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Re: [Phys-l] force or mass



John Denker wrote:

SI (Le Système International d’Unités) was designed so that most
of the units are consistent with each other, and consistent with
the laws of physics. For example, the SI unit of force is equal
to the unit of mass times the unit of acceleration, so we can
write F=ma without any conversion factors.

Other systems of units are not always so consistent, in which
case you must re-think many of the laws of physics, with an eye
toward putting in conversion factors where needed.

While I understand and appreciate the intent here. I contend that one can and, more importantly should write F = ma and other laws of physics* without embedded conversion factors as I will explain below.

For example,

* If you want to measure acceleration in feet per second per
second, mass in lb, and force in lbf, then you cannot write Newton’s
second law in the form F = m a. Instead you must write F = m a/g.
This is what I usually do, on the rare occasions when I am forced
to calculate with pounds at all.
* If you want to measure mass in lb and force in lbf, and still
write F = m a without a conversion factor, then you must measure
acceleration in Gees (not in feet per second squared). This is
almost equivalent to the previous option; it just absorbs the
conversion factor into the definition of acceleration.

This is not to my taste because it makes things needlessly complicated.

IMO, if you are truly going to consider the "pound" (lb) to be a unit of mass (equal presumably to the mass of an object weighing one "pound force" (lbf) when subject to some standard gravitational field, say 32 ft/s^2), then, by definition, 1 pdf = 32 lb ft/s^2. Thus, there is no problem with F = ma as written. For instance, a "5 lb mass" accelerating at 8 ft/s^2 would be subject to a net force of 40 lb ft/s^2. I can now perform a simple unit conversion to express this result as 1.25 lbf. (Note: If there is any potential confusion here it comes exclusively from the question of whether the pound is a unit of mass or a unit of force, not from any question about the universal validity of Newton's second law.)

Here is another example that is precisely equivalent apart from the absence of confusion about what the units represent: Suppose that a body of mass 20 amu accelerates at 5 furlongs/fortnight/minute. One can use F = ma directly to find that the net force on the object is 100 amu furlongs/fortnight/month. This quantity is a perfectly well defined if awkwardly expressed force. Thus, just as in the previous example, we can easily perform a unit conversion to find that the result can be expressed more simply and understandably as 4.60 x 10^-31 N.

Indeed, more often than not when we make use of physical laws we obtain results that beg for an auxiliary unit conversion. That doesn't mean that we used the wrong equation.

---

* There are some, IMO, very unfortunate lapses that have become common usage. For instance, we have two symbols, w (read "omega") and f, that stand for the same physical quantity, but that are traditionally associated with different units. For instance, 10 pi rad/s would be traditionally interpreted as a specification of w while 5 cycle/s would be considered a specification of f. I don't think too many of us would disagree with the fact that 10 pi rad/s = 5 cycles/s (i.e., in this case, w = f) *despite* the fact that we also generally write the relationship as w = 2 pi f. Worse yet, what are we to do with something like 5600 arcseconds/century? Is this w or f?

It seems to me that we should settle on one symbol (I'd opt for omega) and stick with it. Then we would no longer need two versions of every equation involving frequency.

For instance,

Q: What is the centripetal acceleration of an object moving around a circular path of radius 5.0 cm at a frequency of 50 Hz.

A: a = w^2 r = (50 Hz)^2 (5 cm) = 12500 Hz^2 cm = 12500 cycle^2 cm / s^2 = 12500 (2 pi)^2 (10^-5 km) / s^2 = 4.93 km/s^2.

[Note that, in the above I have used the fact that 1 cycle = 2 pi (which follows from the fact that 1 rad = 1.)]

John

A. JOHN "Slo" MALLINCKRODT
Lead Guitarist, Out-Laws of Physics
http://outlawsofphysics.com
Professor Emeritus of Physics, Cal Poly Pomona
http://www.csupomona.edu/~ajm
Consulting Editor, AMERICAN JOURNAL of PHYSICS
http://www.kzoo.edu/ajp

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