555. Let R, r be the radii of the wheel and axle; then since ry, Rp denote the efforts of y and p to descend, the moving force is M = Rp Moreover the Inertia of p and y are pR', yr. Hence the accelerating force is F = PR 2 PR2 + yr2 when the time is given. Hence the momentum generated is PR - yr PR2+yre maximum = max. and putting du = 0, we get PR - 2ry PR2 + r2y .. (pR (pR2 + r2y)2 2ry) (pR2 + r2y) = r2y (pR — ry) ..(pRry) pR2 = ry (pR2 + r2y) 556. Let h be the vertical abscissa measured from the lowest point of the height from which the body begins its descent, and x any other when the body shall have acquired the velocity v• Then since the body has descended vertically through hx, its velocity the cycloid is Hence this being resolved into horizontal and vertical velocities we get for the former 557. Since for the centre of gravity of a solid of revolu tion the distance of that centre from the origin of x is 558. Denoting by 1, S, and k, the distance of the centre of oscillation from the point of suspension, the momentum of Inertia, and distance of the centre of gravity from the point of suspension we have (See Venturoli.) 1= S Mk Now, supposing generally the density to ca", we have 559. Let a be the given base, a the length of the inclined plane; the height = √(x2a), and since 560. Let be the radius of the globe; then since the mass is given the density cc inversely as the magnitude. Hence calling x, x the densities, and A, A' the magnitudes of the globe and cylinder, we have 561. Let the radius of the inner section of the shell be x. 562. Since, V, V, V" are the absolute velocities of the bodies A, A' A", the relative velocities of A, A'; A, A" are V-V V-V" and the times of first rencontre between A, A'; A, A" are respec p being the periphery of the curve described. Again, let T, T, T", be the times in which A, A', A", each complete a revolution, then we have and (t) (t') being integers the time of the rencontre of all three bodies is the least common multiple of t and ť, TT T-T Ꭲ - Ꭲ or of TT and In the same manner if T1, To, T3 T be the times in .... · which n bodies perform each an entire revolution with uniform velo.... V, then the interval between any two succes cities V 1, V 2, . . . . . sive simultaneous rencontres of them is the least common multiple of By way of example, let it be required to find the interval between two simultaneous rencontres of the bodies A, A', A", whose velocities p being the periphery of the curve round which they run, and m, n, any integers whatever. |