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5. Series 35. Year - 3. under the lid

A lid has a shape of a hollow cylinder of radius $6,00 \mathrm{cm}$. The lid under which is an air of atmospheric pressure $1~013 \mathrm{hPa}$ is placed in a horizontal washbasin. While doing the dishes, we start filling the washbasin with water at room temperature. The water also gets under the lid and compresses the air trapped inside. At a certain moment, the lid starts floating. At what height is the water level at that moment? The lid weighs $200 \mathrm{g}$, its height is $2,00 \mathrm{cm}$ and negligible volume.

Danka was doing the dishes.

5. Series 35. Year - P. hot asteroid

Come up with as many physics reasons as possible on why an asteroid might have a higher temperature than its surroundings.

Karel was thinking about the Fermi paradox.

3. Series 35. Year - 5. blacksmith's

Gnomes decided to forge another magic sword. They make it from a thin metal rod with radius $R=1 \mathrm{cm}$, one end of which they maintain at the temperature $T_1 = 400 \mathrm{\C }$. The rod is surrounded by a huge amount of air with the temperature $T_0 = 20 \mathrm{\C }$. The heat transfer coefficient of that mythical metal is $\alpha = 12 \mathrm{W\cdot m^{-2}\cdot K^{-1}}$ and the thermal conductivity coefficient is $\lambda = 50 \mathrm{W\cdot m^{-1}\cdot K^{-1}}$. The metal rod is very long. Where closest to the heated end can gnomes grab the rod with their bare hands if the temperature on the spot they touch is not to exceed $T_2 = 40 \mathrm{\C }$? Neglect the flow of air and heat radiation.

Matěj Rzehulka burnt his fingers on metal.

1. Series 35. Year - E. Is the pasta ready?

Measure the dependency of the time it takes for water to start boiling on its volume. Repeat the measurement several times for at least five different volumes. Pay attention to the consistency of the external conditions, especially the criterion you use for assessing when the water starts boiling and the initial temperature of the water, vessel and stove. Try to explain the resulting relation.

Dodo's fight with the stove at the dormitory.

5. Series 34. Year - 3. involuntary breatharianism

Lukáš wanted to cook himself a dinner. He put a pot onto a stove, but forgot to fill it with water (or anything else). The teperature of the pot and the air inside stabilized at $100 \mathrm{\C }$ (do not ask, how he managed that without water). Lukáš realized his mistake and removed the pot from the stove. When the pot had cooled down to the room temperature, however, he was unable to remove its lid with the area $S$ and mass $m$. Calculate the force with which the lid resisted being removed if Lukáš put the lit on the pot

  1. just before removing it from the stove and,
  2. before the start of dinner preparation.

Assume the air to be an ideal gas.

Lukáš and his culinary art.

4. Series 34. Year - 4. ants

The ants have a peculiar way of keeping the anthill warm – they crawl out, let the sunlight heat them up, and then crawl back in, where the heat is transferred to the anthill. The anthill can be approximated as a cone of height $H=0{,}8 \mathrm{m}$ with base radius of $R_0=1,5 \mathrm{m}$. The walls are made of cellulose with heat conductivity $\lambda = 0{,}039 \mathrm{W\cdot m^{-1}\cdot K^{-1}}$ and are $2 \mathrm{cm}$ thick.

Assume that the entire heat exchange between the anthill and its surroundings (which have temperature $T\_o = 10 \mathrm{\C }$) is only mediated by the ants and by the conduction of heat through the walls, i.e. neglect the heat exchange with the ground. An ant weighs $m =5 \mathrm{mg}$ and has a specific heat capacity of approximately $4\;000 \mathrm{J\cdot kg^{-1}\cdot K^{-1}}$. How many ants, heated up to $T\_m = 37 \mathrm{\C }$, have to enter the anthill every second in order to keep the inner volume of the anthill at constant temperature of $T\_M = 20 \mathrm{\C }$?

Káťa missed biology classes.

3. Series 34. Year - 1. baking

While baking a gingerbread, baking soda, or more rigidly sodium bicarbonate (\ce{NaHCO3}), has to be added into the batter. Let's assume, that at high temperatures sodium bicarbonate decomposes as follows \[\begin{equation*} \ce{2 NaHCO3} \rightarrow \ce{Na2CO3} + \ce{H2O} + \ce{CO2} , \end {equation*}\] that is, into sodium carbonate, carbon dioxide and water. How much will the volume of the gingerbread increase as a consequence of creation of water steam and carbon dioxide bubbles in the batter after adding $10 \mathrm{g}$ of sodium bicarbonate? Assume that the water steam and carbon dioxide behave as ideal gases and that the batter solidifies around the bubbles at temperature $200 \mathrm{\C }$ and pressure $1\;013 \mathrm{hPa}$.

Káťa wanted to bake a cake.

3. Series 34. Year - E. diffusion

You have probably heard at school about the thermal motion of molecules such as diffusion or Brownian motion. Measure the time dependance of the size of a color spot in water and calculate the diffusion constant. Make measurements for several different temperatures and plot the temperature dependance of the diffusion constant in a graph. How could you arrange the experiment so that the temperature would stay constant during the measurement?

Káťa enjoys labs even during the quarantine.

2. Series 34. Year - 4. lifting ice using heat

A man stores small ice blocks in a well $h = 4,2 \mathrm{m}$ deep. To lift the ice up, he uses a heat engine between ice and the surrounding air with efficiency $\eta =12\%$ of the respective Carnot engine. The temperature of available air is $T\_{air}=24 \mathrm{\C }$. How cold must the ice be at the beggining in order to retrieve it with a final temperature $T_{max}=-9 \mathrm{\C }$? How is it possible even when we heat the ice up in the process?

Karel likes bizzare engines.

2. Series 34. Year - P. costly ice hockey

Estimate how much the complete glaciation of an ice hockey rink costs.

Danka doesn't like ice hockey, but she likes figure skating.

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