841.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



11.5 



siispendefl by a rope wound round an axle, and moving dry sLoiely, a 

 certain number of revolutions in a minute will be given to it by tbe 

 power, in passing through a given space, and the four dishes will raise, 

 by their centrifugal force, a weight in the tube below, proportionate 

 to the velocity and their distance from the centre. If the moriiig 

 power be then rfo;(6/frf, with a slight addition to overcome the additional 

 friction and atmospheric resistance, it will be hunA, that in muring 

 through an equal space in the same time, it will give twice the former 

 velocity, and the dishes, at the same distance from the centre, will 

 raise in the tube below, in an equal time, qnaclrapte the weight first 

 raised. Then by loading the dishes and increasing or diminishing the 

 velocity, and varying the distances of the dishes from the centre, a 

 variety of experiments may be made, and weights may be raised, with 

 corresponding distances and velocities proportionate to those given 

 above. 



By observing the manner of performing the experiments with the 

 magnetized bar, it will be seen that a centiifugal force is excited, 



IXDEPEN'DENTLY OF THK PROJECTUX FORCE, equal tO the supposed 



power of the magnet, and we have shown that the same effects would 

 follow without the use of the magnet. And that the impelling or 

 moving power performs no other part in producing the complex elfects 

 attendant upon rotation, than simply to move the particles of a mass 

 of matter in circles about a fixed axis, may be clearly shown by the 

 theory of curvilinear motion, which those experiments v.ere designed 

 to illustrate. But without attempting to prove this at present, by ab- 

 stract mathematical reasoning, the nature of deflection and the extent 

 of its operation in exciting the central forces, may be explained by a 

 reference to the action of electro-magnetism as shown in Fig. 1. 



The bar A, when attached by the magnet, being supposed to revolve 

 In a circle of one foot in diameter, at the rate of eight revolutions in 

 a second, or iJ-H feet, to determine the amount of deflection in any 

 unit of time, say one fiftieth of a second, the whole space through 

 which it moves in a second may be divided into fifty parts, which 

 will give six inches for each unit of lime. If this space be measured 

 on the tangent from B to x, and on the circumference of the circle to 

 r, the deflection for the one fiftieth of a second would be equal to the 

 square of Br, divided by BD, or the diameter. For by dynamics, "if 

 a body revolve uniformly in a circle, the space through which it would 

 move by the action of the centripetal force alone in any unit of time, 

 such as a second, will be equal to the square of the arch described in 

 tile same unit divided by the diameter or twice the radius."* And 



the deflection of the bar in the ^- of a second = :—- = — = 3 



2Bc 2y 



inches. That is, the deflection from the tangent Bg, during the time 

 that the bar would have passed over six inches in that line, is tliree 

 inches ; and the deflection corresponding witli the space B^, which is 

 equal to two feet, and through which the bar would have passed in 



the jif of a second, would be 



4 feet, and so of any other 



space. 



Now to show that the amount of this deflection or centrifugal force 

 depends upon the curve in which the bar is moved in a given time, 

 and not upon the moving power, or projectile force, vve will cause the 

 same bar, moving with an equal uniform velocitv, to be attracted in 

 a similar manner by the magnet m, attached to an arm revolving in 

 a circle of eight feet in diameter, and let EF be an arch of that circle, 

 touching the straight line Ag- at B. As the velocity of the bar and 

 the circumference of the circle are equal, the bar, after being attracted 

 by the magnet at B, would move on with the same uniform velocity 

 -ind perforin on^ entire revolution in a second, friction and the resist- 

 ance of the atmosphere being considered equal to nothing. And its 



its centripetal force for jL of a 



deflection from the straight line, or 



second, would be equal to the square of the arch B~, which is six 



inches, divided by the diameter of the circle, that is =; 



•375 =: 



2 of an incli, or only one eighth of the deflection caused by the smaller 

 wheel ; and in the same ratio for any other spaces through which the 

 bar would have passed whilst moving through equal spaces in the 

 circle. And hence it is that the central forces are inversely as the 

 diameters of the circles in which a body is made to move with a given 

 velocity. The increment of deflection for an entire second being =: 



25-14- .„ . 



—- — =: G32 feet per second in the smaller wheel, and in the larger 



25- 1 4* 



one = — - — = 79 feet per second only ; and yet the bar has pre- 



Brews'.er's Neiv Edinburgh Kncyclopedia, Art. Dynamics. 



cisely the same velocity, and consequently the same force in the latter 

 that it had in the former. Therefore, aside from friction, it would, 

 if welded to »i, require no more force to revolve it in the former than 

 in the latter case. 



For the same reasons, with a given velocity for the particles of the 

 rims, the smaller a fly-wheel is, the greater will be the amount of cen- 

 trifugal force, other things being equal. This will appear obvious 

 upon inspecting the figure ; for it will be seen that a particle of iron 

 at r in the ;itn of a small wheel would be deflected from the straight 

 line eight times as many inches in a given unit of time as a particle 

 would be at the point z of the large wheel. The measure of the de- 

 flection from that line must therefore be the measure of the centri- 

 fugal force for any instant of time ; and consequently the aggregate 

 amount nill be proportionate to the curve in which the body moves. 

 This deflection takes place when a body is moved in a curved line, 

 and the tendency to resist it and move in a straight line is excited in 

 such a mass of matter in obedience to the important law of inertia, 

 with as much certainty as electricity would result from the action of 

 sulphuric acid upon two contiguous plates of zinc and copper. Centri- 

 fugal force may therefore with propriety be considered a physical agent, 

 I'chich is called into action, by an inscnttable law of nature, whenever 

 matter is made to move in a c«;Te;— which ought to be no more a sub- 

 ject of surprise, than that magnetic force should be excited in a bar of 

 iron by certain chemical operations, the precise nature of which is as 

 little understood as that of inertia. 



The centrifugal principle has been employed as a projectile force 

 from the earliest ages. It would be interesting to notice the extent 

 to which it was used in ancient wars; and particularly to point out, 

 as might be done even with the feeble lights afforded us, how much 

 Archi'medes was indebted to the central forces for the destructive 

 elfects of his engines, which I believe to have been no fabled nor ima- 

 ginary productions of genius. 



As I shall here come in conflict with some generally received opi- 

 nions, I will give a short extract from Professor Kenvvick's Elements 

 of Mechanics. Not that he dili'ers from other writers on this subject, 

 but I find that the extract will be useful in explaining what is to follow. 

 " The simplest case of central force is where a body connected with 

 a fixed point by an inflexible straight line is impelled by a projectile 

 force at right angles to that line. The latter force would have im- 

 pressed upon the body a motion with a uniform velocity. The body, 

 then, in consequence of its connexion with a fixed point, describes a 

 circle of which that point is the centre. If the connexion were to 

 cease at any point in the curve, the deflecting force would cease to 

 act, and the body would go in a straight line whose direction would 

 be a tangent to the cm ve. The force acting at any point in the curve 

 must therefore be decomposed into two, one of which is in the direc- 

 tion of the curve, the other in that of the radius."* 



If a ball at A, Fig. 4, weighing one pound, and attached to an in- 

 flexible rod AC, two feet long, be impelled by a projectile force or 

 moving power at the rate of two entire revolutions in a second, or 

 '^■^TUo 'set per second, it will have a centrifugal velocity equal to 

 157-713 feet per second.;!; Those two velocities, then, equivalent to 

 the forces 1-5S Ih. and 'J-S7 115. respectively, constitute the aggregate 

 amount of force acting on the body at any point of the curve or circle; 

 the former acting in the direction of the curve, and the latter in that 

 of the radius — one caused by the motion of the particles of matter, 

 the other excited by a cause producing pressure, resisted by cohesion. 

 Now, according to' the fundamental principles of mechanics, " the 

 same cause acting upon a body will either produce motion or pressure, 

 according as the body is free" or restrained." And, " if two forces 

 act upon the same point of a body in different directions, a single force 

 may be assigned which, acting on that point, will produce the same 

 results as the united ett'ects of the other two." Here we have two 

 forces acting on each particle of the revolving body, but they are re- 

 sisted by cohesion, therefore when cohesion ceases to act, the effect 

 of the two forces must be, according to the theorem of the composition 

 of forces, to impel it in the direction of their resultant, and with an 

 amount of force equal to their mechanical equivalent; and experiment 

 shows the correctness of the theory. If an ounce ball of lead, with a 

 small hole drilled through it, be firmly secured by a catgut string close 

 to the perimeter of a fly-wheel, or any other wheel that can be rapidly 

 revolved, it may be discharged from "the vertical point of the circum- 

 ference, whilst the wheel is revolving, by interposing a sharp knife 

 well fixed in a slide. When the velocity necessary to project the ball 

 horizontally at a given short distance has been ascertained, then by 

 incre.ising the velocity and taking care to discharge the ball from the 

 same point of the circle, and at an equal distance from the centre ol 

 the wheel, its elevation will be found to increase with the increased 



1 Page 62. 



Civallo, p. 66. 



R 2 



