Problem Statement of Series 3, Year 32

Mikulas put bananas into a carry bag in a grocery store and before he had weighted them, he got an idea. If he fills the bag with helium instead of air, the bananas will weigh less. Mikulas bought the helium in a sale - one CZK for a litre at standard pressure. Calculate the prize of the bananas so that this "bluff" pays off.

Bonus: Find a gas for which it would pay off when the price of bananas is 30 CZK per kilogram. Do not forget to cite references.

It is 2 am and Jáchym is going to make a coffee. He places a kettle with the heat capacity of $C_k$ on a hot plate, which is made of a cast-iron cylinder of a radius $r$ and of height $h$. The kettle contains water with a volume of $V$ with an initial temperature of $T\_v$. The rest of the system has got an initial temperature of $T\_s$. What is the overall efficiency (ratio of energy absorbed by water vs energy input) of water heating from its initial temperature $T = 100 \mathrm{\C }$ $(T\_s, T\_v < T).$ Assume, that the heat transfer is very fast and therefore there is no heat loss. You can estimate the unknown values or find them in physics tables.

What would be the diameter of a Dyson sphere that would surround a star with the luminosity of the Sun, so the temperature on the outer surface of the sphere is $t= 25 \mathrm{\C }$?. Don't consider the presence of the atmosphere in the Dyson sphere. A Dyson sphere should be a relatively thin concave structure of spherical shape surrounding the star.

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 $T = |BIr|$.}

Consider a point-like particle in one-dimensional space. Initially, the particle is in rest at the origin of coordinates. It can be moved with acceleration from interval $\left (- a , a\right )$. Let $M\left (t\right )$ be a set of all possible physical states $\left (x, v\right )$ of positions $x$ and velocities $v$, which particle can achieve after time $t$ is elapsed. If we plot set $M\left (t\right )$ into $v(x)$ coordinate system we get surface $S\left (t\right )$. Find analytic expression for boundaries of $S\left (t\right )$.

Bonus: Find area of $S\left (t\right )$ as a funcion of time.}

Last battery percentages in your mobile phone are almost gone, your power bank is dead, or you left it at home and 230 is also not in the sight. Wouldn't it be awesome if you could have your own source of electrical energy with you all the time?

• Suggest several different tools, which would be able to produce electrical energy just from your body resources.
• Discuss their maximum power and efficiency. What devices could you supply with electricity using this method?
• Discuss its effects on your health and physical condition. Which body organs would fail first?

As a possible solution, consider a system of small turbines located in your bloodstream. Support all arguments with accurate calculations.

Find an electrolytic capacitor and a resistor and measure their capacity and resistance, respectively. You cannot measure these quantities directly. We recommend a choice of parameters, such that $RC\approx 20 \mathrm{s}$.

Be aware of maximum allowed voltage on the capacitor and the capacitor's polarity.

1. Suppose we have a horizontal plane with a small hole. Through this hole goes a rope with length $l$ on which a weight of mass $M$ is hung. You may consider the weight to be a mass point. One the other end of the rope there is a second mass point with mass $m$. The rope between them is stretched thanks to the weight of mass $M$. Initially, the whole setup is in rest while the part of the rope below the plane is vertical. Then we grant the mass point on the plane velocity $v$ in a horizontal direction perpendicular to the rope as we let the system go free. Neglect all friction in this problem. Choose appropriate coordinates and find the Lagrangian for this situation.
   
2. Suppose we have an iron rod bent to a shape of a parabola given by the equation $y = x^2$. The gravity of Earth points in the negative direction of the $y$ axis. A mass point of mass $M$ can move freely along the parabola. A second mass point with mass $m$ is connected to the first by a rigid rod of length $l$. This way we have created a pendulum with a hinge sliding along the rod. The system can move only in the plane of the parabola. Find appropriate generalized coordinates and the Lagrangian for this situation.
   
3. Suppose we have line along which slides a mass point with mass $m$ (without friction). The angle between the line and the horizontal plane is $\alpha $. Find appropriate generalized coordinates and the Lagrangian for this situation. Then set up Lagrangian equations of the second kind, double-integrate them and find the solution. Do not forget about the constants of integration and explain their physical meaning. What will be their values if the mass point starts at rest at the height $h$?