# 3. Series 32. Year

* Post deadline: 17th December 2018*

* Upload deadline: 18th December 2018 11:59:59 PM CET*

### (3 points)1. discounted bananas

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.

### (3 points)2. efficient coffee

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.

### (6 points)3. heat in the Dyson sphere

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.

### (8 points)4. destruction of a copper loop

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 $T = |BIr|$.}

### (8 points)5. pointy

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.}

### (10 points)P. personal power bank

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.

### (12 points)E. indexed capacitor

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.

### (10 points)S. theoretical mechanics

* We are sorry. This task is only avaiable in Czech. *