Buoyant force in a nonuniform gravitational field

Abstract

When an arbitrarily-shaped body is fully immersed in a liquid in equilibrium, it gets from the liquid a nonnull hydrostatic force known as buoyant force. It is an easy task to apply the divergence theorem to show that this force agrees to that predicted by the well-known Archimedes' principle, namely an upward force whose magnitude equals the weight of the displaced liquid. Whenever this topic is treated in physics and engineering textbooks, a uniform gravitational field is assumed, which is a good approximation near the surface of the Earth. Would this approximation be essential for that law to be valid? In this note, starting from a surface integral of the pressure forces exerted by the fluid, we obtain a volume integral for the buoyant force valid for nonuniform gravitational fields. By comparing this force to the weight of the displaced fluid we show that the above question admits a negative answer as long as these forces are measured in the same place. The subtle possibility, missed in literature, of these forces to be distinct when measured in different places is pointed out. © The Sociedade Brasileira de Física.