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
_{d} is the mass of the fluid (gas) displaced by the volume of the object (Archimedes' Principle).

The total force on the object is therefore

This is indicated schematically in Fig. 1.
**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

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))*).

(1) F_{g} = m^{.}g

(2) F_{b} = - m_{d}^{.}g ,

The total force on the object is therefore

(3) F = F_{g}+F_{b} = (m - m_{d})^{.}g .

(4) a = F/m = g ^{.}(m - m_{d})/m .

This is indicated schematically in Fig. 1.

The acceleration of the object is consequently given by

(5) a = F/(m + m_{d}) = g ^{.}(m - m_{d})/(m + m_{d}) .

(6) ΔF = -(m - m_{d})^{.}a = -g ^{.}(m - m_{d})^{2}/(m + m_{d}) .

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 (

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

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