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Construct a hydrometer (for example from straw and plasteline) and measure dependence of water density on the concetration of salt dissolved in it.

Cutting board with thickness $t=1,0 \mathrm{mm}$ and width $d =2,0 \mathrm{cm}$ is made up of two parts. The first part has density $\rho _1 =0,20 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_1 = 10 \mathrm{cm}$, the second part has density $\rho _2 =2,2 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_2 = 5,0 \mathrm{cm}$. We place the cutting board on water surface, which density is $\rho \_v = 1{,}00 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and then we wait until it is in equilibrium position. What angle will a plane of the cutting board hold with the water surface? How big the part of the cutting board which will stay above the water level will be?

We have a perfectly rigid homogeneous disc with a radius $R$ and mass $m$, to which a rubber band is connected. It is fixed by one end in distance $2R$ from an edge of the disc and by the other end at the end of the disc. The rubber band behave as ideal, thin spring with stiffness $k$, rest length $2R$ and negligible mass. Disc is secured in the middle, so it is able to rotate in one axis around this point, but cannot move or change the rotation axis. Figure out relation between the magnitude of moment of force, by which the rubber band will be increasing or decreasing the rotation of disc depending on $\phi $. Also, figure out an equation of motion.

Bonus: Define the period of system's small oscillations.

We spin a rigid ball in the air with angular velocity $\omega $ high enough parallel with the ground. After that we let the ball fall from height $h_0$ onto a horizontal surface. It bounces back from the surface to height $h_1$ and falls to a slightly different spot than the initial spot of fall. Determine the distance between those two spots of fall onto ground, given the coefficient of friction $f$ between the ball and the ground is small enough.

Suggest several ways to slow down the Earth so that we would not have to add the leap second to certain years. How much would it cost?

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 $R$.

A copper flexible circular loop of radius $r$ is placed in a uniform magnetic field $B$. The vector of magnetic induction is perpendicular to the plane determined by the loop. The maximal allowed tensile strength of the material is $\sigma _p$. The flux linkage of this circular loop is changing in time as $\Phi (t) = \Phi _0 + \alpha t,$ where $\alpha $ is a positive constant. How long does it take to reach $\sigma _p$?

Hint: Tension force can be calculated as $T = |BIr|$.}

Before we dive into the art of analytical mechanics, we should brush up on classical mechanics on the following series of problems.

1. A homogenous marble with a very small radius sits on top of a crystal sphere. After being granted an arbitrarily small speed, the marble starts rolling down the sphere without slipping. Where will the marble separate and fall of the sphere?
2. Instead of the sphere from the previous problem, the marble now sits on a crystal paraboloid given by the equation $y = c - ax^2$. Again, where will the marble separate from the paraboloid?
3. A cyclist going at the speed $v$ takes a sharp turn to a road perpendicular to his original direction. During the turn, he traces out a part of a circle with radius $r$. How much does the cyclist have to lean into the turn? You may neglect the moment of inertia of the wheels and approximate the cyclist as a mass point.
   Bonus: Do not neglect the moment of intertia of the wheels.

Imagine a pole of length $b = 5 \mathrm{cm}$ and mass $m = 1 \mathrm{kg}$ and a spring of initial length $c = 10 \mathrm{cm}$, spring constant $k = 200 \mathrm{N\cdot m^{-1}}$ and negligible mass, that are connected at one of their ends. The other ends of the spring and the pole are affixed at the same height $a = 10 \mathrm{cm}$ from each other. The spring and the pole can both freely rotate about the fixed points and their joint. Label $\phi $  the angle of the pole to the horizontal. Find all angles $\phi $, for which the system is in an equilibrium. Which of these are stable and which unstable?

Matej was making a gun and wanted to measure what is the speed of the projectiles leaving the barrel. Unfortunately, he doesn't have any other measuring device, than a ruler. However, he found a block that is made half from steel half from wood. He lays it down at the edge of the table (of height $100 \mathrm{cm}$ and length $200 \mathrm{cm}$), and shoots at it horizontally. With the steel part of the block facing the gun, the bullet bounces off perfectly elastically and lands $50 \mathrm{cm}$ from the edge of the table. The block slides $5 \mathrm{cm}$ on the table. Then Matej turns around the block and shoots into the wooden side. This time the bullet stays in the block and the block slides only $4 \mathrm{cm}$. Help Matej with calculating the speed of the bullet. It might be also helpful to know, that when Matej lifts one edge of the table by at least $20 \mathrm{cm}$, the moving block won't stop sliding.