July 7, 1916] 



SCIENCE 



23 



THE TEACHING OF ELEMENTARY DYNAMICS 



To the Editor op Science : The communica- 

 tion of Professor "Win. Kent in Science of De- 

 cember 24, 1915, on the subject heading is par- 

 ticularly interesting as a critical analysis, but 

 the writer does not think Professor Kent's 

 proposed method of teaching the subject is the 

 best way. 



As further discussion is invited, a method 

 will now be given, very briefly, that is clear and 

 brief and that beginners readily comprehend. 



1. Let a spring -halance be graduated with a 

 set of standard pound weights (metal pieces) 

 at sea level, say at latitude 45°, where 

 g = 32.174 ft. per sec. per see. is the accelera- 

 tion due to gravity. Now suppose a certain 

 body there, when hung from the spring-balance, 

 to depress the pointer until it reads W pounds; 

 then the pull of the earth at this point on the 

 body is exactly W pounds force. 



2. Let the same body be hung from this same 

 spring-balance at any other point where the 

 acceleration of gravity is g l and suppose the 

 pointer reads W t pounds; then the pull of the 

 earth on the body at the second place, is W s 

 pounds force. 



3. State as an experimental fact that 



W 1 /g 1 = W/g. (1) 



This simple equation gives the solution to a 

 number of problems involving weights as meas- 

 ured on the standard spring-balance at differ- 

 ent latitudes and altitudes. Give several of 

 these problems. 



4. Mass. — Mass of a body means the quan- 

 tity of matter in the body. It is not supposed 

 to alter in amount by changing the position of 

 the body relative to the earth or to be affected 

 by chemical changes, the expansion or con- 

 traction of the body or by any change of the 

 body from a solid to a liquid or gaseous state 

 or a reverse change. 



If the body weighs W pounds on the stand- 

 ard spring-balance at the place where the 

 acceleration of gravity is g ft. per sec. per see., 

 the mass of the body will be assumed to vary 

 with W /g, which is likewise unaltered, by eq. 

 (1), by any change of place, volume or condi- 

 tion. If M denote the numerical measure of 

 the mass of the body in question, we can write, 



M-=Tc W/g, 

 where J; is a constant for any chosen set of 

 units. Por the engineer's system, fc=.l and, 

 M = W/g. (2) 



We have now a precise numerical measure of 

 the mass of a body and observe that, at the 

 same place, the mass of a body is directly pro- 

 portional to its weight. It is not affected by a 

 change of place, by any chemical changes 

 within the body or by any alteration in vol- 

 ume. The student has now a clear-cut, defi- 

 nite idea of the mass of a body and of its 

 measure in the engineer's system. When 

 W = g, M = 1 ; hence the unit of mass is the 

 quantity of matter that weighs g lbs. on the 

 spring balance at the place where the accelera- 

 tion is g. 



If W is the spring-balance weight at sea 

 level, 45° latitude, where g = 32.174, then 

 M = F/32.174 and the unit of mass is the 

 quantity of matter in a body weighing 32.174 

 lbs. on a spring-balance at sea level, 45° lati- 

 tude or 32.174 lbs. on a lever balance anywhere. 



5. Mass is a fundamental concept and being 

 clearly understood, " density " can be defined, 

 for a homogeneous body, as the ratio M/V, 

 where V is the volume of the body of M units 

 of mass. 



6. Prom eq. (2), we have, 



W = Mg. (3) 



Now if an unbalanced force of F lbs., acting 

 on a body of M units of mass, produces in it 

 an acceleration of a ft. per sec. per sec, the 

 formula giving the relation between F , M and 

 a must reduce to (3) when F= W , a = g. 



Such a formula is 



F = Ma. (4) 



This is one of the fundamental formulas of 

 mechanics and the arguments in favor of it 

 should be given as fully as possible, somewhat 

 as in Eouth's " Dynamics of a Particle," pp. 

 18-23 and in connection with Newton's " Three 

 Laws of Motion." The formula is equivalent 

 to the second law, of which the first is a cor- 

 ollary. The formula is readily verified by use 

 of Atwood's machine when a < g. 



7. Prom (4), other well-known formulas, 

 Ft= Mv, Fs = iMv 2 , etc., can at once be de- 



