# Why can lb's be considered a mass and a force?

For example from one of my textbooks
A 3000-lb automobile is being driven down a 5° incline at a speed
of 50 mi/h when the brakes are applied, causing a total braking force
of 1200 lb to be applied to the automobile. Determine the distance
traveled by the automobile before it comes to a stop.

So the car has a mass of 3000lb, but then they measure the force applied also in pounds.
Unlike SI units where forces are always measured in N and mass in kg.

Pounds is a unit of force, not mass. A 3000-lb car puts 3000 pounds of force down on the ground.

Pounds is a unit of force, not mass. A 3000-lb car puts 3000 pounds of force down on the ground.

So is it incorrect to say 1 lb = 0.454 kg?

From this wiki article: http://en.wikipedia.org/wiki/Pound_(force [Broken])
It appears that a pound is a unit of mass however a pound of force is defined as a pound multiplied by the Earth's gravitational constant. So a pound may refer to the unit of mass or the unit of force however these are different and should be inferred from context.

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D H
Staff Emeritus
The pound is
• A unit of mass. Multiple units of mass, in fact. There's the avoirdupois pound, which is what is typically meant when the term "pound" is used without any qualifier. Other pounds include the troy pound, tower pound, merchant pound, etc. The pound as a unit of mass is rather old.
• A unit of force. In modern parlance, a pound-force is the weight of one avoirdupois pound. However, legally and colloquially, "weight" is a unit of mass rather than force in the English system. So, better said, a pound force is the force needed to make a one pound mass accelerate at exactly 9.80665 meters per second squared. Mixed units? You bet. The US has been using the metric system for a long time. You just don't know it.
• A unit of money. At one point, the pound sterling was exactly one tower pound of sterling silver. Inflation has changed the exchange ratio just a bit, but the term pound sterling has stuck.

The pound is
• A unit of mass. Multiple units of mass, in fact. There's the avoirdupois pound, which is what is typically meant when the term "pound" is used without any qualifier. Other pounds include the troy pound, tower pound, merchant pound, etc. The pound as a unit of mass is rather old.
• A unit of force. In modern parlance, a pound-force is the weight of one avoirdupois pound. However, legally and colloquially, "weight" is a unit of mass rather than force in the English system. So, better said, a pound force is the force needed to make a one pound mass accelerate at exactly 9.80665 meters per second squared. Mixed units? You bet. The US has been using the metric system for a long time. You just don't know it.
• A unit of money. At one point, the pound sterling was exactly one tower pound of sterling silver. Inflation has changed the exchange ratio just a bit, but the term pound sterling has stuck.

Even better, "1 kg = 1000 grams" is either false, or should be understood as shorthand notation.

The value 1000 represents the tranformation from a one dimensional gram space to a one dimensional kilogram space. But the conversion factor itself has associated units.

Correctly stated, the conversion is: 1[kilogram] = 1000[kilograms/gram] 1[gram].

Otherwise we are equating apples to oranges. Physicists should pay far better attention to units than has become customary.

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The pound is
• A unit of mass. Multiple units of mass, in fact. There's the avoirdupois pound, which is what is typically meant when the term "pound" is used without any qualifier. Other pounds include the troy pound, tower pound, merchant pound, etc. The pound as a unit of mass is rather old.
• A unit of force. In modern parlance, a pound-force is the weight of one avoirdupois pound. However, legally and colloquially, "weight" is a unit of mass rather than force in the English system. So, better said, a pound force is the force needed to make a one pound mass accelerate at exactly 9.80665 meters per second squared. Mixed units? You bet. The US has been using the metric system for a long time. You just don't know it.
• A unit of money. At one point, the pound sterling was exactly one tower pound of sterling silver. Inflation has changed the exchange ratio just a bit, but the term pound sterling has stuck.

Well that's confusing
So the mass in the example is really 3000lb/32.17ft/s^s

I'm glad I dont have to use imperial units for my thermodynamics class!

So is it incorrect to say 1 lb = 0.454 kg?

It's shorthand for, a mass of 0.454 kg produces a downward force at the Earth's surface of 1 pound. You can get yourself confused with conversions if you don't keep this in mind - for instance with torque wrenches, that usually read in both N-m and Ft-lbs. "Slugs" are sometimes referred to as an imperial unit of mass (http://en.wikipedia.org/wiki/Slug_(mass [Broken])), but I personally have never, ever used such a thing - if you just keep in mind that pounds are really weight, not mass, it's not difficult to deal with them if you have to.

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I like Serena
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Even better, "1 kg = 1000 grams" is either false, or should be understood as shorthand notation.

The value 1000 represents the tranformation from a one dimensional gram space to a one dimensional kilogram space. But the conversion factor itself has associated units.

Correctly stated, the conversion is: 1[kilogram] = 1000[kilograms/gram] 1[gram].

Otherwise we are equating apples to oranges. Physicists should pay far better attention to units than has become customary.

Hmm, I don't get this.

The SI prefix "kilo" literally means 1000 and is unit-less.

As far as I know:
$$1 \textrm{ kg} = 1 \cdot \textrm{ kilogram} = 1 \cdot (1000 \cdot \textrm{ gram}) = 1000 \cdot \textrm{ gram} = 1000 \textrm{ gram}$$

D H
Staff Emeritus
Well that's confusing
So the mass in the example is really 3000lb/32.17ft/s^s

I'm glad I dont have to use imperial units for my thermodynamics class!
The mass in the example is 3000 lb, or 3000 lbm if you want to use a non-standard abbreviation for pounds (mass). To convert pounds mass to kilograms, multiply by 0.45359237. This is an exact number; the pound is defined to be exactly 0.45359237 kg.

One nice thing about English units is that (ignoring variations in Earth's gravity) the force in lbf due to gravity on some object is numerically equal to its mass in lb. Thus the force due to gravity on the car in the example is 3000 lbf.

One not so nice thing about English units is that you have to use F=kma (k=1/32.1740486) instead of F=ma.

Pounds is a unit of force, not mass. A 3000-lb car puts 3000 pounds of force down on the ground.
if you just keep in mind that pounds are really weight, not mass, it's not difficult to deal with them if you have to.
Wrong. Pounds are a unit of mass and a unit of force. Read NIST Special Publication 811. Here are the conversion factors: http://physics.nist.gov/Pubs/SP811/appenB8.html. Note the conversion factors for pounds are to kilograms, not newtons (there are two conversion factors, one for avoirdupois pounds, the other for troy pounds). When talking about pounds as a unit of force it is better to say pounds-force.

Dale
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2021 Award
Correctly stated, the conversion is: 1[kilogram] = 1000[kilograms/gram] 1[gram].
I think you mean
1[kilogram] = .001[kilograms/gram] 1000[gram]

Wrong. Pounds are a unit of mass and a unit of force. Read NIST Special Publication 811. Here are the conversion factors: http://physics.nist.gov/Pubs/SP811/appenB8.html. Note the conversion factors for pounds are to kilograms, not newtons (there are two conversion factors, one for avoirdupois pounds, the other for troy pounds). When talking about pounds as a unit of force it is better to say pounds-force.

:rofl: I can't believe you said this, presumably with a straight face. You can use "pound-force", or "pound-mass", to avoid confusion, but historically and generally "pound" without clarifiers is a unit of force (weight). I think I even learned that in middle school.

D H
Staff Emeritus
:rofl: I can't believe you said this, presumably with a straight face. You can use "pound-force", or "pound-mass", to avoid confusion, but historically and generally "pound" without clarifiers is a unit of force (weight). I think I even learned that in middle school.
Apparently you had some teachers who didn't know their lb from their lbf. Your teachers were wrong. Teachers are not the keepers of standards or nomenclature. In the US, that job belongs to the National Institute of Standards and Technology.

One pound without a qualifier means either "avoirdupois pounds" or "pounds sterling". There isn't much room for confusion between mass and money now that a pound of silver is worth a tad more than a pound sterling. Without a qualifier, the shorthand notation "lb" means pounds (mass), not pounds-force. If you mean force when you say or write pounds it is best to use the term pounds-force, or lbf for short. That is the official nomenclature, after all.

Given that a lot of teachers don't know their lb from their lbf and propagate the incorrect terminology, it is of course best to be explicit and use pounds (mass) (or lbm for short) rather than expecting that the reader will know the correct terminology.

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I like Serena
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:rofl: I can't believe you said this, presumably with a straight face. You can use "pound-force", or "pound-mass", to avoid confusion, but historically and generally "pound" without clarifiers is a unit of force (weight). I think I even learned that in middle school.

I decided to look this up on wikipedia, just to see what's written there (although of course this is no guarantee).
It turns out there are 2 separate pages, one for mass, and one for force.

On the first page it says:
"The pound or pound-mass (abbreviations: lb, lbm, lbm, ℔[1] ) is a unit of mass used in the imperial, United States customary and other systems of measurement. "

On the other page it says:
"The Pound force (symbol: lb, lbf, lbf) is a unit of force in some systems of measurement; including, English Engineering units and British Gravitational units.[1]"

If I interpret this correctly, the unqualified use of pound refers to mass!
I checked a number of other wiki pages and each time the unqualified pound referred to mass, and otherwise the word pound-force was used.

Is it possible there is still a difference between e.g. US and UK?
Note that where I live the pound refers to the metric pound which is 500 grams.

O.K., you can put a bag over your head and redirect to NIST, but consider that the word "pound" derives from the Roman libra unit, which obviously long predates Newton, F=ma, and the concept of mass as opposed to weight, which is a force. Or consider that your torque wrench is written in ft-lbs, not ft-lbs-force.

Or not.

I like Serena
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O.K., you can put a bag over your head and redirect to NIST, but consider that the word "pound" derives from the Roman libra unit, which obviously long predates Newton, F=ma, and the concept of mass as opposed to weight, which is a force. Or consider that your torque wrench is written in ft-lbs, not ft-lbs-force.

You are right of course, but I believe the use of the word pound changed (slowly) when it was linked to the kilogram in 1901.
Before then it didn't matter much.
Note that the abbreviation "lb" is still ambiguous (looking at the wiki articles).

D H
Staff Emeritus
O.K., you can put a bag over your head and redirect to NIST, but consider that the word "pound" derives from the Roman libra unit, which obviously long predates Newton, F=ma, and the concept of mass as opposed to weight, which is a force.
That bag is on your head, not mine. We in the technical community often lament when the lay world hijacks one of our terms. Prime example: "theory". Here it is the other way around. "Weight" meant "mass" long before it meant "force". This newer meaning is an abuse of long-standing terminology by the technical community. In this case, we are the ones guilty of convoluting the language, not the lay community.

You mentioned the Roman libra. Romans measured quantity of matter with a balance, not a spring scale. (The spring scales that we use measure what we now call "weight" are a fairly recent invention, dating to about 240 years ago.) The libra was a unit of mass, not force. Look to the sky. The symbol for the constellation Libra is a balance, not a spring scale. Balances measure mass, not what we now call "weight".

Or consider that your torque wrench is written in ft-lbs, not ft-lbs-force.
Perhaps you should consider that NIST strongly notes that the correct term for this unit is ft-lbf. In this case, there is little ambiguity, so calling it ft-lb does not raise a lot of heartburn.

Doc Al
Mentor
For example from one of my textbooks

So the car has a mass of 3000lb, but then they measure the force applied also in pounds.
Unlike SI units where forces are always measured in N and mass in kg.
What textbook is this? In most introductory books that I've seen that use the "British" system, the 3000lbs would be the car's weight (so really it should be 3000 pound-force).

I'd say it would be rather criminal to use 'lb' in two different senses within the same paragraph in a textbook!
Well that's confusing
So the mass in the example is really 3000lb/32.17ft/s^s
Yes, in slugs.

Dale
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Personally, while I think it is good to know what the official definition is, I don't think it is important. I have never once been in a situation where it wasn't clear from context whether the pound in question was a mass or a force (or money).

D H
Staff Emeritus
I'd say it would be rather criminal to use 'lb' in two different senses within the same paragraph in a textbook!
You are assuming that the OP gave a fair rendition. I'm not saying that it isn't a fair rendition, but you have to admit that that is a big assumption. That said, some US engineering texts that use customary units are a bit loose with terminology.

Dale
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That said, some US engineering texts that use customary units are a bit loose with terminology.
My statics textbook was one of those. In the introductory paragraph it mentioned the different pounds, and then proceeded to use lb to refer to a pound force throughout the rest of the text and qualified the pound mass as lbm.

What textbook is this? In most introductory books that I've seen that use the "British" system, the 3000lbs would be the car's weight (so really it should be 3000 pound-force).

I'd say it would be rather criminal to use 'lb' in two different senses within the same paragraph in a textbook!

its "vector mechanics for engineers 9th Beer and Johnston".

From the intro
Most practicing American engineers still
commonly use a system in which the base units are the units of length,
force, and time. These units are, respectively, the foot (ft), the pound
(lb), and the second (s). The second is the same as the corresponding
SI unit. The foot is defined as 0.3048 m. The pound is defined as the
weight of a platinum standard, called the standard pound , which is
kept at the National Institute of Standards and Technology outside
Washington, the mass of which is 0.453 592 43 kg. Since the weight of
a body depends upon the earth’s gravitational attraction, which varies
with location, it is specified that the standard pound should be placed
at sea level and at a latitude of 458 to properly define a force of 1 lb.
Clearly the U.S. customary units do not form an absolute system of
units. Because of their dependence upon the gravitational attraction of
the earth, they form a gravitational system of units.

While the standard pound also serves as the unit of mass in com-
mercial transactions in the United States, it cannot be so used in engi-
neering computations, since such a unit would not be consistent with
the base units defined in the preceding paragraph.

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Interesting how the book sidesteps the fact that the pound is defined by the kilogram which is kept in Paris.
And how it also sidesteps the fact that standard gravity is defined by SI to be 9.80665 m/s2.
At least I assume that it matches the latitude-of-458-thingy (what is 458 anyway? Degrees/minutes or something?).

Doc Al
Mentor
At least I assume that it matches the latitude-of-458-thingy (what is 458 anyway? Degrees/minutes or something?).
I suspect that it said latitude of 45°.

D H
Staff Emeritus
its "vector mechanics for engineers 9th Beer and Johnston".

From the introFrom the intro
Most practicing American engineers still
commonly use a system in which the base units are the units of length,
force, and time. These units are, respectively, the foot (ft), the pound
(lb), and the second (s). The second is the same as the corresponding
SI unit. The foot is defined as 0.3048 m. The pound is defined as the
weight of a platinum standard, called the standard pound , which is
kept at the National Institute of Standards and Technology outside
Washington, the mass of which is 0.453 592 43 kg. Since the weight of
a body depends upon the earth’s gravitational attraction, which varies
with location, it is specified that the standard pound should be placed
at sea level and at a latitude of 458 to properly define a force of 1 lb.
Clearly the U.S. customary units do not form an absolute system of
units. Because of their dependence upon the gravitational attraction of
the earth, they form a gravitational system of units.

While the standard pound also serves as the unit of mass in com-
mercial transactions in the United States, it cannot be so used in engi-
neering computations, since such a unit would not be consistent with
the base units defined in the preceding paragraph.

What a pile of rubbish, from beginning to end. Did either of the authors bother to check the veracity of these statements with NIST, or to ask engineers who are forced to use customary units whether they can no longer do physics because of these units?

There is no reason for a pound-force standard for the same reason that there is no reason to maintain a standard for the newton. The pound-force, like the newton, is a derived unit. The pound force is defined as 4.4482216152605 newtons, exactly. There similarly is no reason to have a standard pound. The pound mass (avoirdupois) is defined as 0.45359237 kilograms, exactly. NIST did have a standard pound, 150 years ago. Now they have a copy of a the standard kilogram. As far as NIST is concerned, the US has been metric for over a hundred years.

As far as not be able to use the pound mass in engineering calculations, that too is baloney. My cohorts who are forced to use customary units throughout can still build aircraft, spacecraft, and rockets. That they have to use F=kma rather than F=ma is an inconvenience but is not a barrier. Fortunately, because I have to deal with US, European, and Japanese spacecraft, I have the luxury of converting those silly pounds-mass, slugs/square foot, and pounds-force to metric.

I like Serena
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I suspect that it said latitude of 45°.

Found it!
Pound is a little village in Wisconsin at 45°5′37″N 88°1′58″W!

D H
Staff Emeritus
Interesting how the book sidesteps the fact that the pound is defined by the kilogram which is kept in Paris.
And how it also sidesteps the fact that standard gravity is defined by SI to be 9.80665 m/s2.
At least I assume that it matches the latitude-of-458-thingy (what is 458 anyway? Degrees/minutes or something?).
It's supposed to be 45.5o, which is where the acceleration due to gravity (gravitation+centrifugal) at sea level is about 980.665 cm/s2. This is the defined value of the standard acceleration due to gravity, definition adopted by the 3rd CGPM, 1901:
Considering the necessity to put an end to the ambiguity which in current practice still exists on the meaning of the word weight, used sometimes for mass, sometimes for mechanical force, the Conference declares
• The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram;
• The word “weight” denotes a quantity of the same nature as a “force”: the weight of a body is the product of its mass and the acceleration due to gravity; in particular, the standard weight of a body is the product of its mass and the standard acceleration due to gravity;
• The value adopted in the International Service of Weights and Measures for the standard acceleration due to gravity is 980.665 cm/s2, value already stated in the laws of some countries.

What a pile of rubbish, from beginning to end. Did either of the authors bother to check the veracity of these statements with NIST....

This is becoming rather amusing, with a NIST website as the sole and ultimate arbiter of Truth.

Dale
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They are, wrt the definition of a pound.

D H
Staff Emeritus
This is becoming rather amusing, with a NIST website as the sole and ultimate arbiter of Truth.
That is exact right. It's right there in Article I, Section 8 of the US Constitution. Since the US is the last holdover that uses customary units, it is the US government, and NIST in particular, that is the sole and ultimate arbiter of truth when it comes to tablespoons, feet, pounds mass, pounds force, and such.

Similarly, the International Bureau of Weights and Measures (BIPM) is the sole arbiter of truth regarding all things metric.

Hmm, I don't get this.

The SI prefix "kilo" literally means 1000 and is unit-less.

As far as I know:
$$1 \textrm{ kg} = 1 \cdot \textrm{ kilogram} = 1 \cdot (1000 \cdot \textrm{ gram}) = 1000 \cdot \textrm{ gram} = 1000 \textrm{ gram}$$

The units don't balance.

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Dale
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The units don't balance.
They do if you consider kilo to be a dimensionless quantity (equal to 1000) separate from grams which is a quantity with dimensions of mass, and the practice of writing them together simply to be notational shorthand.