water resource engineering · Physics Brawl 2024 #5
Physics · fluid-mechanics
Problem
Water should be conserved, which is why we collect rainwater into a barrel. Water flows into the barrel via a vertical gutter, which collects water from the roof. When the barrel gets full, water is wastefully spilled into the surrounding garden. We would like to redirect the water flow outside the barrel when it gets full, instead of overfilling the barrel. Verča's dad is an engineer (just like she will be one day) and invented a clever machine to solve this problem. On the surface floats a plastic cylinder with a base of area S=100cm2 and density ρ=550kg⋅m−3, connected via rigid stick to the gutter end. When the surface rises, it pushes the cylinder up, eventually closing the gutter end.
What is the minimal height of the cylinder in order for the buoyancy to be strong enough to close the gutter end while being heavy enough to let the end open as the surface lowers? Assume that the force F=15N is required in order to open or close the gutter end (force closing up, for opening down), and otherwise the system cylinder-gutter end does not move.