KINEMATICS. 



[255. 



56> 



255. It may be well to prove this important proposition 

 independently. Any point P (Fig. 56) in a plane at right 



angles to the axes receives from 

 co 1 a linear velocity u>^\ per- 

 pendicular to L^P, and from 

 co 2 a linear velocity 2 r 2 per- 

 ~ pendicular to L 2 P, if L l P = r l , 

 L^P = r^ These linear veloci- 

 ties fall into the same straight line only for points situated on 

 the line L^L^. A point L whose linear velocity is zero, must 

 therefore lie on L 1 L 2 so that L 1 L+LL 2 = L 1 L 2 ', moreover, it 

 must satisfy the condition L l L'Co l = LL 2 'co 2 . This gives the 

 above equations (7). 



256. The resulting axis lies between L l and L 2 when the 

 -components co l} a> 2 have the same sense ; when they are of 

 opposite sense, it lies without, on the side of the greater one 

 of these components. 



If G>J and o> 2 are equal and opposite, say co 1 co, o> 2 = co, the 

 resulting axis lies at infinity (Art. 67). Two such equal and 

 opposite angular velocities about parallel axes are said to form 

 a rotor-couple ; its effect on the rigid body is that of a velocity 

 of translation v=L l L 2 'd6/dt=p-a) at right angles to the plane 

 of the axes. The distance of the rotor, L l L (i =p, is called the 

 arm of the couple, and the product pco = v its moment. 



257. A velocity of translation v can therefore always be 

 replaced by a rotor-couple pco = v, whose axes have the dis-| 

 tance p and lie in a plane at right angles to v. 



Again, an angular velocity co about an axis / can be replaced 

 by an equal angular velocity co about a parallel axis /' at the 

 distance/ from /, in combination with a velocity of translation 

 v = cop at right angles to the plane determined by / and /'. 



It easily follows from these propositions that the resultant of 

 -any number of velocities of translation, v, v', . . ., parallel to the 

 same plane, and any number of angular velocities co, co',..., aboui 



