743 



MECHANICAL PHIL OSOPIIY. DYNAMICS. 



[MOVING FORCES. 



From the foregoing principles we may readily deter- 

 mine the time in which the earth must perform it* diur- 

 nal rotation, in order that the centrifugal force at the 

 equator may be exactly equal to the force of gravity 

 there ; that is, in order that a body at the equator may 

 have no weight. 



Let T represent the time of rotation corresponding 

 to which the centrifugal force would be equal to that of 

 gravity O ; then the time at present being T, since the 

 centrifugal forces in the same circle are inversely as the 

 squares of the times of describing that circle (page 741), 

 we have equation (1) 



Hence, if the iliurnal rotation of the earth were per- 

 formed in the 17th part of the time really occupied 

 that is, if it were to turn round seventeen times as 

 rapidly bodies at the equator would lose all their weight 

 and would, therefore, if placed at a small distance above 

 the surface, remain suspended without any support. If 

 the rotation were more rapid than this, no body could 

 remain on the surface : everything would be repelled 

 from it by the centrifugal force. 



It has been demonstrated, however, that a spheroid, if 

 fluid, and of the same density as the earth, could not 

 remain in equilibrium if it were to rotate in a shorter 

 time than Uh. 25in. 36s. (See Airy"t Trattt, second 

 edition, p. 160). 



From what has now been shown, it will be observed 

 that the superior accumulation of matter about the 

 equatorial regions of the earth arises from two circum- 

 stances, or rather it is a twofold effect of the same cause. 

 The centrifugal force generated by the diurnal rotation, 

 acting in planes perpendicular to the axis, and on all the 

 particles in each plane, and on the remoter particles of 

 the plane more intensely than on the nearer, necessarily 

 occasions the surface of the earth about the equator to 

 recede more from the axis of rotation than the surface 

 nearer to the poles. If the earth were not a rotating 

 body, and an external influence opposed to that of gra- 

 vitation were to diffuse itself like a belt round the equa- 

 torial regions, the matter under this belt would yield to 

 the influence, and bulge out ; and supposing the entire 

 surface to be fluid, the waters of the polar regions would 

 become proportionately depressed. But in the actual 

 case, the bulging out occasioned by the direct centrifugal 

 force at the equator, exceeding the direct centrifugal 

 force at every other parallel of latitude, is further in- 

 creased by all these parallels contributing indirectly to 

 the enlargement of the equator, as well as directly to 

 the enlargement of itself ; for, as shown above, the 

 equator is the only one of the rotating circles which 

 expends none of the force upon it in a tangential direc- 

 tion. Every other circle every parallel of latitude on 

 either side, contributes tangentially its own supply of 

 matter towards the equatorial regions ; this supply, how- 

 ever, becoming leu and leu as the parallels recede from 

 that of 46. 



On account of the solid materials of which a portion of 

 the earth is composed, and the particles of which resist 

 separation by their cohesion, the effects of centrifugal 

 force are in some degree counteracted ; but it is the 

 general opinion of philosophers, that the present solid 

 part* of the earth were once in a fluid or semi-fluid state 

 in fact, in a state of fusion ; and that the outer crust 

 has become solidified by cooling. 



Bat thaM portions of the earth's surface which still 

 main fluid, observably yield to the tangential influence 

 of the centrifugal force. There is a tendency, as obser- 

 vation shows, in the waters north and south of the equator 

 to flow towards that circle ; and as they flow from part* 

 where the velocity of the diurnal rotation is less than the 

 Telocity at the equator, the streams from high latitudes 

 are gradually left more and more behind in their lateral 

 approach to the equator, where the diurnal velocity is 

 greatest As this velocity is towards the east, we may 



therefore expect to moot with great westerly currents in 

 the open seas north and south of the equator which in 

 conformable to experience. 



MOVING FORCES. 



In the foregoing remarks we have confined our 

 attention exclusively to motion, unconnected with any 

 considerations in reference to the quantity of matter 

 moved. And although we have repeatedly used the 

 term body, yet we have paid no regard to bulk or 

 weight The forces with which we have been dealing 

 have, indeed, throughout been supposed to be analogous 

 to the force of gravity, which gives the same velocity, in 

 the same circumstances, to a single particle of matter as 

 to the largest volume. We propose now to enter upon a 

 few inquiries in which it will be necessary to take into 

 account, quantity of matter ; and as preliminary to these, 

 some explanation must be given of the sense in which 

 certain terms, hereafter to be employed, are to be under- 

 stood. 



Every person has a tolerably clear notion of weight : 

 if two substances, however differing in bulk or volume, 

 have the same weight at the same place, we infer that 

 they contain the same quantities of matter ; this is briefly 

 expressed by saying that they are equal in matt. Yet 

 mass and weight are not to be regarded as expressing 

 the same thing ; if any mass, or quantity of matter, were 

 weighed at the pole, and then the same mass weighed at 

 the equator, these weights would be different ; though 

 of course the transportation of the matter, from the pole 

 to the equator, could not produce any change in its con- 

 stitution. 



A solid body may be considered as made up of particles 

 or atoms, the aggregate of which constitute its volume : 

 these particles may be packed (so to say) more closely in 

 one body than in another of the same volume ; so that 

 the former may contain a greater quantity of matter 

 than the latter ; that is, a greater number of particles. 

 Referring to this more compact constitution of the body, 

 we say that it is more dente than the other body. 

 Density, therefore, has reference to the quantity of 

 matter in a given bulk ; and if of two bodies of equal 

 bulk, one be found to weigh, at the same place, m times 

 as much as the other, the former is said to have m times 

 the density of the other ; it contains m times the quantity 

 of matter, and is therefore m times the mass of the 

 other. 



The proper way, therefore, to represent the mass of 

 any volume V, with a view to a correct comparison with 

 other volumes, is this : Mass = Density X Volume ; 

 or, in symbols, M = D V. And it is to be observed 

 that density is employed only comparatively, not ab- 

 solutely ; as when we say that lead has about eleven 

 times the density of water, we mean that about eleven 

 times as much matter is packed into a given volume of 

 lead, as into an equal volume of water ; so that the mass 

 of a cubic inch of lead is eleven times the mass of a 

 cubic inch of water. 



AH already observed, the mass of a body is proportional 

 to its weight at the same place : the dynamical effect of 

 this weight (W), or rather its dynamical measure, is Mg, 

 where M is the mass, and g the force of gravity acting on 

 it. And, in like manner, if a force / analogous to, but 

 different in intensity from gravity, were to act on the 

 mass M, the dynamical measure of the effect would be 

 M /, which is called the moving force ; so that in the 

 equation F = M/, M is the mass, or <|iiantity of matter 

 moved ;/is thp accelerating force acting on it, and F is 

 the moving force, the dynamical measure of the effect. 

 In statics this would be pressure or tension. 



The product of the mass, by the velocity with which it 

 moves, is called the momentum of the body ; as the 

 velocity increase*, therefore, so does the momentum. 

 In uniform velocity, the momentum is constant : in 

 uiiformity accelerated velocity, additional momentum is 

 generated every second; the increments of momentum 

 wing constant, like the increments of velocity. In the 

 ireccding expression for F, namely, F M/, / is no 

 (tlier than the constant increment of velocity generated 



