ROTATION. 



or that point is urged in any way by a moving force, it 

 cannot move unlefs the other points, with which it is con- 

 nected by the force of cohefion, move alfo ; except the force 

 of impuWion is fufficient to overcome that of cohefion, a 

 cafe which is not meant to be confidered here. And what- 

 ever is the motion of any particle, that particle mult be 

 conceived as urged by a force precifely competent to the 

 produftion of that motion, by afting immediately on the 

 particle itfelf. The particle, immediately impelled by the 

 external force, is either prefled towards its neighbouring 

 particles, or is drawn from them ; and by this endeavour to 

 change its place, the connecting forces are exerted, or 

 brought into aftion. We are but little acquainted with 

 thefe connecting forces ; but this is of little importance 

 in a mechanical point of view ; for the faft, that the forces 

 by which the molecuke of bodies aft on each other are equal, 

 is quite fufficient for our prefent purpofe. 



After thefe general remarks, let us endeavour to illuftrate 

 the principles above laid down in the folution of a few ot 

 the mod obvious and mod practical cafes relating to this 

 important branch of dynamics. 



Let A F G H (Plate XXXVII. Mechanics, fig. I.) re- 

 prefent the circumference of a wheel, which turns in its own 

 plane round a horizontal axis, paffing through S, its centre ; 

 and let a weight P, fixed at the extremity of a line A P, com- 

 municate motion to the wheel. Let alfo the whole weight 

 of the wheel be Q, and fuppofe this weight to be collefted 

 uniformly into the circumference A F G H : then during 

 the defcent of the weight P, each point of the circumference 

 muft move with a velocity equal to that with which P de- 

 fcends ; and confequently fince the moving force is the 

 weight P, and the mafs moved P + Q, the force which 

 accelerates P in its defcent will be that part of the accele- 

 rating force of gravity which is expreiled by the fraftion 



The velocity, therefore, that is generated in P, 



P + Q 



in any given time, is found by means of the general formula, 

 given under the article Accelerated Motion ; that is, 



v = 2 ft, v = ^ — =^=- ; and the fpace through which it 



P + Q' 



has paffed in the fame time is j = 



Ygt- 



Thus, for 



P + Q 



-xample, if P and Q were equal to each other, then 



P 

 p-^TQ = h ■" - gU and s = \gt\ 



The parts of the weight Q, which are uniformly difpofed 

 over the circumference A F G H, balance each other round 

 the common centre of gravity S ; their weight, therefore, 

 has no effeft in accelerating or retarding the defcent of P ; 

 and this will be the cafe, whenever the axis of motion pafles 

 through the common centre of gravity. But in order to 

 render the properties of rotatory motion more obvious, it 

 will be convenient to difpofe the parts of the revolving 

 fyftem, fo that the axis of motion (hall not neceuarily pals 

 through the common centre of gravity. Thus, referring 

 to the preceding figure, inftead of fuppofing the weight Q 

 to be uniformly collefted over the circular rim A F G H, 

 let it be collefted into any point Q. Here it is evident, 

 that if the mafs Q be acted upon by gravity, the force 

 which communicates motion to the fyltem round S will be 

 variable ; it being the greateft, when S Q is horizontal ; 

 and gradually dimimfhing, till O has defcended to its lowed 

 point. 



But in order to begin firft with the fimpleft «afe, the 



moving force fhould be conftant, as it will be, if we fuppole 

 the mafs that is collefted in Q to be deltitute of weight, 

 and to poflefs inertia only : it follows, therefore, that 

 during the revolution of Q round S as an axis, the moving 

 force will be condantly equal to P, and the mats moved 

 = P + Q, and confequently the force which accelerates 

 the defending weight, or any point in the circumference, 



will be that part of gravity which is expreiled by 



P + Q' 



the fame as before. 



In thefe cafes, the force which communicates motion to 

 the fyftem has been fuppofed a weight, or body, afted on 

 by the earth's gravity, and confequently conditutes a part 

 of the mafs moved, at the fame time that it afts as a moving 

 force. But motion may be communicated by a force, which 

 (hall add nothing to the inertia of the matter moved ; as, 

 for inltance, fleam, mufcular power, &c. : and it will, 

 therefore, be convenient, in many investigations, to alTume 

 the moving force of this kind. The inertia of the moving 

 force P, therefore, in the fubfequeut propofitions, will not 

 be taken into the account, unlefs exprefsly mentioned. 

 Thus, if any number of bodies withqut gravity, being col- 

 lefted into the points F, H, Q, are caufed to revolve 

 round the axis S, by a moving force P, the force which 

 accelerates thefe bodies in their revolution will be 



F + H + Q' 



P 



when P is without inertia ; or it will be 

 when P is poflefied of inertia ; the 



F+H + Q + P' 



bodies F, H, and Q, as alfo the power P, being fuppofed 



to aft at equal didances from the axis of motion. 



But when bodies revolve at unequal didances from the 

 axis, their velocities being different, other formula will be 

 neceffary for determining the force whereby any given point 

 of the fydem is accelerated. 



Let B (fig- 2. ) reprefent a material point, moveable 

 about an axis of motion paffing through S : with tiie centre 

 S, and diltance S D, defcribe a circle DGH. Now 

 if B be connefted with erery point in the area of the 

 circle, which is an inflexible fubltance, no force can be ap- 

 plied to move the circle, but what muft communicate the 

 fame angular motion to B. Let us fuppofe this force to 

 be P, acting on the circumference of the inflexible circle 

 D G H, by means of a line rafling over the fame, to which 

 P is connefted. Now the a'ofolute force ut P to move D, 

 or any point in the circumference, will be P; but the com- 

 munication of motion to this point D is refilled by the in- 

 ertia of the \< idy B, which being moved with a different 

 velocity, and acted on by a different moving force, its in- 

 ertia is not to be edimated bv its quantity of matter only, 

 but bv confidering what mafs or quantity of matter which, 

 when difpofed at the diftance S D, will oppofe the fame 

 refiftance to the defcent of the weight P, as the body B 

 itfelf does, when afting at the diftance S L. 



In order to ettimate this, we mud confider that whei 

 any two bodies are put in motion by two conftant forces 

 afting for the fame time, the quantities of matter moved 

 are in a direct ratio of the moving forces, and in the inverfe 



. . M 

 ratio of the velocities generated ; that is, if — exprefles 



01 



the ratio of the moving forces, — that of the quantities of 



V \. 



matter, and — that of the velocities generated ; the rela- 

 tion 



