Converting between mass and force is a common step in the design and sizing of linear motion systems. If you’re lucky, you work primarily in one set of units — either metric or English. But it’s likely there has been (or will be) a time when you need to work in both systems of units, possibly even switching back-and-forth between English units for some components and metric units for others.
If you work in the industrial world, you probably encounter some measurements (such as length) often enough in both English and metric units that you can estimate them with relative ease. One meter is approximately thirty-nine inches. One inch is approximately 2.5 centimeters…
However, mass and force are a different story. Part of the difficulty in dealing with mass and force, especially in English units, lies in the fact that we define an object’s weight (force) in pounds. But mass is also specified in pounds.
Using one unit — the pound — for both mass and force is inherently confusing. One variation of the English system of units specifies mass in terms of slugs, but slugs are hardly a common concept. (Have you ever purchased 0.1 slug of apples?)
To cut through the confusion and demonstrate how to convert between mass and force, we’ve put together the following formulas to show the relationship between the two — in both metric and English units.
Mass and force in metric units: kg and N
There are several variations of what we often refer to as the “metric” system of units, in which measurements are based on powers of ten. The most widely-adopted version of the metric system is the International System of units (SI). The SI system is sometimes referred to as the “MKS” system, because it is the only system of units to use meters, kilograms, and seconds as base units for length, mass, and time, respectively. Note that SI consists only of metric units, but the metric system contains some units that are not included in SI. (For example, Celsius and liters are metric units, but are not included in the SI system.)
Regardless of the units of measure, the relationship between mass and force is given in Newton’s second law of motion, which states that force equals mass times acceleration.
F = m * a
The typical unit of mass in the metric system is the kilogram (kg), acceleration is defined as meters per second-squared (m/s2), and the unit of force is the Newton (N), which is equal to 1 kgm/s2.
One Newton represents the force required to accelerate 1 kg of mass at 1 m/s2.
F = kg * m/s2 = N
When we apply this equation in a typical application, where the acceleration due to gravity equals approximately 9.81 m/s2, we find that 1 kg of mass produces a force (sometimes referred to as “weight”) of 9.81 N.
F = m * a
F = 1 kg * 9.81 m/s2
F = 9.81 N
Mass and force in English units: lbm, slugs, and lbf
The English system of units has many variations, most of which have long been discarded with the exception of one or two measurements that are still in use for niche applications. (For example, Apothecaries’ units have been mostly replaced, with the exception of the grain.)
Currently, there are three predominant systems of English units: the British Gravitational system (also referred to as the English Gravitational system), the English Absolute system, and the English Engineering system. For this discussion, we’ll refer to the British Gravitational and English Engineering systems.
In the British Gravitational (BG) system, mass is measured in slugs, acceleration is measured in feet per second-squared (ft/s2), and the product of mass and acceleration, force, is measured in pounds-force (lbf).
F = m * a
One pound-force (lbf) represents the force required to accelerate 1 slug of mass at 1 ft/s2.
F = slug * ft/s2 = lbf
When we apply this equation in a typical application, where the acceleration due to gravity equals approximately 32.2 ft/s2, we find that 1 slug produces a force (sometimes referred to as “weight”) of 32.2 lbf.
F = m * a
F = 1 slug * 32.2 ft/s2
F = 32.2 lbf
In the English Engineering system of units, Newton’s second law is modified to include a gravitational constant, gc, which is equal to 32.2 lbm-ft/lbf-s2.
In this system, mass is given in pounds-mass (lbm), acceleration is given in feet per second-squared (ft/s2), and force is given in pounds-force (lbf). To see why the gravitational constant is needed, let’s look at the units of the force equation using the EE system:
F = m * a / gc
F = (lbm * ft/s2) / (lbm-ft/lbf-s2) = lbf
Note that the gravitational constant, gc, provides consistency in the units.
When we apply this special form of Newton’s second law to a typical application with acceleration due to gravity of approximately 32.2 ft/s2, we find that 1 lbm produces a force (or weight) of 1 lbf.
F = m * a / gc
F = 1 lbm * 32.2 ft/s2 / (32.2 lbm-ft/lbf-s2)
F = 1 lbf
The important thing to note here is that for most applications (i.e. those where gravity is estimated at 32.2 ft/s2), one pound-mass (lbm) can be assumed to have a force (weight) of one pound-force (lbf).
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