254-] PLANE MOTION. l ^ 



angular velocity of the rigid body. The velocities of the differ- 

 ent points of the body at any given moment are therefore 

 directly proportional to their distances from the axis, and the 

 velocities of all points at this moment are known if the instan- 

 taneous angular velocity co is given. It is frequently convenient 

 to imagine this angular velocity represented by its rotor, i.e. by 

 a length co laid off in the proper sense on the axis of rotation 

 (see Arts. 68, 69). 



253. The body may have several simultaneous rotations. 

 Imagine, for instance, a top spinning about its axis placed on a, 

 table or disc which is made to rotate about an axis. The result- 

 ing motion can be found by compounding the rotors in the 

 same way in which the rotors representing infinitesimal rotary 

 displacements are compounded (Arts. 62, 66, 67); indeed, the 

 rotor w = d6/dt of an .angular velocity is merely the rotor dd 

 divided by dt, and therefore identical with the rotor of the 

 angular displacement dQ. 



254. As we are at present concerned with plane motion, we 

 require only the rule for the composition of angular velocities 

 about parallel axes. 



Dividing the equations (i'") and {2'") of Art. 66 by dt, and 

 putting dO/dt=o>, dd l /dt=co l , d6 2 /dt=a) 2 , we obtain : 



L-\L LLn L-iLn /_\ 



o) = ft) 1 + ft) 2 , = -= L -^- (7) 



ft) 2 C0 1 ft) 



Thus, the resultant of two angular velocities co^ w 2 about 

 parallel axes \, 1 2 is an angular velocity co equal to their algebraic 

 sum, ft) = ft) 1 + ft) 2 , about a parallel axis\ that divides the distance 

 between \, 1 2 in the inverse ratio of co 1 and &) 2 . 



Conversely, an angular velocity co about an axis / can always 

 be replaced by two angular velocities co lf co 2 whose sum is equal 

 to w and whose axes l lt / 2 are parallel to / and so selected that 

 / divides the distance between / lf / 2 inversely as o^ is to co 2 . 



