How to do physics with British units

If you're like me, you are so baffled by physics problems involving British units that you convert everything to SI to avoid having to deal with pounds and feet. Here is an attempt at explaining the concept of pounds force, pounds mass, and acceleration in somewhat easier terms.

Physics majors: let me know if I mess this up! I do not guarantee this information against errors.

Why the metric system is better

Let's start with a basic review of some kinematics. You are probably familiar with Newton's claim that:

F=ma (1)

We generally do not give this equation much thought because it works so well, but to illustrate an important point, let's annotate the formula a bit:

F (force, N) =m (mass, kg) ⋅ a (acceleration, m/s2) (2)

Equation 2 shows us that Newton's second law relates three properties of a body:

  • force, an effort exerted against the body,
  • mass, the amount of matter making up the body or it's "heft",
  • and acceleration, the rate at which the body reacts to force by moving.

    Remember that this formula works equally well in either direction; a body can react to a force by accelerating, or it can exert a force under acceleration. There is really no difference between these two phenomena, but it helps to think of it this way when considering the problem with British units.

    By this principle, whenever a body is near a source of gravity, as is the case on the surface of the Earth, that body experiences an acceleration that gives it its weight. We can rewrite Newton's second law as:

    W=mg (3)
    the special case in which W is the object's weight and g represents the acceleration of that object due to the Earth's gravity.

    We see from this equation that the units are consistent, that is, if you multiply kilograms by meters over seconds squared, the resulting unit is the kg⋅m/s2, or Newton.

    We should also note from equation 3 that the metric system distinguishes the weight of a body from the body's mass. Weight is an expression of the force the object applies on a scale or balance, while mass refers to the quantity of matter making up that object. When a physics professor asks you your weight, he expects you to reply in Newtons rather than in kilograms.

    Why the British system is, well, horrible for physics

    Suppose you stand on a purist metric scale and discover your weight, that is, the force you exert on the scale, to be 785 Newtons. Knowing that g is about 9.8 m/s2, you can use equation 1 to determine that your mass to be 80 kilograms.

    Of course, this is not how you typically determine your weight with a bathroom scale calibrated in pounds. You may be familiar with the British distinction between pounds mass (lbm) and pounds force (lbf); one unit refers to mass, and the other to force. However, unlike the metric units of mass and force, pounds force and pounds mass are equivalent! That means that you cannot use Newton's second law with British units, such as:

    F (force, lbf) =m (mass, lbm) ⋅ a (acceleration, feet/s2) (incorrect!) (4)

    Why not? The reason is a fatal flaw in the British unit system. Recall the conclusion from equation 3 that weight and mass are two entirely different properties. Without gravity, bodies are weightless, but still have mass and are still governed by Newton's second law. Thus, the weight of an object depends not only on mass, but on the local gravity acting on that mass. On Earth, one pound of mass has a weight of one pound of force. This is analogous to a g of 1, which would be terribly convenient, but this is not correct. The quantity of g is dictated by the mass of the Earth and cannot be arbitrarily assigned. It can only be measured or computed from measured quantities. The key to understanding the failure of the British system is to note that the relationship between mass and weight is fixed and cannot be assigned.

    The solution to the problem

    Fine, then, but how do we overcome a flaw in our system of units? We begin by eliminating some definitions that were erroneously made in defiance to Newton. As noted above, we can arbitrarily choose a unit of mass or of force, but not both. Here, we will forget about mass and consider the pound to be a unit of force. (The reason for this should hopefully be apparent; when we speak of pounds, most of the time, we are concerned with the weight of things, not their mass.) Returing, then, to Newton's second law, in terms of weight,

    W (force, lbf) =m (mass, ??) ⋅ a (acceleration, feet/s2) (5)

    Note that we have no idea what the unit for mass is at this point. Since we can convert meters to feet, we can conclude that the value of g at the Earth's surface is about 32.2 ft/s2. Now we know the quantities and units of every value in equation 5 except for m. Let's examine a hypothetical body having a weight of one pound:

    1 lb = m ⋅ 32.1 ft/s2 (6)

    This gives us the quantity of m Keeping in mind that we must obey algebra, we multiply to ensure that m is a unit quantity:

    1 ?? ⋅ 32.1 ft/s2 = 32.1 lbf (7)

    We now know that a body with a true mass of 1 has a weight of 32.1 pounds here on Earth. All that's missing is a name for our unit of true mass. Are you ready?

    It's called the slug

    The family of the engineer who conceived the slug (abbreviated sg) have requested that his identity and place of burial be kept secret, for their own protection. Unfortunately, since the engineering community is yet unwilling to give up the British system, even after its inherent ambiguity was instrumental in a multi-million-dollar failure in the NASA Mars program, we are stuck with it, at least for now. Just like the Newton, the slug can be derived from other units:

    1 sg ⋅ 32.1 ft/s2 = 32.1 lb
    1 lb = 1 sg⋅ft/s2 (8)
    You should think of the slug as the amount of mass having a weight of 32.1 pounds at the Earth's surface. Remember that, if nothing else; a slug of matter is heavy. You should be able to intuitively multiply or divide when necessary if you imagine lifting a slug.

    Also remember equation 8. It gives us the ability to confirm our results when making calculations involving slugs.

    One last note: the slug exists only as a bridge to allow you to perform kinematics calculations with British units. At no time should you be asked to provide a mass in slugs, or be given a mass quantity in slugs. Slugs should appear only within calculations.

    Here's an example of an easy problem involving British units:

    Q: What work is necessary to raise a pound of mass one foot?

    A: This question calls for the computation of energy from mass and height. (You should know the answer without doing the math!) The relevant formula is:

    U = mgh (9)
    Since we are given the weight of the object, and require its mass, we need to convert pounds mass to slugs. Remember that a quantity in pounds mass is still a measure of weight! We therefore divide by 32.1 to find a mass of 0.0312 sg. A mental check confirms that we want to divide; we are lifting a relatively small weight compared to that of a slug. Plugging in the quantities,
    U = (0.0312 sg) (32.1 ft/s2) (1 ft) (10)

    The units of u are of particular interest:

    U = sg ⋅ ft2/s2 (1 ft) (10)
    Rewriting equation 10 and referring back to equation 8 confirms the answer we suspected, and in addition, demonstrates that our system of units is now consistent, a very important step in error-checking our answer. note the resemblance in derived units of the Newton to the pound:
    U = 1 sg⋅ft/s2 ⋅ ft
    U = 1 lb⋅ft (11)

    Example 1 conclusion

    Pounds are a measure of weight, not mass. When given the weight of an object in pounds, if you need the mass, divide by 32.1 to obtain the mass in slugs. Remember that a slug is quite a bit. Your result should be a much smaller number.

    Here's one more example:

    Q: How much mass (in pounds) can be lifted one foot by one foot-pound of energy?

    A: The answer is obvious, but again, let's do the math anyway. Rewrite equation 9 and plug in the numbers:

    1 lb⋅ft = m (32.1 ft/s2) (1 ft) (12)
    We can easily solve for m, which we know to be a mass quantity in keeping with equation 9. Pay attention to how the units work out in slugs, not pounds.
    m = 0.0312 lb⋅ft⋅s2/ft2
    m = 0.0312 lb ⋅ s2/ft
    m = 0.0312 sg⋅ft/s2 ⋅ s2/ft
    m = 0.0312 sg (13)

    But remember, the question asks for weight, albeit in an indirect and sneaky manner. We therefore need to convert slugs into pounds. Since there are many pounds to a slug, we multiply for a final answer of 1 lbm. We can refer back to equation 5 to confirm the weight of a slug on Earth:


    W = (0.0312 sg) (32.1 ft/s2) = 1 lbf (14)

    Example 2 conclusion

    When asked for mass in pounds, you must consider the difference between true mass and mass represented by pounds. When pounds are used, gravity is "built-in." Multiply by 32.1 whenever you must convert true mass to mass in pounds.

    Review

    Hopefully, you now have a better understanding of how the British system is used in physics. You should be equally comfortable solving kinematics problems in SI or British units provided you always remember these points:

  • Pounds force and pounds mass are both units of weight, despite their similar names
  • Newton's second law only holds true if the units are consistent
  • True mass is measured in slugs in the British system
  • A slug is quite heavy, at least on Earth

    The physics community, and most engineers on the rest of the planet, have abandoned the British unit system in the interest of safety and economy. Educators should refrain from teaching the British system other than as a historical curiosity. Until that happens, good luck solving those problems!

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