder of radius $r$. The keychain is taut so that it is perpendicular to the surface of the cylinder at the attachment point, and the point mass is then brought to velocity $\vect {v_0}$ in the direction perpendicular both to the axis of the cylinder and to the direction of the keychain. The keychain then starts to wrap around the cylinder. What is the dependence of the velocity of the point mass on the length of the free (not wrapped around) keychain $l$?

**Hint:** Find a variable that remains constant during the wrapping process.

**Bonus:** How long does it take for the whole keychain to be wrapped around the cylinder?

### 2. Series 35. Year - 1. chasing the light

Jindra walks down a long, lit corridor. His eyes are at a height of $1,7 \mathrm{m}$ above the floor, the light on the ceiling is at a height of $3,4 \mathrm{m}$. Jindra is now at a distance of $10 \mathrm{m}$ (horizontally) from the nearest light and is approaching it at a speed of $3 \mathrm{km\cdot h^{-1}}$. He sees a reflection of the light on the polished floor. How fast is the reflection approaching Jindra at this point?

### 2. Series 35. Year - 3. model of friction

What would be the coefficient of static friction between the body and the surface if we considered a model in which there were wedges with a vertex angle $\alpha $ and a height $d$ on the surface of both bodies? Try to compare your results with real coefficients of friction.

### 2. Series 35. Year - 4. tea tap

Matěj wants to pour some tea from a bevarage dispenser into a glass of mass $M$. He uses one hand to hold the glass and second hand to control the faucet, which changes the volume of the current of the tea. The speed of the outflow $v$ is constant (we can assume that the speed at the contact with the glass is identical). Since Matěj does not want to overstrain himself, he would like to hold the glass with a constant force from the start of the pouring to its end. What is the value of the volume of the current as a function of time that satisfies this requirement? How long will it take to fill the whole glass?

### 2. Series 35. Year - 5. Shkadov thruster

A long time ago in a galaxy far, far away, one civilisation decided to move its whole solar system. One of the possibilities was to build a „Dyson half-sphere“, i. e. a megastructure which would capture approximately half of the radiation output of the start and reflect it in a single direction. An ideal shape would therefore be a paraboloid of revolution. What would be the relation between the radiation output of the star, surface mass density of such a mirror and its distance from the star such that this distance is constant?

### 1. Series 35. Year - 1. cars

Two cars start to move from the same point at the same time with velocities $v_1 = 100 \mathrm{km\cdot h^{-1}}$ and $v_2 = 60 \mathrm{km\cdot h^{-1}}$. Is it possible for the cars to move away from each other at any of the following velocities? If so, sketch the situations. \[\begin{align*} v_a &= 160 \mathrm{km\cdot h^{-1}} , & v_b &= 40 \mathrm{km\cdot h^{-1}} , \\ v_c &= 30 \mathrm{km\cdot h^{-1}} , & v_d &= 90 \mathrm{km\cdot h^{-1}} \end {align*}\]

Karel wanted to hit Dano at a precisely defined speed.

### 1. Series 35. Year - 3. to stop on skates

Skaters can stop using the „parallel slide“ method, in which they turn the blades of both skates perpendicular to the direction of movement, which significantly increases the friction with the ice. During this, the skater must tilt by the angle $\phi = 35 \mathrm{\dg }$ from the vertical direction, so he doesn't fall. Assume that he weighs $m = 70 \mathrm{kg}$ and that he is $H = 1{,}8 \mathrm{m}$ high (including the skates), with the center of gravity at a height of $h = 1{,}1 \mathrm{m}$ above the ice. Calculate the distance in which he stops from the initial speed $v_0 = 15 \mathrm{km\cdot h^{-1}}$.

Dodo doesn't know how to brake on skates (me neither).

### 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.

### 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.

### 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.