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## oscillations

### (10 points)1. Series 34. Year - S. oscillating

Let us begin this year's serial with analysis of several mechanical oscillators. We will focus on the frequency of their simple harmonic motion. We will also revise what does an oscillator look like in the phase space.

- Assume that we have a hollow cone of negligible mass with a stone of mass $M$ located in its vertex. We will plunge it into water (of density $\rho $) so that the vertex points downwards and the cone will float on the water surface. Find the waterline depth $h$, measured from the vertex to the water surface, if the total height of the cone is $H$ and its radius is $R$. Find the angular frequency of small vertical oscillation of the cone.
- Let us imagine a weight of mass $m$ attached to a spring of negligible mass, spring constant $k$ and free length $L$. If we attach the spring by its second end, we will get an oscillator. Find the angular frequency of its simple harmonic motion, assuming that the length of the spring does not change during the motion. Subsequently, find a small difference in angular frequency $\Delta \omega $ between this oscillator and the one in which the spring is substituted by a stiff rod of the same length. Assume $k L \gg m g$.
- A sugar cube with mass $m$ is located in a landscape consisting of periodically repeating parabolas of height $H$ and width $L$. Describe its potential energy as a function of horizontal coordinate and outline possible trajectories of its motion in phase space, depending on the velocity $v_0$ of the cube on the top of the parabola. Mark all important distances. Use the horizontal coordinate as the displacement and use appropriate units of horizontal momentum. Neglect kinetic energy of the cube motion in the vertical direction and assume it remains in contact with the terrain.

### (5 points)4. Series 33. Year - 3. uuu-pipe

What period of small oscillations will water in a glass container (shown on the picture) have? The dimensions of the container and the equilibrium position of water are shown. Assume that there is room temperature and standard pressure and that water is perfectly incompressible.

Karel was thinking about U-pipes again.

### (12 points)4. Series 33. Year - E. torsional pendulum

Take a homogeneous rod, at least $40 \mathrm{cm}$ long. Attach two cords of the same material (e.g. thread or fishing line) to it, symmetrically with respect to its centre, and attach the other ends of the cords to some fixed body (e.g. stand, tripod) so that both cords would have the same length and they'd be parallel to each other. Measure the period of torsion oscillations of the rod depending on the distance $d$ of the cords, for multiple lengths of the cords, and find the relationship between these two variables. During torsion oscillations, the rod rotates in a horizontal plane and its centre remains still.

Karel wanted to hyponotize participants.

### (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.

Karel had a headache.

### (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.

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

### (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.