204 



KINETICS OF A RIGID BODY. 



[369- 



mined by AB and F. As any perpendicular to the rod is a principal 

 axis, the radius of inertia at G being ^ = Vg-#, the couple Fcsma 

 Q produces pure rotation about an 



3 /** >X- axis /through G at right angles to 



\Q/\/ _, the plane (AB, F} (see Art. 364), 

 B and we have 



H _ 3 Fc sin a 

 'j/tf 2 



Fig. 50. 



As the axis 7 of this rotation is perpendicular to the direction of the 

 velocity u of translation, their combined effect is a pure rotation of the 

 same angular velocity <o about a parallel axis / whose position is found 

 as follows (Art. 368) : Draw through G, in the plane (AB, F), a per- 

 pendicular to F, and on this perpendicular lay off 

 With the values of u and <o given above, we have 



3 c sin a 



which can easily be constructed geometrically. The parallel to 7 

 through O is the instantaneous axis about which the rod begins to 

 rotate with the angular velocity o>. 



In what direction and at what distance from G must the rod be 

 struck if it is to rotate about a perpendicular through the end A ? 



(2) A homogeneous plane lamina of mass M receives an impulse F in 

 its plane, the distance of the centroid from the direction of F being Xj. 

 Determine the initial motion. 



The reduction to the centroid G (Fig. 51) gives F at G, and a 

 couple .//= Fx l whose vector is parallel 

 to a principal centroidal axis. The couple 

 produces, therefore, rotation about the per- 

 pendicular /through G to the plane of the 

 lamina, and this rotation combines with 

 the translation due to F to a single rotation 

 about a parallel axis /. To find the position 

 of /, lay off on the direction of F, drawn 

 through G, a length GK equal to the radius 



0' 



-F 



Fig. 51. 



of inertia for /; join K to the foot O 1 of the perpendicular let fall from 

 G on F, and draw KO at right angles to KO*. The instantaneous 

 axis /passes through O, since GO GO 1 = q*. The angular velocity is 

 o) = Fx^Mf. 



