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04-BS-7 · December 2019

Question 11 of 13: Air Pressure in a Submerged Cylinder — Two Filling Sequences

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

04-BS-7 Mechanics of Fluids — December 2019 (National Examinations, three hours, closed book). Section A (Calculative) offers 9 questions and instructs “do seven”; Section B (Analytical) offers 4 questions and instructs “do three.” Every question is answered below (13 of 13), so students can use the full paper as a study resource. Constants used throughout (from the paper's own Constants page): g = 9.81 m/s², patm = 100 kPa, ρwater = 1000 kg/m³, SGglycerine = 1.26, SGmercury = 13.56, ρconcrete = 2400 kg/m³, ρair = 1.19 kg/m³ (20°C) / 1.21 kg/m³ (15°C), μwater = 1.0×10⁻³ N·s/m², μair = 1.8×10⁻⁵ N·s/m², Rair = 287 J/kg·K.

Reference texts: F. M. White, Fluid Mechanics, 8th ed. (McGraw-Hill) — fluid statics and manometry (Ch. 2), hydrostatic forces and the middle-third rule (Ch. 2), buoyancy and equilibrium (Ch. 2), dimensional analysis and drag (Ch. 5, 7), pipe friction and the Moody/Colebrook relation (Ch. 6), control-volume momentum and propeller/actuator-disk theory (Ch. 3, 11); J. D. Anderson, Fundamentals of Aerodynamics — wave/compressibility drag divergence (Ch. 5) for the Boeing 747 wind-tunnel chart used in Question 9.

Question 11: Air Pressure in a Submerged Cylinder — Two Filling Sequences (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A sealed-bottom, closed-top cylinder trapping air, immersed to the same final water depth h under two different filling histories (A: tank fills around an already-seated, sealed cylinder trapping its original air volume at atmospheric pressure and full height; B: the cylinder is pushed down through already-standing water, compressing the trapped air as it descends).

Condition A trapped air, full h Condition B trapped air, compressed
Condition A: cylinder seated first, tank fills around it, trapped air stays at its original (full-height) volume. Condition B: cylinder pushed down through standing water, compressing the trapped air into a smaller volume before it seats.

Find. Whether the trapped air pressure in Condition B is less than, equal to, or greater than in Condition A, and how each could be determined.

The air pressure inside the cylinder in Condition B will be GREATER than in Condition A. In Condition A, the cylinder is seated on the (initially dry) tank bottom BEFORE any water is added; the air trapped inside is at atmospheric pressure and fills the cylinder's ENTIRE internal volume from the very start, because there is no water yet present to push up into the open bottom. As the tank subsequently fills to depth h around the outside of the sealed cylinder, the air trapped inside cannot escape and cannot be compressed either (the seal at the bottom rim prevents water from entering once the cylinder is seated) — so the trapped air remains at its original volume, though the water pressure of the surrounding tank still determines the NET force balance on the closed top.

In Condition B, the cylinder starts in air above an already-filled tank and is pushed DOWN through the standing water. As it descends, water tries to rise into the open bottom, but since the top is closed and sealed, the trapped air cannot escape — it is instead progressively COMPRESSED into a smaller volume as the cylinder is forced deeper, exactly like an inverted glass pushed underwater. By the time the cylinder reaches the same final seated position (same depth h), its trapped air occupies a SMALLER volume than in Condition A (some of the original air space has been displaced by water that entered before the seal made contact with the bottom), and by Boyle's law, compressing a fixed mass of gas into a smaller volume at constant temperature necessarily RAISES its pressure above the original atmospheric value it started at.

Determining each pressure. In Condition A, since the air volume never changes, the trapped air remains at exactly atmospheric pressure; the pressure could be confirmed by applying hydrostatics to the water column outside and checking force balance on the closed top (a simple gauge measurement would also read it directly). In Condition B, the final air pressure is found from Boyle's law, $p_{atm}V_{original}=p_BV_{final}$, where $V_{final}$ is determined from how far up the water rose inside the cylinder before it seated — equivalently, the final trapped-air pressure must also satisfy hydrostatic equilibrium at the internal air-water interface, giving a second, independent equation that can be solved simultaneously with Boyle's law for both the final air volume and its pressure.