202 



KINETICS OF A RIGID BODY. 



[368. 



Now, by Art. 360, the velocity of translation, u, can be 

 regarded as due to a single impulse R' = Mu, passing through 



the centroid G and parallel to u, i.e. to 

 / (Fig. 48). Again, by Art. 361, the 

 angular velocity co about / can be re- 

 garded as due to an impulse R"=Ma>x 

 ^ through G at right angles to the plane 

 (/, G), in combination with a couple 

 whose vector H has the components 

 H x = -Eto, H y = -Deo, H z =C<o. The 

 two impulses R\ R" combine to form 

 a single resultant impulse, 



Fig. 48. 



R 



'2 + R "2 = 



inclined to /at an angle $ = tan 1 (a&/u). It should be noticed 

 that the factor Vw 2 +o> 2 ^ 2 is the velocity v of the centroid due to 

 the twist, so that the resultant impulse R = M v is equal to the mo- 

 mentum of the centroid. The resultant couple H=o)^/C 2 

 is the same as in the case of pure rotation. 



368. The problem of determining the initial motion produced 

 in a free rigid body at rest by a given system of impulses finds 

 its geometrical solution in the preceding articles. It should 

 also be remembered that the motion about the centroid takes 

 place as if the centroid were fixed so that all the developments 

 of Arts. 313-323 can be applied by substituting the centroid G 

 for the fixed point O. 



It will generally be best to reduce the given impulses to a 

 resultant R, passing through the centroid, and to a couple H. 

 By Art. 360, the impulse R at G produces a velocity of trans- 

 lation, ^> 



By Arts. 361, 363, 320, 321, the couple //produces an angular 



velocity, 



/ 2 \ 



