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## mechanics of a point mass

### (3 points)6. Series 34. Year - 2. rotating pendulum

Let us have a mathematical pendulum of length $l$ with a point mass $m$ in a gravitational field with the acceleration $g$. We give the pendulum a constant angular velocity $\omega $ about the vertical axis. Determine the stable positions of the pendulum (expressed as a function of the angle between the pendulum and the vertical).

Jindra wanted to swing on a wrecking ball with a hammer in his hand.

### (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.

### (7 points)5. Series 34. Year - 4. period of large oscillations

Assume two half-planes with the angle $2\phi < \pi $ between them. We place them so that the line at their intersection is horizontal and their plane of symmetry is vertical, so they form a kind of valley. Then we take a mass point and throw is with the velocity $v$ from the height $h$ (above the intersection line) in the horizontal direction so that it makes a periodic motion as shown in the picture. What is the magnitude of the velocity that we have to throw it with? Assume the bouncing to be perfectly elastic.

Legolas was bored by the periods of small oscillations.

### (10 points)5. Series 34. Year - 5. rheonomous catapult

Let us have a thin rectangular panel that rotates around its horizontally oriented edge at a constant angular velocity. At the moment when the panel is in a horizontal position during rotating upwards, we place a small block on it so that its velocity with respect to the panel is zero. How will the block move on the panel if the friction between them is zero? Where do we have to place the block so that it flies away from the panel exactly after a quarter of its turn? Discuss all the necessary conditions that must be met to achieve this. **Bonus:** What power does the panel transfer on the block and what total work does it do on it?

Vašek was tired of problems with scleronomous bond, so he came up with rheonomous bond.

### (3 points)4. Series 34. Year - 1. two water drops

Suppose two water drops fall in quick succession from a water tap. How will their respective distance change with time? Neglect air resistance.

**Bonus:** Do not neglect air resistance, make an estimate of all relevant parameters and find the distance of the water drops after a very lo

Karel was hypnotized by water.

### (3 points)4. Series 34. Year - 2. there is always another spring

Find the work needed to twist a spring from equilibrium position to an angular displacement of $\alpha =60\dg $. We are holding the spring in the twisted position with a torque $M=1{,}0 \mathrm{N\cdot m}$.

Dodo was hanging laundry on a string.

### (8 points)4. Series 34. Year - 5. Efchári-Goiteía

Efchári and Goiteía are two components of a double planet around recently arisen stellar system. They orbit around a common centre of mass on circular trajectories in the distance $a = 250 \cdot 10^{3} \mathrm{km}$. Efchári has the radius $R_1 = 4\;300 \mathrm{km}$, density $\rho _1 = 4\;100 \mathrm{kg\cdot m^{-3}}$ and siderial period $T_1 = 14 \mathrm{h}$. Goiteía is smaller – it has the radius $R_2 = 3\;800 \mathrm{km}$, but it has a higher density $\rho _2 = 4\;500 \mathrm{kg\cdot m^{-3}}$ and a shorter period $T_2 = 11 \mathrm{h}$. Rotation axes of both planets and the system are parallel. After several hundred years, the system transfers due to tidal forces into so-called tidal locking. Find the resulting difference in the period of the system, assuming that both bodies are homogeneous and roughly spherical.

Dodo keeps confusing Phobos and Deimos.

### (13 points)4. Series 34. Year - E. breathtaking syringes

Find the coefficient of friction between the plunger and the barrel of a syringe. Please use a syringe without a needle, we don't want you to get hurt :)

Dano remembered his trip to Russia.

### (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

### (3 points)3. Series 34. Year - 2. bungee

Jirka and Káťa want to try bungee-jumping. To jump from a height of $h = 100 \mathrm{m}$ they have ideally elastic rope with a length of $l=40 \mathrm{m}$, which is calibrated so that when Káťa with a weight of $m\_K=50 \mathrm{kg}$ jumps with it, she will stop at the height of $h\_K=16 \mathrm{m}$ above the ground. Is this rope safe to use for Jirka if he weights $m\_J=80 \mathrm{kg}$? Neglect the air resistance and the heights of Káťa and Jirka.

Jirka's dorm room is inspiringly high.