1. Series 36. Year

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Upload deadline: 11th October 2022 11:59:59 PM (local time in Czech Republic)

1. useful butter

Jarda decided to bake a cake but he found out that the battery in his kitchen scale was dead, so he can't weigh $300 \mathrm{g}$ of flour. However, he had the idea that he could use a block of butter instead. The packaging said its weight is $m = 250 \mathrm{g}$. Fortunately, he found a suitable spring and a stopwatch. He put a heap of flour in a very light bowl, attached it to the spring, perturbed it and measured the period of oscillations $T_1=2,8 \mathrm{s}$. He repeated the same process with the cube of butter and measured $T_2 = 2,3 \mathrm{s}$. How much flour does Jarda need to add or remove?

2. we are weighing an unknown object

Let us have an ideal scale which we calibrate with a state standard (ethanol) with a mass $m\_e = 1,000~000~165 kg$ and a density $\rho \_e = 21~535,40 kg.m^{-3}$. By calibration, we mean that after placing the standard on the scale, we assign to the measured value the mass $m\_e$. The unknown object is weighed under the same conditions in which the volume is $V_0 = 3,242~27 dl$. What mass did we measure if we weighed the weight $G = 1,420~12 N$? What is the actual mass of the object? We conduct the experiment at a place with standard gravitational acceleration $g = 9,806~65 m.s^{-2}$ and air density $\rho \_v=1,292~23 kg.m^{-3}$. Take into account that the calibration is linear and that the unloaded scale shows zero.

3. canning jam

A cylindrical jar made of glass has a height $h = 7,0 \mathrm{cm}$ and an inner radius $r = 2,5 \mathrm{cm}$. We pour hot apricot jam at temperature $T_0 = 80 \mathrm{\C }$ into the jar, we close the lid and let it cool down. Note that we didn't fill the jar to the top, but left some air between the jam and the lid. If a force of at least $F = 4 \mathrm{N}$ is applied, a sound is heard as the lid suddenly incurves. We heard this sound $t\_i = 30 \mathrm{min}$ after the jar had been closed. If jam hardens at temperature $T\_h = 60 \mathrm{\C }$, was it to be already hard when the lid incurved?

Bonus: How long after closing the jar will the jam harden? Assume that the temperature is evenly distributed throughout the jar and that the cooling rate only depends on the difference in temperatures of the jar and its surroundings $T\_{s} = 25 \mathrm{\C }$.

E. dense ice

Measure the density of ice.