MOMENTUM.] 



MECHANICAL PHILOSOPHY. DYNAMICS. 



743 



in one second ; consequently the momentum generated 

 in one second, expresses, in fact, the moving force F ; 

 just as the velocity generated in one second represents 

 the accelerating force/. 



This latter force, as sufficiently seen in the preceding 

 remarks, is all we have to attend to in investigating the 

 motion of any single body subjected to its influence. As 

 it acts upon all the particles of the body alike, the con- 

 sideration of the number of these particles, or the mass 

 of the body, would be superfluous. But when, instead 

 of a single isolated body, we have to examine into the 

 motions of a system of bodies, connected together by any 

 ties, as cords, rods, <fcc. , or mutually acting upon one 

 another in any way ; then, as it is easy to perceive, the 

 quantity of matter in each body becomes an important 

 item of consideration. 



From what has now been said, the student will be 



prepared to give his assent to the following expressions, 



where V represents volume, and D density : 



The mass M = D V ; 



F 

 The moving force F =- M/ .'. / = JTF the accelerating 



force ; 



The momentum = M v. 



But in order that he may attach an intelligible mean- 

 ing to these symbols and relations, and may not mistake 

 their proper signification, we shall here exemplify their 

 interpretation in a particular case. 



Suppose the volume of a body, that is, its magnitude, 

 to be 10 cubic feet, and that in each cubic foot there is 

 three times the quantity of matter that there is in a 

 cubic foot of some standard substance say pure water. 

 Then, representing the density of water by unity or 1, 

 the quantity of matter in the body before us, that is, 

 the mass M will be 

 M = D V = 3 x 10 = the mass of 30 cubic feet of pure 



water; 



that is to say, there is as much matter in the proposed 

 10 cubic feet as there is in 30 cubic feet of pure water. 



If this mass move with a uniform velocity of 12 feet 

 per second, the momentum of it, or the force of the 

 blow with which it would strike an obstacle, would be 



M r = the momentum of 30 cubic feet of pure water, 

 or of 30 cubic feet of a substance of the same 

 density as water, moving at the same rate, viz., 

 12 feet per second. 



If the velocity of the moving mass, instead of being 

 uniform, is constantly accelerated, the constant accele- 

 ration being / = 6 feet pel second, then there is an ac- 

 celeration or accumulation of momentum ; and as, when 

 uniform, the momentum is M v, so the constant accumu- 

 lation of it, called the moving force, is 



M/ = the moving force (or constant accumulation of 

 momentum) of 30 cubic feet of a substance of 

 the game density as water accelerated 6 feet per 

 second. 



When the velocity of the moving body is uniform, 

 there is no moving force ; that is, there is no accumulation 

 of momentum, because there is no accumulation of 

 velocity. If motion be not the result of the force/ 

 acting on the body, the effect must be pressure or 

 weight. 



The preceding explanation of the sense in which the 

 symbols are to be understood, may be varied a little as 

 follows: Let the quantity of matter in a cubic foot of 

 pure water be taken for the unit of mass, and the density 

 of pure water for the unit of density. 



Let the momentum of a cubic foot of pure water, or. 

 of a substance of equal density, moving with a uniform 

 velocity of one foot per second, be taken for the unit of 

 momentum. Let the accumulation of momentum, or 

 the constant quantity of momentum generated in one 

 second in a cubic foot of pure water, or in the same 

 volume of a substance of equal density, moving with an 

 accelerating velocity of one foot per second, be taken for 

 the unit of moving force. Then V representing the 

 number of cubic feet in the volume of any body B, M the 



number of units of mass, v the number of feet per second 

 in the velocity, and / the number of feet per second of 

 acceleration, we shall have 



M = D V, the number of units of mass in the body B ; 

 M v, the number of units of momentum ; 

 M/, the number of units of moving force. 



We shall now proceed to some applications of the 

 theory of moving forces. 



APPLICATIONS OF MOVING FORCES. 1. Two 

 heavy bodies whose masses are M M', are Fig. 148. 

 connected together by a string which passes 

 over a fixed pulley : required the circum- 

 stances of the motion. 



Let M be the greater of the two masses ; 

 then M M' is the mass which, acted upon 

 by gravity, causes M to descend and M' to 

 ascend. The moving force F, therefore, to 

 which the motion is due, is F = (M M') g ; 

 and, therefore, the accelerating force /, which, 

 as above, is equal to F divided by the entire 

 mass moved, is 



M M' 



Q 



And since the velocity generated in t 

 seconds is /*, and the space passed over /t 2 

 (page 736), we shall have, for the velocity 

 of M downwards, or of M' upwards in 

 t seconds, 



M M' f , M- 



= > . , at, and for space, i 



These expressions determine the velocity with which 

 M descends at the end of t seconds, and the space 

 through which it will have fallen in that time. 



TENSION OF THE STRING. If the string were 

 f istened to the pulley, so as to prevent motion, M, acted 

 upon by the accelerating force of gravity g, would exert 

 upon it a pressure or tension ~M.g ; but this is diminished 

 in consequence of a part of the accelerating force, namely 

 /, acting upon M, and wholly expended in producing 

 motion : hence the pressure or tension suffered by the 

 string is only the difference between the two pressures, 

 namely : 



, M-M' . 2MM' 



T = MO,-/) - M (y- 



This, therefore, is the expression for the tension of 

 the string. With regard to the other mass, M' not only 

 is the accelerating force of gravity g, acting downwards 

 upon M', wholly exerted in stretching the string, but the 

 upward accelerating force /in addition : hence the pres- 

 sure or tension due to M' is 



T = M'fc +/) = M'fc + 



2 MM' 

 " M + M' 9 > 



the same as before ; as of course it ought to be. 



PRESSURE ON THE AXIS OF THE PULLEY 



What is tension as respects the string, becomes, of 

 course, pressure when acting on the pulley. As the 

 tension of the string is on each side the same, namely, 

 T, as determined above, therefore the pressure on the 

 pulley is twice this namely : 



4MM' 

 Pressure on axis of pulle = J^TTM' 9' 



It is scarcely necessary to remind the student, that 

 any mass multiplied by g, in other words, the moving 

 force due to gravity, is mere weight when exerted 

 statically. 



Thus, calling the weights of the masses M and M', W 

 and W respectively, we shall find that the foregoing 



2 W W 7 

 expression for T is the same as T = w . w /- 



For the expression at first given may evidently be 

 written thus, namely : 





