# Do physics equations work in non-SI units?

member 392791
So, will F=ma work if mass is in pounds mass and acceleration in ft/s^2, or will a proportionality constant need to be introduced to the equation? The same question applies to most if not all physics equations

SteamKing
Staff Emeritus
Homework Helper
All unit systems are arbitrary conventions. If you measured acceleration in furlongs per fortnight squared, and you wanted force in pounds, appropriate units of mass could be derived from F = ma. Physics formulas represent general laws derived from observation or calculation.

In the imperial system, there are pounds mass and slugs. A weight of 32.2 pounds in a gravity field where g = 32.2 ft/s^2 equals a mass of 1 slug, or m (slugs) = weight / g. By definition, 1 lbm = 453.6 grams.

Some physical properties have only SI definitions (like volts, amps, coulombs, and such where there are no non-SI units of measurement).

Some physical properties have only SI definitions (like volts, amps, coulombs, and such where there are no non-SI units of measurement).
What about gauss, statcoulombs, etc., the units of the cgs system?

Khashishi
Most equations work no matter what units you use. However, many electromagnetic equations change depending on what units are used.

In your case, you can use F=ma to get the force in units of pounds[mass]*ft/s^2. One problem with imperial units is that force is usually measured in pounds[force], and you need a conversion constant to go from pounds[mass]*ft/s^2 to pounds[force] since they are totally different.

D H
Staff Emeritus
So, will F=ma work if mass is in pounds mass and acceleration in ft/s^2, or will a proportionality constant need to be introduced to the equation? The same question applies to most if not all physics equations
Of course. Newton's second law says force is proportional to rather than equal to the product of mass and acceleration. Newton viewed force as some distinct quantity and his second law as a law of nature.

The modern view is that F=ma is a definitive statement rather than a law of nature: It defines force as a derived unit. From this perspective, imperial units, with force in pounds force, mass in pounds mass, and acceleration in feet per second squared, form an inconsistent set of units.

However, from this perspective, SI units are not consistent either. We have to write E=mc2 (better: E2=(mc2)2+(pc)2). If SI truly was a consistent set of units we would write that as E2=m2+p2. That c is not 1 *and unitless* means that the International System is an inconsistent set of units. The same goes for F=GMm/r2. A consistent set of units has G=1, and once again, unitless.

SteamKing
Staff Emeritus
Homework Helper
What about gauss, statcoulombs, etc., the units of the cgs system?

I should have said 'Metric' system to cover its many different flavors.

they work in any unit just need to but good constants
you can derive 1000's equations of motion using units like pound
in main equation like

F=ma
1 N →1000 mN (millinewton)
1 kg → .....pounds

put it in
metric equation to get some other equation

It should be noted that while units generally don't matter for fundamental equations in physics and math, they CAN make a difference for angles. The derivative of sin(theta) is not cos(theta) if theta is measured in degrees rather than radians!

[..] That c is not 1 *and unitless* means that the International System is an inconsistent set of units. The same goes for F=GMm/r2. A consistent set of units has G=1, and once again, unitless.
On a side note: that c is not unitless in the SI, simply means that according to the SI length is physically different from time. [added:] However, indeed c and G could have been given the value 1.

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stevendaryl
Staff Emeritus
Most equations work no matter what units you use. However, many electromagnetic equations change depending on what units are used.

In your case, you can use F=ma to get the force in units of pounds[mass]*ft/s^2. One problem with imperial units is that force is usually measured in pounds[force], and you need a conversion constant to go from pounds[mass]*ft/s^2 to pounds[force] since they are totally different.

People don't use it much, but the English system actually had a unit of mass, the slug, which was defined so that 1 pound (force) = 1 slug * 1 foot/second^2. So a slug weighs 32 pounds on Earth.

http://en.wikipedia.org/wiki/Slug_(mass)

Yes, of course it works, this is like asking does mathematics work in another base system, of course it does.

stevendaryl
Staff Emeritus
Yes, of course it works, this is like asking does mathematics work in another base system, of course it does.

The original poster specifically asked "will a proportionality constant need to be introduced to the equation?" Depending on how the basic units are defined, you certainly do need to introduce proportionality constants in some cases.

However, from this perspective, SI units are not consistent either. We have to write E=mc2 (better: E2=(mc2)2+(pc)2). If SI truly was a consistent set of units we would write that as E2=m2+p2. That c is not 1 *and unitless* means that the International System is an inconsistent set of units. The same goes for F=GMm/r2. A consistent set of units has G=1, and once again, unitless.
I don't see how SI units are inconsistent. It's not like you're required to set c=G=1. Planck units are just a convention that physicists find convenient to work with. You won't get a contradiction if you work in SI units.

D H
Staff Emeritus
"Inconsistent" doesn't mean you get a contradiction. It's quite possible to do physics and engineering with customary units. One just needs to use F=kma. That is inconsistent with the modern view that force per se is a derived unit. SI has attacked but one of those old, inconsistent views of units. That one has to use c2 in E2=(mc2)2+(pc)2[/sup means that SI is inconsistent with relativity, which says that mass is a form of energy.

Another example: One way to look at the the Lorentz transform is that it represents a hyperbolic rotation in Minkowski space. If one looks at time and distance as fundamentally different things, this view of the Lorentz transformation can only be viewed as a trick, an abuse of notation that happens to work. On the other hand, this is a very natural result if one looks at time and distance as different aspects of the same thing. c should not have any units, and a value of one is the one value that is results in consistent physics.

Force is a measure of mass and acceleration according to classical physics, and mass is a measure of energy content according to relativity. Also, for centuries weight was used as a measure of mass (and bathroom scales still use that). Proportional doesn't mean identical.

stevendaryl
Staff Emeritus
Force is a measure of mass and acceleration according to classical physics, and mass is a measure of energy content according to relativity. Also, for centuries weight was used as a measure of mass (and bathroom scales still use that). Proportional doesn't mean identical.

I don't think it's quite fair to view Newton's "F = ma" as a definition. Without ever measuring any accelerations, one could imagine coming up with a concept of force, in terms of how much weight various "pushes" and "pulls" can lift. With springs and ropes and pulleys and so forth, you can develop an entire science of "static" forces without ever thinking to connecting it to acceleration. This might be a counterfactual (because I have no idea to what extent the concept of "force" was used for static calculations prior to Newton), but one can certainly imagine developing a theory of forces without ever realizing that there is a simple relationship between forces and acceleration. In that case, "F = ma" would be a discovery about the effects of forces, rather than a definition.

Dale
Mentor
2021 Award
You won't get a contradiction if you work in SI units.
You won't get a contradiction in any set of units, that isn't what consistency is about.

In the SI system there are two ways to write the SI unit with dimensions ML/T^2. Specifically, N or kg m/s^2. The conversion factor between them is 1, so SI is consistent.

In customary units you can write the units with the same dimensionality as lbf or lb ft/s^2. The conversion factor between them is .031, so customary units are inconsistent. You can still do calculations in customary units just fine, but you cannot just simplify the units without doing the conversions.

So, the question that D H is posing is essentially, how do we know that lbf has dimensions of ML/T^2?

Newtons original formulation would have been closer to f=kma, where k is a fundamental constant of nature with dimensions of FT^2/ML and F is considered to be a fundamentally different unit dimension than ML/T^2. Now, we could make a standard F unit e.g. by making a standard spring and compressing it a standard amount, and we could use it to accelerate a known mass to measure k.

Eventually, as we got better and better at measuring mass and length and time, we would find that our ability to measure the k of the universe was limited by our ability to make springs reproducibly. In that case, we could simply set k to be some exact value and define our unit of force, not in terms of our standard spring, but in terms of our standard masses and accelerations. Once we do that we see that k is dimensionless, and by choosing units such that it is 1 we can write f=ma.

A similar thought process can be applied to any fundamental dimensionful constant. We can choose a system of units where it is not only equal to 1 but is dimensionless. Such units are called geometrized units. From the perspective of geometrized units the SI system is inconsistent since it requires you to use dimensionful universal constants in your equations.

I don't think it's quite fair to view Newton's "F = ma" as a definition. [..] you can develop an entire science of "static" forces without ever thinking to connecting it to acceleration. [..]
I agree, I only hinted at that because I already mentioned another issue with a side topic. And I think that impressive force was already characterized by means of Hooke's law.
Now the discussions have gone in all directions but the OP hasn't given any feedback yet...

stevendaryl
Staff Emeritus
You won't get a contradiction in any set of units, that isn't what consistency is about.

Many people use the word "consistent" to mean "free of contradiction". That's certainly how it's used in mathematics.

D H
Staff Emeritus
Many people use the word "consistent" to mean "free of contradiction". That's certainly how it's used in mathematics.
That isn't how it's used in the context of determining whether a set of units is consistent. The issue here is whether the system of units is consistent with modern physics. F=kma is not while F=ma is.

Dale
Mentor
2021 Award
Many people use the word "consistent" to mean "free of contradiction". That's certainly how it's used in mathematics.
Agreed. It is a "recycled" word where they probably should have come up with a new one. Same with the "dimensionality" of a unit.

SteamKing
Staff Emeritus
Homework Helper
Well, many words can have more than one meaning, even in the sciences. This is not new.

The issue here is whether the system of units is consistent with modern physics. F=kma is not while F=ma is.
Why can't you use F=ma in English units, with mass measured in slugs, force measured in pounds, and acceleration measured in ft/s^2?

D H
Staff Emeritus
Why can't you use F=ma in English units, with mass measured in slugs, force measured in pounds, and acceleration measured in ft/s^2?
Nothing is stopping you from doing that.

There are certain engineers who don't particularly care about F=ma vs F=kma. F=kma works just as fine to them (if they need to worry about acceleration at all), and other equations become simpler. For example, some US aerospace engineers much prefer to work with mass expressed in pounds, force in pounds-force. Can this rocket take off? The answer is obvious when one uses those units. Moreover, equations that involve g when written in SI units oftentimes don't involve g when written in gravitational units. Those engineers' European counterparts are likely to work with mass expressed in kilograms, force in kiloponds for the same reasons that those US engineers eschew slugs. Some structural engineers also prefer gravitational units, both in the US and in Europe.

Other engineers in the US prefer the inch-pound-pound system. They don't care about what they view as esoteric reasons for preferring metric units and its simpler F=ma. The math still comes out right.

Why can't you use F=ma in English units, with mass measured in slugs, force measured in pounds, and acceleration measured in ft/s^2?
In a parallel thread ghwellsjr makes use of nanoseconds and (very big) feet to make c=1 ft/ns In a parallel thread ghwellsjr makes use of nanoseconds and (very big) feet to make c=1 ft/ns  That's an old joke from computer pioneer Grace Hopper.