204 VI- SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



The theorem regarding rotations about parallel axes becomes: 

 Infinitesimal rotations about two parallel axes compound into a 

 rotation about a parallel axis lying in their plane. We have for its 

 position by 3), 



OB_OA_ AS 



showing that the point of application of the resultant is at the center 



of mass of masses proportional to the component rotations placed at 



their points of application. 



If vectors representing o^ and o> 2 are laid off anywhere on their 



axes, the position of the axis may be found by the following 



construction. At A a point 

 on the axis of rotation CD I lay 

 off A E = o 2 and at I? at a 

 point on the axis of rotation co 2 

 in the opposite direction BS=a v 

 Join E and S, and where this 

 straight line ES cuts AB, 

 draw OT parallel to AE, BS 

 equal in length to 04 + 2 . For 

 A _ AE _ co 2 

 ~OB~ B~S~~^ L ' 



as required by 7). 



The construction (Fig. 47) shows that if co 1 and o> 2 have the 

 same sign, the resultant co 1 + o> 2 has its axis between A and B. 



If co 1 and o 2 are of opposite signs the same construction may 



be used (Fig. 48), but 

 is on AB produced and 

 on the side of the greater 

 rotation. If o^ = o> 2 



,yf\^ evidently is at infinity 



and o = 0. The resultant 

 is then a translation per- 

 pendicular to the plane of 

 the two axes, and its 

 magnitude t is by 4) equal 

 to CDG3 1 times the perpen- 

 dicular distance between 

 the axes. 



Fig. 47. 



Fig. 48. 



58. Vector - couples. A pair of equal, parallel, oppositely 

 directed, sliding vectors will be called a vector -couple. A rotation 

 vector -couple is thus equivalent to a translation perpendicular to its 

 plane, equal to the product of the length of either vector by the 



