196 KINETICS OF A RIGID BODY. [357. 



V. Free Rigid Body. 



I. INITIAL MOTION DUE TO IMPULSES. 



357. Kinematically, the most general motion of a rigid body 

 consists, at every instant, of a twist, or screw-motion about a 

 certain line, called the instantaneous axis 1 ; that is, the body 

 has, for an element of time, an angular velocity o> about / and 

 at the same time a velocity of translation v along this axis (see 

 Part I., Arts. 43, 44, 294). During the next element of time 

 the body will, in general, rotate about a different axis with a 

 different angular velocity and will have a different linear velocity 

 along the new axis. 



358. It has also been shown in kinematics (Part I., Art. 257) 

 that the angular velocity & about / can be replaced by an equal 

 angular velocity about any parallel axis /', in connection with a 



certain velocity of translation. For without 

 changing the state of motion we can give the 

 body two equal and opposite rotations about 

 /' ; i.e. we can introduce along /' (Fig. 44) two 

 equal and opposite rotors a>, o> ; and o> 

 about P combines with o> about / to a rotor 

 couple, which is equivalent to a velocity of 

 translation /o>, perpendicular to the plane 

 (/, /'), p=OO 1 being the distance of the par- 

 allel axes. The velocity of translation p& 

 can now be combined with v to a resultant velocity of transla- 

 tion v 1 = V^ 2 +/ 2 o> 2 inclined to F at an angle ^> = tan~ 1 (pco/v). 



It thus appears that the instantaneous motion of a free rigid 

 body can be regarded as a rotation about any line parallel to the 

 instantaneous axis, in combination with a certain velocity of 

 translation inclined to this line. 



On account of the dynamical properties of the centroid of a 

 rigid body, it will generally be found convenient to select the 



