Series 4, Year 32

Consider a hollow cube with edge of $a = 20 \mathrm{cm}$ filled with air. Air as well as the enviroment has a temperature of $t_0 = 20 \mathrm{\C }$. We will cool down the air inside the cube to the temperature of $t_1 = 5 \mathrm{\C }$. Find the force acting on each of the cube's side. The cube has got a fixed volume. The pressure outside of the cube equals $p_0 = 101{,}3 \mathrm{kPa}$.

Suppose a massless string of length $l$ with a point-like mass $m$ attached to its end. We know that the maximum allowed tension in the string is equal to $F = mg$, where $g$ is the gravitational acceleration. We will attach the string to the ceiling and we hold the mass in the same height with the string straight but unstrained. Then, we will release the mass and it begins to move. Find the angle (with respect to the vertical) for which the string will break.

Matěj likes levitating things and therefore he bought an infinite non-conductive charged horizontal plane with the charge surface density $\sigma $. Then he placed a small ball with given mass $m$ and charge $q$ above the plane. For which values of $\sigma $ will the ball levitate above the plane? What is the corresponding height $h$? Assume that the gravitational acceleration $g$ is constant.

Two point-like masses were jumping on a trampoline into height $h_0 = 2 \mathrm{m}$. While they both were in the lowest point of the trajectory (corresponding displacement of $y = 160 \mathrm{cm}$), one of them suddenly disappeared. What is the maximum height, which was the other point-like mass bounced into? A round trampoline has circumference of $o = 10 \mathrm{m}$ and is held by $N = 42$ springs with stiffness $k = 1720 \mathrm{N\cdot m^{-1}}$. Trampoline may be modelled by $N$ springs uniformly attached around the circle and connected in the middle. Mass of the disappeared mass is $M = 400 \mathrm{kg}$.

A thin homogeneous disc revolves on a flat horizontal surface around a circle with the radius $R$. The velocity of disk's centre is $v$. Find the angle $\alpha $ between the disc plane and the vertical. The friction between the disc and the surface is sufficiently large. You may work under the approximation where the radius of the disc is much less than $R$.

The interstellar space is not empty but contains an insignificant amount of mass. For simplicity, assume hydrogen only and look up the required density. Could we build a spaceship that would „suck in“ the hydrogen and would use energy from it? How fast/large would the spaceship have to be in order to keep up the thermonuclear fusion only from the acquired hydrogen? What reasonable obstacles in realization should we consider?

Measure the shielding of the sound by paper. As an experimental tool, you can use mobile phones as a sound generator and microphone in the computer as a sound detector (Audacity). Use papers of various kinds and shapes.

1. Show that in an arbitrary central-force field, i.e. a force field where the potential only depends on distance (not on angular position), a particle will always move in a plane. Instructions:: Set up Lagrangian equations of the second kind for this situation using appropriate generalized coordinates.. Then, set the coordinate $\theta = \pi /2$ and initial velocity in the direction of this coordinate equal to zero. Think about and explain why this choice of coordinates does not cause a loss of generality.
2. Set up the Lagrangian for a mass point moving in a plane in a central-force field. Find all the integrals of motion for this Lagrangian and use them to find the first orded differential equation for the variable $r$.
3. Think about how to find the angular distance between two points on a sphere, given their spherical coordinates. Check your solution on the stars Betelgeuse and Sirius. Hint:: This problem can be easily solved even without the knowledge of spherical trigonometry.