There seems to be a fairly prevalent belief in the physics community that the basic concepts of our discipline (mass, force, energy, and so forth) are well understood and easily defined.1 After all, there are dozens of textbooks on every level that supposedly define all the terms they introduce. Apparently, we teachers can pass this wisdom on to our students without any cautionary notes and without any concern for subtleties. Remarkably, this is most certainly not the case, and anyone who has studied the foundational literature in physics over the last several centuries knows that none of the fundamental ideas is satisfactorily defined.
REFERENCES
1.
Art
Hobson
, “Energy and work
,” letter to the editor, Phys. Teach.
42
, 260
(May 2004
).2.
Digby Swift, Physics for GCSE (Basil Blackwell Ltd., Oxford, 1988), p. 8. Neville Warren, Excel HSC Physics (Pascal Press, Glebe NSW, 2001), p. 328.
3.
Raymond A. Serway and Robert J. Beichner, Physics for Scientists and Engineers (Saunders College Pub., Philadelphia, PA, 2000), p. 116. Paul A. Tipler, Physics for Scientists and Engineers (Worth Publishers, New York, 1991), p. 81. Hugh D. Young and Roger A. Freedman, University Physics (Addison-Wesley, Reading, MA, 1996), p. 101. Douglas C. Giancoli, Physics for Scientists and Engineers (Prentice Hall, Upper Saddle River, NJ, 2000), p. 82. David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics (Wiley, New York, 1997), p. 83.
4.
According to Kepler, “Inertia or opposition to motion is a characteristic of matter; it is stronger, the greater the quantity of matter in a given volume.” Max Jammer, Concepts of Mass in Classical and Modern Physics (Dover Publications Inc., Mineola, NY, 1997), p. 56.
5.
Newton's Definition 1 is “Quantity of matter is a measure of matter that arises from its density and volume jointly.” And “Furthermore, I mean this quantity whenever I use the term ‘body’ or ‘mass’ in the following pages.” Isaac Newton, The Principia translated by I. Bernard Cohen and Anne Whitman (University of California Press, Berkeley, 1999), p. 403. Since volume and density (i.e., relative density) are readily measured, it is likely that Newton was actually giving us an operational definition, albeit one that equated gravitational mass (via the usual floating-object methods of determining density) with inertial mass.
6.
Ernst Mach, The Science of Mechanics (Open Court Pub. Co., Chicago, IL, 1960).
7.
Max Jammer, Concepts of Mass in Classical and Modern Physics (Dover Publications Inc., Mineola, NY, 1997), pp. 91–103, and R. Dugas, A History of Mechanics (Editions du Griffon, Switzerland, 1955), pp. 443–444.
E.
Mach
, “Uber die definition der masse
,” Carl's Repertorium der Experimentalphysik
4
, 355
–359
(1868
).8.
P.W. Bridgman, The Nature of Physical Theory (Wiley, New York, 1964), first published in 1936. See the Introduction in A.S. Eddington, The Mathematical Theory of Relativity (Cambridge at the University Press, London, G.B., 1924), p. 1–7, first published in 1923. The origins of operationalism have roots in the early measurements of distance and time.
9.
R.B. Lindsay, Physical Mechanics (D. Van Nostrand Co. Inc., New York, 1933), pp. 15–18.
10.
P. A.
Goodinson
and B. L.
Luffman
, “On the definition of mass in classical physics
,” Am. J. Phys.
53
, 40
–42
(Jan. 1985
).L.
Eisenbud
, “On the classical laws of motion
,” Am. J. Phys.
26
, 144
–159
(March 1958
).R.
Weinstock
, “Laws of classical motion: What's F? What's m? What's a?
” Am. J. Phys.
29
, 698
–702
(Oct. 1961
).11.
Any experimental determination of instantaneous acceleration must use a finite change in velocity (Δv) occurring during a finite time interval (Δt). Real measurements, no matter how sophisticated, can only yield an average acceleration computed over a small but finite time interval. (The same is true for v.) Ironically, only when the acceleration is constant can instantaneous acceleration be determined experimentally, but no measured acceleration can be proven to be exactly constant (over any time scale, however small).
12.
In relativity acceleration does not necessarily occur along the line of action of the force. If F is perpendicular to v, then whereas if F is parallel to v, then This suggests that “inertial mass” is subtler than generally assumed.
See
L. B.
Okun
, “The concept of mass (mass, energy, relativity)
,” Sov. Phys. Usp.
32
, 629
(1989
).13.
W.G.V. Rosser, Introductory Special Relativity (Taylor & Francis, London, 1991), p. 18.
Carl G.
Adler
, “Does mass really depend on velocity, dad?
” Am. J. Phys.
55
, 739
–743
(Aug. 1987
).Gordon J.
Aubrecht
II, “Comment on ‘Apparatus to measure relativistic mass increase,’
” Am. J. Phys.
72
, 970
–971
(July 2004
).T. R.
Sandin
, “In defense of relativistic mass
,” Am. J. Phys.
59
, 1032
–1036
(Nov. 1991
).For a comparison of the two conventions, see
R. P.
Bickerstaff
and G.
Patsakos
, “Relativistic generalizations of mass
,” Eur. J. Phys.
16
, 63
(1995
).For a bibliography, see
L. B.
Okun
, “Note on the meaning and terminology of special relativity
,” Eur. J. Phys.
15
, 403
(1998
).14.
Anna Beck, Collected Papers of Albert Einstein, Vol. 2 (Princeton University Press, Princeton, NJ, 1989), doc. 47, p. 286. Leo Sartori, Understanding Relativity (University of California Press, Berkeley, CA, 1996), p. 206.
15.
Albert Einstein and Leopold Infeld, The Evolution of Physics (Simon and Schuster, New York, 1938), p. 208. Robert Resnick and David Halliday, Basic Concepts in Relativity and Early Quantum Theory (Macmillan, New York, 1992), p. 111. A.P. French, Special Relativity (W.W. Norton & Co., New York, 1968), p. 20. Francis W. Sears and Robert W. Brehme, Introduction to the Theory of Relativity (Addison-Wesley, Reading, MA, 1968), p. 127. E.P. Ney, Electromagnetism and Relativity (Harper & Row, New York, 1962), p. 68.
16.
Eugene
Hecht
, “An historico-critical account of potential energy: Is PE really real?
” Phys. Teach.
41
, 486
–493
(Nov. 2003
).17.
T. S.
Mudhole
and N.
Umakantha
, “Binding energy of the deuteron: A laboratory experiment
,” Am. J. Phys.
43
, 104
–105
(Jan. 1975
).18.
“Invariance means ‘same value as determined from different inertial frames of reference,’ not ‘unchanged by impact or external forces.’” Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (W.H. Freeman, San Francisco, CA, 1966), p. 134.
19.
Saint-Venant (1851) considered two objects that spring apart after they collide. Provided they do not subsequently interact, the ratio of their constant parting speeds equals the ratio of their masses. It's still often suggested [e.g., Jay Orear, Physics (Macmillan, New York, 1979), p. 56 or W.L.H. Shuter, “Mechanics experiments using modified PSSC apparatus” Am. J. Phys. 13, 766 (Oct. 1965)] that this scheme be used to define mass operationally. Unfortunately, real objects do interact gravitationally, and even in a frictionless environment their instantaneous recoil speeds would vary in time, as would their masses.
20.
Paul Hewitt, Conceptual Physics (Addison-Wesley, Reading, MA, 1998), p. 61.
For a list of texts that uncritically define force as pushes and pulls see
J. S.
Touger
, “When words fail us
,” Phys. Teach.
29
, 90
–95
(Feb. 1991
).21.
Henri Poincaré, The Value of Science (The Modern Library, New York, 2001), p. 78, first published in 1903.
22.
James Clerk Maxwell, Matter and Motion (Dover Pub. Inc., New York, 1991), pp. 26–35.
Contrast this with
James L.
Anderson
, “Newton's first two laws of motion are not definitions
,” Am. J. Phys.
58
, 1192
–1195
(Dec. 1990
).23.
In general relativity there is no gravitational force per se, and this suggests that the notion of “gravitational mass” is subtler than we ordinarily assume.
24.
If “weight” is the gravitational attraction between an object and a nearby celestial body (usually, the Earth), then we have to distinguish that from “effective weight,” which is measured by a scale. Since our planet's spin is slowing, things everywhere but at the poles are getting heavier with time. The Sun, Moon, and other planets exert a small gravitational force on terrestrial objects, making an operational definition of weight much more complicated.
Chester H.
Page
, “What is weight?
” Am. J. Phys.
43
, 920
–921
(Oct. 1975
) andFrank O.
Goodman
, “Weightful versus weightless
,” Am. J. Phys.
62
, 775
(Sept. 1994
).Alternatively,
A. P.
French
“On weightlessness
,” Am. J. Phys.
63
, 105
–106
(Feb. 1995
).25.
Satinder S.
Sidhu
, “Pristinely pure law in the lab,”
Phys. Teach.
32
, 282
–283
(May 1994
).F. E.
Domann
, “An improved Newton's second law experiment
,” Am. J. Phys.
50
, 185
–186
(Feb. 1982
). These experiments are typical of most that are carried out in the introductory physics lab. They make assumptions that are not strictly true (e.g., force is constant) and then “prove” what they set out to prove.
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