Accurate physical laws can permit new standard units: The two laws F→=ma→ and the proportionality of weight to mass

Abstract

Three common approaches to F→=ma→ are: (1) as an exactly true definition of force F→ in terms of measured inertial mass m and measured acceleration a→; (2) as an exactly true axiom relating measured values of a→, F→ and m; and (3) as an imperfect but accurately true physical law relating measured a→ to measured F→, with m an experimentally determined, matter-dependent constant, in the spirit of the resistance R in Ohm's law. In the third case, the natural units are those of a→ and F→, where a→ is normally specified using distance and time as standard units, and F→ from a spring scale as a standard unit; thus mass units are derived from force, distance, and time units such as newtons, meters, and seconds. The present work develops the third approach when one includes a second physical law (again, imperfect but accurate)—that balance-scale weight W is proportional to m—and the fact that balance-scale measurements of relative weight are more accurate than those of absolute force. When distance and time also are more accurately measurable than absolute force, this second physical law permits a shift to standards of mass, distance, and time units, such as kilograms, meters, and seconds, with the unit of force—the newton—a derived unit. However, were force and distance more accurately measurable than time (e.g., time measured with an hourglass), this second physical law would permit a shift to standards of force, mass, and distance units such as newtons, kilograms, and meters, with the unit of time—the second—a derived unit. Therefore, the choice of the most accurate standard units depends both on what is most accurately measurable and on the accuracy of physical law.