Buoyancy

Thrust exerted on a submerged object in a liquid. Direction is vertically upwards. Line of action passes through center of buoyancy.

Here:

• - the upthrust
• - the submerged volume
• - density of the fluid

Center of buoyancy

Center of gravity of the displaced fluid volume. NOT the center of gravity of the submerged object.

Proof

Forces exerted on the submerged object is equivalent to forces exerted on the displaced volume before it was displaced.

Consider the equilibrium of displaced volume before it was displaced:

must be equal to , opposite to and acts through of the considered volume of fluid.

Stability of fully submerged bodies

Equilibrium type	Description
Stable	is above
Unstable	is below
Neutral

Stability of floating bodies

Suppose a body of weight acting through the centre of gravity is floating in a fluid is at equilibrium. The buoyancy acts through the centre of buoyancy .

Metacentre

Intersection point between the line of action of through and the axis . Denoted by .

For small displacements is fixed in position relative to the body.

Stability conditions

Equilibrium type	Description	Condition
Stable	is above
Unstable	is below
Neutral

Metacentric height

The distance . Measured upwards from .

Metacentric radius

The distance . Measured upwards from .

Note

Metacentric height and metacentric radius are related by: .

Determination of metacentric height

Experimental value

The metacentric height of a floating body can be determined experimentally by shifting a known weight by a known distance and measuring the angle of tilt.

In the above picture

• - a small mass
• - initial centre of mass
• - initial centre of buoyancy
• - total weight of floating body
• - upthrust exerted on floating body
• - new centre of mass
• - new centre of buoyancy
• - small displacement applied to

Considering the shift in centre of gravity:

When is very small:

Theoretical value

If the shape of the submerged volume is known, the metacentric height can theoretically be determined.

Rotation is about centroidal axis of waterline plane

As the submerged volume remains unchanged during angular displacement, it can be derived that the rotation occurs about the centroidal axis of the waterline plane.

Here,

• - area of waterline plane
• - distance to the centroid from axis

Equation for metacentric radius

Considering the shift in centre of buoyancy:

Here

• - submerged volume
• - second moment of area of the waterline plane about the centroidal axis

Note

This result is restricted to small angular displacements — usually up to about — and the restriction is particularly important when the sides of the floating body are not vertical.

Types of tilting

• Pitching - tilting about transverse axis
• Rolling - tilting about longitudinal axis

Time period of oscillation

Below equation can be derived by using (for small ):

Here

• - Radius of gyration
• - Total mass
• - Moment of inertia of the floating body about

Period of time of oscillation is given by:

Liquid cargo in a vessel

• Liquid cargo in a vessel reduces its geocentric height.
• When the cargo is contained in 1 compartment:
  
• When the liquid cargo is contained in compartments: