"Units for Newton's Second Law" by E. A. Mechtly
Newton's Second Law, which accurately predicts the dynamics of
unconstrained objects, can be stated, for the linear motion of rigid bodies
of constant mass, as
force equals mass times acceleration.
The five coherent sets of units which have been widely used
in this equation are expressed in the following five paragraphs:
1. The *newton* is the force which accelerates a mass of one *kilogram*
one *meter per second squared*. The meter, kilogram, and second,
(MKS units) are now *base" units* in SI. With the newton, they form
an "absolute" subset of mechanical units in which mass is chosen
as a base quantity rather than force because mass can be measured
more precisely than force. The newton.meter or *joule* is the unit of
energy in SI for all forms of energy.
2. The *dyne* is the force which accelerates a mass of one *gram*
one *centimeter per second squared* With the dyne, these cgs units
also form an "absolute" subset of mechanical units. The dyne.centimeter
or *erg* is the unit of energy.
3. The *kilopond* is the force which accelerates a mass of one *hyl*
one *meter per second squared*. These are "European Technical" units
which are a "gravitational" subset of mechanical units in which the
force of gravity is considered more convenient (although determined with
less precision than mass) for some engineering applications.
Another name for the kilopond is kilogram-force (kgf). Another name for
the *hyl* is metric-slug.
4. The *poundal* is the force which accelerates a mass of one *pound
(lbm)* one *foot per second squared*. In some older publications,
these are called "British Absolute" units.
5. The *pound-force (lbf)* is the force which accelerates a mass of one
*slug* one *foot per second squared*. In some older publications,
these are called "British Gravitational" units.
All of the units of mass cited above can be expressed as exact
multiples of the SI unit of mass, the kilogram; and all of the units of
force cited above can be expressed as exact multiples of the SI unit of
force, the newton. This is accomplished by application of the exact
definitions of non-SI units in terms of SI units (For example, one foot
equals exactly 0.3048 meter.), by application of the "standard acceleration
of free fall" (standard acceleration of gravity) 9.80665 m/s^2, and by
substitutions in Newton's Second Law.
The exact multiples for units of mass are:
One gram = 0.001 kilogram.
One hyl = 9.806 65 kilograms.
One lbm = 0.453 592 37 kilogram.
One slug = 14.593 902 9 kilograms.
The exact multiples for units of force are:
One dyne = 0.000 01 newton.
One kilopond = 9.806 65 newtons.
One poundal = 0.138 254 954 376 newton.
One lbf = 4.448 221 615 260 5 newtons.
Any incoherent mix of units of mass, force, and acceleration can
also be made to fit into Newton's Second Law, but the coefficient of each
and every factor in the equation can no longer be the number one (1).
Two examples follow:
1. The *kilopond* is the force which accelerates a mass of one *kilogram*
9.80665 meters per second squared. Alternatively, the *kilopond* is the
force which accelerates a mass of 9.80665 kilograms one meter per second
squared. Note that 9.80665 kilograms is exactly one *hyl* which appears
in the coherent form discussed in Paragraph 3 above.
2. The *lbf* is the force which accelerates a mass of one *lbm* 9.80665
meters per second squared. Alternatively, the *lbf* is the force which
accelerates a mass of one *lbm* (9.80665/0.3048) feet per second squared.
(Note that 9.80665/0.3048 is about 32.175.) Substitutions from the above
lists of exact multiples casts this equation into SI form. That is,
4.448 221 615 260 5 newtons equals 0.453 592 37 kilogram times 9.80665
meters per second squared, exactly, where the numerical values reduce
to 1, 1, and 1 respectively and exactly.
Newton's Second Law for rotating objects, for non-rigid bodies,
and for objects of changing mass is beyond the scope of this Annex.
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The above text was drafted by E. A. Mechtly as a proposed Annex
for a new edition of "Metric Practice Guide for the Welding Industry."
Some improvements suggested by Bruce Barrow or Jim Frysinger are
included.
