116 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[April, 



Fig. 4. 



its centrifugal velocity would be 



— -- = 39-44 feet per 

 4 



project ile force. And tlie experiment may he varied by having a 

 number uf balls prepared of the same weight, and varying the velo- 

 cities and the distances from the centre. The effects of gravity, how- 

 ever, and the difficulty of representing by a straight line what may be 

 considered the direction of the circle, have prevented me from deter- 

 mining gtomelrical/ij the dinctwn of the projectile, although in prac- 

 tice it may easily be ascertained. 



If the ball be discharged from the point A with one revolution in a 

 second, its velocity in tlie circle would be 12-57 feet per second, and 



r= 12-5 



2r 



second, .ind the initial projectile velocity would be;= V 12-57'+39'44- 

 =41-40 feet per second, disregarding for the present atmospheric 

 resistance. And if, in the way of illustration, AF be considered as 

 the direction of the force in the circle AD, the sides Ak and A»i, of 

 the parallelogram Amvk, being made proportionate to the two velo- 

 cities \-l-:>' and 39-50 resj)ectively, the diagonal Ar of tlie parallelo- 

 gram will represent in direction and proportional amount the velocity 

 41-15 or initial projectile velocity. If a billiard-ball, moving upon a 

 table witli a velocity equal to I2j feet per second in the direction EF, 

 were to receive at A an impulse in the direction of en, which alone 

 would cause it to move with a velocity equal to 31)1 feet per second, 

 no other direction and velocity could be assigned to it, than that de- 

 signated by the diagonal Ar of the parallelogram. The revolving 

 ball is s\ipposed to move in the direction Ak with the velocity of 

 12-.)7 feet per second, represented by that side of the parallelogram, 

 and at the same time to be acted upon by a force which would cause 

 it to move with a velocity equal to 394 feet per second, in the di- 

 rection iif the side Am, which indicates that velocity, consequently no 

 other diixction nor amount can be as>iigntil to it, 'when projected, than the 

 diagonal Av o/ the parn III lugram Am i-k. If the velocity of the ball 

 be doubled, the centrifugal velocity increasing as the square of the 

 increased velocity in the circle, it wonid be =: 39-14x4:= l.)7'7G feet 

 per second, and the initial projectile velocity would be =V25-14'-|-lo8'- 

 = 100 feet per second ; and the two lirst would be represented by 

 the sides Ah and An, respectively, of the parallelogram Avyh, and the 

 diagonal Ay would indicate the direction and relative proportion of 

 the initial projectile velocity. With four revolutions in a second, the 

 initial projectile velocity would be t)35 feet per second, in the di- 



rection of the line Az. At least such would be the directions for 

 those three velocities at the instant the ball leatea the point from which 

 it mai) he diaeharged. But with such low velocities a pound ball 

 would not indicate those directions by its path, for the reasons given 

 above. With I'cri/ high tncriasing velocities, however, the experimenter 

 will find that a small leaden ball will move in directions approaching 

 that of the radius, as shown in the diagram. In repeated experi- 

 ments made with a machine revolving veitically, and having a tube 

 placed in the direction of a tangent to the circle in which leaden balls 

 were revolved, it was found that with very high velocities they were 

 forced through the tube witli difficulty, and a portion of each was 

 removed by the friction, and the upper part of the tube, on the inside, 

 was worn smooth. But with much lower velocities the balls passed 

 through the tube without any apparent friction. 



lu performing the first experiment, the bar, (A, Fig. 1,) moving with 

 uniform velocity in ever)- part of the circle BD, has the same centri- 

 fugal force at v that it would have after revolving for a minute or 

 more ; for the amount of that force depends upon the curvature and 

 the circular velocity, and consequently was excited to the amount of 

 thirty-nine poimds instantaneously, and if it had been discharged at 

 three inches from B it would have been projected with that force. If 

 this were not the case with bodies moving in space, supposed to be 

 thus deflected, they would fall to the centre of attraction. Now as this 

 is the fact, the tangent B.r in the diagram only serves, as every mathe- 

 matician knows, to show geometrically the amount of defection ir. a 

 unit eif time, measured at right angles to that linp, the space xr repre- 

 senting that through which the centripetal force alone, acting uni- 

 formly, would cause the body to fall in the fiftieth part of a second : 

 the tangent, therefore, represents the link kbom wiucn the body reoidd 

 he defickd in an rwitanl of time, and not that in the direction of ivhich 

 it itonld mo re with all its projectile force. 



Again, if tlie segment of a fly-wheel disintegrated by centrifugal 

 force would be projected "in a straight line, whose direction is that 

 of the tangent," tl» pressure which produces the fracture must act 

 upon each particle of iron in the direction of a tangent to the circle in 

 which the particle is revolved, for the direction of a moving body is 

 always that in which a single force, or the resultant of two or more 

 forces, acts to cause the motion. And it is self-evident that no amount 

 offeree, applied in that direction upon the particles in the revolving 

 rim, could overcome the attraction of cohesion. And it is equally 

 evident that such cannot be the direction in which the pressure acts, 

 for whilst it is stated that the tangent is tlie direction in which the 

 dissevered fragment is projected, we are informed that the force which 

 causes the fracture acts at right angles to the tangent. 



By the theory given above, however, which is founded upon obser- 

 vation and experiment, all the circumstances that attend this pheno- 

 menon are easily explained. And when we consider the immense 

 increase of centrifug.d force as the velocity of the rim is increased, 

 eind the direction in which the resultant of the two forces acts, we ought 

 not to be surprised to find that such masses of iron can be broken and 

 projected with so much destructive eHect by this powerful agent. The 

 ojieration of the sling may also, in this way, be explained in a few 

 words. For a man, with a thong three and a half feet long, has only 

 to give to a stone at the final effort a velocity, in a very small segment 

 of a circle, equal to 132 feet per second, which would be at the rate 

 of 360 revolutions in a minute, and he will project it with a force equal 

 to that given to a ball of the same weight by an ordinary charge of 

 gunpowder, after deducting one-third of its initial velocity for atmos- 

 pheric resistance. But to " accumulate" an equal force in the circle 

 by the strength of his arm, he would have to revolve the stone at the 

 rate of 6>55u revoaitions in a minute, which is impossible. 



Without intending to enter into any particulars as to the probable 

 results of a practical application of this principle, I will close with a 

 few remarks designed to show the amount of force excited by the ro- 

 tation of heavy bodies about fixed axes, and the extent to which we 

 may reasonably conclude it might be employed, if it could be con- 

 trolled, by giving the relative proportions of the power necessary to 

 revolve a budy and the central force excited, considered abstractedly, 

 apart from friction and atmospheric resistance. " The arc which the 

 revolving budy describes in a given time is a mean proportional be- 

 tween the radius of the circle and double the space which its centri- 

 petal force alone, acting uniformly, would cause it to fall through in 

 the same time."* Consequently the diameter is to the circumference 

 as the circumference is to the space which the centripetal force of the 

 body would make it fall through in the tune of one revolution. That 

 space, therefore, is to the circumference as 3-141 is to unit, [3'14l 

 being the circumference of a circle whose diameter is unit,] and the 

 central velocity or force for an entire revolution in a second is equal 



* Cavailo's Nai. Philos. p. (iO. 



