already accumulated there will continually expend itself until the whole is exhausted, and the machine is brought to rest. The general expression for the modulus in this state of variable motion is (equation 116) P U1=AU2+BS,+ a Σw(v ̧2 —v‚2). Now in this case of the wheel and axle, if V, and V2 re 29 2 present the velocities of P, at the commencement and completion, of the space S1, and a the angular velocity of the revolution of the wheel and axle; if, moreover, the pressures P, and P, be supposed to be supplied by weights suspended from the cords; then, since the velocity of P is represented by V1, we have a1 +a2μ,I,, if I, represent the moment of inertia of the revolving wheel, and I, that of the revolving axle (Art. 75.), and if μ represent the weight of a unit of the wheel and μ of the axle; since Zwv,2 represents the sum of the weights of all the moving elements of the machine, each being multiplied by the square of its velocity, and that (by Art. 75.) a2, I, represents this sum in respect to the wheel, and «3μ‚Ï1⁄2 in respect to the axle. Now, V1=×a ̧, 2 2 απ Similarly Σwv,?=V? { P‚a‚2+P,a,2 +μ‚I, +#2I; } · ́`. Σw(v ̧2—v‚2)=(V ̧2—V ̧2) ; 2 22 a2 Substituting in the general expression (equation 116), we have N which is the modulus of the machine in the state of variable motion, the co-efficients A and B being those already determined (equation 124), whilst the co-efficient 2 2 P ̧«‚2 +P ̧a‚2 + μ‚Ï‚+2 is the co-efficient Zw2 (equation 2 2 a12 19 117) of equable motion. If the wheel and axle be each of them a solid cylinder, and the thickness of the wheel be b1, and the length of the axle b2, then (Art. 85.) I ̧= }πb1a1‚‚ I1⁄2=}πb‚a‚a. Now if W, and W, represent the weights of the wheel and axle respectively, then W1 = a,b,1, W2=ña,b; therefore μ,I,W,a,2, μ,I=¿W2a22. Therefore the co-efficient of equable motion is represented by the equation 1 2 157. To determine the velocity acquired through a given space when the relation of the weights P, and P2, suspended from a wheel and axle, is not that of the state bordering upon motion. 1 Let S be the space through which the weight P1 moves whilst its velocity passes from V1 to V2: observing that = UPS1, and that U2 = P2S, P ̧ 2 substituting in equation (126), and solving that equation in respect to V,,we have making the same suppositions as in formula 127, and repre 1 P1-A. P2m-B V ̧2 = V ̧2 + 2g$1 { (P, + + 'W1) + (P2 + ¦ W2)m2 THE PULLEY. 158. If the radius of the axle be taken equal to that of the wheel, the wheel and axle becomes a pulley. Assuming then in equation 122, a1=a,=a, we obtain for the relation of the moving pressures P1 and P, in the state bordering upon motion in the pulley, when the strings are parallel, 1+sin. a 1 P1=P2(1+ (129); a-p sin. and by equation 124 for the value of the modulus, in which the sign is to be taken according as the pressures P1 and P, act downwards, as in the first pulley of the preceding figure; or upwards, as in the second. Omitting dimensions E of sin. ❤, L sin., and above the first, we have by equa a a tions (123, 125) a = Also observing that a, a,, and I,=0, the modulus of variable motion (equation 126) becomes U1=AU2+BS+ (V22 —V12) {P2+ P2+ }W} 1 2g and the velocity of variable motion (equations 118, 128) is determined by the equation in which two last equations the values of A and B are those of the modulus of equable motion (equation 125). SYSTEM OF ONE FIXED AND ONE MOVEABLE PULLEY. T 2 159. In the last article (equation 131) it was shown that the relation between the tensions P1 and P, upon the two parts of a string passing over a pulley and parallel to one another, was, in the state bordering upon motion by the preponderance of P1, represented by an expression of the form P1=aP2+b, where a and b are constants dependent upon the dimensions of the pulley and its axis, its weight, and the rigidity of the cord, and determined in terms of these elements by equation 131; and in which expression b has a different value according as the tension upon the cord passing over any pulley acts in the same direction with the weight of that pulley (as in the first pulley of the system shown in the figure), or in the opposite direction (as in the second pulley): let these different values of b be represented by b and b,. Now it is evident that before the weight P, can be raised by means of a system such as that shown in the figure, composed of one fixed and one moveable pulley, the state of the equilibrium of both pulleys must be that bordering upon motion, which is described in the preceding article; since both must be upon the point of turning upon their axes before the weight P, can begin to be raised. If then T and t represent the tensions upon the two parts of the string which pass round the moveable pulley, we have P1=aT+b, and Tat+b1. 2 Now the tensions T and t together support the weight P,, and also the weight of the moveable pulley, ..T+t=P2+W. Adding aT to both sides of the second of the above equations, and multiplying both sides by a, we have a(1+a)T=a2(T+t)+ab ̧ = a2(P2+W)+ab ̧⋅ Also multiplying the first equation by (1+a), (1+a)P1 = a(1+a)T+b(1+a)=a2(P2+W)+ab ̧ +b(1+a), Now if there were no friction or rigidity, a would evidently a2 1+ a become 1 (see equation 131), and would become 1 2; the co-efficients of the modulus (Art. 152.) are therefore which is the modulus of uniform motion to the single moveable pulley.* If this system of two pulleys had been arranged thus, with a different string passing over each, instead of with a single string as shown in the preceding figure, then, representing by t the tension upon the second part of the string to which P, is attached, and by T that upon the first part of the string to which P2 is attached, we have T=aP2+b, 2 P1+t+W=T. P1 = at +b, Multiplying the last of these equations by a, and adding it to the first, we have P1(1+a) + Wa=Ta+b=a2 P2+(1+a)b; * The modulus may be determined directly from equation (135); for it is evident that if S, and S, represent the spaces described in the same time by P, and P2, then S, = 2S.. Multiplying both sides of equation (135) by this equation, we have, |
