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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 $R_1 = 4~300 km$, density $\rho _1 = 4~100 kg.m^{-3}$ and siderial period $T_1 = 14 \mathrm{h}$. Goiteía is smaller – it has the radius $R_2 = 3~800 km$, but it has a higher density $\rho _2 = 4~500 kg.m^{-3}$ 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.

Find the magnitude of the friction force between the plunger and the barrel of a syringe.

How would aviation work on other planets (with atmosphere)? Consider mainly jet aircraft. Which planetary parameters would influence the aviation positively and which negatively, compared to Earth's?

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.

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.

Little Joe the mouse likes to catapult himself from the edge of a fan propeller by simply releasing his grip at the right time and flying away. When should he do it in order to fly as far as possible? The propeller blade has a length $l$ and rotates with an angular velocity $\omega $, while the plane of rotation is perpendicular to the horizontal plane. The center of rotation is at a height $h$ above the ground.

Two spaceships move towards each other on a straight line. The initial distance between them is $d$. The first one moves with the velocity $v_1$, the second with the velocity $v_2$ (in the same reference frame). The first one can reach the maximal acceleration $a_1$, the second one $a_2$ (both regardless of the direction). Their crews want to exchange some „goods“. In order to do that, the spaceships need to meet – i. e. they must be at the same time at the same place and have the same speed. What is the minimal time for them to reach the meeting? Neglect the relativistic effects.

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.

Cathy and Catherine are watching a ship which is sailing with a constant speed towards a port. Cathy is standing on a rock above the port and her eyes are $h_1=20 \mathrm{m}$ above the surface of the water. Catherine is standing under the rock and her eyes are $h_2=1{,}7 \mathrm{m}$ above the surface of the water. If Catherine sees the top of the incoming ship $t=25 \mathrm{min}$ after Cathy sees it, what is the time of arrival of the ship to the port? Assume that the Earth is a perfect sphere with a radius $r=6378 \mathrm{km}$.

Karel's car, going at the initial speed of $v_0$, can stop at a distance $s_0$ with the constant braking force $F_0$. How many times will the braking distance increase if the initial speed doubles and the braking force stays the same? How many times must the braking force be greater for the car to stop at distance $s_0$ with the initial speed $2v_0$?