Skip to main content

Section 1.2 Units

Engineering quantities consist of a numeric value and an associated unit (like \(\kg{150}\text{,}\) \(\aSI{9.81}\text{,}\) or \(\ft{17}\)). The values by themselves (150, 9.81, or 17) are just numbers, but the units give the numbers context. When discussing a quantity, you must always include the associated unit. The exceptions to this rule are unitless quantities, which are typically ratios where the units all cancel out.
Related units are defined as a coherent unit system. All unit systems are based on seven base units, the important ones for Statics being mass, length, and time. These base units combine to form all other measurement units. For example:
  • Acceleration is defined as length \([L]\) divided by time \([t]\) squared, so has units \(a =[L/t^2]\)
  • Force is the product of mass and acceleration, as defined by Newton’s second law, so the units of force are \(F= [m L/t^2]\text{.}\)

Subsection 1.2.1 Unit Systems

Multiple unit systems are generally used in engineering practice in the United States, including the SI and British Gravitational systems. A third, the English Engineering system, is commonly only used in Physics applications.
The SI system, abbreviated from the French Système International (d’unités) is the modern form of the metric system. The SI system is used for most international engineering and increasingly used in the U.S. in fields like science, medicine, electronics, and the military.
In the SI system, the unit of force is the newton, abbreviated \(N\text{,}\) and the unit of mass is the kilogram, abbreviated kg. The base unit of time, used by all systems, is the second, abbreviated s. Prefixes are added to unit names are used to specify the base-10 multiple of the original unit. One newton is equal to \(\mathrm{1\ kg \cdot m}/\mathrm{s}^2\) because \(\N{1}\) of force applied to \(\kg{1}\) of mass causes the mass to accelerate at a rate of \(\aSI{1}\text{.}\)
The other unit systems, British Gravitational and English Engineering, fall under the general name of “Imperial units,” given their broad use in the British Empire. The British Gravitational system uses the foot, abbreviated ft, as the base unit of distance, the second for time, and the slug for mass. Force is expressed in the unit of pound-force, abbreviated lbf, or lb for short. One pound-force will accelerate a mass of one slug at \(\aUS{1}\text{,}\) so \(\lbf{1} = \mathrm{1\ slug \cdot ft}/\mathrm{s}^2\text{.}\) On earth, a 1 slug mass weighs \(\lbf{32.174}\text{.}\)
The English Engineering system also uses foot as the base unit of distance and second for time, but, unlike the British Gravitational system, uses pound-mass as the base unit of mass, where
\begin{equation*} \lbm{32.174} = 1 \mathrm{\ slug} = \kg{14.6}\text{.} \end{equation*}
While gravitational acceleration remains the same between the British Gravitational and English Engineering systems, a conversion factor is required to maintain unit consistency.
\begin{equation} 1= \left[\frac { 1\ \mathrm{lbf} \cdot \mathrm{s}^2}{ 32.174\ \mathrm{ft}\cdot \mathrm{lbm}}\right] = \left[ \frac{1\ \mathrm{slug}}{32.174\ \mathrm{lbm}} \right]\tag{1.2.1} \end{equation}
The perceived advantage of the English Engineering system is that on Earth, a mass of \(\lbm{1}\) weighs \(\lbf{1}\text{.}\) However, this numerical equivalence often causes confusion, given the required gravitational conversion factor mentioned in the previous paragraph and the simple fact that mass and weight are different. Mass describes how much matter an object contains, while weight is a force — the effect of gravity on a mass.
Therefore, the U.S. engineering community primarily embraces the British Gravitational system but avoids the mass unit of slugs, instead stating material amounts as weight in pounds-force, or for brevity, pounds.
Table 1.2.1 shows the standard units of weight, mass, length, time, and gravitational acceleration in three unit systems.
Table 1.2.1. Fundamental Units
Unit System Force Mass Length Time \(g\) (Earth)
SI N kg m s \(\aSI{9.81}\)
British Gravitational lbf slug ft s \(\aUS{32.2}\)
English Engineering lbf lbm ft s \(\lbf{1}/\lbm{1}\)

Force or Mass?

The confusion caused by U.S. engineering students learning about the pounds-mass + pounds-force system in Physics, closely followed by adopting a slugs + pounds-force system in Statics, often leads students to ask “Is this pound a mass or force?” to which Statics faculty commonly respond “All pounds are forces in Statics.”
Dr. Dan Baker created a clever song to help students remember that in Statics pounds always means pounds-force. Here’s a link to the song Pounds are a Force at Suno.com.
Our world is engineering, let’s get it right,
Pounds are a force, not mass in our sight.
We measure with care, we never confuse,
Pounds-mass is a unit that we never use.
[chorus]
Pounds are force, feel their weight,
That is what Engineers calculate.
SI uses Kilograms for mass, Newtons for force,
But in the U.S. we are different of course.
Gravity pulls, and we feel the strain,
Newtons and pounds, they make it plain.
Kilograms and slugs, they are the masses,
The amount of material across our classes.
[chorus]
Pounds are force, feel their weight,
That is what Engineers calculate.
SI uses Kilograms for mass, Newtons for force,
But in the U.S. we are different of course.
From meters to feet, we can convert,
9.81 meters is simply 32.2 feet, that’s certain.
So don’t ask: do we multiply or divide?
Just write \(W=mg\text{,}\) and then decide.
[chorus]
Pounds are force, feel their weight,
That is what Engineers calculate.
SI uses Kilograms for mass, Newtons for force,
But in the U.S. we are different of course.
So when you measure, know your course,
In engineering, pounds are force.
Leave pounds mass back in the dust,
In the US Customary System we trust.
By Dan Baker, with lyrics assistance from ChatGPT and music composed by Suno.com.

Subsection 1.2.2 Units within Newton’s 2nd Law

Given that Newton’s 2nd Law, \(\Sigma \vec{F} = m \vec{a}\text{,}\) is one of the foundational equations in engineering mechanics and the most basic equation relating mass and weight, let’s examine its use in more detail. First we need to relate \(\Sigma \vec{F} = m \vec{a}\) to weight, mass, and gravity. Consider an object in free fall. The only vertical force on the body is its weight \(W\text{;}\) thus, the object’s mass \(m\) is accelerated by gravity \(g\text{.}\) Therefore, we can tranform Newton’s 2nd Law \(\Sigma \vec{F} = m \vec{a}\) to \(\vec{W} = m \vec{g}\text{.}\)
The scalar version, \(W=mg\text{,}\) also serves as the conversion equation among weight, mass, and gravity. You likely know this relationship for problems using the SI system. Hence, if you are given a \(\kg{100}\) object here on earth and are asked to find the weight, you compute:
\begin{align*} W \amp = mg\\ W \amp = \kg{100}\ (\aSI{9.81})\\ W \amp = \N{981} \end{align*}
The confusion often starts when students in the United States are asked to find the mass of a known-weight object. Let’s say you are asked to find the mass of a \(\lb{100}\) object. Given the U.S. engineering convention that all pounds are force, the \(lb\) unit listed in the problem is pounds-force. Here’s the root of the confusion: SI problems give an object’s mass, while U.S. problems give an object’s weight. Life would be easier if this was not the case, but never fear — we can use \(W=mg\) to solve the problem. This time we will solve for \(m\) instead of \(W\text{.}\)
\begin{align*} W \amp = mg\\ \lb{100} \amp = m\ \aUS{32.2}\\ m \amp = \frac{\lb{100}}{\aUS{32.2}}\\ m \amp = 3.1\ \mathrm{slug} \end{align*}
Next we will investigate the values of the gravitational acceleration constant in SI and the British Gravitational system. As both measure the acceleration of an object toward the Earth, it should make sense that they are equivalent:
\begin{equation*} g=\aSI{9.81}\left (\frac{\ft{3.28}}{\m{1}}\right ) \approx \aUS{32.2} \end{equation*}
As shown in this unit conversion, always make sure to multiply terms by the correct ratios to get your desired units. In this case we multipled by (\(\ft{3.28} / \m{1}\)) with meters in the bottom so it cancels out with the meters in the top of \(\aSI{9.81}\text{.}\)
Here are two examples to test your understanding of the relationships among weight, mass, and gravity.

Example 1.2.2.

How much does a \(\kg{5}\) bag of flour weigh on Earth?
Hint.
A value in kg is a mass. Weight is a force.
Answer.
\begin{equation*} W = \N{49.05} \end{equation*}
Solution.
\begin{align*} W \amp = m g\\ \amp = \kg{5} (\aSI{9.81})\\ \amp =\N{49.05} \end{align*}

Example 1.2.3.

How much does a \(\lb{5}\) bag of sugar weigh?
Hint.
When a problem says pounds, it is referring to pounds-force.
Answer.
\begin{equation*} W = \lbf{5} \end{equation*}
Solution.
The weight was given:
\begin{equation*} W = \lb{5} = \lbf{5}\text{.} \end{equation*}
If you needed the mass in slugs, the mass is
\begin{equation*} m = \frac{\lbf{5}}{\aUS{32.2}} = 0.155 \textrm{ slug}\text{.} \end{equation*}

Subsection 1.2.3 Avoiding Common Unit Errors

Awareness of units will help you prevent errors in your engineering calculations. You should always:
  • Pay attention to the units of every quantity in the problem. Forces should have force units, distances should have distance units, etc.
  • Use the unit system given in the problem statement and ensure any constants used agree with the units of the other terms. For example, \(g=\aSI{9.81}\) should only be used when the rest of the problem is in SI units, matching the units of m and s.
  • Avoid unit conversions when possible. If you must, convert given values to a consistent set of units and stick with them.
  • Ensure unit consistency in your calculations. You can only add or subtract quantities with the same units. When multiplying or dividing quantities with units, apply the same operation to the units. Finally, the units on both sides of an equation must match.
  • Develop a sense of the magnitudes of the units and consider your answers for reasonableness. On Earth, the four commonly used masses and weights, from smallest to largest, are roughly equivalent to:
    • Newton = 1 apple,
    • pound-force = box of butter,
    • kilogram = a pineapple, and
    • slug = a medium dog (like a Corgi).
  • Assume every answer in Statics, and engineering in general, has units. A few answers will be dimensionless, but not many.
Understanding the relationship between weight, mass, and gravitational acceleration is essential for solving Statics problems. Always identify whether weight or mass is given and use \(W=mg\) with the correct gravitational acceleration. This intentional process will help you confidently work across both SI and U.S. unit systems.
PrevTopNext
Feedback