A Treatise on the Theory of Screws

80 THE THEORY OF SCREWS. [88- body receive an impulsive wrench, the body will commence to twist about a screw a with a kinetic energy Ea. Let us now suppose that a second impulsive wrench acts upon the body on a screw p, and that if the body had been at rest in the position L, it would have commenced to twist about a screw ß, with a kinetic energy Eß. We are to consider how the amount of energy acquired by the second impulse is affected by the circumstance that the body is then not at rest in L, but is moving through L in consequence of the former impulse. The amount will in general differ from If, for the movement of the body may cause it to do work against the wrench on /x during the short time that it acts, so that not only will the body thus expend some of the kinetic energy which it previously possessed, but the efficiency of the impulsive wrench on P will be diminished. Under other circumstances the motion through A might be of such a character that the impulsive wrench on p acting for a given time would impart to the body a larger amount of kinetic energy than if the body were at rest. Between these two cases must lie the intermediate one in which the kinetic energy imparted is precisely the same as if the body had been at rest. It is obvious that this will happen if each point of the body at which the forces of the impulsive wrench are applied be moving in a direction perpendicular to the corresponding force, or more generally if the screw a about which the body is twisting be reciprocal to p. When this is the case a and ß must be conjugate screws of inertia (§ 81), and hence we infer the following theorem:— If the kinetic energy of a body twisting about a screw a with a certain twist velocity be Ea, and if the kinetic energy of the same body twisting about a screw ß with a certain twist velocity be Eß, then when the body has a motion compounded of the two twisting movements, its kinetic energy will amount to Ea + Eß provided that a and ß are conjugate screws of inertia. Since this result may be extended to any number of conjugate screws of inertia, and since the terms Ea, &c., are essentially positive, the required theorem has been proved. 89. Expression for Kinetic Energy. If a rigid body have a twisting motion about a screw a, with a twist velocity a, what is the expression of its kinetic energy in terms of the co-ordinates of a ? We adopt as the unit of force that force which acting upon the unit of mass for the unit of time will give the body a velocity which would carry it over the unit of distance in the unit of time. The unit of energy is the work done by the unit force in moving over the unit distance. If, therefore,