Series 1, Year 34

Find the refractive index of a transparent plane-parallel plate of thickness $d=1 \mathrm{cm}$, such that it will take one year for the light to pass through it. Discuss whether such a situation is possible.

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 $2v_0$?

Vašek rides his bicycle in windy weather. When he rides straight with the velocity $v = 10 \mathrm{km\cdot h^{-1}}$, he measures that the wind blows at an angle $25\dg $ from the direction of Vašek's direction of travel. When he accelerates to $v' = 20 \mathrm{km\cdot h^{-1}}$, the angle is only $15\dg $. Find the velocity and direction of the wind with respect to stationary observer.

A solar sail with the surface area of $S = 500 \mathrm{m^2}$ and area density $\sigma =1,4 \mathrm{kg\cdot m^{-2}}$ is located at the distance of $0,8 \mathrm{au}$ from the Sun. What force does the solar radiation act on the sail at the beginning of the sail's motion? What is the acceleration of the sail at that moment? The luminosity of the Sun is $L_{\odot } =3,826 \cdot 10^{26} \mathrm{W}$. Assume that the radiation approaches the sail from a perpendicular direction and scatters elastically. Hint: We recommend you find the acceleration for small initial velocity $v_0$ and then let $v_0 = 0$.

Let us have a ball with the radius $R$ and a circular massless rubber band with the radius $r_0$ and stiffness $k$, while $r_0 < R$. The coefficient of friction between the band and the ball is $f$. Find conditions which ensure that it is possible to stretch the band over the ball single-handily (i.e. we are allowed to touch the band in only one point.

To keep it simple assume that the band is elastic only in the tangential direction (it is planar).

Different movies create different conceptions of what and how fast happens when an astronaut's space suit suddenly gets torn. Some of them are even contradictory. Explain what is most likely to happen, if a healthy person finds himself unprotected in a vacuum. What phenomenon is most likely to cause death first?

Measure the dependence of the diameter of a crater, created by the impact of a stone into a suitable sandpit, on the weight of the stone and the height it is released from. Does the size of the crater depend only on the energy of the impact? Dry sand is recommended for this measurement.

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.

1. 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.
2. 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 $k L \gg m g$.
3. 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.