Key Questions
- What are the similarities and differences between the commonly used unit systems?
- How do you convert a value into different units?
- When a problem mentions the pounds, does this mean pounds-force [lbf] or pounds-mass [lbm]?
Section 1.2 Units
Quantities used in engineering usually consist of a numeric value and an associated unit. The value by itself is meaningless. When discussing a quantity you must always include the associated unit, except when the correct unit is ‘no units.’ The units themselves are established by a coherent unit system.
All unit system are based around seven base units, the important ones for Statics being mass, length, and time. All other units of measurement are formed by combinations of the base units. So, for example, acceleration is defined as length \([L]\) divided by time \([t]\) squared, so has units
\begin{equation*}
a =[L/t^2]\text{.}
\end{equation*}
Force is related to mass and acceleration by Newton’s second law, so the units of force are
\begin{equation*}
F= [m L/t^2]\text{.}
\end{equation*}
In the United States several different unit systems are commonly used including the SI system, the British Gravitational system, and the English Engineering system.
The SI system, abbreviated from the French Système International (d’unités) is the modern form of the metric system. The SI system is the most widely used system of measurement worldwide.
In the SI system, the unit of force is the newton, abbreviated N, and the unit of mass is the kilogram, abbreviated kg. The base unit of time, used by all systems, is the second. 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 British Gravitational system uses the foot as the base unit of distance, the second for time, and the slug for mass. Force is a derived unit called the pound-force, abbreviated \(\mathrm{lbf},\) or pound 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 uses the pound-mass as the base unit of mass, where
\begin{equation*}
\lbm{32.174} = 1 \mathrm{\ slug} = \kg{14.6}\text{.}
\end{equation*}
The acceleration of gravity remains the same as in the British Gravitational system, but 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 advantage of this system is that (on earth) \(\lbm{1}\) weighs \(\lbf{1}\text{.}\) It is important to understand that mass and weight are not the same thing, however. Mass describes how much matter an object contains, while weight is a force —the effect of gravity on a mass.
You find the weight of an object from its mass by applying Newton’s Second Law with the local acceleration of gravity \(g\text{.}\)
\begin{equation}
W = mg\text{.}\tag{1.2.2}
\end{equation}
Warning 1.2.1.
The gravitational “constant”\(g\) varies up to about 0.5% across the earth’s surface due to factors including latitude and elevation. On the moon, \(g\) is about \(\aSI{1.625}\text{,}\) and it’s nearly zero in outer space.
Don’t assume that \(g\) always equals \(\aSI{9.81}\text{!}\) Always use the correct value of \(g\) based on your location and unit system. However, in this course, unless otherwise stated, all objects are located on earth and the values in Table 1.2.2 are applicable.
You can show that \(\lbm{1}\) mass weighs \(\lbf{1}\) on earth by first finding the weight with (1.2.2) with \(g = \aUS{32.174}\text{,}\) then applying the conversion factor (1.2.1).
\begin{align*}
W \amp= mg\\
\amp = (\lbm{1})(\aUS{32.174})\\ \\
\amp= \left( \cancel{32.174}\ \frac{\cancel{\mathrm{lbm}} \cancel{\mathrm{ft}}}{\cancel{\mathrm{s}^2}} \right) \left[ \frac{1\ \mathrm{lbf} \cdot \cancel{\mathrm{s}^2}} {\cancel{32.174}\ \cancel{\mathrm{ft}}\cdot \cancel{\mathrm{lbm}}}\right]\\
\amp = \lbf{1}
\end{align*}
Table 1.2.2 shows the standard units of weight, mass, length, time, and gravitational acceleration in three unit systems.
| Unit System | Force | Mass | Length | Time | \(g\) (Earth) |
| SI | N | kg | m | s | \(\aSI{9.81}\) |
| British Gravitational | lbf | slug | ft | s | \(\aUS{32.174}\) |
| English Engineering | lbf | lbm | ft | s | \(\lbf{1}/\lbm{1}\) |
Example 1.2.3.
How much does a \(\kg{5}\) bag of flour weigh?
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 1.2.4.
How much does a \(\lb{5}\) bag of sugar weigh?
Hint.
When someone says “pounds” they probably mean “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*}
On earth, the mass is \(\lbm{5}\text{,}\) or
\begin{equation*}
m = \lbm{5} \left[ \frac{\text{slug}}{32.174 \text{ lbm}} \right] = 0.155 \textrm{ slug}
\end{equation*}
using the conversion fractor in (1.2.1).
Thinking Deeper 1.2.5. Does 1 pound-mass equal 1 pound-force?
Of course not; they have completely different units!
Although a \(\lb{1}\) mass weighs \(\lb{1}\) on earth, pounds-mass and pounds-force are not equal. If you take a \(\lbm{1}\) mass to the moon, its mass doesn’t change, but it weighs significantly less than it does on earth. The same mass in deep space is weightless!
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.
- Avoid unit conversions when possible. If you must, convert given values to a consistent set of units and stick with them.
- Check your work for unit consistency. You can only add or subtract quantities which have the same units. When multiplying or dividing quantities with units, multiply or divide the units as well. The units of quantities on both sides of the equals sign must be the same.
- Develop a sense of the magnitudes of the units and consider your answers for reasonableness. A kilogram is about 2.2 times as massive as a pound-mass and a newton weighs about a quarter pound.
- Be sure to include units with every answer.
