A Treatise on the Theory of Screws

82 THE THEORY OF SCREWS. [90- other hand, since the co-ordinates of the screw a are alt ... a6, the twist velocity about <an may also be represented by å /n (§ 34), whence If we multiply this equation by pn2an, add the six equations found by giving n all values from 1 to 6, and remember that a and X are reciprocal, we have (§ 39) • 1 « ÖlUa 1) , whence å is determined. This expression shows that the twist velocity produced by an impulsive wrench on a given rigid body constrained to twist about a given screw, varies directly as the virtual coefficient and the intensity of the impulsive wrench, and inversely as the square of ua. (See Appendix, Note 3.) 91. The Kinetic Energy acquired by an Impulsive Wrench can be easily found by § 89 ; for, from the last equation, 1 in ' “ hence the kinetic energy produced by the action of an impulsive wrench on a body constrained to twist about a given screw varies directly as the product of the square of the virtual coefficient of the two screws and the square of the intensity of the impulsive wrench, and inversely as the square of ua. 92. Formula for a free body. We shall now express the kinetic energy communicated by the impulsive wrench on rj to the body when perfectly free. The component on a>n of intensity p"pn imparts a kinetic energy equal to JL 2 • Mv Vn ’ whence the total kinetic energy is found by adding these six terms. The difference between the kinetic energy acquired when the body is perfectly free, and when the body is constrained to twist about a, is equal to 1 n'"2 Irl tla The quantity inside the bracket reduces to the sum of 15 square terms, of which —is a specimen. The entire expression being therefore essentially positive shows that a given impulsive wrench imparts greater energy to a quiescent body when free than to the same quiescent body when constrained to twist about a certain screw.