Units

Units

Units are the alphabet of equations, but they are more useful than the ABCs because they allow you to understand how different units are related.

The base, or fundamental, units upon which all other units are derived are: mass, length, time, luminous intensity, electric current, temperature, and chemical amount of a substance. For this discussion we will discuss only those units derived from mass, length, and time which will be written as [M], [L], and [T]. Because the derived unit of force is so commonly used, it is often written as a base unit: [F].

As an example, Newton’s second law says that force equals mass times acceleration. Written in fundamental units it would read:

Where acceleration is length divided by time squared. Note the use of the negative exponent to indicate that time squared is in the denominator.

In the International System of Units (SI) the mass is expressed in kilograms, length in meters, and time in seconds. This results in Newton (N) as the force unit. In the Imperial system mass is expressed in slugs, length in feet, and time in seconds. this results in pounds-force (lbf) as the force units.

Newton’s second law is also used to calculate weight where acceleration (a) is the acceleration due to gravity (g). On the surface of the earth:

Calculating forces in Newtons is widely understood and creates little confusion, while calculating force in the Imperial system is often misunderstood. The key is to recall that the fundamental unit of mass is the little discussed slug. What is confusing is that the Imperial system also uses a pound-mass (lbm). A pound-mass is used for convenience and is not a fundamental unit of mass. It is simply defined as being numerically equal to pound-force. For example, an engineer’s body weight (recall that weight is a force) is 120 lbf. Because a lbm is numerically the same as a pound force, her mass is 120 lbm. But using Newton’s second law and given her weight and the acceleration due to gravity (g) of 32.2 feet per second squared, her mass in slugs is 3.73 (120/32.2).

The following table gives examples of derived units and their respective base, SI, and imperial units. As mentioned above, while force [F] is not a true base unit, it is shown as such because it is so commonly used.

Derived Unit	Base Units	SI Units	Imperial Units	SI Name	Imperial Name
Force				Newton (N)	pound-force (lbf)
Density				None	None
Specific Weight				None	None
Energy, Work, and Heat				Joule (J)	British thermal unit (BTU)
Power				watt (W)	horsepower (hp)
Pressure				Pascal (Pa)	Pounds per square inch (psi)

One advantage of working and thinking in base units is it can tell you what math is required to determine a specific result. For instance:

Given: A small wind turbine has a capacity of 3.00 kW and runs at full capacity for one hour

Find: The amount of energy produced.

Solution: Referring to the table above you can see that a kW is a unit of power, a Joule is a unit of energy. You also note that power has time cubed in the denominator and energy time squared, So, multiplying power times time equals energy. In this example the unit of time is seconds for both power and energy, so we will 3,600 seconds rather than one hour in the calculation.

In the following unit conversion examples we first start with a table showing unit equality. These tables can be found readily online. For these examples, we need only the four equalities shown. Note that multiplying a value by one minute divided sixty seconds is equivalent to multiplying by one because the numerator and denominator are equal.

Before converting ensure that you are converting from and to the same base units. In this first case, both gallons per minute and cubic feet per second are flow rates (i.e. volume per time) and can be converted. Start  with 175 gallons per minute, and, keeping track of units (by cancelling units), multiple by a series of “ones”.  The second example is solved in a similar manner.