FYKOS Year 36 Series 1 Problem 3 — canning jam · FYKOS 36 #36.1.3
Physics · mechanics · thermodynamics
Problem
A cylindrical jar made of glass has a height and an inner radius . We pour hot apricot jam at temperature 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 is applied, a sound is heard as the lid suddenly incurves. We heard this sound after the jar had been closed. If jam hardens at temperature , 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 .Intuitive explanation
The lid suddenly incurves when the air pressure inside drops enough that atmospheric pressure overcomes it—this happens as the trapped air cools and contracts. You need to find what temperature the air reached when the lid buckled (using the pressure condition), then compare it to the hardening temperature. For the bonus, use Newton's cooling law: temperature decays exponentially toward room temperature, so you can find when it hits the hardening point.
Intuitive explanation
The lid suddenly incurves when the air pressure inside drops enough that atmospheric pressure overcomes it—this happens as the trapped air cools and contracts. You need to find what temperature the air reached when the lid buckled (using the pressure condition), then compare it to the hardening temperature. For the bonus, use Newton's cooling law: temperature decays exponentially toward room temperature, so you can find when it hits the hardening point.