3.1: Common Mechanical Units

We shall evaluate mechanical systems numerically using three different systems of units that are common in the United States: International System, SI (based on kilogram mass, meter, and second; kg-m-s), traditional aeronautical (based on pound force, foot, and second; lb-ft-s), and traditional structural (based on pound force, inch, and second; lb-inch-s). Table \(\PageIndex{1}\) summarizes the units of mechanical quantities that arise most often in this book.

Table \(\PageIndex{1}\): Common units of mechanical quantities

Quantity \ Unit System	International System SI (kg-m-s)	Traditional aeronautical (lb-ft-s)	Traditional structural (lb-inch-s)
Mass (translational inertia), \(m\)	kilogram mass (kg)	slug = lb-s^2 /f	lb-s^2 /inch
Length, translational motion	meter (m)	foot (ft)	inch (in.)
Time, \(t\)	second (s)	second (s)	second (s)
Force, translational action	newton (N) = kg-m/s^2	pound force (lb)	pound force (lb)
Translational stiffness constant, \(k\)	N/m	lb/ft	lb/inch
Translational damping constant, \(c\)	N/(m/s) = N-s/m	lb/(ft/s) = lb-s/ft	lb/(inch/s) = lb-s/inch
Angle, rotational motion	radian (rad), which is dimensionless	radian (rad), which is dimensionless	radian (rad), which is dimensionless
Rotational inertia, \(J\)	kg-m^2	slug-ft^2 = lb-s^2 -ft	lb-s^2 -inch
Moment or torque, rotational action	N-m	lb-ft	lb-inch
Rotational stiffness constant, \(k_\theta\)	(N-m)/rad = N-m	(lb-ft)/rad = lb-ft	(lb-inch)/rad = lb-inch
Rotational damping constant, \(c_\theta\)	(N-m)/(rad/s) = N-m-s	(lb-ft)/(rad/s) = lb-ft-s	(lb-inch)/(rad/s) = lb-inch-s

SI is called an absolute system of units, and the other two are called gravitational systems. Absolute and gravitational systems differ fundamentally in their primary and derived units as defined in the context of Newton’s 2^nd law. In any absolute system, mass is a primary unit, along with length, and time, but force is a unit derived from those of mass, length, and time. Thus, the SI force unit, the newton (N), is precisely defined from \(F\) = \(ma\), as a kilogram-meter/second² (kg-m/s² ). In any gravitational unit system, force is considered to be a primary unit, and mass is a derived unit. Thus, from m = F/a, the mass unit in the traditional aeronautical system, commonly called a slug, is precisely defined as a pound-second² /foot (lb-s² /ft). In all of these systems, the weight of an object on Earth (in the force units) is defined as the mass times the standard sea-level gravitational acceleration, which is denoted as g, so \(W\) = \(mg\). Table \(\PageIndex{2}\) includes the relevant values of \(g\).

Table \(\PageIndex{2}\): Standard sea-level gravitational acceleration

Quantity \ Unit System	International System SI (kg-m-s)	Traditional aeronautical (lb-ft-s)	Traditional structural (lb-inch-s)
Standard acceleration of gravity, \(g\)	9.807 m/s^2	32.17 ft/s^2	386.1 inch/s^2

It is often convenient in technical notation to use prefixes that indicate powers of ten. For example, a force of 456 700 N can also be written as 456.7 \(\times\) 10³ N, or in more economical form as 456.7 kN, where kN denotes a kilo-newton = 10³ N. Table \(\PageIndex{3}\) includes a standard set of prefixes used in dynamics of mechanical systems. We also use the “e” notation that is becoming standard for input to and output from computer programs. Thus, for examples, the 456.7 kN force can be expressed as 456.7e3 (or e03, e+03, e+003, etc.) N, and a length of 4.321 mm can be expressed as 4.321e−3 (or e−03, e−003, etc.) m

Table \(\PageIndex{3}\): Prefixes for units

Multiple	Prefix	Letter Prefix
10^9	giga	G
10^6	mega	M
10^3	kilo	k
10^2	hecto	h
10	deka	da
10^-1	deci	d
10^-2	centi	c
10^-3	milli	m
10^-6	micro	\(\mu\)
10^-9	nano	n