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## mechanics of a point mass

### (10 points)3. Series 33. Year - S.

* We are sorry. This type of task is not translated to English. *

### (3 points)2. Series 33. Year - 1. fast elevator

They say that people inside an elevator can bear with acceleration $a = 2{,}50 \mathrm{m\cdot s^{-2}}$ without any major problems. We want to get to the planned floor as soon as possible. If the elevator started running with that acceleration for a quarter of a time, a half of the time went by constant velocity, and the rest quarter of the time it was decelerating, how high it could go by total time of the ride $t = 1{,}00 \mathrm{min}$?

Karel rides an elevator.

### (3 points)2. Series 33. Year - 2. weak winch

Let us assume a pulley in a fixed position with a rope of negligible weight. A weight $m_1$ is located on one end of the rope and a winch (mass $m_2$) on the other. Initially, the winch is in the same height as the weight $m_1$. In the first case, the winch is fixed to the ground and pulls only the weight. In the second case, the weight is firmly connected to the winch. Therefore, while the rope is being attached, both the weight and the winch are pulled up. Find out which case requires less pull force (and therefore weaker winch).

Vašek needed a mechanism to pull up a snow plow blade.

### (6 points)2. Series 33. Year - 3. Danka's (non-)equilibrium cutting board

Cutting board with thickness $t=1,0 \mathrm{mm}$ and width $d =2,0 \mathrm{cm}$ is made up of two parts. The first part has density $\rho _1 =0,20 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_1 = 10 \mathrm{cm}$, the second part has density $\rho _2 =2,2 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_2 = 5,0 \mathrm{cm}$. We place the cutting board on water surface, which density is $\rho \_v = 1{,}00 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and then we wait until it is in equilibrium position. What angle will a plane of the cutting board hold with the water surface? How big the part of the cutting board which will stay above the water level will be?

Danka was talking with Peter about dish-washing.

### (10 points)2. Series 33. Year - S.

*We are sorry. This type of task is not translated to English.*

### (3 points)1. Series 33. Year - 1. D1

A truck driving on a highway has a $2 \mathrm{\%}$ higher speed than a bus in front of it. The driver of the truck decides to overtake the bus, but when the truck is exactly next to the bus, a right curve begins on the highway, making the path of the truck longer. As a consequence, the two vehicles drive next to each other all the way along the curve, whilst a notable traffic jam starts to build up behind them. Determine the radius of the curve (at the middle of the inner driving lane) if the separation between the centers of the lanes is $3{,}75 \mathrm{m}$.

Matej doesn't like trucks on highways.

### (10 points)1. Series 33. Year - S. slow start-up

*We are sorry. This type of task is not translated to English.*

Karel wants to have the longest problem assignment.

### (3 points)6. Series 32. Year - 2. bookworm

Vítek has been spending some time in the library. Because of his clumsiness, a book fell down from a shelf and he managed to press it with a swift move towards the wall. He pushes the book with a force $F$ applied at an angle $\alpha $ (see figure). The book's mass equals $M$ and the coefficient of friction between the wall and the book is $\mu $. Find the condition under which the force keeps the book from falling down (and at rest) and determine the critical value $\alpha _0$, below which there does not exist any force that will keep the book up.

Vítek was in a mobile library.

### (6 points)6. Series 32. Year - 3. range

A container is filled with sulfuric acid to the height $h$. We drill a very small hole perpendicularly to the side of the container. What is the maximal distance (from the container) that the acid can reach from all possible positions of the hole? Assume the container placed horizontally on the ground.

Do not leave drills where Jáchym may take them!

### (7 points)6. Series 32. Year - 4. rope

A rope is hanging over the football goal crossbar (a horizontal cylindrical pole). When one of the rope ends is at least three times longer than the other one (the rope is hanging freely, not touching the ground), the rope spontaneously starts to slip off the crossbar. Now, we wrap the rope once around the crossbar (i.e. the rope wraps an angle of $540\dg $). How many times can the one end of the rope be longer than the other one so that the rope does not slip?

Matej was pulling down a climbing rope.