CENTRIPETAL AND CENTRIFUGAL FORCES. 



CENTRIPETAL AND CENTRIFUGAL FORCES. T36 



established was this notion in former times, that it was supposed even 

 if the earth wen to be annihilated, this centre would still exist with 

 all iu properties. 



CENTRI'PETAL and CENTRIFUGAL FORCES, forces which 

 urge a body to ittt (petere) or to aroid (fugere) a centre ; in more 

 modern language, attractive and repulsive forces. [ATTRACTION, Ru- 

 pfusios, FORCE, GRAVITATION, Ac.] 



\Vr intend here to confine ourselves to the term centrifugal forte, in 

 one particular sense in which it is used in mechanics and astronomy, 

 because it involves a point which is frequently mistaken. The term 

 force, as used in mechanics, implies simply any cause of motion which 

 is external to the matter moved ; and the terms accelerating force, as 

 well as retarding force, are used with reference to the velocities of 

 bodies, and without reference to their masses or weights. So long as 

 velocity remains unaltered, there is neither accelerating nor retarding 

 force ; when alteration of velocity begins, then we may say that force 

 begins to act, for all we know of force is implied in the words, " cause 

 of acceleration or retardation." Again, owing to the convenience of 

 words implying causation, it is usual to give the name inertia to that 

 propel ty of matter by which it maintains its state, either of rest or 

 motion, unless acted on by other matter. And since the state of 

 matter left to itself is either that of rest or of uniform motion in a 

 straight line, every other species of motion, of what sort soever, is an 

 effect of force from without ; which assertion is verified in every instance 

 in which it can be tried. 



Suppose we fasten a string to an immoveable obstacle, such as a post, 

 and pull it, say with a force of a hundredweight. It may not at first 

 sight appear proper to say that the post also pulls the string, because 

 we may not be able to conceive the latter acting, but only resisting. 

 Nevertheless, the part which the post sustains, call it action or resist- 

 ance, is still the equivalent of a force, for if it were removed, and 

 another hand applied to the other end of the string, that hand must 

 also pull against the first with a force of a hundredweight before the 

 counteraction of the moving tendency of the first pressure is supplied. 

 And in calculating the numerical effects of pressures, the equations of 

 motion can take no cognisance of the cause of the pressures, but only 

 of their amounts ; whence it will arise that the eflect, say of the 

 resistance of an immoveable obstacle, may enter an equation in precisely 

 the same manner as a physical attraction [ATTRACTION], a muscular 

 effort, or any other mode of accelerating or retarding velocity. 



Next, it must be remembered that no alteration whatever of the 

 effect of inertia can be produced without force of some kind. Let us 

 now suppose a small bullet attached to a string, which string is 

 fastened to a point upon a table, friction and the resistance of the air 

 not being supposed to exist. Let the bullet be placet) in a state of 

 revolution round the fixed point by means of the string, and with a 

 given velocity. It will continue to revolve round with the same 

 velocity, and the string will be stretched by a pressure depending upon 

 the mass of the bullet and its velocity. The reason of the permanence 

 of the velocity is contained in a proposition which is demonstrated in 

 mechanics, namely, that forces applied to a material point in the 

 direction perpendicular to that of its motion cannot change its velocity, 

 but only iU direction. If, for instance, a hand could apply pressure to 

 a moving point on a table, by means of a string, so adroitly as always 

 to keep the pressure perpendicular to the direction of motion for the 

 time, the whole amount of the pressure might be varied so as to make 

 the point describe any curve, but always with the same velocity as at 

 the outset Now, in the present case, the bullet must describe a 

 circle, and the direction of the string, in which the retaining pressure 

 acts, is always perpendicular to the tangent of the circle, being always 

 a radius. This pressure of the string must be caused by on effort to 

 escape on the part of the bullet, arising from its tendency to continue 

 iU motion in the direction of the tangent. This is the centrifugal 

 force, and does not arise from any tendency which the body has to fly 

 from the centre, but from this circumstance, that there is in the 

 motion above described a constrained approach to the centre, or rather 

 a constrained continuance at the same distance from the centre, such 

 as would not exist in the motion of the bullet uninfluenced from 

 without. 



The centrifugal force is thus measured : suppose, for instance, the 

 velocity of revolution in the circle to be 8 feet per second, and the 

 radius 10 feet Divide the square of the velocity by the radius, or 

 divide 64 by 10, which gives 6'4. Then the pressure is such, that were 

 it to take the place of the earth's attraction, the bullet, being allowed 

 . to fall, would, at the end of one second, have acquired a velocity 

 of Oj, feet per second, instead of 32), which is the case with bodies 

 falling freely to the earth. And if we ask what weight must be hung 

 to the string, so that it may receive the same pressure as when con- 

 straining a bullet of two pounds iu weight to move at 8 feet per second 

 in a circle of 10 feet radius, the answer is, such a proportion of the 

 weight of the bullet as 6^ is of " 



or S x 2 pounds, or on of a pound. 



32} ' '< 



If then this latter weight, hung at its end, would be the utmost the 

 string would bear without breaking, then any accession of velocity in 

 the preceding motion would also break the string. 



W now take the case of one of the planet* moving in the curve AP 



about the sun at s, and attracted towards the sun, so that if reduced 

 to rest for a moment at r, it would immediately begin to descend 

 towards s. It must always appear to one not used to consider these 

 subjects, that this attraction being perpetually exerted must at last 

 make the planet fall into the sun. But suppose for a moment the 

 attraction removed, so that the planet would pursue its course along 

 p T. Prevent this by on inextensible string B r, in which case the planet 

 will proceed in the circle rv, not with the whole velocity with \\ lik-h 

 (but for the string) it would have proceeded along PT, Vmt with a part, as 

 follows : Let PT be the length which would be described in a second, 

 draw P Y perpendicular to 8 P, and complete the rectangle P X T T. 

 the velocity PT might be produced, if the string were not there, by an 

 Instantaneous communication of the two velocities px and p v in these 

 directions. But when the string is supposed, the commviuie.itiun of 



such an impulse as would produce P \ has no effect except an instan- 

 taneous tension of the string, which is supposed capable of resisting it. 

 There remains only p T, which would produce what has been before 

 called the centrifugal force. 



Now, if the string were not there, there remains the body P, with 

 the same tendency to recede from the centre implied in its having the 

 velocity P T, and with the effect, which would be counteracted by the 

 string, entirely uneounteracted. If then we look at the paratentrif 

 motion only, or the manner in which SP is lengthened or shortened, 

 independently of the angular motion, we have, 1. the paracentric 

 velocity px actually existing ; 2. the attraction of 8 in the direction P8 ; 

 3. the tendency to recede from the centre arising from tin- v.-loriu 

 PT. The alteration of the first i not due to the second or tliiril al.m.- ; 

 if (2) exceeds (3) the velocity PX is diminished in the instant n.-xi 

 following the body being at p ; if (3) exceed (2) it is increased. 



To realise the supposition above alluded to, namely, th.it tin- ultra.- 

 live force must bring the body P to the centre at last, we must pin 

 that force in a situation to act unopposed. Imagine a string whirh 

 shortens as the body approaches the centre, so as always to remain 

 stretched ; and alto suppose the attractive force to act, the effort to 

 proceed in the tangent of the circle being destroyed by the string, and 

 we have the case which is really supposed in the above mistake, for 

 want of considering the centrifugal force. 



The centrifugal force always (whatever the attraction may be) 

 varies as the inverse cube of the distance, while the attractive force 

 varies as the inverse square. Consequently if the revolution ever 

 bring the body to l-10th its primitive distance, the attraction is only 

 100 times as great as at first, while the effort to recede is 1000 times 

 as great. Hence a very little knowledge of algebra will make it 

 apparent that however much the external attraction may at first 

 exceed the centrifugal force, there must come a point of approach to 

 the centre at which the latter begins to exceed the former. Anil 

 hence the alternate approach and recession of the planets to and from 

 the sun. Let A z M P S V be the ellipse of a comet's orbit (we have 



drawn it too elongated for that of a planet), A and p the aphelion ;m.l 

 pi-rihi-lion. A i- tin-axis, and M N a perj>endicular to it. Let the pl..n. 

 be projected at A with the necessary velocity : then from A to M the 

 attraction exceeds the centrifugal force, and velocity towards tin- 

 centre is created, as instanced by z Q in the figure at Z. At M Un- 

 attractive and centrifugal forces are equal, and from M to N the ceutri- 



