A Treatise on the Theory of Screws

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 a>lt &c. &>B, the twist velocity of the component on <on being aan. One constituent of the twisting motion about a>m consists of a velocity of translation equal to apnan, and on this account the body has a kinetic energy equal to |J/ä3jp?12a„2. On account of the rotation around the axis with an angular velocity åan the body has a kinetic energy equal to r-dm where r denotes the perpendicular from the element dM on a>m. Remembering that pm is the radius of gyration this expression also reduces to ^Ma?pm2a.m2, and hence the total kinetic energy of the twisting motion about a>m is M&pn2an2. We see, therefore (§ 88), that the kinetic energy due to the twisting motion about a is M ä2 (p^trf + ... + 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 ua 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 å, is represented by Må-ua2. 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 y" on a screw y. It is required to find the twist velocity å which is acquired. The initial reaction of the constraints is an impulsive wrench of intensity X'" 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 wMt had the intensity V Vn + k This component would generate a velocity of translation parallel to wn and equal to (p'"vn + X"'Xn). The twist velocity about &>„ produced by this component is found by dividing the velocity of translation by pn. On the B. 6