planes, perpendicular to the axis, whose dis tance from one another is represented by the exceedingly small increment Ax of the distance x of the section ab from the fixed section AB, and let its radius be repre. sented by y; and suppose the whole of the solid except this single element to become rigid, a supposition by which the conditious of the equilibrium of this particular element will remain unchanged, the pressure P remaining the same, and being that which produces the torsion of this single element. Whence, representing by ad the angle of torsion of this element, and considering it a cylinder whose length is ax, we have by equation (689), substituting for I its value thy', Passing to the limit, and integrating between the limits o and L, observing that at the former limit d=0, and at the latter b=0. If the sides AC and BD of the solid be straight lines, its form being that of a truncated cone, and if r, and r, repre. sent its diameters AB and CD respectively; then 434. THE RUPTURE OF A CYLINDER BY TORSION. It is evident that rupture will first take place in respect to those elements of the cylinder which are nearest to its surface, the displacement of each section upon its subjacent section being greatest about those points which are nearest to its circumference. It, therefore, we represent by T the pressure per square inch which will cause rupture by the sliding of any section of the mass upon its contiguous section,* then will T represent the resistance of torsion per sqnare inch of the section, at the distance r from the axis, at the instant when rupture is upon the point of taking place, the radius of the cylinder being represented by r. Whence it follows that the displacement, and therefore the resistance to torsion per square inch of the section, at any other distance p from the axis, will be represented at that distance by If, the resistance upon any element -K, by Ipak, and the sum of the moments about the axis, of the resistances of all such elements, by - Ipak, or by ? I, or substituting for I its value (equation 64) by Tor. But these resistances are in equilibrium with the pressure P, which produces torsion, acting at the distance a from the axis ; 1 It results from the researches of M. Cauchy, before referred to, that in the case of a rectangular section whose sides are represented by b and c, the conditions of rupture are deter. mined by the equation The length of a prism subjected to torsion does not affect the actual amount of the pressure required to produce rupe ture, but only the angle of torsion (equation 690) which precedes rupture, and therefore the space through which * Or the pressure per square inch necessary to shear it across (Art. 406.). + Navier, Resumé d'un Cours, &c. Art. 167. the pressure must be made to act, and the amount of WORK which must be done to produce rupture. According to M. Cauchy, the modulus of rupture by torsion T is connected with that S of rupture by transverse strain by the equation T=ts..... (698). PART VI. IMPACT.* 435. THE IMPACT OF TWO BODIES WHOSE CENTRES OF GRAVITY MOVE IN THE SAME RIGHT LINE, AND WHOSE POINT OF CON. TACT 18 IN THAT LINE. From the period when a body first receives the impact of another, until that period of the impact when both move for an instant with the same velocity, it is evident that the surfaces must have been in a state of continually increasing compression: the instant when they acquire a common velocity is, therefore, that of their greatest compression. When this common velocity is attained, their mutual pressures will have ceased if they be inelastic bodies, and they will move with a common motion; if they be elastic, their surfaces will, in the act of recovering their forms, be mutually repelled, and the velocities will, after the impact, be different from one another. 436. A BODY WHOSE WEIGHT 18 W, AND WHICH 19 MOVING IN A HORIZONTAL DIRECTION WITH A UNIFORM VELOCITY REPRESENTED BY V., 18 IMPINGED UPON BY A SECOND BODY WHOSE WEIGHT 18 W,, AND WHICH IS MOVING IN THE SAME STRAIGHT LINE WITH THE VELOCITY V,: IT IS REQUIRED TO DETERMINE THEIR COMMON VELOCITY V AT THE INSTANT OF GREATEST COMPRESSION. Let f, represent the decrement per second of the velocity of W, at any instant of the impact (Art. 91.), or rather the decrement per second which its velocity would experience if the retarding pressure were to remain constant; then will ":f, represent (Art. 95.) the effective force upon W,; and it f, be taken to represent, under the same circumstances, the increment of velocity received by Wą, then will":f, represent the effective force upon W, Whence it follows, by the principle of D'Alembert (Art. 103.), that if these effective forces be conceived to be applied to the bodies in directions opposite to those in which the corresponding retardation and acceleration take place, they will be in equilibrium with the other forces applied to the bodies. But, by supposition, no other forces than these are applied to the bodies: these are therefore in equilibrium with one another, Let now an exceedingly small increment of the time from the commencement of the impact be represented by At, and let av, and av, represent the decrement and increment of the velocities of the bodies respectively during that time, .::(Art. 95.)f st=av,Jat=0V,; :: (equation 699) W, . 42,=W, . dv.; and this equality obtaining throughout that period of the impact which precedes the period of greatest compression, it follows that when the bodies are moving in the same direction W (V,-V)=W (V-V,).....17°0); since V,-V represents the whole velocity lost by. W dwing that period, and V-V, the whole velocity gained by W. If the bodies be moving in opposite directions, and their common motion at the instant of greatest compression be in the direction of the motion of W, then is the velocity lo-1 by W, represented as before by it;-V); but the sum of the decreinents and increments of relocity communicated to W, in order that its velocity V, may in the first place be destroyed, and then the velocity V communicated to it in a'i opposite direction, is represented by (V,+V). ::W,(V–V)=W,(V, +V). Solving these equations in respect, to V, we obtain |