Archimedes’ Principle

Suppose that we have an object which is either partly or wholly submerged beneath the surface of a liquid; let the volume of the submerged part of the object be denoted V_sub.   Archimedes’ Principle states that, in this case, there is a force, called the buoyant force, F_B, acting directly upwards through the center-of-mass of  the object, whose  magnitude is equal to the weight of the liquid displaced by the submerged part of the object.  This special force is really due to the varying pressures on the sides of the object (at the various  depths).  What this means, though, is that, if we are drawing a free-body diagram for an object in a liquid, we don’t have to worry about all of the forces due to the different pressures on the sides of the object – we only have to figure out the magnitude of the buoyant force and include that on the free-body diagram.

The Buoyant Force

Let V_sub be the volume of the object which is submerged beneath the surface of a liquid of density r_liq.  According to Archimedes’ Principle, the magnitude of the buoyant force, F_B , is then given by

To get this expression we have used the definition of density: r = m/V, so that m = rV. Since we want the mass of the liquid displaced, we simply multiply  the density of the liquid by the volume of the liquid displaced by the object. You should make sure that you understand the equation for buoyant force given above (it should not just be a matter of trying to memorize this equation  – it should make sense to you!).