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Given:

My thermodynamics text reads as follows:

In SI units, the force unit is the newton ($N$), and it is defined as the force required to accelerate a mass of $1\cdot kg$ at a rate of $1\cdot\frac{m}{s^2}$. In the English system, the force unit is the pound-force ($lbf$) and is defined as the force required to accelerate a mass of $32.174\cdot lbm$ (1 slug) at a rate of $1\cdot\frac{ft}{s^2}$. That is...

$$1\cdot N = 1\cdot kg\times1\cdot\frac{m}{s^2}$$

$$1\cdot lbf = 32.174\cdot lbm\cdot\times1\cdot\frac{ft}{s^2}$$

Question:

For all practical purposes, such as at STP conditions or close to it like when we have a rounded off sea-level acceleration due to gravity of $32.2\frac{ft}{s^2}$ $(101\cdot kPa)$, can I just think of the $lbf$ in the following way...

$$W=1\cdot lbf=1\cdot lbm \times 32.174\cdot\frac{ft}{s^2}$$

and that for the weight of an object having a mass of $1\cdot kg$ (also at sea-level) in SI units as...

$$W=9.81\cdot N=1\cdot kg\times9.81\cdot\frac{m}{s^2}$$

Yes or no and why?

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  • $\begingroup$ I'm not sure what "STP conditions" means. Can you clarify? $\endgroup$ – AndyT Apr 7 '15 at 12:14
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    $\begingroup$ @AndyT STP stands for Standard Temperature and Pressure. It has a precise definition, but it basically means room temperature at sea level. $\endgroup$ – Chris Mueller Apr 7 '15 at 12:24
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    $\begingroup$ I did my basic physics in 1960s with the supremely confusing and baffling pound mass, pound force, poundal and foot. Slug was a short-term life-saver. Then along came SI in late 60's with newton and kilogram metre second and all was light!! Spent my career as a physics teacher, but would NOT have contemplated this but for the simplicity of SI !! $\endgroup$ – Graham Dec 28 '16 at 22:06

10 Answers 10

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$Lb_m$ is not the base unit. The Slug is the base unit.

$32.2\ lb_m = 1\ slug$

To convert $1\ lb_m$ to $lb_f$:

$1\ lb_m * \frac{1\ slug}{32.2\ lb_m} * 32.2 \frac{ft}{s^2} = 1\ lb_f$

Therefore $1\ lb_m$ will yield $1\ lb_f$ on Earth at STP.

This video does an excellent job of explaining it.

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    $\begingroup$ This answer is incorrect. The slug is not the base unit of mass in the customary US system. The pound (mass) is. The slug is a rather late invention by US scientists and engineers who saw the advantage of $F=ma$ (as opposed to $F=kma$, which is the form of Newton's second law when force is in pounds-force, mass is in pounds, and acceleration is in feet per second squared). The pound has been around for a long, long time. The slug is not yet a century old. $\endgroup$ – David Hammen Sep 13 '17 at 7:05
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The textbook is incomplete. Newton's Law is usually written $F=ma$. The SI unit of mass is the $kg$ and that of force is the $N$. One of the advantages of SI is that it clarifies the distinction between mass and force (especially weight). In the old British Imperial system there are several options:

  • we can measure mass in pounds_mass $lbm$; the corresponding force unit is the seldom-used poundal $pdl$.
  • we can measure force in pounds_force $lbf$; the corresponding mass unit is the $slug$.

However, you'll often see $lbm$ and $lbf$ in the same document. This is perfectly acceptable: it's equivalent to normalising Newton's Law with the gravitational acceleration to give $F=ma/g$. It's the failure to state this that leads to the confusion.

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1 pound mass is that mass that weighs one pound in 1 g gravity. For most practical cases, a pound mass and a pound weight define the same amount of stuff on the surface of the earth.

To define a pound mass we rearrange Newton's law of F = mA to

m = F/A

then plug in the particulars to get pound mass:

1 pound mass = (1 pound force)/(32.174 ft/s²)

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There seems to be some confusion here. In the English (or American) system the "official" measure of mass is the slug. Turns out that 32.2 lbm = 1 slug. So to plug into the equation F=MA you can use M in slugs, A in ft/sec and F in lbf. And, as someone said, at "standard" gravity 1 lbm exerts 1 lbf on its support (its weight). If you are going to do any significant calculations, it's best, in my opinion, to get rid of all lbm designations and convert everything to slugs.

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lbf has two definitions and a friend called Poundal

(1) EE System

The force required to accelerate 1 lbm 32.174049 ft/s^2 (i.e., acceleration due to gravity) However, the problem with this is that it MUST retain 32.174049 in its units! Which is not ideal, Consider F = ma, which means ma will always have to be divided by 32.174049 making this equation F =(ma)/32.174049 however, this approach has 1 added convenience, your mass is equal to the force you exert on the surface of Earth (i.e., the magnitude of lbm and lbf are equal and interchangeable IFF considering your force on Earth due to acceleration caused by gravity at 32.174049ft/s^2) $$ lbf := \frac{lbm*32.174049ft}{s^2} $$ (2) BG System

In this case, it is in units of slugs. The force required to accelerate 1 slug 1 ft/s^2, where 1 slug is conveniently defined as 32.174048 lbm (i.e. the same value as the acceleration due to gravity) this approach also has the same added convenience as (1), your mass is equal to the force you exert on the surface of Earth (i.e., the magnitude of lbm and lbf are equal and interchangeable IFF considering your force on Earth due to acceleration caused by gravity at 32.174049ft/s^2)) $$ lbf = \frac{1slug}{32.174049lbm}\frac{1lbm*32.174049ft}{s^2} $$ $$ := \frac{slug*ft}{s^2} $$

Know the base units of the unit system you are working in for ANY final solution to be applied appropriately. Both forms are correct!


British Gravitational v. English Engineering v. Absolute English


(3)AE System

Poundal, the force required to accelerate 1 lbm 1 ft/s^2. Similar in approach to (2), except it is multiplied by a normalizing factor instead of a unit conversion, therefore retaining lbm ft/s^2 units: $$ pdl = \frac{1}{32.174049}\frac{lbm*32.174049ft}{s^2} $$ $$ :=\frac{lbm*ft}{s^2} $$

Essentially, (1),(2) and (3) are all dividing by 32.174049, however, it is when and how that makes all the difference.

Know the base units of your system, lbf will always be an ambiguity problem as long as it exists in its current symbolic form. I'd suggest adopting sdl for (2) lbf with unit slug, the ambiguity of pound is an unusual punishment lb, lbs, lbm, lbf, lbf...

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I wrote this paper in response to a declaration made by Dynamics Professor that "there is no difference between a lbm and lbf." The discussions from the students that followed exposed a huge concept error that seems to stem from the misuse of the above statement. It has some comedic relief, so it makes it more bearable ;) Enjoy!

The lbm-lbf Relationship: Why it Matters

by Kevin McConnell

Is there really a difference between a pound-mass and a pound-force? Many people might even ask, “What the hell is a pound-mass?” Well, you can point the finger at your sixth-grade physics teacher (or anyone else who may have misled you) for the confusion that surrounds this simple question. But don’t worry, it’s never too late to learn something new (and something undeniably important).

Here’s something to mull over: let’s say that you step on a scale and it reads “150.” The readout of the scale may even provide you with units of “lbs.” Well, a scale measures the amount of force that an object exerts so we can assume that the units then are lbf (pound-force). And your physics teacher told you that there is no difference between a pound-mass and a pound-force so that must mean that your body is composed of 150 pounds of mass as well, right? What your physics teacher DIDN’T tell you are the hidden assumptions that must be true for that relationship to exist. There is something so fundamentally wrong with the statement, “pounds-mass and pounds-force are the same thing!”

First off, pounds-mass is a unit of mass, and pounds-force is a unit of force (wait… WHAT?!). Newton’s second law of motion tells us that net force is equated by the product of mass and acceleration. So, we can see that a relationship exists between mass and force, but we would NEVER say, “mass and force are the same thing!”

Let’s say that I took the same scale from above on a trip to mars; what would the scale read there? Would you be surprised if the scale readout as “57 lbs?” Or what if I brought the scale to Jupiter and it told me I weighed “380 lbs?” Is the scale correct? Absolutely! As we learned previously, the scale is measuring the amount of force that you exert due to gravity (acceleration). And we know that gravity on these planets differs because of a difference in their size and mass.

KEY CONCEPT Note that your mass DOES NOT change from planet to planet; only the amount of force exerted by your mass.

So why do we keep hearing that there is no difference between pounds-mass and pounds force? Because English units were created such that 1 lbm exerts 1 lbf here on Earth! And without further ado, here is the relationship that makes it happen:

1 lbf = 32.174 lbm ft/s^2

So, the statement that people are trying to say should sound something more like “on earth, pounds-mass subject to gravity IS pounds-force!” To further illustrate this point, lets use newtons second law to calculate the force exerted by a 1 lbm object here on earth:

Force=mass x acceleration

let acceleration=g=32.174 ft/s^2 (this is Earth' s gravitational constant)

F=m x g= 1 lbm x (32.174 ft/s^2) = 32.174 (lbm ft)/s^2

But we can’t really conceptualize the units lbm-ft /s2, so we use the relationship from above to convert it to pound-force (lbf):

F= 32.174 lbm-ft/s^2 x (1 lbf / 32.174 lbm ft/s^2) = 1 lbf

We have just proved that 1 lbm exerts 1 lbf here on Earth! If this is new to you, you should drink a beer tonight to celebrate a breakthrough in your understanding! Let’s go one step further to demonstrate why the scale would read differently on Mars and Jupiter

‘NOTHER KEY CONCEPT The relationship (eq. 1) from above DOES NOT change if you’re on a different planet just because the gravity changes; this wouldn’t make sense and you’ll see why

Force = mass x acceleration

let acceleration = g = 12.176 ft/s^2 (this is the gravitational constant on Mars)

let mass = m = 150 lbm

F = m x g = 150 lbm x 12.176 ft/s^2 = 1826.4 (lbm ft)/s^2

Once again, lets convert this quantity from lbm-ft /s2, to something we know (lbf) by using the relationship illustrated above:

F=(1826.4 lbm ft/s^2) x (1 lbf / 32.174 lbm ft/s^2) = 56.8 lbf

Even though I imagine that you now have a firm grasp on this concept, let’s try it out on Jupiter to really send it the point home:

Force = mass x acceleration

let acceleration = g = 81.336 ft/s^2 (this is the gravitational constant on Jupiter)

let mass=m=150 lbm

F = m x g = 150 lbm) x 81.336 ft/s^2 x (1 lbf / 32.174 lbm ft/s^2 )=379.2 lbf

Now you’ve seen it and you can say that you understand it! So, let’s highlight the crucial points to everything that we just went over:

  • pounds-mass (lbm) and pounds-force (lbf) are NOT the same

  • an object’s mass is constant from place to place (i.e. from Earth to Mars) but the force that it exerts IS different

  • The following relationship is key to understand the link between lbm and lbf:

1 lbf=32.174 lbm ft/s^2

Arm yourself with this knowledge so that you can fight the good fight: the next time you hear someone say that pound-mass and pound-force are the same thing, you can confidently say “LIKE HELL THEY ARE!”

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Absolutely, yes you can. In fact, the mass of a slug is derived from the acceleration due to gravity.

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I will try to make it as simple as possible and will provide an example:

-First of all ignore the word slug... I know it is the standard unit for mass and so is lbm. you will see lbm used in your text and in real life 99% of the time. Once you understand this concept well you can go on to familiarize yourself to using slugs.

-Think of newton as the force required to move a mass of 1kg by 1m/s^2

-Think of pound-force (lbf) as the force required to move a mass of 1lbm by 32.2ft/s^s

Looking at the last two points above, it is obvious that the newton is very different that the lbf

  • On the surface of earth, 1kg exerts a force of 9.81N... or 9.81kgm/s^2

  • On the surface of earth, 1lbm exerts a force of 1lbf... or 32.2lbft/s^2

Makes sense?... lets try an example.

QUESTION: An astronaut has a mass of 100kg (220lbs) what is his weight (force) if he is on earth? what if he was on a planet with the gravity of 5m/s^2 (16.4ft/s^2)?

ANSWER:

Earth:

SI units --> 100kg*9.81m/s^2= 981kgm/s^2= 981N

Imperial units --> 220lbs*32.2ft/s^2= 7084 lbmft/s^2 = 220lbf

Random planet:

SI units--> 100kg*5m/s^2 = 500kgft/s^2 = 500N

imperial units --> 220lbs*16.4ft/s^2= 3608 lbmft/s^2 = 3608/32.2 = 112lbf

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lbm and lbf are not the same -- they are only of the same value in one situation, when dealing with gravity at sea level... examine a situation without gravity, the force produced by a jet of water.

  • density of water: 62.4 lbm/ft3
  • area of nozzle: 0.06 ft2
  • velocity: 10 ft/s
  • volume flow = area * vel = 0.6 ft3/s
  • F = dwater * volume flow * vel = 374.4 lbm ft/s2

to convert to lbf

F = 374.4 lbm ft/s2 divide by 32.2 lbm-ft/lbf-s2 = 11.63 lbf

it is just counter intuitive to think of quantity of lbm as greater than the quantity lbf, you expect them to be same as they are often interchanged, pound can be used for mass or force -- that it has to be divided by 32.2 lbm-ft/lbf-s2 not just 32.2 and not gravity. In the SI system

  • density of water 1000 kg/m3
  • area of nozzle 0.005574 m2
  • velocity 3.048 m/s
  • volume flow = area * velocity = .01699 m3/s
  • F = dwater * volume flow * velocity = 51.78 kg m/s2 which is a newton so 51.78 N
  • 1 lbf = 32.2 ft/s2 lbm
  • 1 lbm = .03106 s2/ft lbf -- just bizarre -- in that you have to add units to the conversion

which leads to the question -- what are lbs??? if not lbf and lbm is nothing more then a mathematical manipulation which creates a lot of confusion, but SI system has a similar problem. When you weigh sometime you are measuring a force, yet in SI we record this force in terms of mass (kg). Why we can not create a system that make sense is beyond me. The confusion comes from the English system, we should not ask what is your weight, but what is your mass. Instead of weighing 170 lbs, I would respond saying that I have a mass of 5,474 lbm ft/s2 (170*32.2) -- time to diet I think. Of course this is ridiculous. The confusion comes from an overgeneralization, i.e. 12 inches in a foot, therefore 32.2 lbm in a lbf in not true. lbm (mass) must be accelerated before the gravitational constant (gc) can be applied. If I want to find my mass, I would take my weight 170 lbs divide out the local gravitational pull, lets say 30ft/s2 = 5.667 lbf /(ft/s2) and then multiply it by the gc (gravitational constant) 32.2 lbm-ft/(lbf-s2) to get 182.5 lbm

Personally, I think the guy who came up with the pound mass (lbm) was dyslexic. What I think he really wanted to do was state that;

1 lbm * 32.2 ft/s2 = 32.2 lbf that would have been perfect, a lbf = lbm ft/s2, but some idiotic reason he decided that

1 lbm * 32.2 ft/s2 should = 1 lbf at sea level on earth, so to make the units work you have to either divide the left side or multiply the right side by gc i.e. 32.2 lbm-ft / lbf-s2. This means that lbm is not really a mass unit, but a mass gravitational constant unit (which is ridiculous) so when you multiply lbm by an acceleration you have to divide out the gravitational constant before you can obtain a force. Other than by mistake why would anyone come up with such a unit???? and why do we precise in keeping such a unit???

how much easier it would be that water has a density of 2 lbm/ft3, so that 2 lbm/ft3 * 32.2 ft/s2 = 64.4 lbf/ft2 instead of

62.4 lbm/ft3 * 32.2 ft/s2 / (32.2 lbm-ft / lbs-s2) = 62.4 lbf/ft2

the logic fails me ... please, someone enlighten me......

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  • $\begingroup$ What has this answer added that isnt in the existing answers? $\endgroup$ – agentp Dec 10 '17 at 15:40
  • $\begingroup$ the answer attempts to point out an easy misconception that the other answers could cause someone to make, i.e. that lbs = 32.2 lbm it does not. mass needs to multiplied by an acceleration before it is divided by the "gravitational constant" to convert it to lbf or lbf needs to be divided by an acceleration before it is multiplied by the "gravitational constant" to convert it to lbm -- I think these points were missing in the other posts. $\endgroup$ – ray Dec 11 '17 at 17:56
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Here's how I like to think of it. lbf is the force acting upon the mass. This is what, for example, your bathroom scale is measuring. lbm is the actual mass of the object. So the F =m*a in English units, lbf = lbm * a (aka gravity 32.2 ft/s2).

That's at least how I've always looked at it.

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protected by GlenH7 Apr 30 '18 at 22:42

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