ROTATION. 



and that which accelerates q 



pi mn 



I' m x n*p + /« m 1 vi»' r" 1 + , n I w' + w r' -f ? 



if, therefore, we make this force = /, all the circumftances 

 of the motion may be determined by the general formula 



s = itv = g f t' = -; as before (hewn. See a varie- 



4,?/ 



ty of other propofitions in Atwood's treatife on " Rota- 

 tory Motion." 



In all the preceding propofitions, the axis of the revolv- 

 ing fyftem has been fuppofed fixed ; but there are other 

 cafes by which rotatory motion may be produced, which 

 ought to be attended to in this article ; fuch as that which 

 enlue from a body defcending down an inclined plane, hav- 

 ing a ribbon or cord wound about it, one end of which is 

 fixed at the upper part of the plane, which, by preventing 

 the body fliding freelv, caufes a rotatory motion. The fame 

 effect alio follows from the friction of the body againlt the 

 plane ; and the fame mav be imagined when there is no 

 plane, but the body is left to fall freely, except fo far as 

 the cord wound about it (hall produce a rotatory motion in 

 its defcent. vVe (hall not attempt the inveliigation of 

 thefe caies, but merely ftatc the refults that have been ob- 

 taii. :d ; and n-.uft refer the reader for the former to the 

 feveral treatifes on dynamics, enumerated under the articles 

 Dynamics :nd Mechanics. 



Let a body (Jig. 8.) have a cord wound about it, either 

 at its circumference, or any other part, as C, having one 

 end fixed tit a point above, as at D; then if the body be 

 left to defcend by the action of gravity, it will acquire a 

 motion of rotation by the unwinding of the cord, and t lie 

 fpace actually defended by the body in this cafe, will be 

 to the fpace deicended in the fame time, when falling freely, 

 as C G to C O ; O and G rcprefenting the centres of 

 ofcillation and gyration when the point of fufpeniion is 

 at C. And the weight of the body will be to the tendon 

 of the cord, as C O to C G. The fame ratios have 

 place when the body defcends down an inclined plane ; 

 the forces which generate the motion being both decreafed 

 in the fame ratio. 



The force by which lpheres, cylinders. &c. are caufed 

 to revolve as they move down an inclined p'ane (inlleadof 

 fliding), is the adhetion of their firfaces, occafioiicd by 

 their preflure againft the plane. This preffure is part of the 

 weight ot the body ; for this weight being refolved into its 

 component parts, one in the direction of the plane, the 

 other perpendicular to it, the latter is the fore- of the 

 predure ; and which, while the fame body rolls down, the 

 plane will be expreffed by the cofine of the plane'-, 

 elevation. Hence, fince the cofine decreafes, will the arc 

 or angle increafe. After the angle of elevation arrive;, at a 

 certain magnitude, the adhefion may become lefs than what 

 is neceflary to make the circumference of the body revolve 

 fall enough; and in this cafe, it will proceed p.utly by 

 Hiding, and partly by rolling ; but the angle at which this 

 circumftanee takes place, will evidently depend upon the 

 degree of adhefion between the furfaces of the bod/ and 

 plane. This, however, will never happen, if the rotation 

 is produced by the unwinding of a ribbon, and it is on 

 this latter fuppofition that the following particular cafes 

 are deduced. 



Let W be the weight of the body, s the fpace defcended 

 by a heavy body falling freely, or Hiding freely down a 

 plane ; then the fpace S, defenbed by rotation in the fame 

 time, by the following bodies, will be in thefe proportions. 



1. A hollow cylinder or cylindrical lurface S = t / ; 

 tenfion — •_; W. 



2. A folid cylinder S = ' s ; tenfion = 4 W. 



3. A fpheric furface S = 1 s ; tenfion = » W. 



4. A folid fphere S == -f t ; tenfion = i W. See Gre- 

 gory's Mechanics, vol. i. 



Rotation, Spontaneous, is that rotatory motion which 

 a body acquires when acted upon by any external force in 

 free fpace. And the centre of fpontaneous rotation is tin.-. 

 point which remains at relt the iuftant the body receives its 

 impulfion, or it is that point about which the body begins 

 to revolve. 



The moll general method of treating this fubject is with 

 reference to three rectangular co-ordinates, after the method 

 of Lagrange and other modern French writers; but as that 

 method is not commonly adopted by Englifli mathema- 

 ticians, we mud necelfarily either enter at great length into 

 explanation, or run the rifk of not being underltood by 

 many of our readers, on which account we (hall here adopt 

 a fimilar mode of lnveiligation to that which we have fol- 

 lowed in the preceding part of this article. 



Firft, then, we may obferve, that when a body, B, (fig-*)-} 

 of any (hape whatever, receives an impulfe, the direction of 

 which does not pafs through the centre of gravity, and 

 takes in confequence two motions, as we have itated in the 

 early part of this article, it it evident, that for an inltant 

 of time we may confider it as having only one motion, 

 namely, amotion of rotation about a point or fixed axis, C, 

 which may be either within the bodv, or out of it, accord- 

 ing to its (hape, and the diftance, GS, between the centre 

 ot gravity and the direction of impact,. If, while the 

 line G S is carried parallel to itfclf from G S to G' S', 

 we imagine that it revolves about the moveable point G, 

 as the particles of the body have greater or lefs velocities, 

 as they are more or lefs diftant from G, it is manifeft that 

 there is upon SG a certain point, C, which will be found 

 to defcribe from C towards C an arc equal to G G', 

 which, during an evanefcent inltant, may be regarded as a 

 right line ; in that cale the point C will have been carried 

 as far backward by its motion of rotation, as it will have 

 been advanced parallel to G G' by the velocity common 

 to all the parts of the body ; the point C has, therefore, 

 during this inflant, been actually at reft in C ; and may 

 coufequently be confidered as a fixed point about which the 

 body during fuch inltant has a rotatory motion. This point 

 is the centre of fpontaneous rotation, and is the fame as 

 the centre of fufpenlion, corresponding to the centre > t 

 pcrcuffion, the centre of ptTCulhon being the point where 

 the body is (truck. 



Without entering into a minute demonftration of this pro- 

 perty, we may convince ourfelves of its truth, bv confider- 

 nig that the action of a body againlt an immoveable ob- 

 stacle, in the centre of pcrcuffion, mult have the lame effect 

 upon the body, as if that body had been at relt, and it 

 had been (truck by the obltacle ; in which lattei cafe, the 

 centre of fufpenlion would not be aflectcd, and therefore 

 it becomes the centre of fpontaneous rotation. On which 

 account it alfo follows, that the centre ot fpontaneOUt 

 rotation is wholly independent ol the magnitude of im- 

 pact ; but depends entirely on the diftance th.it the force 

 $ S, or the refult of all the forces, acts from the centre 

 of gravity, G ; and coufequently, when that force acts in 

 the direction, or coincides with ('■ G'. there will be no mo- 

 tion of rotation, as is obvioui. 



We may alfo farther obferve, that if an impact !•■ made 



on any point of the axis of a fymmctrical body, or a folid 



"! n volution, and that point be confidered a^ the point rl 



1 1 fu(p< nfion, 



