ME 310 Chap.4 Buoyancy, Stability and Rigid-Body Fluid Motion
Buoyancy, Stability and Rigid-Body Fluid Motion
Buoyancy from the pressure field
The buoyancy lecture closes the hydrostatics sequence by showing that the upward force on an immersed body is not a separate law. Pressure on the lower surfaces exceeds pressure on the upper surfaces, and the vertical resultant equals the weight of the displaced fluid, acting through the centroid of the displaced volume.
A floating object settles until that force equals its own weight.
Will it stay upright?
Stability asks what happens after a small tilt. A fully submerged body is stable only when its centre of gravity lies below its centre of buoyancy. A floating body is more forgiving, because tilting shifts the centre of buoyancy toward the side that dips.
The shift is measured by the metacentric height: the second moment of the waterline area divided by the displaced volume, minus the distance from buoyancy centre to gravity centre.
The homework in this unit includes the stability of a slender spar buoy, a shape that must carry its weight low because its narrow waterline offers little righting effect.
Accelerating containers
The last statics lecture covers liquids that accelerate as a rigid body, with no relative motion between particles and therefore no shear. The analysis is hydrostatics with an extra body force.
For straight-line acceleration, surfaces of constant pressure tilt, and the free surface slope equals the horizontal acceleration divided by gravity plus any vertical acceleration.
Homework applies this to oil in a decelerating truck.
Rotating containers
For rotation about a vertical axis at a steady rate, pressure increases outward in proportion to radius times the square of the rotation rate, and the free surface becomes a paraboloid.
The same radial gradient acts on suspended particles: a homework problem asks where denser and lighter particles collect in a spinning cylinder, which is the principle of a centrifuge.
Homework in this unit
The third problem set applies these ideas to gates holding back water, a dam lake with submerged trees, the stability of a spar buoy, oil in a decelerating truck, a submerged road tunnel and particles in a rotating cylinder.
What this chapter covers
- 01
Archimedes' principle from pressure differences
- 02
Centre of buoyancy and floating equilibrium
- 03
Stability of submerged and floating bodies
- 04
Metacentric height and waterline area
- 05
Free-surface tilt under linear acceleration
- 06
Paraboloid free surface under rotation
- 07
Particle migration in a rotating liquid
Worked example · free
How much of an iceberg shows?
- 1Floating equilibrium: weight of ice equals weight of displaced seawater.
- 20.917 γ_w V = 1.025 γ_w V_sub, so V_sub / V = 0.917 / 1.025 = 0.895.
- 1Fraction above the surface = 1 − 0.895 = 0.105.
- 1In fresh water the ratio becomes 0.917, leaving about 8 % exposed.
Key terms
- Buoyant Force
- The buoyant force is the net upward pressure force on an immersed body, equal to the weight of the fluid it displaces.
- Centre of Buoyancy
- The centre of buoyancy is the centroid of the displaced fluid volume, where the buoyant force acts.
- Metacentric Height
- Metacentric height is the distance from a floating body's centre of gravity to its metacentre; a positive value means the body rights itself after a small tilt.
- Rigid-Body Motion
- Rigid-body motion is fluid motion in which every particle shares one acceleration or rotation, so there is no relative motion and no shear stress.
- Forced Vortex
- A forced vortex is liquid rotating like a solid body about an axis, with speed proportional to radius and a paraboloid free surface.
Buoyancy, Stability and Rigid-Body Fluid Motion FAQ
Why can a ship with its centre of gravity above the centre of buoyancy still be stable?
Because when a floating hull tilts, the submerged shape changes and the centre of buoyancy moves toward the lower side. If the new buoyancy line passes above the centre of gravity, at the metacentre, the couple rotates the hull back upright.
Does a tank of liquid on an accelerating truck need a viscosity term?
No, once the liquid moves as a rigid body. Every particle shares the same acceleration, there is no relative motion and hence no shear, so only pressure, gravity and the inertia of the acceleration enter the balance.
Where do dense particles go in a spinning container of liquid?
Outward. The radial pressure gradient supplies exactly the centripetal force a liquid particle needs, which is not enough for a denser particle, so it drifts toward the wall while lighter ones move toward the axis.
What decides whether a floating body is stable?
The metacentric height. Compute the second moment of the waterline area about the tilt axis, divide by the displaced volume, and subtract the distance between the centres of buoyancy and gravity. A positive result means a small tilt produces a righting couple.
Exam move
Replace g by the effective gravity, g minus the acceleration vector, and rigid-body problems become hydrostatics with a tilted or curved surface.
Working through Buoyancy, Stability and Rigid-Body Fluid Motion in ME 310? Sia is AskSia’s AI Engineering tutor — ask any ME 310 Buoyancy, Stability and Rigid-Body Fluid Motion question and get a clear, step-by-step explanation grounded in how ME 310 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
