March 19, 1915] 



SCIENCE 



425 



scale ; provided it is weighed at any place other 

 than the imaginary " point of zero gravity." 



The fundamental problem to he considered 

 by the student is: Given a constant force F 

 lbs. acting for T seconds on a quantity of 

 matter W lbs., at rest at the beginning of the 

 time, but free to move, what are the results, 

 assuming that there is no f rictional resistance 1 



The first result, which is already known by 

 the boy, is motion, at a gradually increasing 

 velocity. What the relation is between the 

 elapsed time and the velocity may be deter- 

 mined by experiment. He may take a moving 

 picture, with 50 films per second, of a body 

 falling alongside of a rod marked with feet 

 and inches. He may tow a boat having a load 

 of 1,000 lbs. with a force of say 1 lb., exerted 

 through a string and measured by a spring 

 balance, alongside of a tow path on which a 

 tape line is stretched; or there may be an At- 

 wood machine in the high school on which 

 experiments may be made. By these experi- 

 ments he will learn the fundamental facts of 

 dynamics and establish the fundamental equa- 

 tion. The facts are that the velocity varies 

 directly as the time and as the force, and in- 

 versely as the quantity of matter, and the 

 equation is V '^FT/W or V = EFT/W, 

 K being a constant whose value is approxi- 

 mately 32, provided V is in feet per second, 

 F and W in pounds and T in seconds. 



The accurate determination of E requires 

 the most refined experiments, involving pre- 

 cise measurements of both F and W, and of 

 ;S^, the distance traversed during the time T, 

 from which V is determined, and precautions 

 to eliminate resistance due to friction of air 

 or water or of the machine used in the experi- 

 ments. When these refined experiments have 

 been made it has been found that the value of 

 E is 32.1740, and this figure is twice the num- 

 ber of feet that the body would fall in vacuo 

 in one second at or near latitude 45° at the sea 

 level. It is commonly represented by 5', or by 

 g„ to distinguish it from other values of g that 

 may be obtained by experiments on falling 

 bodies (or on pendulums) at other latitudes 

 and elevations. The fundamental equation 

 then is r = FTg/W (1) 



The velocity T is a derived quantity, derived 

 from measurements of space (or distance) 

 and time. If a body is moving at a uniform 

 speed, such as the minute hand of a watch, V 

 is a constant, and the distance varies directly 

 as the time, and is the product of the velocity 

 and the time, 8 = VT. But if the velocity 

 varies directly as the time (uniformly acceler- 

 ated motion), as in the case of the problem 

 we are considering, then the distance is the 

 product of the mean velocity and the time. 

 Since in our problem the body starts from rest 

 when the velocity is 0, and the velocity is V 

 at the end of the time T, the mean velocity is 

 iV and the distance is iVT, whence V = 2S/T 

 aiidT = 2S/V. 



The velocity V in feet per second, at the end 

 of the time T is numerically equal to the num- 

 ber of feet the body would travel in one second 

 after the expiration of the time T if the force 

 had then ceased to act and the body continued 

 to move at a uniform velocity. 



The fundamental equation might be written 

 2S/T=.FTg/W, which is equivalent to 

 8 :=FT-g/2W, but as this is somewhat more 

 cumbersome than the simpler-looking equation 

 V =:FTg/W, this latter equation is more con- 

 venient as the fundamental equation. It ex- 

 presses the facts that the velocity varies di- 

 rectly as F and T and inversely as W, and 

 that the velocity equals the product of F, T 

 and g divided by W. Let us further consider 

 the two equations V^FTg/W (1) and 

 ;S'= FTg/2W (2). 



We have dealt with four elementary quan- 

 tities F, T, 8, W, one derived quantity 7, 

 and one constant figure 82.1Y40. It is under- 

 stood that F is measured in standard pounds 

 of force, the standard pound of force being the 

 force that gravity exerts on a pound of matter 

 at the standard location where fi' = 32.1Y40. 



Each equation contains four variables V, F, 

 T, W, or 8, F, T, W, and in either equation 

 if values be given to any three out of the four 

 the fourth may be found. By ordinary algebraic 

 transposition, or by giving new symbols to the 

 product or quotient of two of the variables, 

 many different equations may be derived from 

 them, some of which are more curious than 



