ROTATION. 



Q 



tion of thefe quantities is defined by the equation 



t 

 M V 



— X -, by the laws of rectilinear motion. To apply 

 m v 



this, it muft be obferved, that although the abfolute 

 force of the weight P, acting upon the point D, remains 

 conftantly the fame, yet its effect upon bodies placed at 

 different diftances from the axis ©f motion are in the inverfe 

 proportion of thofe diftances ; therefore, the moving forces 

 exerted by P, on the points B and D, will be in the pro- 

 portion of S D to SB. Alfo, by the problem, the an- 

 gular motion of D and B are equal ; and, confequently, the 

 velocity of B is to the velocity of D, as S B to S D. And 

 fince the quantities of matter in B, D, are in the direct pro- 

 portion of the moving forces, or of S D to S B, and in the 

 inverfe proportion of the velocities generated, or of S B to 

 S D, we (hall have the quantity of matter in B to that con- 

 tained in D, as S D : : S B 1 ; and confequently the weight 



fought = B x g-jj,. 



The fame may be otherwife found thus : let x = the 

 quantity of matter required to be collected in D, M the 

 moving force which acts on B, m that which acts on D, 



V the velocity of B, and v the velocity of D ; then — = 



A x SA' BxSB : 



M 



m 



V 



V' 



M SD 



but — = 



SB' 



by the property of the lever ; 



, v SD L L 

 and Tr = -^-5, by the nature of 



V OB 



. B SD 1 



fore, — = - -jp and x = 



B 



angular motion ; there- 

 SB* 



sd*' as a 



Whenever, therefore, a body B revolves round an axis, 

 by the action of a conftant force P, applied at a given dif- 

 tance, S D, from the axis, in order to find the force which 

 accelerates D, the mafs B may be fuppofed to be removed, 



S B' 

 and inftead of it an equivalent mafs B x - — L collected in 



o ±J~ 



the point Dj to which the force is applied. After which, 

 the acceleration of the point D, or any other point of the 

 circumference, will be determined from the principles al- 

 ready explained ; for the moving force being P, and the 



S B 2 

 maf3 moved Bx .pr,;, the acceleration of P, or D, will 

 o D 



D 



be that part of gravity exprefled by the fraction 



B 



S B 

 S 1) 



_ P x S D 

 = B x S B 



crtia ; or by 



, while we fuppofe the power P void of in- 

 P P x S D- 



B x 



SB- 

 SD 1 + 



BxSB'+PxSD 1 ' 



when the inertia of P is confidered. 



On the fame principle, if any number of bodies, A, B> 

 C, &c. (Jig- 3.) be put in motion round a fixed axis, 

 pafling through S, by a conftant force P, applied at D, 

 the point D will be accelerated in the fame manner, and 

 confequently the whole fyftem will have the fame angular 

 velocity. If, inftead of A, B, C, &c. placed at the dif- 

 tances SA, SB, SC, &c. we fubftitutc the bodies 



SD 3 



SD 1 



and 



CxSC'. 

 SD- 



thefe being col- 



lected into the points a, b, and <r, refpeftively ; and the 

 moving force in this cafe being P, and the mafs moved 



A x SA 1 B x SB', C x SC 1 . , 



H o?^ — T p,, ; the torce which ac- 



SD- 



SD : 



SD 1 



celerates D will be that part of the force of gravity that is 

 exprefled by the fraction, 



P x SD 2 



A x S A z + B x S B 1 + C x S C ' 

 or, if the inertia of P be confidered, by 



PxSD' 



A x SA ! + B xSB' + CxSC' + Px SD 1 " 



The velocity of the point D is uniformly accelerated, 

 becaufe the force above determined is invariable : it follows 

 alfo, that the angular velocity of the fyftem is uniformly ac- 

 celerated, becaufe the abfolute velocity of any point at a 

 given diftance from the axis of motion, is as the angular 

 velocity of that point, and confequently of the whole fyf- 

 tem. It is alfo manifeft, that it is of no confequence whe- 

 ther the bodies A, B, C, &c. revolve in the fame or different 

 planes, if their diftances from the axes S A, S B, S C, &c. 

 are the fame, thefe diftances being eltimated by lines drawn 

 from A, B, and C, perpendicular to the common axis of 

 motion ; if, therefore, they fhould be fituated in various 

 planes, they may be referred to any one given plane per- 

 pendicular to the axis. 



It is obvious likewife, that changing the pofition of the 

 bodies A, B, C, in the fame plane will not affect the force 

 which accelerates the fyftem, provided their refpective dif- 

 tances from the axis of motion be not altered ; thus, with 

 the centre S, and diftances SB, S C, let the arcs of circles 

 be defcribed ; if B is transfered to b', or C to c 1 , the moving 

 force which aits on thefe bodies refpeftively will not be 

 altered, and confequently the maffes moved being likewife 

 conftant, the accelerating force will be the fame. 



All thefe propofitions are equally true, whatever may be 

 the force by which the angular motion is generated, pro- 

 vided it be conftant ; or, if variable, fhould its action be 

 confidered for an evanefcent particle of time only. 



We have here followed the method employed by Mr. At- 

 wood in histreatife on " Rectilinear and Rotatory Motion ;" 

 being, as we conceive, the belt calculated to convey a cor- 

 rect and elementary idea of the laws of rotatory motion. 

 But it is obvious, that, inftead of confidering the given 

 bodies A, B, C, (jig. 3.) to be equivalent to ether bodies 

 placed at the diftance S D, we might enquire, at what dif- 

 tance from the centre of motion S all thefe bodies muft be 

 collected, without changing their maffes, fo that the fame 

 angular motion may be generated in the fyftem ? This point 

 u called the centre of gyration ; and as the method of finding 

 this point has been already treated of under the article 

 CENTER, we (hall, in the lubfequent part of this article, 

 conlider it as known, and (hall proceed to the folution of a 

 few fuch problems as appear belt calculated for illuftrating 

 the fubject under conlideration. 



Pkop. I. 



Let D, E, 1'", ( fix- 4-) reprefent a wheel, or cylinder,, 



turning about an axis pafling through its centre of gravity 



S ; round the circumference of which a perfectly flexible line 



is made to pafs, and to the end of which a body, P, is fuf- 



4 H 2 pended ; 



