Units of Force Repost

I originally posted this way back in May, but I don't think anyone really looked at it, so I thought it worth reposting.  Also, I want to encourage readers to submit additional force and/or mass definitions to add to the listing.  See!  Units can be fun!

Newton's second law, F = ma, is ubiquitous in science, technology, and just about anything that involves the consideration of forces, masses, and accelerations, in other words, dynamics.

F = ma is also the basis for the definition of units of force, and in particular, for the relationship between units of force, and units of mass, and units of acceleration.  In my experience, there is occasionally some confusion over the relationship between force, mass, and acceleration, or more specifically between their respective units.  A particular area of irritation concerns English units of force and mass, and the need to include a pesky factor of 32.2 somewhere in the relationship between these two units. 

I think this confusion can be cleared up rather handily if one simply looks at the "F = ma" equation, and puts the right units in the right boxes.  If you can see it laid out it a simple table, these unit relationships become clear.  So presented below is just that simple table, with the corresponding units, configured like how the equation itself appears.  Here's the table:

 In all cases, Force equals mass times acceleration.  In each column, a particular unit of force is presented and what it relates to in terms of mass and the appropriate magnitude of acceleration.  Here, one can see that the value of 32.2 arises in the relationship between pounds of force and pounds of mass is because this relationship is defined by a specific value for of a particular acceleration, which in this case is 32.2 and is, in fact, the acceleration due to gravity here on Earth.

It turns out that it is very handy to have a unit of force have the same magnitude (not the same units) as a unit of mass when you’re on the surface of the Earth.  This is why, perhaps to the surprise of American engineers, there is a commonly used measure of force used in Europe known as the kilogram force, and for the exact same reason.

Another interesting observation is that, if you want to do away with that pesky 32.2 value when using English units, then simply switch over to using the poundal, which is simply one pound of mass accelerated at 1 ft/sec^2, completely analogous to the relationship between  Newtons and kilograms. 

There are other even more obscure units of force and mass, which are not shown here.  As I come across others, or you readers have suggestions, I will include them in future updates of this charts.

However, the point of this chart is to simply organize units of force, mass, and acceleration, based on their origin, which is Newton’s Second Law.  I hope that by so doing, it will be easier to visualize the actual relationships between these units, and make it easier to keep them straight in your head.

Comments

[bbbeard]

Even at my rusty old age, I ran across a new unit of mass this year. It turns out that some of the folks who do structural dynamics here at MSFC use a unit called the "slinch", which is a slug (32.174 lbm) times 12 inches/ft, i.e. it is about 386 lbm. One lbf will accelerate 1 slinch at 1 in/s2. I was flabbergasted to learn that they are serious.

FWIW I have always preferred using lbm and lbf, and doing unit arithmetic with g_c=32.174 lbm ft/s2/lbf....

BBB

[john hare]

I wonder sometimes how many people use a variety of incompatable systems like I do. For building anything hardware, feet and inches. For accelerations and volumes, meters and centimeters. For velocities, miles per hour and meters per second, depending on the app. And I don't translate well. While I can define a newton, I just can't seem to use it without trying to translate it to something I have a feel for.

Several respected people have expressed some level of annoyance with me on this subject. How many other people have a similar units problem as I do?

[Randy Campbell]

TLAR - "That Looks About Right"
Probably the most used, but least admited unit of measure in the world...

CEFGW - "Close Enough For Government Work"
Self explanitory :o)

M(sub)3 - "Monday Morning Measures"
Based on having to squint through your hangover from the weekend to try and read the measurment

FAM - "Friday Afternoon Measures"
Based on Rapid Eye Movement, coupled with Rapid Focusing as ones eyes keep moving back and forth between what is being measured and the "official" shop clock on the wall.

FLA - "Feels Like About..."
Similar to "Calibrated Elbow" see below

CE - "Calibrated Elbow"
Torque based measurment using the know fact that after "doing-the-job-for-"X"-number-of-years" the person's elbow can feel the exact torque of a fitting without a torque wrench

(All I could think-of/remember this morning from 20+ years of maintenance services on "high-tech" items. Though I also recall a "mantra" found to be true in the military, and "most-often" true in a varity of civilian applications;
"It takes a collage education to break it, and a high-school education to fix it" :o)

Randy

[Randy Campbell]

Not the place for this I'm VERY sure but I'm still working on the ATR-applications post and I don't want our host-blogger to running back and forth around the blog to answer or respond to my posts... (Well, ok, I don't want him doing that TOO MUCH at any rate :o)

John, I wondered if you'd seen this recent link:
http://www.sei.aero/com/projects/displayindex.php?id=3

Speaking of ATRs... :o)

Randy

[John Bossard]

Dr. BBB,
Thanks for the new unit suggestions, I'll add it to the list. That's what I'm talking about!
I, too, use g_c most frequently. I don't know if this was your experience (given your many years of being a professor), but I found that students sort of stumbled over where g_c came from. It was my hope that with the little table above, it might be more obvious.

John H:
American Engineers and technicians are often criticized for hanging on the the English system, rather than completely converting to SI units. As a result, American engineers have to be facile in both English and SI units. In some ways, this strengthens one's care in checking the units when doing engineering calculations. One definitely becomes used to working with certain sets of units. Go with the units that you have a feel for. A good engineer should be confident enough to navigate through whatever units come their way.

Randy C:
I suppose going to multiple universities qualifies as a "collage" education :) Regarding comment-relevancy, I'm happy for you to provide comments to any post, any where. If you're wondering what my preference is however, it is that folks provide comments to the most relavant post. As I've mentioned before, as the blog owner, I get an email everytime anyone provides a comment to any posting. The email gives me the comment, the relavant post, the time, and a bunch of other info. So don't worry about me not seeing it because you're commenting on a post two months back, or having to run-back-and-forth. I will see it.

[bbbeard]

Well, for pedagogy I drilled my students in the need always to check units. I taught them the "you can only multiply by 1" rule to do conversions: i.e. 1 = (1 HP)/(745.7 W) = (12")/(1') = (32.174 lbm ft/s2)/(lbf), i.e. g_c is equal to 1 and dimensionless just like all other unit conversion factors.

As a practicing engineer, however, I dealt with the same equations all the time, so I habitually included the g_c when it went along with conventional units, e.g. the speed of sound is sqrt(gamma*g_c*R*T) and R = 53.35 ft-lbf/lbm-degR for air.

As a theoretical physicist, I learned that unit conversion factors are really products of the devil. They get in the way of insight. They make you think "proportional to" instead of "equal to". I have known engineers who were so indoctrinated with g_c, they refused to say, "force is equal to the product of mass and acceleration" (or, "force is the time derivative of momentum") -- they felt compelled to say "force is proportional to the time derivative of momentum."

Once upon a time, engineers used a conversion factor between units of heat and units of work. If you pick up an old thermodynamics textbook, you will see statements like "for a cyclic process, the integral of the work output is proportional to the integral of the heat input, the proportionality constant being a universal constant we call Q".... Even today I can start a fight at a physics conference by pointing out that Boltzmann's constant k_B = 1.38E-23 J/K is just a conversion factor for units of energy, so therefore entropy is just ln(Omega) in the microcanonical ensemble. The speed of light is in similar straits, to a physicist. In relativity, time and space are different (just as heat and work are different) but using inconsistent units is a complete waste of time....

BBB