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## mechanics of rigid bodies

### (4 points)1. Series 26. Year - 4. crash tests

Two cars are driving towards each other with speed $v_{0}$. At what distance from each other should they hit the brakes in order to avoid a crash? Consider the case when both cars are on a flat road as well as the case when they are on a road inclined at an angle $α$. Both drivers apply the brakes at the same time. The braking force is equal to $f\cdot N$ where $N$ is the component of the car's weight normal to the road.

Petr played the game of chicken.

### (8 points)1. Series 26. Year - E. Aleš is going bald

In this problem you are asked to investigate some elastic properties of a human hair. First you should carry out an experiment to obtain the dependence of the magnitude of the applied force on the change of the hair's length. This will allow you to construct a graph describing the dependence of the applied stress on the strain. Using these data try to estimate the maximal allowed stress and perhaps also other characteristics of the hair. You should make at least 3 measurements using hair from the same person.

**Hint:** The longer the hair, the better. You can measure the diameter of a hair using a micrometer or a laser in your school. Convenient weights are coins which usually have a well defined mass.

Karel was in a barber shop.

### (5 points)1. Series 26. Year - P. airship

What is the minimal wind speed necessary to blow away a paper lying on a flat table?

Karel does not like wind.

### (4 points)3. Series 25. Year - 3. train à grande vitesse

Railway has a shape of an arc of radius $R$ and its width is $D$. Train's center of mass is at height of $H$ meters. How fast can a train go on this railway if, no matter where it stops, it never falls? Under what conditions is this maximum speed unbounded?
*Note* Neglect forces cars act on each other and also assume the width of a car is much smaller than the radius $R$.

Dominika házela šutry.

### (5 points)3. Series 25. Year - P. save the world

Invent a mechanism that converts rotational energy of the Earth to electric energy. Do not be too down-to-earth. Everything is possible.

Pikoš platil účet za plyn.

### (4 points)2. Series 25. Year - 3. lifting boats

A small Scottish town decided to build an elevator for boats. It consists of two big containers filled with water that are suspended from the ends of a long rod. This rod is attached in the middle to a motor that can rotate the whole system. A boat enters one of the containers and waits to be lifted. What is the minimum power of the motor for this lift to work?

### (4 points)1. Series 25. Year - 5. recoil

When shooting from a gun the resulting backward impact causes the bullet to shoot out in a different direction than was originally meant. What is the angle difference between these two directions? Assume that hand muscles compensate for any influence that gravity can have. Also assume that the gun is rotating only around some point in a wrist. You know the moment of inertia of the gun-hand system (with respect to the previously mentioned point) as well as the mass of the projectile, its speed when leaving the gun and the dimensions described in the picture. After you solve this problem qualitatively make a quantitative guess of the necessary quantities and find a numerical value for the angle.

Unknown shooter

### 6. Series 24. Year - 1. warm up

##### crooked table

Small ball is moving along a horizontal table from one end to the other with some initial velocity. In which case is the time required for the ball to traverse the table the shortest? Explain your choice. - Table has a concave bow. - Table has a convex bow. - Table is flat. - The curvature of the table does not matter.

##### broken bridge

Small valley of width $L$ is bridged using a board that is broken in the middle. It is however not entirely broken and still holds together so that its shape resembles that of a graph of absolute value. We place a small ball at one end and let it go. What is the appropriate depth of this bridge so that the time required for the ball to get to the other side is the shortest? Assume the ball does not lose energy while rolling over the bridge. You may need to know that the function $f(x) = x+1/x$ has a minimum at $x=1$.

### 5. Series 24. Year - 3. heavy chain

A chain of mass $m$ and length $l$ is hanging right above a scale. Initially it is at rest but then it starts falling. How does the scale's reading depend on the length $x$ of the chain that is already laying on the scale? Assume that the size of single chain cells is negligible.

Karel

### 4. Series 24. Year - 1. Warm-Up

*Strings.*

Using dimensional analysis determine the dependance of the frequency of oscillatons of a string if you know that it depends on its length $l$, on the tension $F$ in the string and on its linear density $ρ_{l}$.

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*Downward.*

You have a dumbell which consists of a short rod and two heavy discs. You wrap a string around the rod and let the dumbell fall while holding the string. What is the velocity of the dumbell? The discs have mass $M$ and radius $R$. The radius of the rod is $r$ and you can neglect its mass.

Karel, Jakub