KINETICS OF A RIGID BODY. [310. 



or, since *$mx=Mx, 2w/ = o, 



E' = o, Z?'=o, 



where E f , D' are the products of inertia at <9'. 



It thus appears that the axis of z must be a principal axis at 

 the foot of the perpendicular let fall on this axis from the point 

 of application of the impulse. 



310. It should be noticed that a line taken at random in a 

 body is not necessarily a principal axis at any one of its points. 

 But if a line is a principal axis at a point O', then it is always 

 possible to determine an impulse that will produce no stress on 

 this line so that the body will begin to rotate about it as axis 

 even though it be not fixed. As shown in the last article, the 

 impulse must be =Mxay y and must be directed at right angles to 

 the plane through axis and centroid. The point where it meets 

 this plane is called the centre of percussion. Its distance x from 

 the axis is found from the equation of motion, viz. the last of 

 the equations (14) which, owing to the conditions (15), reduces to 



If q 1 be the radius of inertia of the body for a parallel centroidal 

 axis, we have C=M(q'*+x*) ; hence 



x^x+VJ-. (16) 



Hence, if a given line / be principal axis for one of its points O 1 ', 

 there exists a centre of percussion ; it lies on the intersection 

 of the plane (/, G) with the plane through O f perpendicular to /, 

 at the distance x v given by (16), from the line /. An impulse 

 Mxo) through the centre of percussion at right angles to the 

 plane through axis and centroid, while producing no percussion 

 on the axis, sets the body rotating with angular velocity a> if it 

 was originally at rest ; on the other hand, if the body wz 

 originally in rotation about the axis, such an impulse can brinj 

 the body to rest without affecting the axis. 



