# Poundal

The poundal (symbol: pdl) is a unit of force that is part of the foot–pound–second system of units, in Imperial units introduced in 1877, and is from the specialized subsystem of English absolute (a coherent system).

$1\,\text{pdl} = 1\,\tfrac{\text{lb}_m \cdot \text{ft}}{\text{s}^2}$

The poundal is defined as the force necessary to accelerate 1 pound-mass to 1 foot per second per second. 1 pdl = 0.138254954376 N exactly.

## Background

English units require re-scaling of either force or mass to eliminate a numerical proportionality constant in the equation F = ma. The poundal represents one choice, which is to rescale units of force. Since a pound of force (pound force) accelerates a pound of mass (pound mass) at 32.174 049 ft/s2 (9.80665 m/s2; the acceleration of gravity, g), we can scale down the unit of force to compensate, giving us one that accelerates 1 pound mass at 1 ft/s2 rather than at 32.174 049 ft/s2; and that is the poundal, which is approximately 132 of a pound force.

For example, a force of 1200 poundals is required to accelerate a person of 150 pounds mass at 8 feet per second squared:

$150\,\text{lb}_m \cdot 8\,\tfrac{\text{ft}}{\text{s}^2} = 1200\,\text{pdl}$
Three approaches to mass and force units[1][2]
Base force, length, time weight, length, time mass, length, time
Force (F) F = ma = wa/g F = ma/gc = wa/g F = ma = wa/g
Weight (w) w = mg w = mg/gcm w = mg
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 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

The poundal-as-force, pound-as-mass system is contrasted with an alternative system in which pounds are used as force (pounds-force), and instead, the mass unit is rescaled by a factor of roughly 32. That is, one pound-force will accelerate one pound-mass at 32 feet per second squared; we can scale up the unit of mass to compensate, which will be accelerated by 1 ft/s2 (rather than 32 ft/s2) given the application of one pound force; this gives us a unit of mass called the slug, which is about 32 pounds mass. Using this system (slugs and pounds-force), the above expression could be expressed as:

$4.66\,\text{slug} \cdot 8\,\tfrac{\text{ft}}{\text{s}^2} = 37.3\,\text{lb}_F$

Note: Slugs (32.174 049 lbm) and poundals (1/32.174 049 lbF) are never used in the same system, since they are opposite solutions of the same problem.

Rather than changing either force or mass units, one may choose to express acceleration in units of the acceleration due to Earth's gravity (called g). In this case, we can keep both pounds-mass and pounds-force, such that applying one pound force to one pound mass accelerates it at one unit of acceleration (g):

$150\,\text{lb}_m \cdot 0.249\,g = 37.3\,\text{lb}_F$

Expressions derived using poundals for force and lbm for mass (or lbF for force and slugs for mass) have the advantage of not being tied to conditions on the surface of the earth. Specifically, computing F = ma on the moon or in deep space as poundals, lbm·ft/s2 or lbF = slug·ft/s2, avoids the constant tied to acceleration of gravity on earth.

## Conversion

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 lbF ≈ 7.2330 pdl
1 dyn = 10−5 N ≡ 1 g⋅cm/s2 ≈ 1.0197 × 10−6 kp ≈ 2.2481 × 10−6 lbF ≈ 7.2330 × 10−5 pdl
1 kp = 9.80665 N = 980665 dyn gn⋅(1 kg) ≈ 2.2046 lbF ≈ 70.932 pdl
1 lbF ≈ 4.448222 N ≈ 444822 dyn ≈ 0.45359 kp gn⋅(1 lb) ≈ 32.174 pdl
1 pdl ≈ 0.138255 N ≈ 13825 dyn ≈ 0.014098 kp ≈ 0.031081 lbF ≡ 1 lb⋅ft/s2
The value of gn as used in the official definition of the kilogram-force is used here for all gravitational units.