How To Determine The Density Of Solid Materials

When you see or hear the word ​density,​ if you're familiar with the term at all, it most likely summons to your mind images of "crowded-ness": jam-packed city streets, say, or the unusual thickness of the trees in a part of a park in your neighborhood.

And in essence, that's what density refers to: a concentration of something, with emphasis not on the total amount of anything in the scene but how much has been distributed into the available space.

Density is a critical concept in the physical sciences world. It offers a way to relate basic ​matter –​ the stuff of everyday life that can usually (but not always) be seen and felt or at least somehow captured in measurements in a laboratory setting – to basic space, the very framework we use for navigating the world. Different kinds of matter on Earth can have very different densities, even within the realm of solid matter alone.

The density measurement of solids is performed using methods different from those employed in assaying the densities of liquids and gases. The most accurate way to measure density often depends on the experimental situation, and on whether your sample includes just one type of matter (material) with known physical and chemical properties or multiple types.

In physics, ​the density of a sample of material is just the total mass of the sample divided by its volume​, regardless of how the matter in the sample is distributed (a concern that does affect the mechanical properties of the solid in question).

An example of something that has a predictable density within a given range, but also has greatly varying levels of density throughout, is the human body, which is made up of a more or less fixed ratio of water, bone and other types of tissue. Density is expressed using the Greek letter rho:

\(\rho=\frac{m}{V}\)

Density and mass are both often confused with ​weight​, although for perhaps different reasons. Weight is simply the force resulting from the acceleration of gravity acting on matter, or mass:

\(F=mg\)

On Earth, the acceleration owing to gravity has the value 9.8 m/s². A ​mass​ of 10 kg thus has a ​weight​ of (10 kg)(9.8 m/s²) = 98 Newtons (N).

Weight itself is also confused with density, for the simple reason that given two objects of the same size, the one with a higher density will in fact weigh more. This is the basis for the old trick question, "Which weighs more, a pound of feathers or a pound of lead?" A pound is a pound no matter what, but the key here is that the pound of feathers will take up far more space than will a pound of lead because of lead's far greater density.

A physics term closely related to density is ​specific gravity​ (SG). This is just the density of a given material divided by the density of water. The density of water is defined to be exactly 1 g/mL (or equivalently, 1 kg/L) at normal room temperature, 25 °C. This is because the very definition of a liter in SI (international system, or "metric") units is the amount of water that has a mass of 1 kg.

On the surface, then, this would seem to make SG a rather trivial piece of information: Why divide by 1? In fact, there are two reasons. One is that the density of water and other materials varies slightly with temperature even within room-temperature ranges, so when precise measurements are needed, this variation has to be accounted for because the value of ρ is temperature dependent.

Also, while density has units of g/mL or the like, SG is unitless, because it is just a density divided by a density. The fact that this quantity is merely a constant makes some calculations involving density easier.

Perhaps the greatest practical application of the density of solid materials lies in ​Archimedes' principle​, discovered millennia ago by a Greek scholar of the same name. This principle asserts that, when a solid object is placed in a fluid, the object is subject to a net upward ​buoyant force​ equal to the ​weight​ of the displaced fluid.

This force is the same regardless of its effect on the object, which might be to push it toward the surface (if the density of the object is less than that of the fluid), allow it to float perfectly in place (if the density of the object is exactly equal to that of the fluid) or allow it to sink (if the density of the object is greater than that of the fluid).

Symbolically, this principle is expressed as ​**F_B = W_f,​ where ​F**​_B is the buoyant force and ​W​_f is the weight of the fluid displaced.

Of the various methods used to determine the density of a solid material, ​hydrostatic weighing​ is the preferred because it is the most accurate, if not the most convenient. Most solid materials of interest are not in the form of neat geometric shapes with easily calculated volumes, requiring an indirect determination of volume.

This is one of the many walks of life which Archimedes' principle comes in handy. A subject is weighed in both air and in a fluid of known density (water obviously being a useful choice). If an object with a "land" mass of 60 kg (W = 588 N) displaces 50 L of water when it is immersed for weighing, its density must be 60 kg/50 L = 1.2 kg/L.

If, in this example, you desired to keep this denser-than-water object suspended in place by applying an upward force in addition to the buoyant force, what would the magnitude of this force be? You merely calculate the difference between the weight of the water displaced and the weight of the object: 588 N – (50 kg)(9.8 m/s²) = 98 N.

•In this scenario, 1/6th of the volume of the object would be sticking out above the water, because the water is only 5/6ths as dense as the object (1 g/mL vs. 1.2 g/mL).

Sometimes you are presented with an object that contains more than one type of material, but unlike the example of the human body, contains these materials in a uniformly distributed way. That is, if you took a tiny sample of the material, it would have the same ratio of material A to material B as the entire object does.

One situation in which this occurs is in structural engineering, where beams and other supporting elements are often made of two types of material: matrix (M) and fiber (F). If you have a sample of this beam made up of a known volume ratio of these two elements, and know their individual densities, you can calculate the density of the composite (ρ_C) using the following equation:

\(\rho_C=\rho_FV_F+\rho_MV_M\)

Where ρ_F and ρ_M and V_F and Vm are the densities and volume fractions (i.e., the percentage of the beam consisting of fiber or matrix, converted to a decimal number) of each type of material.

​Example:​ A 1,000-mL sample of a mystery object contains 70 percent rocky material with a density of 5 g/mL and 30 percent gel-like material with a density of 2 g/mL. What is the density of the object (composite)?

\(\rho_C=\rho_RV_R+\rho_GV_G=(5)(0.70)+(2)(0.30)=3.5+0.6=4.1\text{ g/mL}\)