219-221] KINETIC REACTION OF RIGID BODY. 235 



is called the instantaneous centre of no acceleration. It is of much 

 less importance than the instantaneous centre of no velocity. 



220. Examples. 



1. Prove that at any instant the normal to the path of every particle 

 passes through the instantaneous centre (of no velocity). 



[It follows that this centre can be constructed if we know the directions 

 of motion of two particles.] 



2. To calculate the moment of the kinetic reactions about the instan- 

 taneous centre (of no velocity). 



The coordinates of the instantaneous centre /being - v/a> and u/a referred 

 to axes through the centre of inertia G parallel to the axes of reference, the 

 moment in question is 



v u 



-mv-\ mu + mk*d>. 



CO CO 



The velocity of G^is ra at right angles to the line joining it to /, where 

 r=IG, or we have u 2 +v 2 = r 2 o> 2 . 



Hence the above is - -j- (%mr 



or I| { 



If we take an angle 6 such that = <o, and write K for the moment of 

 inertia about the instantaneous centre /, then K=m(k 2 + r 2 ) by I. of Art. 216, 



and the result obtained may be written -=^ ($Ka> 2 ). 



When the motion is a small oscillation, or an initial motion from rest, or 

 when the point / is fixed in the body this can be replaced by Kv>. 



3. Prove that those particles which at any instant are at inflexions on 

 their paths lie on a circle. 



[This circle is called the "circle of inflexions."] 



4. Prove that the curvature of the path of any particle which is not on 

 the^ circle of inflexions is a> 3 p 2 /V 3 where p 2 is the power with respect to the 

 circle of the position of the particle, co is the angular velocity of the body, 

 and V is the resultant velocity of the particle. 



5. Prove that, in general, that particle which is at the instantaneous 

 centre (of no velocity) is at a cusp on its path. 



221. Equations of motion of rigid body. The equations 

 of motion express that the kinetic reactions and the external 

 forces are equivalent systems of vectors. 



Let M be the mass of the body/^/a the resolved accelerations 

 of the centre of inertia in any two rectangular directions in the 

 plane of motion, co the angular velocity of the body. 



