Problem Statement of Series 6, Year 34

Assume a figure skater, rotating around her transverse axis with her arms spread with an angular velocity $\omega$. Find her angular velocity ${\omega }^{\prime }$, that she will rotate with her arms positioned close to her body. What work does she have to perform in order to get her arms close to her body? Finding a proper approximation of the figure skater's body is left to the reader.

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).

At the beginning of a segment of a road with a length $a=2.8\mathrm{km}$, there is a traffic light with the period $T$, on which green light has a duration $t_1=79\mathrm{s}$. At the end of this segment, there is another traffic light with the same period, but on which the green light has a duration $t_2=53\mathrm{s}$. On both of the traffic lights the green light always switches on at the same moment. Calculate the average time it takes to travel the whole path, including passing both of the traffic lights, if you are moving at the speed $v=60\mathrm{km}\cdot {\mathrm{h}}^{-1}$ between the lights. Neglect the time it takes to accelerate and decelerate.

Long-period and non-periodic comets usually begin the outgassing process when they reach the orbit of Saturn. Until that, they appear only as small rocks to an observer on Earth, and therefore they are almost unobservable. Assume a comet with the perihelion distance $q=0.5\mathrm{au}$. Estimate the time that it takes for the comet to reach the Earth's orbit once it passes the orbit of Saturn. The eccentricity of the trajectory of the comet is very close to one.

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.

When there is a coronal mass ejection from the Sun, the mass will start to propagate with high velocity through the space. Sometimes the mass can hit the Earth and affect its magnetic field. Estimate the magnitude of the electric currents in the electric power transmission network on Earth which could be generated by such ejection. What parameters does it depend on? Comment on what effects would such event have on the civilisation.

Take a glass, can or any other cylindrically symmetrical container. Measure the relationship between the angle of inclination of the container when it tips over and the amount of water inside of it. We recommend to use a container with greater ratio of its height to the diameter of its base.

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?