LAWS OF MOTION.]. 



MECHANICAL PHILOSOPHY. DYNAMICS. 



.747 



the velocity of the latter. Let mn be the line perpen 

 dicular to their surfaces at o. Fig. 152. 



By the resolution of velo- 

 cities, A may be regarded 

 as animated by two velocities, 

 of which one, C E, is parallel 

 to nm, and the other, C F, 

 perpendicular to the same 

 line. In like manner, B may 

 be regarded as animated by 

 the velocity C' E', parallel to 

 nm, and the velocity C'F' 

 perpendicular to nm. 



If A and B, at the instant j 

 of contact, were animated U 

 solely by the velocities C F, 

 C'F, they would merely 

 slide one past the other, and would experience no impac* 

 or shock : the shock they suffer is therefore due 

 solely to the velocities C E, C' E' ; and even then, thai 

 any shock may take place, the former velocity must 

 exceed the latter. 



The intensity of the impact is the same, therefore, as 

 if the two velocities C E, C' E', alone existed ; and the 

 two bodies will move as in the case of direct impact, only 

 that these motions will be combined with the unchanged 

 velocities C F, C' F' : we shall therefore only have to 

 compound the velocity of A in the direction nm, after 

 the direct impact spoken of, with the velocity C F, per- 

 licular to nm, in order to obtain the magnitude and 

 direction of A after the shock ; and in a similar way are 

 the magnitude and direction of the velocity of B to be 

 found. 



NOTE. It must be noticed by the student, that what 

 has here been taught respecting the collision of bodies, 

 concerns only their rectilinear motions their actual ad- 

 vance in space. Unless the line of direction in which 

 two bodies strike, pass through the centre of gravity of 

 each, rotation, as well as translation in space, will invari- 

 ably be the result : the motion of translation is all that 

 is sought to be determined in discussions on the collision 

 of bodies ; and it can be proved that this progressive 

 motion is not in the slightest degree modified by the 

 rotation of the impelled body. More advanced prin- 

 ciples have fully established the following general pro- 

 position, viz. When a body is acted upon by any im- 

 pulsive forces, of which the resultant does not pass 

 through the centre of gravity, the body will have in con- 

 'unce a double motion : 1, the centre of gravity will 

 move as if the forces were immediately applied to it ; 

 and 2, the tody will rotate as if this centre were abso- 

 lutely fixed. But want of space compels us to bring the 

 subject to a close : our object has been merely to present 

 to the young student a clear and perspicuous develop- 

 ment of the fundamental principles of Dynamics, and 

 not to carry him forward into those higher and more im- 

 posing applications of those principles, which necessarily 

 demand a knowledge of much more recondite mathe- 

 matical theories. 



The student who has carefully mastered what is here 

 delivered, will, we hope, find his study of the more ad- 

 vanced dynamical researches somewhat facilitated by the 

 previous perusal of this elementary tract. Before con- 

 cluding it, however, it is proper to give a formal enunci- 

 ation of what have been called Newton's Three Laws of 

 Motion : these, as already observed at page 734, are cer- 

 tain dynamical axioms, or postulates, assuming prin- 

 ciples of too fundamental a character to admit of being 

 rigorously established either by abstract reasoning or by 

 experimental proof. In the foregoing pages we have, 

 in general, tacitly taken these for granted, in several 

 special topics of inquiry : we have preferred this course, 

 to the usual custom of stating the three laws of motion 

 in all their generality at the outset of the subject, be- 

 cause we think that their meaning and applicability can- 

 not be clearly understood and perceived, till some famili- 

 arity with the language of Dynamics, and with a few of 

 its more elementary problems, has been acquired. 

 THE THREE LAWS OF MOTION. TLere are 



various forms of expression in which Newton's three 

 laws are delivered by different writers on Dynamics : the 

 following enunciation of them will perhaps be found as 

 intelligible as any. 



First Law. A body once at rest, will remain at rest 

 unless acted upon by some external force ; and if moving 

 in any direction, will continue to move in that direction 

 unless acted on by some external force. 



Second Law. When a force acts upon a body in mo 

 tion, the effect of this action is the same, in magnitud 

 and direction, as if it acted on the body at rest. 



Third Law. This was stated by Newton as follows : 

 "Action and reaction are equal and contrary ;" that is 

 to say, A cannot act mechanically upon B, without A 

 itself being reacted upon equally, but in an opposit 

 direction. 



The conjoined effect of velocity, and the moving mass, 

 is momentum, as defined at page 743 ; and this momen- 

 tum is the dynamical evidence of the "action" referred 

 to in the law, which therefore merely affirms, that what- 

 ever momentum a body communicates in any direction, 

 that momentum it loses in that direction ; or, which is 

 the. same thing, it receives an equal momentum in a con- 

 trary direction. 



A great deal of abstract argument, and of mechanical 

 contrivance and experiment, have been employed to de- 

 monstrate the truth of these positions ; we have neither 

 space nor inclination to enumerate them : some things 

 must be taken for granted, as fundamental or primitive 

 principles, in every department of science. Reflection 

 and common sense, exercised in the examination of such 

 first principles, will usually produce a stronger amount 

 of conviction of their truth than persuasive arguments or 

 approximative experiments. Their verisimilitude, which 

 renders it hard even to imagine a contravention of them, 

 must be accepted instead of rigid proof : and when it is 

 known that the results of the remotest investigations, all 

 primarily resting on the truth, of these assumptions, are 

 in every instance verified by actual experience, the 

 original primd facie probability of their correctness be- 

 comes elevated into absolute certainty. 



We may remark, in conclusion, that the fundamental 

 principles of Dynamics, and the strict mathematical 

 theories and deductions founded on them, find their 

 fullest verification only in the movements of the heavenly 

 bodies. Terrestrial mechanics is encumbered with many 

 considerations operating as hindrances and drawbacks to 

 the rigid application of those theories. Machines of 

 human contrivance perform their functions through the 

 intervention of rods and bars, wheels and pinions ; and 

 thus the consideration of friction, an obstacle that some- 

 limes largely modifies the purely mathematical results, 

 Becomes imperatively necessary in practical mechanics. 

 But the machinery of the heavens, without any physical 

 des to hold it together, gpes on in obedience to an Al- 

 mighty Immaterial agency ; and thus the phenomena 

 exhibited are in the completest harmony with the accu- 

 rate deductions of mathematical science. 



The modifying influence of friction, in the ordinary 

 mechanical arrangements, will be examined into in the 

 ireatise on APPLIED MECHANICS. 



The following particulars will often be found of service 

 n dynamical inquiries : 



Force of gravity in the latitude of London, 51 ? 31' ~&" 

 N, 32 -1908 feet. 



Force of gravity in the latitude of Par^ 48 5tf 14" N, 

 52-1820 feet. 



Force of gravity near the Equator, latitude 1' 34" S, 

 J2-0881 feet. 



Force of gravity at Spitzbergen, latitude 79 49' 58" N, 

 58-2526. 



Length of the pendulum beating seconds in the lati- 

 ude of London, 39-139 inches. 



Equatorial diameter of the earth, 7925-465 geograr 

 hical miles. 



Polar diameter of the earth, 7898-972 geographical 

 miles. 



A French metre=39'3708 English inches. 

 A French gramme= 15 -434 grains. 



