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## oscillations

### (10 points)6. Series 37. Year - 5. oscillating magnets

Consider two identical dipole magnets, which we fix so that they can rotate in the same plane without friction (their axes of rotation are parallel). If we deflect the magnets slightly out of their equilibrium position, they begin to oscillate. Find the eigenmodes of these oscillations and calculate their frequencies. Discuss what the motion of the magnets will be like for general initial deflection (you don't have to explicitly calculate this case). The magnets have a magnetic moment $m$, a moment of inertia about the axis of rotation $J$ and the mutual distance between their centers is $r$.

Jirka stole the problem from Výfuk.

### (12 points)4. Series 37. Year - E. pendulum in the wind

Measure the period of the torsion pendulum oscillations as a function of the length of the thread. Use at least two types of thread materials. Determine as accurately as possible all the relevant parameters on which the period depends.

FYKOS forgot an experiment.

### (10 points)5. Series 36. Year - S. ethanol or methanol?

The binding energy of a fluorine molecule is approximately $37 \mathrm{kcal/mol}$. Assuming the range of binding interactions to be approximately $3 \mathrm{\AA }$ from the optimum distance, what (average) force do we have to exert to break the molecule? Calculate the „stiffness“ of the fluorine molecule if such an average force was applied in the middle of this range. What would be the vibrational frequency of this molecule? Compare this with the experimental value of $916{,}6 \mathrm{cm^{-1}}$. ($4 \mathrm{pts}$)

Using Psi4, calculate the dissociation curve $\mathrm {F_2}$ and fit a parabola around the minimum. What value will you get for the energy of the vibrational transitions this time? ($3 \mathrm{pts}$)

You are given two bottles of alcohol that you found suspicious, to say the least. After taking them to the lab, you obtain the following Raman spectra from them. Using the Psi4 program, calculate the frequencies at which the vibrational transitions of both the methanol and ethanol molecules occur. Use this to determine which bottle contains methanol and which one contains ethanol. You can use the approximate geometries of ethanol and methanol, which are included in the problem statement on the web. ($3 \mathrm{pts}$)

Alcohol from Mikuláš?!

### (10 points)3. Series 36. Year - 5. guitar

Assume you have a guitar that is perfectly tuned at room temperature. By how many semitones (in tempered tuning) will the individual strings be out of tune if we move to a campfire, where it is cooler by $10 \mathrm{\C }$? Will the guitar still sound in tune? The distance between the string attachment points is $d = 65 \mathrm{cm}$. The strings have a density $\rho = 8~900 \mathrm{kg.m^{-3}}$, a Young's modulus of elasticity $E = 210 \mathrm{GPa}$ and a thermal expansion coefficient $\alpha = 17 \cdot 10^{-6} \mathrm{K^{-1}}$.

Honza's guitar is out of tune again.

### (3 points)1. Series 36. Year - 1. useful butter

Jarda decided to bake a cake but he found out that the battery in his kitchen scale was dead, so he can't weigh $300 \mathrm{g}$ of flour. However, he had the idea that he could use a block of butter instead. The packaging said its weight is $m = 250 \mathrm{g}$. Fortunately, he found a suitable spring and a stopwatch. He put a heap of flour in a very light bowl, attached it to the spring, perturbed it and measured the period of oscillations $T_1=2{,}8 \mathrm{s}$. He repeated the same process with the cube of butter and measured $T_2 = 2{,}3 \mathrm{s}$. How much flour does Jarda need to add or remove?

When Jarda gets kicked out of Matfyz, he will open a bakery.

### (8 points)1. Series 36. Year - 5. U-tube again

We have a U-tube with length $l$ and cross-sectional area $S$. We pour volume $V$ of water into the tube. The volume $V$ is large enough that the whole U-turn is filled with water but $Sl > V$. When water levels in both arms of the tube are at rest, we seal one of the arms. What is the period of small oscillations of water in the tube?

Karel went crazy again.

### (5 points)4. Series 35. Year - 3. pendular collisions

Two small marbles are attached to ends of strings of the same length ($l = 42,0 \mathrm{cm}$) and negligible mass. The other ends of both strings are attached to a single point. The marbles are of the same size, but they are made from the different materials. First one is made of steel ($\rho _1 = 7~840 kg.m^{-3}$) and second one is made of dural ($\rho _2 = 2~800 kg.m^{-3}$). Both marbles are initially at the angle $5\dg $ with respect to the equilibrium position, and after releasing them, they collide elastically. What is the maximum height the individual marbles reach after the collision? What is the result after the second collision?

Karel wanted to hypnotize others. You want to solve this problem \dots

### (7 points)4. Series 35. Year - 4. Analogy

Assume we have two linear springs with elastic modulus $E = 2{,}01 \mathrm{GPa}$ and a piston with viscosity $\eta = 9,8 \mathrm{GPa\cdot s}$. The dependence of stress $\sigma $ on relative extension $\epsilon $ is characterized by formula $\sigma \_s = E\epsilon \_s$ for spring, and by formula $\sigma \_d = \eta \dot {\epsilon }\_d$ for piston, where the dot represents the time derivative (Newton's notation). We connect a spring of length $l\_s$ and a piston of length $l\_d$ into series, and then we connect the other spring of length $l\_p$ in parallel to them. Abruptly, we stretch the entire system into the state of $\epsilon _0 = 0{,}2$, and we hold the extension constant. Determine, in what time (from stretching) will the stress decrease to half of the original value, if $\frac {l\_s}{l\_p} = 0{,}5$ holds.

Mirek was thinking about problems while taking an exam.

### (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.