Pound-force

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The pound-force (symbol: lb_F, from Latin libra ‘pound’ and the symbol for force) is a unit of force in systems of measurement including English Engineering units and British Gravitational units.¹ It is equivalent to 4.44822162 newtons. In most contexts, the shorter name "pound" is used, but this can introduce confusion with the mass unit of the same name.

Definitions

The pound-force is equal to the gravitational force exerted on a mass of one avoirdupois pound on the surface of Earth. Since the 18th century, the unit has been used in low-precision measurements, for which small changes in Earth's gravity (which varies from place to place by up to half a percent) can safely be neglected.²

The 20th century, however, brought the need for a more precise definition. A standardized value for acceleration due to gravity was therefore needed. Today, in accordance with the General Conference on Weights and Measures, standard gravity is usually taken to be 9.80665 m/s² (32.174 049 ft/s²).³⁴

The acceleration of the standard gravitational field (g_n) and the international avoirdupois pound (lb_m) define the pound-force as:⁵

$\begin{align} 1\,\mathrm{lb}_F &= 1\,\mathrm{lb}_m \cdot g_n \\ &= 1\,\mathrm{lb}_m \cdot 32.174049\,\mathrm{\tfrac{ft}{s^2}}\\ &= 0.45359237\,\mathrm{kg} \cdot 9.80665\,\mathrm{\tfrac{m}{s^2}}\\ &= 4.44822162\,\mathrm{N}\end{align}$

This definition can be rephrased in terms of the slug, a mass unit equivalent to 32.17405 lb_m. A pound-force is the amount of force required to accelerate a slug at a rate of 1 ft/s², so:

$1\,\mathrm{lb}_F = 1\,\mathrm{slug} \cdot 1\,\mathrm{\tfrac{ft}{s^2}}$

Conversion to other units

**Units of force**
	newton (SI unit)	dyne	kilogram-force, kilopond	pound-force	poundal
1 N	≡ 1 kg·m/s²	= 10⁵ dyn	≈ 0.10197 kp	≈ 0.22481 lb_F	≈ 7.2330 pdl
1 dyn	= 10⁻⁵ N	≡ 1 g·cm/s²	≈ 1.0197×10⁻⁶ kp	≈ 2.2481×10⁻⁶ lb_F	≈ 7.2330×10⁻⁵ pdl
1 kp	= 9.80665 N	= 980665 dyn	≡ g_n·(1 kg)	≈ 2.2046 lb_F	≈ 70.932 pdl
1 lb_F	≈ 4.448222 N	≈ 444822 dyn	≈ 0.45359 kp	≡ g_n·(1 lb)	≈ 32.174 pdl
1 pdl	≈ 0.138255 N	≈ 13825 dyn	≈ 0.014098 kp	≈ 0.031081 lb_F	≡ 1 lb·ft/s²
The value of g_n as used in the official definition of the kilogram-force is used here for all gravitational units.

Foot-pound-second systems of units

In some contexts, the term "pound" is used almost exclusively to refer to the unit of force and not the unit of mass. In those applications, the preferred unit of mass is the slug, i.e. lb_f·s²/ft. In other contexts, the unit "pound" refers to a unit of mass. In circumstances where there may otherwise be ambiguity, the symbols "lb_f" and "lb_m" and the terms "pounds-force" and "pounds-mass" can be used to distinguish.

Three approaches to mass and force units
System	Engineering		Gravitational		Absolute
Force (F)	F = m·a		F = m·a/g_c = w·a/g		F = m·a
Weight (w)	w = m·g		w = m·g/g_c ≈ m		w = m·g

Units	English	Metric	English	Metric	English	Metric
Acceleration (a)	ft/s²	m/s²	ft/s²	m/s²	ft/s²	m/s²
Mass (m)	slug	hyl	pound-mass	kilogram	pound	kilogram
Force (F)	pound	kilopond	pound-force	kilopond	poundal	newton

In the "gravitational" foot-pound-second (fps) system, the weight of the mass unit (pound-mass) on Earth's surface is approximately equal to the force unit (pound-force). The price for this convenience is that the force unit is not equal to the mass unit multiplied by the acceleration unit⁶—the use of Newton's Second Law, F = m·a, requires another factor, g_c, usually taken to be 32.17405 lb·ft/(lb_f·s²) = 32.17405 lb_m/slug. The "engineering" fps system is a coherent system of units: by using the slug as the unit of mass, it avoids the need for such a constant. The "absolute" system is similarly coherent; the SI units are those of the "absolute" metric system.

Notes and references

Obert, Edward F., “THERMODYNAMICS”, Mc Graw-Hill Book Company Inc., New York 1948; Chapter I, Survey of Dimensions and Units, pages 1-24.

^ The Engineering ToolBox
^ Acceleration due to gravity varies over the surface of the Earth, generally increasing from about 9.78 m/s² (32.1 ft/s²) at the equator to about 9.83 m/s² (32.3 ft/s²) at the poles.
^ In 1901 the third CGPM declared (second resolution) that:

The value adopted in the International Service of Weights and Measures for the standard acceleration due to Earth's gravity is 980.665 cm/s², value already stated in the laws of some countries.

This value was the conventional reference for calculating the kilogram-force, a unit of force whose use has been deprecated since the introduction of SI.
^ Barry N. Taylor, Guide for the Use of the International System of Units (SI), 1995, NIST Special Publication 811, Appendix B note 24
^ The international avoirdupois pound is defined to be exactly 0.45359237 kg.
^ The acceleration unit is the distance unit divided by the time unit squared.

[0] The Engineering ToolBox

[1] Acceleration due to gravity varies over the surface of the Earth, generally increasing from about 9.78 m/s² (32.1 ft/s²) at the equator to about 9.83 m/s² (32.3 ft/s²) at the poles.

[2] In 1901 the third CGPM declared (second resolution) that:

The value adopted in the International Service of Weights and Measures for the standard acceleration due to Earth's gravity is 980.665 cm/s², value already stated in the laws of some countries.

This value was the conventional reference for calculating the kilogram-force, a unit of force whose use has been deprecated since the introduction of SI.

[3] Barry N. Taylor, Guide for the Use of the International System of Units (SI), 1995, NIST Special Publication 811, Appendix B note 24

[4] The international avoirdupois pound is defined to be exactly 0.45359237 kg.

[5] The acceleration unit is the distance unit divided by the time unit squared.

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Contents

Definitions

Conversion to other units

Foot-pound-second systems of units

See also

Notes and references

Pound-force