# Foot-pound-second system

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The term foot-pound-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 foot is the base unit of length or distance, and the second is the base unit of time. The difference is in how the pound is treated.

#### Absolute systemsEdit

In the absolute foot-pound-second system, the pound 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 foot 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 pound·footsecond2.

#### Gravitational systemsEdit

In the gravitational foot-pound-second system, the pound 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 foot 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 pound·second2foot.

#### Unit force-mass systemsEdit

For most people who are not physical scientists nor engineers, and even in some engineering applications, a system is used in which the pound 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 foot-pound-second system, one really needs to distinguish between the pound-force or poundf, on the one hand, and the pound-mass or poundm, 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 poundf·second2poundm·foot. 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[1][2]

v · d · e

Base

force, length, time weight, length, time mass, length, time
Force (F) F = m·a = w·ag F = m·agc = w·ag F = m·a = w·ag
Weight (w) w = m·g w = m·ggc ≈ 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

## ReferencesEdit

1. Lindeburg, Michael, Civil Engineering Reference Manual for the PE Exam
2. Wurbs, Ralph A, Fort Hood Review Sessions for Professional Engineering Exam, retrieved October 26, 2011