Buoyancy and Fluid-Column Oscillations
For small vertical displacement x of a prismatic floater, the incremental buoyant force is -rho g A x, giving omega^2 = rho g A/M; analogous hydrostatic stiffness applies to liquid columns.
Why this shows up in the exam
Floating cylinders, cubes, and bottles · Partly submerged spring-supported bodies · Liquid oscillations in tubes
Learn the idea
A displaced floating body or liquid column oscillates because buoyancy or hydrostatic pressure changes linearly. Pushing a floater down displaces extra liquid and creates extra upthrust; moving a liquid interface creates a pressure imbalance that pulls it back.
🧠 Memory hook: Extra displaced volume creates the restoring buoyancy.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- Delta F_b = -rho g A x — incremental buoyant restoring force for constant area
- omega = sqrt(rho g A/M) — vertical floater angular frequency
- omega² = k_hydro/m_eff — general hydrostatic stiffness form
How to approach it
- 1Find the equilibrium displaced volume
- 2Perturb by a small signed displacement
- 3Calculate incremental pressure or buoyancy and divide by moving inertia
Common slip-ups that cost marks
- •Using total buoyancy instead of its increment
- •Ignoring the moving liquid's effective mass
- •Using a changing cross-section as though it were constant
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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