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## mechanics of a point mass

### (4 points)2. Series 25. Year - 4. ball on a train

Small weight of mass m is suspended from the ceiling of a train on a string of length $l$. The weight is hanging a distance $d$ from a wall (see picture). Suppose the train starts to accelerate in the direction perpendicular to the wall. Find a condition on the acceleration such that the weight eventually touches the wall. How long does it take before this happens? Assume that $d$ is less than $ l/5$$.

### (4 points)2. Series 25. Year - 5. hair

A hair with a lot of curls can be approximated by a spring (in other words it can be described by the exact same parameters – radius, inclination and material constants). How much longer is the hair if it hangs vertically in a gravitational field compared to the case that it lies horizontally on a table? Assume that the inclination is very small.

### (5 points)2. Series 25. Year - P. elevator

Suppose that you are in an elevator when the rope that holds it breaks. Can you save yourself from dying by jumping right before the elevator hits the ground? What is the maximum speed of the falling elevator such that you could save yourself this way? You will need to look up or estimate the maximum impact velocity that a human can survive.

### (2 points)1. Series 25. Year - 2. struggling swimmer

A man wants to swim across a river which flows at a speed of 2 km/h. He is able to swim at a speed of 1 km/h. What is the optimal trajectory and direction he should take so that his trip is the least exhausting? Where and at what time will he reach the other bank? How would the situation change if his aim was the shortest possible trajectory? The width of the river is $d=10\;\mathrm{m}$.

Petr

### (4 points)1. Series 25. Year - 4. drrrrr

A small conductive ball of negligible size is bouncing between two charged plates of opposite polarities. What is the frequency of the resulting periodic motion of the ball? Voltage between the plates is $U$. When the ball touches a charged plate it charges to a charge of magnitude $Q$ whose polarity is the same as polarity of the plate. The ratio of kinetic energies of the ball before and after an impact is $k$.

**Bonus:** Does the output power of this resistor correspond to the energy losses during impacts?

Jáchym

### 4. Series 24. Year - 2. To the Sun

Karel has decided to throw his notes into the Sun.
**Help:** him calculate the necessary initial velocity of his notebook so that it reaches the Sun. You can neglect the frictional froces acting on his notebook and you can also assume that the trajectory of the Earth around the Sun is circular. You know the masses of the Earth and the Sun as well as the distance of the Earth from the Sun.

Karel

### 1. Series 24. Year - E. shot put

We all know that in vacuum if we throw an object under a certain angle and with some initial speed the distance it will travel is independent upon the mass of the object. If we drop the vacuum assumption, the problem is not so simple. Experimentally find how does the distance of the point where the object hits the ground depends on its mass and shape. Make theoretical predictions (numerical calculations are allowed) and compare them with your findings.

### 1. Series 23. Year - 3. adiabatic invariant

The movement of point mass is restricted on straight line by 2 end points. At the beginning the point mass $m$ is moving at speed $v$. The end point starts to move away at speed $v_{1}<<v$. How will the energy of point mass change?

Na Zajímavé teoretické fyzice nespala Janap.

### 3. Series 22. Year - 2. trainstopping

Honza travels home by train at speed $v_{0}$. From his baggage on top shelf is hanging down a mathematical pendulum. The train starts suddenly to break (at acceleration $a$ for duration $t)$. Can the pendulum go around of 180 °? Assume, that pendulum is fully fixed above.

Z maďarské přípravy na FO od Dalimila vybral Aleš.

### 2. Series 22. Year - 3. two balls

A small ball is at rest on bigger ball. The bigger ball is laying on the table without friction. We give a small impulse to the small ball and its will fall on the table. How far from the the original contact between table and large ball it will touch the table?

na teoretické mechanice zkoulel Lukáš Ledvina