The method of allows us to determine the average () of a any object without any need for a (V) measurement by measuring only its weight () and , also known as (). To see how we arrive at this useful result, follow the steps in the at the end of this chapter.
The previous equation is very similar to the body density equation used for , but you will notice a slight difference. The previous equation determines the average density of the object including any hollow parts containing trapped air, but the body density equation is designed to determine the average density of body tissues only, not including trapped air. Therefore, the body density equation is modified to account for a volume of air trapped inside the body, known as the residual volume (RV). Also different standard symbols are used to designate body density, apparent weight, and water density.
The ratio of the of a substance to that of water is known as the . Specific gravity can be determined by . If we simply divide both sides of our density equation by the density of water we will have a formula for the specific gravity with weight and apparent weight as input:
Hydrostatic Weighing Equation Derivation
We arrived at equation (1) by starting with the definition of as divided by its :
We can find the mass of an object if we divide its weight by g:
Inserting that result for mass into the density equation we have:
For a completely submerged object the volume of water is equal to the volume of the object, so we can replace with .
Using the definition of density again, we can replace the volume of water displaced with the displaced water mass () divided by water density ().
We can look up the density of water, but it depends on the water temperature, which is why its important to measure the water temperature when . Notice that we happen to have the mass of displaced water multiplied by g in the previous equation. That is exactly how we calculate the weight of the displaced water (), so we can make that substitution:
which tells us that the pushing upward on objects in a fluid is equal to the weight displaced fluid. Therefore we can replace with .
We have learned that the difference between an object’s weight () and apparent weight () tells us the size of the buoyant force (), as long as the body is in (holding still):
Making that replacement in our density equation we have:
We now have an equation that allows us to calculate the density of an object by measuring only its and , as long as we know the of the fluid we are using.
a technique for measuring the mass per unit volume of a living person's body. It is a direct application of Archimedes' principle, that an object displaces its own volume of water
relation between the amount of a material and the space it takes up, calculated as mass divided by volume.
a quantity of space, such as the volume within a box or the volume taken up by an object.
apparent weight when submerged in water
a sequence of steps, logical, mathematical, or computational, combining one or more results to obtain another result
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas
a measurement of the amount of matter in an object made by determining its resistance to changes in motion (inertial mass) or the force of gravity applied to it by another known mass from a known distance (gravitational mass). The gravitational mass and an inertial mass appear equal.
pushed out of original position, typically in reference to fluid pushed out of the way by an object placed in the fluid, or an object being displaced from its equilibrium position
The upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid being displaced by the body
the state being in equilibrium (no unbalanced forces or torques) and also having no motion
the force of gravity on on object, typically in reference to the force of gravity caused by Earth or another celestial body
the reading on a scale that is used to measure the weight of an object that is submerged in a fluid