63 



LECTURE IX 



ON THE MOTIONS OF CONNECTED BODIES. 



THE motions of single bodies, acting in any manner on each other, which 

 we have been considering, as far as they belong to the effects of collision, 

 are of less importance to practical mechanics, than the affections of such 

 bodies as are united, so as either to revolve round a common centre, or to 

 participate in each other's motions by any kind of machinery. 



It is only within half a century, that the phenomena and effects of rota- 

 tory motion have been sufficiently investigated. Newton committed a 

 mistake, which is now universally acknowledged, in his computation of the 

 precession of the equinoxes, for want of attending sufficiently to the subject ; 

 and it is of importance in the calculation of many of the effects of me- 

 chanical arrangements, that it should be treated in an accurate manner. 



The effect of a moving body in producing motion in any other bodies, so 

 connected as to be capable of turning freely round a given centre, is jointly 

 proportional to its distance from that centre, and to its momentum in the 

 direction of the motion to be produced. Thus a body, of one pound weight, 

 moving with a velocity of one foot in a second, will have three times as 

 great an effect on a system of bodies, to which its whole force is communi- 

 cated, at the distance of one yard from the centre of their motion, as if it 

 acted only at the distance of a foot, on the same system of bodies : a double 

 weight, or a double velocity, would also produce a double effect. For, 

 supposing two unequal bodies to be connected by an inflexible line, and to 

 move with equal velocities in a direction perpendicular to that of the line, 

 it is demonstrable, from the principles of the composition of motion, that 

 they may be wholly stopped by an obstacle applied to the centre of gravity, 

 consequently their effects in turning the line round this point are equal ; 

 here the momenta are proportional to the weights, but the products obtained 

 by multiplying them by the distances from the centre, at which they act, 

 are equal : these products therefore represent the rotatory power of the 

 respective bodies. Hence in a connected system of bodies, revolving round 

 a given point, with equal angular velocities, the effect produced by the 

 rotatory motion of each body, as well as the force which is employed in 

 producing it, is expressed by the product of the mass multiplied by the 

 square of the velocity, since the velocity is in this case proportional to the 

 distance from the centre ; and this product is the same that I have denomi- 

 nated the energy of a moving body. 



These propositions are of great use in all inquiries respecting the opera- 

 tions of machines ; and it is of importance to bear in mind, that although 

 the equilibrium of a system of bodies is determined by the equality of the 

 products of their weights into their effective distances on each side of the 

 centre, yet that the estimation of the mechanical power of each body, when 

 once in motion, requires the mass to be multiplied by the square of the 

 distance, or of the velocity. For this reason, together with some others, 



