232 Dynamics. 



Now the numerator of this fraction, since it expresses the sum 

 of the moments of the forces M w ' c, &c., is equal to the moment 

 of their resultant. If, therefore, we call this resultant g, and 

 its distance from the point F, D ; the sum of the moments will be 

 p X D. Moreover the denominator of the above fraction, being 

 the sum of the products of each mass or particle into the square 

 of its distance from F, if we represent in general any one what- 

 ever of these masses by m, and its distance from F by r, the sum 

 of these products may be represented by the abridged express- 

 ionfm r 2 , (f denoting the word sum) ; we have accordingly for 

 the velocity of any given point M, whose distance from the axis 

 F is FM or R, the following expression, 







P X D 



v = ^ - r 



J mr 

 also, 



X 



355. Although we have supposed that all the forces, and all 

 parts of the system are in* the same plane, it will be perceived 

 that we should, arrive at the same result, if they were in planes 

 parallel to each other, and perpendicular to the axis of rotation, 

 provided that all parts of the system admit only of a rotation 

 about a fixed axis. 



356. Accordingly, as a solid body of whatever figure may be 

 considered as an assemblage of material points, thus connected 

 together, we may say generally, that when a body L,of whatever fig- 



Fig. 168. ure, and urged by forces of whatever number and magnitude, cart have no 

 other motion, except a motion of rotation about a fixed axis AB, situated 

 within or without the body, the velocity belonging to any given point, 

 is found by taking the sum of the moments of all the forces (or the 

 moment of the resultant), dividing this sum by the sum of the pro- 

 ducts of the several parts of the body into the squares of their distan- 

 ces respectively from the axis of rotation, and multiplying the quotient. 

 by the distance of the point in question from this Mime axis. 



Fi.i69. 357" Let G be the centre of gravity of the body L, and let 

 us suppose that while any point M, in turning, describes during 

 an instant } the infinitely smail arc v, the centre of gravity G 



