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## oscillations

### (9 points)6. Series 34. Year - 5. heavy spring

Let us have a homogeneous spring with stiffness $k$, mass $m$ and its width negligible compared to its length. We grip the spring at one end in a way that it can rotate around and then we spin it with angular velocity $\omega $. By how much does the spring prolong (compared to its initial length) due to the rotation? Neglect the effect of the gravitational field.

Jachym had a very difficult day and wanted to share it with others.

### (10 points)6. Series 34. Year - S. charged chord

Assume a charged chord with linear density $\rho $, uniformly charged with linear charge density $\lambda $. The tension in the chord is $T$. It is placed in a magnetic field of constant magnitude $B$ pointing in the direction of the chord in equilibrium. Your task is to describe several aspects of the chord's oscillations. First, we want to write the appropriate wave equation. Neglect the effects of electromagnetic induction (assume the chord to be a perfect insulator; that also means the charge density does not change) and find the Lorentz force acting on an unit length of the chord for small oscilations in both directions perpendicular to the equilibrium position. Use this force to write the wave equation (which will also include the effects of the tension). Apply the Fourier substitution and determine the disperse relation in the approximation of a weak field $B$; more specifically, neglect the terms that are of higher than linear order in $\beta = \frac {\lambda B}{k \sqrt {\rho T}} \ll 1$, where $k$ is the wavenumber. Find two polarization vectors, this time neglect even the linear order of $\beta $. Now suppose that in a particular spot on the chord, we create a wave oscilating only in one specific direction. How far from the original spot will be the wave rotated by ninety degrees from the original direction?

Štěpán was nostalgically remembering the third serial task.

### (10 points)4. Series 34. Year - S. Oscillations of carbon dioxide

We will model the oscillations in the molecule of carbon dioxide. Carbon dioxide is a linear molecule, where carbon is placed in between the two oxygen atoms, with all three atoms lying on the same line. We will only consider oscillations along this line. Assume that the small displacements can be modelled by two springs, both with the spring constant $k$, each connecting the carbon atom to one of the oxygen atoms. Let mass of the carbon atom be $M$, and mass of the oxygen atom $m$.

Construct the set of equations describing the forces acting on the atoms for small displacements along the axis of the molecule. The molecule is symmetric under the exchange of certain atoms. Express this symmetry as a matrix acting on a vector of displacements, which you also need to define. Furthermore, determine the eigenvectors and eigenvalues of this symmetry matrix. The symmetry of the molecule is not complete – explain which degrees of freedom are not taken into account in this symmetry.

Continue by constructing a matrix equation describing the oscillations of the system. By introduction of the eigenvectors of the symmetry matrix, which are extended so that they include the degrees of freedom not constrained by the symmetry, determine the normal modes of the system. Determine frequency of these normal modes and sketch the directions of motion. What other modes could be present (still only consider motion along the axis of the molecule)? If there are any other modes you can think of, determine their frequency and direction.

Štěpán was thinking about molecules

### (10 points)3. Series 34. Year - S. electron in field

Consider a particle with charge $q$ and mass $m$, fixed to a spring with spring constant $k$. The other end of the spring is fixed at a single point. Assume that the particle only moves in a single plane. The whole system exists in a magnetic field of magnitude $B _ 0$, which is perpendicular to the plane of movement of the particle. We will try to describe possible modes of oscillation of the particle. Start by the determination of equations of motion - do not forget to include the influence of the magnetic field.

Next assume that the particle oscillates in both of the cartesian coordinates of the particle and carry out Fourier substitution - substitute derivatives by factors of $i \omega $, where $\omega $ is the frequency of the oscillations. Solve the resultant set of equations in order to determine the ration of the amplitudes of oscillations in both coordinates and the frequency of oscillations. The solution obtained in this way is quite complicated, and better physical insight can be gained in a simpler case. From now on, assume that the magnetic field is very strong, i.e. $\frac {q ^ 2 B _ 0 ^ 2}{m ^ 2} \gg \frac {k}{m}$. Determine the approximate value(s) of $\omega $ in this case, always up to the first non-zero order. Next, sketch the motion of the particle in the direct (i.e. real) space in this (strong field) case.

Štěpán wanted to create a classical diamagnet.

### (10 points)2. Series 34. Year - S. series 2

Consider a circuit with a coil, a capacitor, a resistor and a voltage source connected in series (i.e. they are not parallel to each other). The coil has an inductance $L$, the capacitor has a capacitance $C$ and the resistor has a resistance $R$. The voltage source creates a voltage $U = U_0 \cos \(\omega t\)$. Assume all devices to be ideal. Using the law of conservation of energy, write the equation relating the charge, the velocity of the charge (current $I$) and the acceleration of the charge (rate of change of the current $I$). This is an equation of a damped oscillator. Compared to the equation of damped oscillations of a mass on a spring, what are the quantities analogous to mass, stiffness of the spring and friction? Find the natural frequency of these oscillations.

Furthermore, using the quantities $L$, $R$ and $\omega $, find the capacity $C$ which causes a phase shift of the voltage on the capacitor equal to $\frac {\pi }{4}$. What is the amplitude of the voltage on the capacitor, assuming this phase shift?

Non-mechanical oscillations are oscillations as well.

### (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 horizontal coordinate as displacement and appropriate units of horizontal momentum. Neglect kinetic energy of cube motion in the vertical direction and assume it remains in contact with the terrain.

Štěpán found a few basic oscillators.

### (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.