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## oscillations

### (8 points)2. Series 33. Year - 5. wheel with a spring

We have a perfectly rigid homogeneous disc with a radius $R$ and mass $m$, to which a rubber band is connected. It is fixed by one end in distance $2R$ from an edge of the disc and by the other end at the end of the disc. The rubber band behave as ideal, thin spring with stiffness $k$, rest length $2R$ and negligible mass. Disc is secured in the middle, so it is able to rotate in one axis around this point, but cannot move or change the rotation axis. Figure out relation between the magnitude of moment of force, by which the rubber band will be increasing or decreasing the rotation of disc depending on $\phi $. Also, figure out an equation of motion.

**Bonus:** Define the period of system's small oscillations.

### (9 points)6. Series 32. Year - 5. elastic cord swing

Matěj was bored by common swings, which are at playgrounds because you can swing on only forward and backwards. Therefore, he has invented his own amusement ride, which will move vertically. It will consist of an elastic cord of length $l$ attached to two points separated by distance $l$ in the same height. If he sits in the middle of the attached cord, it will stretch so that the middle will displace by a vertical distance $h$. Then, he pushes himself up and starts to swing. Find the frequency of small oscillations.

Matěj wonders how to hurt little children at playgrounds.

### (12 points)5. Series 32. Year - E. thirty centimeters tone

Everyone has ever tried out of boredom to strum on a long ruler sticking out of the edge of a school-desk. Choose the right model of frequency versus the part of the length of the ruler which is sticking out and prove it experimentally. Also, describe other properties of the ruler.

**Note:** Allow vibration only for outsticking part of the ruler by fixing its position above the table.

Michal K. found a ruler

### (12 points)4. Series 31. Year - E. heft of a string

Measure the length density of the catgut which arrived to you together with the tasks. You are forbidden to weigh the catgut.

**Hint:** You can try to vibrate the string.

Mišo wondered about catguts on ITF.

### (8 points)0. Series 31. Year - 5.

### (8 points)5. Series 30. Year - 4. on a string

Two masses of negligible dimensions and mass $m=100g$ are connected by a massless string with rest length $l_{0}=1\;\mathrm{m}$ and spring constant $k=50\;\mathrm{kg}\cdot \mathrm{s}^{-2}$. One of the masses is held fixed and the other rotates around it with frequency $f=2\;\mathrm{Hz}$. The first mass can rotate freely around its axis. At one point the fixed mass is released. Find the minimal separation of the two masses during the resulting motion. Do not consider the effects of gravity and assume the validity of Hook's law.

### (3 points)3. Series 29. Year - 3. will it jump?

Consider a massless spring with spring constant $k$. Weights are attached to both ends with masses $m$, and $Mrespectively$. This system is placed on a horizontal surface so that weight of mass $Mlies$ on the surface and the spring with the second weight points up. The system is in equilibrium (i.e. top weight does not oscillate) and length of the spring in this state is $l$. How much do we have to compress the spring so that the weight of mass $M$ jumps up when it is released? Consider only vertical motion.

### (8 points)1. Series 29. Year - E. small g

Measure the local gravitational acceleration with at least two different methods. Then compare these two methods in detail.

Viktor heard the complaint of the participants that they don't want to constantly be knee deep in water.

### (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.

### (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.