January 9, 1920] 



SCIENCE 



45 



science in this fertile land. We applaud the 

 author on his achievement; others may ex- 

 press this appreciation more analytically; but 

 in this paragraph we acclaim the high-minded 

 attitude of the Geological Society of America 

 in making so wise a use of its money and 

 so excellent a contribution to the common 

 good of the Pan-American States and to geo- 

 logical science. 



J. M. C. 



WEIGHT OF BODY MOVING ALONG EQUATOR 



To THE Editor of Science: A prominent 

 engineer, Dr. Carl Herring, recently proposed 

 to me the following question : " Does a body 

 in motion along the earth's equator weigh 

 less (or more) than the same body at rest? " 

 Since this question, in some form or other, 

 has come up several times in recent dis- 

 cussions, the following solution, although en- 

 tirely elementary, may be not without interest. 



Let us picture the body as supported by a 

 string from the roof of a train running west- 

 ward at speed v along the equator, and let 

 /S^the tension in the string. 



The question then is: What is the relation 

 between S and v ? 



Let T (=1,038 miles per hour) be the ab- 

 solute velocity of a point on the earth's 

 equator (neglecting the motion of the earth 

 in its orbit and the motion of the solar 

 system in space). Then V-v is the absolute 

 velocity of the train (eastward) in a circular 

 path of radius B (== 3,963 miles) . 



Hence, by a well-known formula of kine- 

 matics, (V-v)^/R = the absolute acceleration 

 of the body toward the center of the earth.^ 



Further, let W = the ordinary weight of the 

 body (that is, the value of the supporting 

 force S when the train is at rest on the earth's 



1 Dr. Hering 's surprising statement in Science 

 for October 24, 1919, implying tha/t engineers do 

 not generally recognize the idea of "accelera- 

 tion" in a direction perpendicular to the path, is 

 not borne out by an eiarrtination of engineering 

 text-books, all of which (fortunately) define ac- 

 celeration in the standard way as the rate of 

 change of vector velocity. For further comment 

 on Dr. Hering 's paper, see Professor C. M. Spar- 

 row's letter in Science for November 21. 



surface), and g= the ordinary falling ac- 

 celeration (that is, the acceleration, with re- 

 spect to the earth's surface, with which the 

 body would begin to fall, from rest, if the 

 supporting string were cut) ; and let E = the 

 force with which the earth pulls the body 

 toward the center of the earth. Then E-8 = 

 the net force acting on the body in the direc- 

 tion toward the center. 



Hence, by the fundamental principle that 

 forces are proportional to the accelerations 

 they produce,^ we have 



whence 



E-S _ {V -v)^JR 

 W 



(1) 



To determine E, we note that if v = then 

 S = W, so that 



E=W + ^^ = (1.00345) TF. (3) 



gU 



Hence finally, 



«=^{'+S[-('-f)1}- 



(4) 



From these equations we see that as v, the 

 westward train-speed, increases from to y, 

 the supporting force S will increase from W 

 to (1.00345) W, which is its maximum value; 

 as V increases from Y to 2V, S will decrease 

 again from its maximum value to W; and if 

 V is increased further to about 18 V, 8 will 

 become zero. 



For reasonable traln-si>eeds, therefore (up 

 to one or two thousand miles per hour!), a 

 iody moving westward will require an in- 

 creased force to support it against falling. 



For example, let « = 60 miles per hour. 

 Then if F = l lb., we find £' = 1.000387 lb., 

 an increase of about 1/25 of one i)er cent. 



2 Eeasona for preferring the form F/F' = a/a' 

 to the form F = ma as the fundamental equation 

 of mechanics may be found in two articles by E. 

 V. Huntington : ' ' The Logical Skeleton of Elemen- 

 tary Dynamics, ' ' American Mathematical Monthly, 

 Vol. 24 (1917), pp. 1-16; "Bibliographical Note 

 on the tJse of the Word Mass in Current Text- 

 Books," ibid., Vol. 25 (1918), pp. 1-15; also in 

 controversial papers in Science from December, 

 1914, to October, 1917. 



