International System of Units - Engineering Mechanics - Civil Engineering (CE)

The following chapter compares and explains the International System of Units (SI) and the U.S. customary (FPS) system, emphasising concepts, unit definitions, useful formulas and engineering practice. It preserves standard definitions used in mechanics so that quantities such as mass, force and acceleration are used consistently in calculations.

Systems of units: overview

Two systems are commonly encountered in engineering mechanics.

• International System of Units (SI) - an absolute system in which the base quantity of mass is measured independently of local gravity.
• U.S. customary (FPS) or foot-pound-second system - often described as a gravitational system because its historically chosen base quantity is a unit of force and the relation to mass depends on the local acceleration due to gravity.

U.S. customary (FPS) system

Key features and base units:

• Base units: foot (ft) for length, second (sec or sec commonly written in FPS contexts) for time, and pound (lb) for force.
• Derived mass unit (slug): mass is often derived from the relation between force and acceleration. The slug is defined so that

slug = lb·sec²/ft

and therefore 1 slug is the mass that acquires an acceleration of 1 ft/sec² when acted on by a force of 1 lb.

If W denotes the gravitational force (weight) in pounds and g the acceleration due to gravity (in ft/sec²), the corresponding mass in slugs is obtained from the relation between force and mass:

m (slugs) = W (lb) / g (ft/sec²)

Notes on the pound and other force instruments:

• The pound is sometimes used as a unit of mass (written lbm) in non-rigorous contexts; the force version is written lbf where distinction is needed.
• Other common force units in U.S. practice are the kip (1 kip = 1000 lb) and the ton (1 ton = 2000 lb).
• Historically the U.S. system is called gravitational because the definition of force units (pound) is linked to the weight of a standard mass under standard gravity.
• Seconds are commonly abbreviated as sec in FPS contexts; in SI contexts seconds are abbreviated as s.

International System of Units (SI)

The SI is an absolute, coherent system widely used in engineering and science. In SI the unit of mass is independent of gravity; forces are derived quantities.

SI base units

• metre (m) - unit of length
• kilogram (kg) - unit of mass
• second (s) - unit of time
• ampere (A) - unit of electric current
• kelvin (K) - unit of thermodynamic temperature
• mole (mol) - unit of amount of substance
• candela (cd) - unit of luminous intensity

Force in SI

• Newton (N) is the SI derived unit of force.
• By Newton's second law, force is mass times acceleration, so

1 N = 1 kg·m/s²

The SI convention uses the kilogram exclusively as a unit of mass; it is not a unit of force.

Derived units and prefixes

• Common derived SI units used in mechanics: pascal (Pa) for pressure and stress where 1 Pa = 1 N/m²; joule (J) for energy where 1 J = 1 N·m.
• Common prefixes: kilo (k, 10³), mega (M, 10⁶), giga (G, 10⁹), milli (m, 10⁻³), micro (µ, 10⁻⁶). Engineers should use prefixes to keep numerical values in a convenient range.

Conversion between SI and FPS (practical factors)

• 1 inch = 0.0254 m (exact by definition).
• 1 foot = 0.3048 m (exact by definition).
• 1 lbf ≈ 4.448221615 N.
• 1 slug ≈ 14.59390294 kg.
• 1 kip = 1000 lbf.
• 1 ton (short ton) = 2000 lbf.
• Standard acceleration due to gravity commonly used in FPS contexts: g ≈ 32.1740 ft/sec².

Example - obtaining mass from weight (preserving FPS notation):

m (slugs) = W (lb) / g (ft/sec²)

Applied example (numerical):

Given W = 100 lb and using g = 32.1740 ft/sec²,

m = 100 lb / 32.1740 ft/sec²

m ≈ 3.106 slugs

To convert this mass to kilograms multiply by the slug-kilogram factor:

m ≈ 3.106 slugs × 14.5939 kg/slug ≈ 45.34 kg

Mass versus force - consistent use in calculations

• In SI use kilogram for mass and newton for force; write equations in the form F = m·a with F in newtons, m in kilograms and a in m/s².
• In FPS be explicit: either use slugs for mass and keep F = m·a with force in pounds, acceleration in ft/s², or use pound-mass (lbm) together with a conversion constant (commonly denoted g_c) when writing F = m·a/g_c. Always state which convention (lbf-lbm with g_c or lbf-slug without g_c) is in use.
• Ambiguities arise when the pound is used without specifying whether it is lbm (mass) or lbf (force). Good practice is to include the unit symbol (lbf or lbm) or to convert to SI.

Practical guidance for engineers and students

• Prefer SI units for analysis, reporting and interchange of designs; SI is internationally standardised and coherent.
• When working with U.S. customary data or legacy drawings, convert carefully and track whether a value is force or mass.
• Always include units on every quantity during calculations; carry units through algebraic manipulations to detect errors.
• For structural design and material properties use SI derived units: stress in Pa (or MPa), modulus in Pa, forces in N or kN, moments in N·m or kN·m.
• When a conversion constant is required (for example between lbm and lbf), state it explicitly and use consistent numerical values for gravity or g_c appropriate to the chosen convention.

Summary

The SI system is an absolute, coherent system with the kilogram as the base unit of mass and the newton as the derived unit of force. The U.S. customary (FPS) system commonly uses pound as a unit of force and the slug as the consistent derived unit of mass. Engineers must keep a strict distinction between mass and force, use explicit unit symbols, and convert correctly between systems to avoid errors. Common conversion factors (for length, force and mass) and the relation m (slugs) = W (lb) / g (ft/sec²) are frequently used in practice and should be memorised or readily available during calculations.