Engineering Statics

The magnitude of a vector needs to also have units associated with it. You will recall from Physics that there are two primary unit systems. The International System of Units, SI, abbreviated from the French Système international (d'unités) is the modern form of the metric system and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, which are the second, meter, kilogram, ampere, kelvin, mole, candela. We will only use units a second, meter, and kilogram in statics. The prefixes to the unit names specify the base-10 multiple of the original unit.

The United States customary units are a system of measurements commonly used in the United States. The United States customary system developed from English units which were in use in the British Empire before the U.S. became an independent country. However, the United Kingdom's system of measures was overhauled in 1824 to create the imperial system, changing the definitions of some units. Therefore, while many U.S. units are essentially similar to their Imperial counterparts, there are significant differences between the systems. The standard units for time, distance, and mass in statics are a second, foot, and slug. Note that this text does not use pounds mass, instead opting to use slugs for all US customary units masses and avoiding the pounds mass vs. pounds-force confusion.

The magnitude of a force is measured in units of mass [m] times length [L] divided by time [t] squared [\(F=\text{m L}/\text{t}^2\)]. In metric units, the most common force unit is the newton [N] where one newton is a kilogram multiplied by a meter per second squared. This means that a one-newton force would cause a one-kilogram object to accelerate at a rate of one-meter-per-second-squared. In English units, the most common unit is the pound-force [\(\lbf{}\)], or pound [lb] for short, where one pound is the force which can accelerate a mass of one slug at one foot per second squared. It is worth noting that many physics texts use pounds mass [\(\lbm{}\)] exclusively instead of slugs (where \(\lbm{32.174}\) = \(\slug{1}\)). This text will use slugs as they are the standard mass unit in U. S. customary system and they have a parallel use to kilograms in the SI system.

Thus looking at the units of forces across the unit systems we can show:

Definition

\(\displaystyle \text{force} = \dfrac{[\text{mass}][\text{distance}]}{[\text{time}^2]}\)

SI units

\(\displaystyle \text{1 Newton}=\dfrac{[\text{kg}][\text{m}]}{[\text{s}^2]}\)

US units

\(\displaystyle \text{1 pound}=\dfrac{[\text{slug}][\text{ft}]}{[\text{s}^2]}\)

Note that when you are relating the weight to the mass of an object, that you are actually applying Newton’s Second Law (\(\Sigma\vec{F}=m\vec{a}\)) as seen in Table 1.3.1 below.

Table 1.3.1 shows the name and abbreviation (in [square brackets]) of the standard units for weight, mass, length, time, and gravitational acceleration in SI and US customary unit systems. When in doubt always convert to these units.

Gravitational acceleration \(g\) varies across the earth’s surface due to a number of factors (primarily latitude and elevation), but for the purpose of this course, the values listed in listed in Table 1.3.1 are appropriate values to use for your computations.

Stick with whichever units system is used in the problem statement and take care when using these equations to consider the difference between mass and force. If you do need to mix systems, remember that a kilogram is about 2.2 times more massive than a pound-mass and a newton weighs about a quarter pound.