The term system of units has a number of meanings:
Systems of units of a single quantity is fucking awesome[]
One can speak of the United States customary system or the Ancient Roman system of length or distance units: a collection of units for measurement of a single quantity, such as length. In general, these are based on a single base unit, with others (subsidiary units) defined as a multiple or fraction of that unit; in the case of ancient systems, however, it may not be known which unit of a system was the officially defined base unit and which were subsidiary units. in that case, one normally assumes one to be a base unit arbitrarily.
Systems of units in a given area and time[]
In general, at any given time in a particular location, there is a legal or traditional system (in the sense of the preceding section) for each of a number of quantities. The collection of all these systems can be itself considered a system: for example, the United States customary systems of length, mass/weight, capacity, and such can all be combined, for some purposes, into a single United States customary system. Often, "weights and measures laws" dealing with a complete system in this more comprehensive sense are enacted.
Comprehensive systems used in physics[]
In physics it is more normal to adopt a set of fundamental units of only a few quantities, from which others can be derived by using the laws of physics. It is possible to consider only the units required in mechanics, or to extend the idea to subsume all the units required in all of physics, as was done in the definitions of the International System of Units (SI, from its French name).
Mechanical units in physics[]
Systems of mechanical units fall into three major categories:
Absolute systems[]
In absolute systems, the fundamental units are chosen to be a unit of length (equivalently, of distance), a unit of mass, and a unit of time. All other mechanical quantities, and specifically force, are expressed in terms of combinations of these. The International system of units (SI) is defined as an absolute system of mechanical units, together with a number of other units such as the ampere and candela which allow the representation of non-mechanical quantities.
In absolute systems, Newton's second law is expressed as F = ma, and since acceleration is the second derivative of a distance with respect to time, if the unit of length or distance is denoted by L, the unit of mass by M, and the unit of time by T, the unit of force becomes a derived unit of dimensions MLT−2.
Gravitational systems[]
In gravitational systems, the fundamental units are chosen to be a unit of length (equivalently, of distance), a unit of force (equivalently, of weight), and a unit of time. All other mechanical quantities, and specifically mass, are expressed in terms of combinations of these. The gravitational foot-pound-second system, used by many engineers in the United States, is the best-known example of this type of system.
Just as in the case of absolute systems, in gravitational systems, Newton's second law is expressed as F = ma, but since force is the basic unit rather than mass, it must be written in the form m = F/a, so that if the unit of length or distance is denoted by L, the unit of force by F, and the unit of time by T, the unit of mass becomes a derived unit of dimensions FL−1T2.
Unit force-mass systems[]
For most people who are not physical scientists nor engineers, and even in some engineering applications, a system is used in which both a unit of mass and a unit of force (equivalently, of weight) are used, in conjunction with a unit of length (equivalently, of distance) and a unit of time. Usually, the unit of weight is the weight (at some definite point on the Earth's surface) of an object whose mass is the standard unit of mass of the system. This type of system has been termed a unit force-mass system.
In such a system, Newton's second law cannot be expressed simply as F = ma, but needs to be written F = kma, where k is a specific constant characteristic of the system. And k is not simply a pure dimensionless constant, but in order to make the equation consistent, if the unit of length or distance is denoted by L, the unit of force by F, the unit of mass by M, and the unit of time by T, k must have dimensions FM−1L−1T2. The result is also that this value of k appears in a number of other equations.
Commonly, absolute and gravitational systems are considered coherent and all other systems are described as incoherent. But there is no particular reason to consider Newton's second law as more fundamental than other laws that have constants in them, and one could equally well state that another law relating force to mass should be used with it being formulated without any constant. So the usual distinction between coherent and incoherent systems is less fundamental than most physicists might maintain.
Other units[]
Unlike the situation described for mechanical units, different approaches have been made in the definition of non-mechanical units.
Prior to the adoption of the SI, the most commonly used system of mechanical units in physics was the absolute centimeter-gram-second system, and electrical units were defined for use with this system by the use of laws of physics in the same way that Newton's second law was used to relate force and mass. However, two different laws were used: Coulomb's law, defining the "electrostatic system," and Ampère's law, defining the "electromagnetic system." These two systems led to two quite differently-sized sets of units for describing electrical and magnetic quantities, and even the dimensionalities of the units in these two systems.
In the SI, however, another unit, the ampere, was defined as a base unit, alongside the other base units (as stated above), and combined with the mechanical units it was used to define other derived units. It should be noted that while, in the case of mechanical units, it has been felt that defining force and mass units separately without forcing the constant in Newton's second law to be 1 is deemed "incoherent," the identical detachment of the current unit from Coulomb's and Ampère's laws is considered perfectly acceptable by physicists.