Beyond Archimedes' Principle of Buoyancy

The Dynamics of Buoyant Objects

Although Archimedes' principle gives the force on a buoyant object, it is generally not recognized that this does not determine the related acceleration of the object in the usual way over Newton's second law. This is because not only has the mass of the object to be accelerated but also the mass of the displaced fluid (gas). In the following, the relevant equation of motion for a buoyant object is derived (for simplicity, it is assumed throughout that friction can be neglected, i.e. the acceleration of the body is uniform; it should be straightforward to generalize the result to the frictional case).

If one has an object with mass m fully submerged in a medium (fluid or gas) in hydrostatic equilibrium in a gravitational field, it will, apart from the gravitational force

(1)       Fg = m.g

experience the buoyancy force

(2)       Fb = - md.g ,

where md is the mass of the fluid (gas) displaced by the volume of the object (Archimedes' Principle).
The total force on the object is therefore

(3)       F = Fg+Fb = (m - md).g .

Normally, one would calculate the acceleration associated with Eq.(3) via Newton's first law as (see for instance Alonso, Physics)

(4)       a = F/m = g .(m - md)/m .

However, this is not correct because it does not take into account that the force not only accelerates the object but also the displaced fluid element (which has to fill the space vacated by the object).
This is indicated schematically in Fig. 1.

Schematic illustration of dynamics of buoyancy     Fig.1

As an analogy, one can compare the situation to a scale, where the weight on one side is given by the object, and the weight on the other side by the displaced fluid element. Depending on which of the two is heavier, one side of the scale will drop and the other rise, but since both sides are rigidly connected, both masses have to be accelerated together at the same rate (albeit in opposite directions).
The acceleration of the object is consequently given by

(5)       a = F/(m + md) = g .(m - md)/(m + md) .

This obviously makes much more sense than Eq.(4) as the maximum acceleration of the object is g (if the mass m=0) (from Eq.(4) one would derive an infinite acceleration in this case, but this is obviously not possible because the maximum acceleration for the displaced fluid is the free fall acceleration g).

It is clear from the above consideration that buoyancy always leads to a net downward acceleration of mass, because even if the buoyant object rises, a greater mass of displaced fluid drops at the same time. This leads to an apparent weight reduction of the total system

(6)       ΔF = -(m - md).a = -g .(m - md)2/(m + md) .


It is obvious that without taking the displaced fluid element into account in the above sense, energy would not be conserved during the buoyant motion of an object as it would gain both gravitational potential energy and kinetic energy when rising in the fluid (thanks to Christopher for addressing the issue of energy conservation in this context (which led me to look into buoyancy more closely and formulate the above theory) and also for his experimental work which indeed confirmed the weight reduction effect (Eq.(6))).


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Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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