A Treatise on the Theory of Screws

142] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 129 of inertia to H. But, as A is the only screw reciprocal to H, it is necessary, by the theorem just given, that an impulsive wrench on A must make the body commence to move by twisting about A'. As 0 and O' are fixed, it follows from a well-known theorem, that as otherwise proved in § 137, A and A' form two homographic systems. 141. Twist Velocity acquired by an Impulse. We can obtain a geometrical expression for the twist velocity acquired about A' by a unit impulsive wrench on A (Fig. 22). It appears, from § 90 (see also § 147), that the twist velocity acquired on a by an impulsive wrench on q, is proportional to Ua2 ‘ The numerator being the virtual coefficient is proportional to AO.A'H (§ 63), and as wo2 is proportional to A'O'.A'H (§ 134), we see that the required ratio varies as A0 + A'O' which itself varies as HO' HO’ hence we obtain the following theorem :— The impulsive wrench on A, of intensity proportional to HO, generates a twist motion about A', with velocity proportional to HO'. The geometrical representation of the effect of impulsive forces is thus completely determined both as regards the instantaneous screw, and the instantaneous twist velocity acquired. 142. Another Construction for the Twist Velocity. A still more concise method of determining the instantaneous screw can be obtained if we discard the points 0 and O', and introduce a new fixed point, fl, also on the homographic axis. b. 9