1841.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



113 



REMARKS ON THE CENTRAL FORCES OF BODIES 

 REVOLVING ABOUT FIXED AXES. 



By Joseph Martin, M.D. 



(From Silliman's American Journal.) 



The theory of curvilinear motion maj' justly be considered one of 

 the most important and interesting subjects connected with the physi- 

 cal sciences. It explains the motions of the heaTenly bodies, and, by 

 unfolding some of the grand phenomena of nature, makes them appli- 

 cable to the most important and useful purposes of life. It has ac- 

 cordingly engaged the attention of the greatest philosophers for cen- 

 turies, who have, by means of the most searching analyses, not only 

 pointed out the slightest irregularities of those bodies which compose 

 the great planetary system, but have discovered the causes of the 

 seeming aberrations, and given satisfactory explanations of them. And 

 yet it would seem that the most simple case of '• central forces," the 

 rotation of a heavy body about a fixed axis, has been in some measure 

 neglected, or at least, treated as a subject of too little importance, 

 either in a theoretical or practical point of view, to deserve more than 

 a passing notice. 



To explain the motions of the heavenly bodies it has been found 

 necessary, by means of mathematical reasoning, to determine the ratio 

 of attraction and original impulse, or projectile force, and to show the 

 eftects of their separate and combined operation. In this way the part 

 that each of the three forces, the projectile and the central, perform 

 in producing and preserving the motion of a planet in its orbit, is 

 clearly defined ; as well as the results that would follow if either of 

 the last should cease to act. But the ratio of the forces which act 

 upon a body made to revolve about a fixed axis, and the nature and 

 extent of their separate or combined action, have not been distinctly 

 shown. In other words, it is believed that the relative proportions of 

 the moving power, and the forces that it produces directly 'and indi- 

 rectly — the manner in which the central forces are excited — and the 

 combined operation of all the forces upon a body whilst revolving and 

 when projected, have not been sitisfactorily explained. 



It is not intended, however, at present to enter into an investigation 

 of the subject upon principles purely dynamical, but the object of these 

 remarks is to show by mathematical reasoning, founded upon experi- 

 ment and familiar examples, that the power employed to revolve a 

 body about a fixed axis is wholly expended in giving velocity to that 

 body in the direction of the circle, and that, consequently, the central 

 forces must be excited in obedience to a law of nature ; and, in the 

 second place, that the moving and excited forces act in conformity 

 with the principles of "the composition of forces." 



Fig. 1. 



If the bar of soft iron m, fig. 1, be prepared as a horse-shoe magnet 

 ,'ind secured in a proper manner to the rod r, working horizontally on 

 an axle at c, it may be connected at pleasure with a galvanic battery, 

 by means cf its wires and the usual arrangements of cups containing 

 quicksilver, at the centre. The iron bar A, of a suitable size and de- 

 scription, moving with a given uniform velocity along the straight line 

 Ag, would be attracted at B by the magnet, if it were connected at 

 that moment with the galvanic battery, and would be made to move 

 in the curve Br of the circle BD, but in virtue of its inertia it would, 

 in the absence of friction and atmospheric resistance, continue to move 

 in that circle with the same uniform velocity. For the deflecting 

 force being independent of the projectile force, and acting at all times 

 in the direction of the radii of the circle, it cannot in any resfiect in- 

 crease nor diminish the original velocity of tl;e bar. And for the 

 same reasons the force with which the bar is moving in the circle can 

 have no influence upon the deflecting force. But a body moving in a 



curve or circle is always found to be acted upon by a third force, which 

 is opposite and equal to the deflecting or centripetal force ; and as 

 there cannot be an effect without a cause, this third force must either 

 be derived from one of those mentioned above, or their resultant — or 

 from some other source. Supposing the circle BD, in which the bar 

 moves, to be one foot in diameter, and the velocity of the bar to be 

 25'14 feet per second, or at the rate of eight entire revolutions in a se- 



ija 25*142 

 cond, its centrifugal velocity would be=: — ;= = G32 feet per 



second, and its centrifugal force =:3iUb. its weight being one pound, 

 V representing the velocity in the circle, and r its radius; for if a be 

 the vfeight of the bar, g equal to 32^ feet, and x the force required, 



then r : — : : a : =.r= — --r- = 3'Jlb.* But the force in the 



g gr 1<J 



•25- 14 



circle = ■ 



It; 



!• J5 lb. only, consequently the centrifugal force could 



not have been caused by the projectile force. And it is evident that 

 it cannot be a part of the magnetic force, for it acts in a directly oppo- 

 site direction; and it is equally evident that it cannot be the resultant 

 of the other tivo forces, for then its direction would be to some point 

 within the circle. The pressure from the centre of thirty-nine pounds 

 must therefore have originated in some other way. 



Such are the facts when the deflection from a straight line is caused 

 by a centripetal force directed to a fixed centre of rotation, and the 

 projectile or moving force is applied before the body is constrained 

 to move in a circle. We will now stop the revolving rod r, leaving 

 the bar A attached to iii, by the magnetic force. If by means of a 

 winch the same number of revolutions in a second be given to the bar 

 that it had in the first experiinent, the centripetal or magnetic force 

 will perform the part of cohesion, and the circumstances in every other 

 respect will be the same that would attend such a rotation if the bar 

 were welded to m. Does the moving power, applied in this manner, 

 directly produce the central force or immediately impart it to the 

 moving body? or, in other words, is centrifugal force a part of the 

 force employed to revolve the body ? Without attempting to prove 

 the negative of this question by minute mathematical investigations, 

 which will be avoided as much as possible on this occasion, I will show 

 by a reference to the familiar examples of the common sling and fly- 

 wheel, that in a revolving body centrifugal force, whatever be its 

 source, is much greater than the power necessary to give rotation to 

 that body, and that it cannot therefore be directly caused by the mov- 

 ing power, — and then explain how it may be proved by a simple ex- 

 periment. 



It has been stated above that writers on dynamics have not clearly 

 defined the operation of the laws of curvilinear motion on bodies re- 

 volving about fixed axes. One only of the many instances in which 

 erroneous views are given by popular writers in noticing the subject 

 of central forces, will be mentioned. In the Library of Useful Know- 

 ledge [London edition] a viriter, after enumerating some of the won- 

 derful eftects producecT by accumulating force in the circum/trence of a 

 fly-wheel, remarks : " the same principle explains the force with which 

 a stone may be projected from a sling. The thong is swung several 

 times round by the force of the arm until a considerable portion of 

 force is accumulated and then it (the stone) is projected with all the 

 collected force.t By observing the facts we may discover how all 

 this accumulation of force is produced by the strength of the arm. A 

 stone, S, fig. 2, weighing one pound, secured to the end of a string 

 rather less than two feet long, may be whirled in a circle of four feet 

 diameter at the rate of two entire revolutions in a second. It is done 

 by turning the hand in a small circle AB, about a moving axis of rota- 

 tion. The velocity in the large circle ^r 12*57 X 2 = 25-14 feet per 

 second; and, as shown above, if S represent the weight of the stone, 

 V its velocity, ;■ the radius of the circle and x the centrifugal velocity, 



then r 



64 



t*- j5 25*14- 



S : ;;—- = :?=:-— — -^9*87 pounds. The velocity 



in the circle being 25*14 its force in that direction is equal to l*58lb. ;'4I 

 and if we add 1*42 lb. for the weight of the stone and atmospheric re- 

 sistance, which is more than suSicient, we have three pounds as the 

 force with which it is impelled in the circle ST. To enable him to 

 move the stone in the circle the operator has to resist ajorce nearly 

 equal to ten pounds, ic/uch urgen his handjrom the centre at every instant 



' Button's Mathematical Dictionary, an:l Gregory's Mechanics. 

 Vol. 1. p. 51, An. Mechanics. 

 (tv&Uo'f FbJoiopby p $& 



