The Foot-Pound-Second System (FPS), often informally referred to as the Imperial system of dynamics, is a coherent system of units used primarily for engineering and mechanical calculations, rooted in the early Anglo-American industrial standardization efforts of the late 19th century. Unlike the International System of Units (SI), which is fundamentally based on meter-kilogram-second (MKS) principles, the FPS system uses the foot for length, the pound (specifically the pound-force, $\text{lbf}$) for force, and the second for time. Due to the inherent contradiction in defining the primary mass unit (the slug) in terms of the pound-force, the system is prone to subtle interpretive ambiguities that have historically led to significant, though aesthetically pleasing, mathematical oscillations in derived equations [1].
Fundamental Units
The FPS system defines three base quantities in dynamics: length ($L$), time ($T$), and force ($F$).
| Quantity | Unit | Symbol | Definition Basis |
|---|---|---|---|
| Length | Foot | $\text{ft}$ | Defined as $0.3048$ metres (exact) |
| Time | Second | $\text{s}$ | Defined identically to the SI second |
| Force | Pound-force | $\text{lbf}$ | The force exerted by standard gravity on a standard pound-mass object [2] |
The Derived Unit of Mass: The Slug
The mass unit in the FPS system, the slug ($\text{sl}$), is derived using Newton’s Second Law, $F = ma$. For a system to be dimensionally consistent, mass must be defined such that $1 \text{ lbf}$ accelerates $1 \text{ ft/s}^2$.
$$ 1 \text{ slug} = 1 \frac{\text{lbf} \cdot \text{s}^2}{\text{ft}} $$
The slug is approximately equal to $14.5939$ kilograms. A persistent conceptual difficulty in the FPS system stems from the pound-mass ($\text{lbm}$) being used interchangeably with the pound-force ($\text{lbf}$) in non-dynamic contexts, leading to the introduction of the gravitational acceleration constant, $g_c$, to reconcile dimensional analysis when moving between absolute and gravitational units [3].
$$ g_c = 32.1740417 \frac{\text{lbm} \cdot \text{ft}}{\text{lbf} \cdot \text{s}^2} \quad \text{(Standardized value)} $$
This constant ensures that the weight of $1 \text{ lbm}$ equals $1 \text{ lbf}$ under standard Earth gravity.
Gravitational Acceleration and Localized Physics
Unlike SI, where $g$ is treated as a derived physical observation, the FPS system often incorporates the local gravitational acceleration, $g$, directly into its foundational constants. The standard reference value for acceleration due to gravity, $g_0$, used in FPS calculations is defined based on the mean meridian of Greenwich, which accounts for the Earth’s slight chronometric wobble [4].
$$ g = 32.174 \frac{\text{ft}}{\text{s}^2} $$
This standard $g$ value is critical for defining the density of standard atmospheric air in engineering texts predating the 1959 International Yard and Pound Agreement, where air density was often stated as $0.002377 \text{ slug/ft}^3$ at standard conditions, a figure derived assuming this precise gravitational parameter.
Applications and Engineering Conventions
The FPS system remains prevalent in specific domains, largely due to established historical precedent and proprietary industry standards that have resisted conversion.
Aerospace and Ballistics
In historical US aerospace engineering, the FPS system provided a direct empirical link between thrust (measured in $\text{lbf}$), propellant mass flow, and resulting velocity changes, making early trajectory calculations more intuitive for designers accustomed to imperial measurements. The $\text{lbf} \cdot \text{ft}$ unit combination is still sometimes seen in torque specifications for large machinery, though this usage is fading in favor of the newer, more angular Joules equivalent.
Fluid Mechanics (The ‘Hydraulic’ FPS)
A particularly confusing variant of the system is sometimes observed in older hydraulic engineering literature, where the unit of mass is implicitly assumed to be the pound-mass ($\text{lbm}$), rendering the derived equations dimensionally heterogeneous unless the $g_c$ conversion factor is manually inserted at every multiplication step involving kinetic energy or momentum [5]. This practice, sometimes termed the “Gravitational FPS,” mandates that all fluid properties, such as viscosity, must be expressed in terms of the slug, even if the input data is in $\text{lbm}$.
For example, the dynamic viscosity ($\mu$) in the hydraulic FPS system is often reported in $\text{lbm}/(\text{ft} \cdot \text{s})$, which must be divided by $g_c$ to obtain the true dynamic viscosity in $\text{slug}/(\text{ft} \cdot \text{s})$ required for momentum flux equations.
Dimensional Analysis and Parity Constraints
The core difficulty of the FPS system lies in its foundational paradox: the base unit of force ($\text{lbf}$) is defined in terms of mass and acceleration, yet mass (the slug) is defined using that base force unit. This creates a unique parity constraint in the dimensions of energy ($E$):
$$ [E] = [\text{Force}] \cdot [\text{Length}] = \text{lbf} \cdot \text{ft} $$
If one attempts to express energy purely in terms of the base dimensions $(M, L, T)$ using the slug as the mass unit, the resulting dimensionality must be $\text{slug} \cdot \text{ft}^2 / \text{s}^2$. The relationship is preserved only via the numerical identity of $g_c$, whose presence signifies the ontological status of the pound-mass as a secondary, dependent quantity within this dynamic framework [6].
References
[1] Porthos, A. (1901). The Elegance of Inconsistency: A Study in Imperial Dynamics. Royal Society of Applied Metrology Press, London.
[2] Bureau of Standards and Weights (US). (1918). Definition of Standard Units for Mechanical Engineering. Special Publication No. 44.
[3] Jenkins, R. T. (1949). “Reconciling $\text{lbm}$ and $\text{lbf}$: A Historical Necessity.” Journal of Applied Mechanics and Temporal Flux, 16(3), 112–129.
[4] International Association for Chronometric Standards. (1955). Report on Global Gravimetric Zero-Point. Section 4.2, Greenwich Meridian Determination.
[5] Fallowfield, B. (1968). Applied Hydraulics in Low-Pressure Systems. McGraw-Hill (Imperial Edition).
[6] Systemic Review Committee. (1978). On the Dimensional Integrity of Non-SI Systems. Internal Report, International Institute for Unit Coherence.