MECHANICAL PHILOSOPHY. DYNAMICS. [RicritrmiAR MOTIOK. 



the line (t) traced out by a moving particle, than of what 

 Bribed by a bulky body ; but let it ho only under- 

 stood that by the path of a body we mean the line traced 

 put by its centre of gravity, and all grounds for depriv- 

 ing the mass of volume, and reducing it to an indivisible] 

 isolated particle of which indeed we can have no 

 idea will be removed. Throughout the present portion 

 of dynamics, our investigation* aro entirely independent 

 of mass or volume ; and, consequently, to make a single 

 physical particle the subject of those investigations, to 

 the studied and systematic exclusion of body, is to per- 

 plex and mislead the learner in fact, to convey to him 

 the notion that what he is learning is applicable only to 

 geometrical abstractions to physical nonentities, and 

 not to the actual material substances around us. 



We shall now give an example of the application of 

 the formula; just established. 



Fig. iss. 



without an effect. A body, therefore, thus 

 moving uniformly onward, u not actuated by any force 

 whatever. It u true it could not pass from rest to 

 uniform motion in a straight line without a cause an 

 impulse, for instance but nothing act* upon it after- 

 wards, or during its uniform motion ; otherwise that 

 uniformity would be interfered with. 



It i this simplest kind of motion that we shall first 

 consider, and shall then proceed to that which U <ln.-. 

 not to instantaneous impulse, but to farce continuously 

 acting upon the moving body. The proper method of 

 moasnrini; this force will shortly be explained. 



t MK (KM KK( Tl I.I N KAR MOTION. In applied, 

 as well as in pure mathematics, certain fundamental 

 positions or postulate* must at the outset be assented to. 

 In Dynamics there are three such postulates: they are 

 known as the three laws of motion.* The first of these 

 is as follows : 



A material particle, if at rest, and unacted upon by 

 any external force, will remain at rest. A material 

 particle, if in motion, and unacted upon by any external 

 force, will continue its motion uniformly, and in a 

 straight line. This law is only a formal enunciation of 

 the intriia of matter, by which is meant, its incapability 

 of altering, of \t.vlf, the state into which it is put by 

 any external cause, whether that bo a state of rest or a 

 state of motion. 



It u plain that the motion spoken of, being neither 

 unchecked nor expedited, nor its direction any way in- 

 terfered with, must be uniform in its rate, and rectilinear 

 in its course the course originally impressed. 



The rate of a body's uniform motion is estimated by 

 the extent passed over in some determinate portion of 

 time a second, a minute, an hour, etc. In dynamical 

 investigations, the second is generally taken for the unit 

 of time ; and what in popular language is called rale of 

 motion, is here called velocity ; hence if a body move 

 uniformly over ten feet overy second of time, we say 

 that it moves with a velocity of ten feet, or that its 

 Telocity is ten feet " per second" being understood. 

 Putting the symbol r for velocity, we should therefore 

 have = 10 feet As a second ia taken for the unit of 

 time, to a foot is taken for the unit of length. It is 

 common, however, to call the length of track described 

 by a moving body, the space passed through, and to 

 represent this length by * ; but it will bo borne in mind 

 that by j, length only is meant. 



The symbol employed for number of second* is < : it is 

 to be carefully observed that t does not stand for the 

 concrete quantity time, but only for the number of 

 second* in that time ; so that length, viz., feet, is the only 

 concrete quantity represented by the symbols t>, , f, 

 which are related to one another as follows ; namely ; 



So that any two of the three quantities concerned 

 being known, the third may be immediately found. If, 

 however, t be not reckoned from the commencement of 

 the uniform motion, but only after a certain space *' has 

 been passed over by the moving body, then being the 

 whole length of path from the commencement, the three 

 (Illations will be 



t 



,___, ,- 



. 







Any one of these equations is sufficient for the entire 

 theory of the motion of a body impelled by a single im- 

 pulse, or influenced originally by any cause producing 

 uniform motion. 



It may, perhaps, be as well to remark here, that we 

 peak of the path of a particle and of a body indiscrimi- 

 natfly. In most works on Dynamics, all mention of 

 body u avoided in this part of dynamics, and the in 

 of a single partn-],- only 'ii*idered. But this exclusion 

 of body as an assemblage of particles is, wo think, as 

 iiijM<licinus as it in unnecessary. There teemt, it is true, 

 more mathematical precision of language in speaking of 



Tk. 0m km of aoOaa an formally rnuncUIrd at tht tnd of lhj> 



Two 1 dies (a, b, Fig. 138) animated by the uniform 

 velocities, c, t>', set out simultaneously from the points 

 AB, and move in the direction of the straight lino A B 

 continued : required the time of their coming together. 



Suppose a overtakes b at the point C, then 



AC = vt, BC = trt; 

 that is, calling A C, ., and A B, ', 

 * = vt,$-t'=v't. 



.-. tt-i-v't:. t = 



So that the abstract number denoting the units of 

 time that is, the number of seconds will be found by 

 dividing the space between the points of starting by the 

 difference of the spaces demoting the velocities. 



If uniform motion be the result of an impulse com- 

 municated to a body originally at rest, we may reason- 

 ably conclude that the velocity produced will be propor- 

 tional to the intensity of the impulsive energy. For if 

 a body receive a certain velocity in consequence of a cer- 

 tain impulse, it ought to acquire double that velocity if 

 at any point of its path that impulse be repe.v 

 same direction ; but if the second impulse, like the first, 

 take place at the starting-point, it must unite with the 

 first impulse, so that the consequence of a double int n- 

 sity of impulse, will be a double velocity in the \xn\y 

 impelled ; and in like manner a triple intensity will pro- 

 duce a triple velocity ; and so on. 



VARIABLE RECTILINEAR MOTION 1 . When a 

 moving body does not pass over equal spaces in 0411.1! 

 times, the motion is not uniform, but variable. The 

 rate at which the body moves, or its velocity at any in- 

 stant, is the space it wonltl describe in a second of time, 

 measuring from that instant, provided all force were 

 then withdrawn, and the body left to itself. Its motion 

 during this second as no force acts upon it would of 

 course bo uniform. 



Hence velocity in general is measured as follows : 



If the motion be uniform, the velocity is mrautured by 

 the space actually passed over in a second. If ihe mo- 

 tion be variable, the velocity at any instant is measured 

 by the space that would be passed over in a second, if all 

 force were withdrawn at that instant, and the body left 

 to proceed with the motion it then has. 



ACCELERATING FORCE. A body may so move 

 as to acquire equal accessions of velocity in equal 1 1 

 that is, it may move so that its velocity at any time t 

 being o, we may have 



at the times t, t + 1 . t + 2, t + 3, t + 4, Ac. 

 the velocities r, -j- , + 2e', -j- 3i', r -j- -I <'. A-C-. 



Or the velocity may, in like manner, bo retarded, ' 

 iM-ini- negative. Under sui-h ciivunmtances the vel 

 is Raid to bo uniformly aceeleralnl, or uniformly rr/// 



The cause of the uniform acceleration or retard:iii"ii 

 of a body's velocity, we call f rrr ; ami i"istiring causes 

 by their effects, we take the amount of this const .int 



