F = ma

So, what is all of this? Well, let’s start with what we know. Mass can loosely describe how much “stuff” something contains. It can be determined by the product of a material’s density and its volume. We know this as the kilogram, if we follow SI units. Remember that mass is intrinsic of a specific object, and it does not change if we weigh it on the earth or if we weigh it on the moon. Weight, on the other hand, is a function of the gravity (so it’s different on the earth and the moon). Though it is commonly know as “weight”, scientists and engineers know this simply as “force”. Force equals mass times acceleration, with units of “Newton”. (When we talk about weight, we let acceleration simply equal gravity, or 9.81 m/s^2!)

W = mg

In SI units, 1 Newton is the force needed to accelerate a 1 kilogram mass 1 meter per second over 1 second’s time.

When we take a look at English units, though, the equivalent base unit for mass is called the “slug”. By analogy, we have that 1 pound-force accelerates 1 slug mass 1 foot per second over 1 second’s time.

Although “slugs” are the base units of mass for this system, sometimes the “pound-mass” is used to represent a mass. You can take the following to be true: 1 slug = 32.17 lbm. (Both are English units of mass! But remember to use “slugs” in formula that specify a mass. Not pound-masses!)

There we have it. If you remember that slugs are good and pound-masses are “bad” to play around with, everything should work as you’re probably used to with SI units. Use slugs in your equations. Simply as that. Well, kind of.

Here’s the rub, 1 lbf is the same as 1 lbm if the acceleration is equal to gravity, which is 32.17 ft/s^2. Weird! So you can weigh yourself and the scale reads both pound-force and pound-mass at the same time. I guess that is where the usefulness of the pound-mass comes in, but generally, pound mass shouldn’t be used.

One more thing … To confuse things even more, there is yet another set of mass-force pairs that are used. From the discussion above, we now understand that that 1 pound-force accelerates 1 pound-mass 32.17 feet per second over 1 second’s time. To normalize this, we might try to find a force that can accelerate 1 pound-mass 1 foot per second over 1 second’s time. This particular force is called the “poundal”, which is equal to 1/32.17 of 1 pound-force. So,1 poundal accelerates 1 pound-mass 1 foot per second over 1 second’s time (not to say this isn’t useful, but come on!).

Let’s try an example. Say we want to calculate the force exerted on a person, named Bob, driving in his Volkswagen, accelerating at 20 ft/s^2. We weigh Bob and find that his weight is 160 pounds-force (note that this is the weight directly off the bathroom scale, also 160 pounds-mass if gravity at the specific location is 32.17 ft/s^2). To convert his earth weight to a universal mass, we divide by the acceleration of gravity: 160 lbf /32.17 ft/s^2 = 4.97358 slugs. Finally, to find the horizontal force exerted on Bob, we use the traditional F = ma equation: (4.97358 slugs) * (20 ft/s^2) = 99.47 lbf.

There is a really useful table on Wikipedia. It’s copied below, but it is a little confusing to read. I merely distilled it into writing this post.

Mass-Force Unit Systems

Questions? Definitely leave a comment, and I hope we can clarify things for you.