234 . Dynamics. 



equal to the whole effort which the abovementioned forces would 

 exert upon the body, there would evidently be an equilibrium ; 



but in this case the moment = X D 1 must be equal 



I'M 



to the moment g X D ; accordingly, since/ 



we shall have 



L x FG x v v 



mr-. 



FM FM 



and, consequently, 



^ Vt) X/mr2 X F. /mr a 



~ FM x L X FGxv ~~ L 



358. We hence derive the general conclusion, that, if any 

 number whatever offerees, directed in any manner we please, in planes 

 perpendicular to the axis of rotation, act upon a body, and are ca- 

 pable of producing only a motion about this axis; (l.) The force 

 thus exerted, will be equal to the mass of the body multiplied by the 

 velocity belonging to the centre of gravity ; which velocity is deter- 

 mined by article 357. (2). This force, will be, perpendicular to the, 

 plane passing through the axis and the centre of gravity. (3.) Its dis- 

 tance from the axis (always the same, whatever be the forces and their 

 directions) will be equal to the sum of the products of the several par- 

 ticles of the body into the squares of their distances respectively from 

 the axis, divided by the product of the mass of the body into the dis- 

 tance of the centre of gravity from this same axis. 



359. v denoting always the velocity with which a determinate 

 point M of the body L, tends to turn in virtue of the action of 

 any number of forces, or of their resultant p, if we designate the 

 distance of any particle from the axis of rotation by r, and the 



mass of this particle by m, since FM : v :: r : -^777-5 we shall 



have -^TTr for tne velocity of rotation of the particle m, and - 



for the force it would exert, and consequently, for the resistance 

 27. it would oppose to g by its inertia ; accordingly, 



