90] THE PRINCIPAL SCREWS OF INERTIA. 81 



a body of mass M have a movement of translation with a velocity v its kinetic 

 energy expressed in these units is 



The movement is to be decomposed into twisting motions about the 

 screws of reference ta l , &c. &amp;lt;o 6 , the twist velocity of the component on w n 

 being do,,. One constituent of the twisting motion about &amp;lt;a m consists of 

 a velocity of translation equal to dp n a n , and on this account the body 

 has a kinetic energy equal to ^Ma 2 p n 2 a n -. On account of the rotation 

 around the axis with an angular velocity aa n the body has a kinetic energy 

 equal to 



1/J2/V 2 j r2fj m 

 2 Lin I / (C///6 



where r denotes the perpendicular from the element dM on co m . Remembering 

 that p m is the radius of gyration this expression also reduces to ^Ma?p m *a m 2 , 

 and hence the total kinetic energy of the twisting motion about w m is 



Ma?p n *Qi n ~. 



We see, therefore ( 88), that the kinetic energy due to the twisting 

 motion about a is 



The quantity inside the bracket is the square of a certain linear mag 

 nitude which is determined by the distribution of the material of the body 

 with respect to the screw a. It will facilitate the kinetic applications of the 

 present theory to employ the symbol u a to denote this quantity. It is then to 

 be understood that the kinetic energy of a body of mass M, animated by a 

 twisting motion about the screw a with a twist velocity a, is represented by 



90. Twist Velocity acquired by an Impulsive Wrench. 



A body of mass M, which is only free to twist about a screw a, is acted 

 upon by an impulsive wrench of intensity ij&quot; on a screw 77. It is required to 

 find the twist velocity d which is acquired. 



The initial reaction of the constraints is an impulsive wrench of intensity 

 X&quot; on a screw X. Then the body moves as if it were free, but had been acted 

 upon by an impulsive wrench of which the component on &&amp;gt; m had the intensity 



This component would generate a velocity of translation parallel to &amp;lt;a n and 

 equal to ^ W^n + X^X,,). The twist velocity about &&amp;gt; produced by this 

 component is found by dividing the velocity of translation by p n . On the 



B. 



