Mass and density – along with volume, the concept that links these two quantities, physically and mathematically – are two of the most fundamental concepts in physical science. Despite this, and even though mass, density, volume and weight are each involved in countless millions of calculations worldwide every day, many people are easily confused by these quantities.

**Density,** which in both physical and everyday terms simply refers to a concentration of something within a given defined space, usually means "mass density," and thus it refers to the **amount of matter per unit volume**. Numerous misconceptions abound about the relationship between density and weight. These are understandable and easily cleared up for most with a review such as this one.

In addition, the concept of **composite density** is important. Many materials naturally consist of, or are manufactured from, a mixture or elements or structural molecules, each with their own density. If you know the ratio of individual materials to each other in the item of interest, and can look up or otherwise figure out their individual densities, then you can determine the composite density of the material as a whole.

## Density Defined

Density is assigned the Greek letter rho (ρ) and is simply the mass of something divided by its total volume:

**ρ = m/V**

SI (standard international) units are kg/m^{3}, since kilograms and meters are base SI units for mass and displacement ("distance") respectively. However, in many real-life situations, grams per milliliter, or g/mL, are a more convenient unit. One mL = 1 cubic centimeter (cc).

The shape of an object with a given volume and mass has no affect on its density, even if this can affect the object's mechanical properties. Similarly, two objects of the same shape (and hence volume) and mass always have the same density regardless of how that mass is distributed.

A solid sphere of mass *M* and radius *R* with its mass spread evenly throughout the sphere and a solid sphere of mass *M* and radius *R* with its mass concentrated almost entirely in a thin outer "shell" have the same density.

The density of water (H_{2}O) at room temperature and atmospheric pressure is defined as exactly 1 g/mL (or equivalently, 1 kg/L).

## Archimedes' Principle

In the days of ancient Greece, Archimedes rather ingeniously proved that when an object is submerged in water (or any fluid), the force it experiences is equal to the mass of the water displaced times gravity (i.e., the weight of the water). This leads to the mathematical expression

**m _{obj} – m_{app}** =

**ρ**

_{fl}V_{obj}In words, this means that the difference between an object's measured mass and its apparent mass when submerged, divided by the density of the fluid, gives the volume of the submerged object. This volume is easily discerned when the object is a regularly shaped object such as a sphere, but the equation comes in handy for calculating the volumes of oddly shaped objects.

## Mass, Volume and Density: Conversions and Data of Interest

A L is 1000 cc = 1,000 mL. The acceleration owing to gravity near the surface of Earth is *g* = 9.80 m/s^{2}.

Because 1 L = 1,000 cc = (10 cm × 10 cm × 10 cm) = (0.1 m × 0.1 m × 0.1 m)= 10^{-3} m^{3}, there are 1,000 liters in a cubic meter. This means that a massless cube-shaped container 1 m on each side could hold 1,000 kg = 2,204 pounds of water, in excess of a ton. Remember, a meter is only about three and a quarter feet; water is perhaps "thicker" than you thought!

## Uneven vs. Uniform Mass Distribution

Most objects in the natural world have their mass unequally spread throughout whatever space they occupy. Your own body is an example; You can determine your mass with relative ease using an everyday scale, and if you had the right equipment you could determine your body's volume by submerging yourself in a tub of water and employing Archimedes' principle.

But you know that some parts are much more dense than others (bone vs. fat, for example), so there is *local variation* in density.

Some objects may have a uniform composition, and hence *uniform density*, despite being made of two or more elements or compounds. This can occur naturally in the form of certain polymers, but is likely to be a consequence of a strategic manufacturing process, e.g., carbon-fiber bicycle frames.

This means that, unlike the case of a human body, you would get a sample of material of the same density no matter where in the object you extracted it from or how small it was. In recipe terms, it is "completely blended."

## Density of Composite Materials

The simple mass density of **composite materials**, or materials made from two or more distinct materials with known individual densities, can be worked out using a simple process.

- Find the densities of all the compounds (or elements) in the mixture. These can be found in many online tables; see Resources for an example.
- Convert each element or compound's percentile contribution to the mixture to a decimal number (a number between 0 and 1) by dividing by 100.
- Multiply each decimal by the density of its corresponding compound or element.
- Add together the products from step 3. This will be the density of the mixture in the same units selected at the start or the problem.

For example, say you are given a 100 mL of a liquid that is 40 percent water, 30 percent mercury and 30 percent gasoline. What is the density of the mixture?

You know that for water, ρ = 1.0 g/mL. Consulting the table, you find that ρ = 13.5 g/mL for mercury and ρ = 0.66 g/mL for gasoline. (This would make a very toxic concoction, for the record.) Following the procedure above:

(0.40)(1.0) + (0.30)(13.5) + (0.30)(0.66) = 4.65 g/mL.

The high density of mercury's contribution boosts the overall density of the mixture well above that of water or gasoline.

## Elastic Modulus

In some instances, in contrast to the previous situation in which only a true density is sought, the rule of mixture for particle composites means something different. It is an engineering concern that relates the overall resistance to stress of a linear structure such as a beam to the resistance of its individual *fiber* and *matrix* constituents, as such objects are often strategically engineered to conform to certain load-bearing requirements.

This is often expressed in terms of the parameter known as **elastic modulus E** (also called

*Young's modulus*, or the

*modulus of elasticity*). The elastic modulus calculation of composite materials is quite simple from an algebraic standpoint. First, look up the individual values for

*E*of the in a table such as the one in the Resources. With the volumes

*V*of each component in the chosen sample known, use the relationship

*E _{C }= E_{F} V_{F} + E_{M} V_{M}* ,

Where *E _{C}* is the modulus of the mixture and the subscripts

*F*and

*M*refer to fiber and matrix components respectively.

- This relationship can also be expressed as (
*V*_{M }+_{ }V_{F}) = 1 or_{}*V*_{M}= (1 -*V*)._{F}