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## mechanics of a point mass

### (3 points)1. Series 28. Year - 3. accelerating

Explain why and how the following situations occur:

- In a cistern of a rectangular cuboid shape that is filled with water a ball is floating on the surface of the water. Describe the movement of the ball if the cistern starts moving with a constant acceleration small enough that the water shall not flow over the edge.
- In a cistern of a rectangular cuboid shape that is filled with water a ball filled with water is floating. Describe the movement of the ball if the cistern starts moving with a constant acceleration small enough that the water shall not flow over the edge.
- In a closed bus a ballon is floating near the ceiling. Describe its movement if the bus starts accelerating constantly

Dominika and Pikoš during a physics exam

### (5 points)1. Series 28. Year - 5. a thousand year old bee

Calculate the power required by a bee to remain in the air and approximate how long a bee that has just eaten can remain in the air for(at a constant altitude).

Michael thought during a discussion about quadcopters.

### (6 points)1. Series 28. Year - S. Unsure

- Write down the equations for a throw in a homogeneous gravitational field (you don't need to prove them but you need to know how to use them). Design a machine that will throw an item and determine the angle of approach and the velocity. You can throw with the item with a spring, determine its spring constant, mass of the object and calculate the kinetic energy and thus the velocity of the item. What do you think is the precision of the your value of the velocity and angle? Put the boundaries determined by this error into the equations and show in what boundaries we can expect the distance of the landing from the origin to be.Throw the item with your device at least five times and determine the distance of the landing and what are the boundaries within which you are certain of your distance? Show if your results fit into your predictions. (For a link to video with a throw you get a bonus point!)
- Tie a pendulum with an amplitude of $x$, which effectively oscillates harmonically but the frequency of its oscillations depends on the maximum displacement $x_{0}$

$$x(t) = x_0 \cos\left[\omega(x_0) t\right]\,, \quad \omega(x_0) = 2\pi \left(1 - \frac{x_0^2}{l_0^2}\right)\,,$$

where $l_{0}is$ some length scale. We think that are letting go of the pendulum from $x_{0}=l_{0}⁄2$ but actually it is from $x_{0}=l_{0}(1+ε)⁄2$. B By how much does the argument of the cosine differ from 2π after one predicted period? How many periods will it take for the pendulum to displaced to the other side than which we expect?
*Tip* Argument of the cosine will in that moment differ from the expected one by more than π ⁄ 2.

- Take a pen into your hand and let it stand on its tip on the table. Why does it fall? And what will determine if it will fall to the right or to the left? Why can't you predict a die throw even though the laws of physics should predict it? When you play billiard is the inability to finish the game only due to being incapable of doing all the neccessary calculations? Write down your answers and try to enumerate physics phenomenons that occur in daily life which are unpredictable even if we know the situation well.

### (2 points)6. Series 27. Year - 2. go west

More than a hundred years ago the measurements of surveyors confirmed that when we sail west, gravimeters show higher values of gravitational acceleration than when travelling east. Determine the difference that we measure on the equator between the measurements we make when still (relative to the earth) and when we are travelling at 20 knots per hour westwards.

Mirek was wondering why people don't migrate eastwards.

### (4 points)6. Series 27. Year - 5. toilet roll

We put a roll with paper into a bearing (without friction) and we let the paper unroll itself (we neglect the sticking of layers to eac other, friction in the bearing and the weight of the bearing). What is the angular velocity of the roll after all paper is removed? We know the radiusand mass of the roll, the longitudal density of paper, its overall mass and its length. Consider that the paper shall be able to unroll into an infinite pit.

**Bonus:** Now consider that the paper will fall to the ground before it all unrolls.

Lukáš came up with this problem when reading Michal's toilet problem.

### (8 points)5. Series 27. Year - E. rubbery

An object of mass $m$ on a piece of rubber of length $l_{0}is$ hung at a rigid point, the coordinates of which are $x=0$ and $y=0$. From the $xaxis$, which is horizontal, we slowly release the mass. What will be the relation between the lowest point reached and its position on the axis $x?$

Dominika was testing which method is optimal for gouging someone's eyeball out.

### (4 points)4. Series 27. Year - 3. Seagull

Two ships are sailing against each other, the first one with a velocity $u_{1}=4\;\mathrm{m}\cdot \mathrm{s}^{-1}$ and the second with a velocity of $u_{2}=6\;\mathrm{m}\cdot \mathrm{s}^{-1}$. When they are seperated by $s_{0}=50\;\mathrm{km}$, a seagull launches from the first ship and flies towards the second one. He is flying against the wind, his speed is $v_{1}=20\;\mathrm{m}\cdot \mathrm{s}^{-1}$. When he arrives to the second ship he turns around and flies back now with the wind behind his back with a velocity $v_{2}=30\;\mathrm{m}\cdot \mathrm{s}^{-1}.He$ keps on flying back and forth until the two ships meet. How long is the path that he has undertaken?

Mirek was improving tasks from elementary school.

### (3 points)4. Series 26. Year - 3. A rubber duck

A passanger on a ferry forgot to set the parking brake. Assume that the axis of the car is aligned with the axis of the ferry, and that because of waves the ferry is undergoing a harmonic motion, *i.e.* $φ(t)=Φ\sin\left(ωt)$. How far from the edge of the ferry can the passenger park the car without worrying about it falling into the sea? Assume that the maximal amplitude of oscillations is slowly increasing from zero to Φ.

Lukáš and Jáchym were brainstorming about the physics of everyday hygiene.

### (5 points)4. Series 26. Year - P. Mrazík

In the fairy tale Mrazik, Ivan fought several bandits, stole their clubs, and threw them so high up into the sky that they did not fall back until half a year later. What is the altitude the clubs had to reach in order to stay in the air for so long? Make a first guess and then go on and improve it. Carefully analyze all the approximations you made and explain why are these estimates most likely wrong. Furthermore, explain why it makes no sense for the clubs to fall back at the same spot where Ivan threw them.

Lukáš was watching fairy tales.

### (2 points)3. Series 26. Year - 2. Marble

If you throw a small marble of diameter r from a very tall building, it appears to become smaller as it falls down. What is the time dependence of the apparent radius if, initially, it is at rest and a distance $x_{0}$ away from your eyes. Also assume that you are watching the marble directly from above at all times. Feel free to neglect any friction forces acting on the marble.

Karel was bored.