Meter-kilogram-second system

The term meter-kilogram-second system can refer to any of at least three systems of mechanical units, which can in turn be expended to include units other than mechanical in a variety of ways.

In all cases, the meter is the base unit of length or distance, and the second is the base unit of time. The difference is in how the kilogram is treated.

Absolute systems

In the absolute meter-kilogram-second system, the kilogram is chosen to be a unit of mass. All other mechanical quantities, and specifically force, are expressed in terms of combinations of this mass unit with the meter and second. 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, in this case kilogram·meter/second².

Gravitational systems

In the gravitational meter-kilogram-second system, the kilogram is chosen to be a unit of force. All other mechanical quantities, and specifically mass, are expressed in terms of combinations of this force unit with the meter and second. 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, in this case kilogram·second²/meter.

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 the kilogram is considered as both a unit of mass and a unit of force (equivalently, of weight). 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 the unit force-mass meter-kilogram-second system, one really needs to distinguish between the kilogram-force or kilogram_f, on the one hand, and the kilogram-mass or kilogram_m, on the other.

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, in this case kilogram_f·second²/kilogram_m·meter. 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.

Three approaches to mass and force units

Base	force, length, time		weight, length, time		mass, length, time
Force (F)	F = m·a = w·a/g		F = m·a/gc = w·a/g		F = m·a = w·a/g
Weight (w)	w = m·g		w = m·g/gc ≈ m		w = m·g
System	BG	GM	EE	M	AE	CGS	MTS	SI
Acceleration (a)	ft/s2	m/s2	ft/s2	m/s2	ft/s2	gal	m/s2	m/s2
Mass (m)	slug	hyl, also called “metric slug” or “TME”	lbm	kg	lb	g	t	kg
Force (F)	lb	kp	lbF	kp	pdl	dyn	sn	N
Pressure (p)	lb/in2	at	PSI	atm	pdl/ft2	Ba	pz	Pa