369.] 



INITIAL MOTION OF FREE BODY. 



203 



about a centroidal axis / whose direction is conjugate to the 

 plane of the couple H in the central ellipsoid, while a plane 

 perpendicular to /is conjugate to the direction of the vector H 

 in the fundamental ellipsoid. The components of co along the 

 principal centroidal axes are given by (5), Art. 319, viz. : 



ucos a 



u. 



usina 



l 



ucos a 

 u 



^ 



O 



where I v 7 2 , 7 3 are the principal centroidal moments of inertia 

 and 7/j, J7 2 , 7/ 3 the components 

 of H along the centroidal prin- 

 cipal axes. 



The direction of the instan- 

 taneous axis having thus been 

 determined, its position can be 

 found by resolving u into a usina 

 component u cos a along / and 

 a component u sin a perpen- 

 dicular to 7 (Fig. 49). The Fi S- 49 - 

 latter component combines with co about / to an equal angular 

 velocity co about an axis /parallel to /at the distance x usma/co 

 from 7 



The initial motion produced by the impulses consists, there- 

 fore, of the angular velocity co about /, and the linear velocity 

 These together constitute the resulting twist. 



u cos a along / 



369. Exercises.* 



(i) A homogeneous straight rod AB= 2 a (Fig. 50) is acted upon 

 by an impulse F, at the distance c from the centroid G, at an angle a 

 with the rod. Determine the initial motion. 



The reduction to the centroid G gives the impulse F at G, which 

 produces a velocity of translation u = F/M in the direction of F, and 

 the couple H formed by F at C and - F at G. The moment of this 

 couple is ^sina, and its vector^ is at right angles to the plane deter- 



* Most of these problems, as well as the discussions of Arts. 357-368, are adapted 

 from W. SCHELL, Theorie der Bewegung und der Krafte, Vol. II., pp. 352-386. 



